Permanence and the dynamics of biological systems.

REVIEW

Permanence and the Dynamics of Biological Systems VIVIAN HUTSON Department of Applied Mathematics, The University, Sheffield SlO 2UN, England AND

KLAUS SCHMITT Department of Mathematics, Universi& of Utah, Salt Lake City, Utah 84112 Received 11 December 1990; accepted 31 January 1992

ABSTRACT A basic question in mathematical biology concerns the long-term survival of each component, which might typically be a population in an ecological context, of a system of interacting components. Many criteria have been used to define the notion of long-term survival. We consider here the subject of permanence, i.e., the study of the long-term survival of each species in a set of populations. These situations may often be modeled successfully by dynamical systems and have led to the development of some interesting mathematical techniques and results. Our intention here is to describe these and to consider their application to several of the most frequently used models occurring in mathematical biology. We particularly wish to include and cover those models leading to problems that are essentially infinite dimensional, for example reaction-diffusion equations, and to make the discussion accessible to a wide audience, we include a chapter outlining the fundamental theory of these.

1. 1.1.

INTRODUCTION COEXISTENCE

OF SPECIES

For a set of different species interacting with each other in an ecological community, perhaps the simplest and probably the most important question from a practical point of view is whether all the species in the system survive in the long term. An analogous question is equally important in other contexts, for example where the system consists of genotypes, or of strategies in an evolutionary game, or indeed of polynucleotides competing for energy sources. For clarity of M4THEhUTICAL

BIOSCIENCES

1

lll:l-71(1992)

OElsevier Science Publishing Co., Inc., 1992 655 Avenue of the Americas, New York, NY 10010

00255564/92/$5.00

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VIVIAN HUTSON AND KLAUS SCHMIYIT

exposition we shall discuss in this section only a model of the first situation mentioned. A central issue of community ecology, an area that currently is much studied, is the discussion of conditions that ensure the survival or otherwise of the community as a whole, together with an investigation of whether or not addition or deletion of species leads to new communities that exhibit long-term survival. A recent survey [90] presents an interesting evaluation of progress in this area. It is at any rate clear that the very idea of a community as an entity that is preserved depends on some concept of stability. What is perhaps surprising is that there has not been more discussion in recent years among biologists themselves as to what stability concepts should be selected to reflect well the biological concept under investigation. It has become almost a central tenet in ecological theory that asymptotic stability (often known in that context as neighborhood stability) is the “correct” concept. As a result much of the theoretical effort has been based on it, see for example [80], [89], and [86]. There are certainly good reasons for the ecologist to adopt this approach, for asymptotic stability has the great virtue of tractability, and even if other definitions should be more attractive, if they are not tractable then the ecologist cannot adopt them with profit. However, there have been technical advances, and it is we believe time to open up the discussion and to see whether better alternatives are available at least in some circumstances. Our object is to present here another definition of a stability concept, which is somewhat less tractable, but we would argue, more satisfactory, and to outline the very considerable body of mathematical theory that has grown up to deal with this idea. We do not claim that this concept is the “best” in some absolute sense. Indeed, perhaps there is no single best concept, merely concepts that are most appropriate in a certain class of situations. We do believe, however, that some general discussion in this area is long overdue, and we hope that presenting the mathematical theory as it stands now may be helpful in this discussion. We also believe that though models based on ordinary differential equations have been central to the development of theoretical ecology, to achieve further understanding it is now essential to consider more general and hence more “difficult” models. For example, the spatial diffusion of species is clearly important, see [90, pp. 212, 2191. Thus our second main object here is to outline a mathematical theory that is sufficiently powerful to tackle this more difficult class of infinite dimensional models of which systems with diffusion are a representative example.

PERMANENCE 1.2.

3

THE BASIC MODELS.

The model that occurs most frequently in mathematical biology is the system of autonomous ordinary differential equations of the form Lii = z&(u)

(i=l

,..*, n).

(1.1)

In this context (1.1) arose first in the (independent) work of Lotka and of Volterra, where ui represents the total population of species i in a region; see [891, [90]. However, the system has subsequently appeared in many other contexts, notably genetics and evolutionary games. One of the most interesting is the fairly recent idea of an “evolution reactor” due to Nobel Laureate Eigen; in fact mathematical investigations in this area led directly to the idea of permanence; see [54, ch. 121 and [96] for historical remarks and further references. Applications of permanence are treated in Sections 4.2 and 5.1. A very well known special case is the class of Lotka-Volterra equations ~i=~i[~;-~~~ij~j]

(i=l,...,n).

(1.2)

We shall see in Section 4.5 that the discussion of permanence is particularly fruitful in this case. When the generations are nonoverlapping, a common model is the system of difference equations

u; = Uifi(

u)

(i=l

,...,n),

(I-3)

where U[ denotes the value of ui in the next generation; see for example [891 or [SO] for the background. These models have also become well known for exhibiting exotic behavior even in one dimension. In Section 4.3 we discuss what progress has been made with permanence for (1.3). Again the special class, which we call Lotka-Volterra difference equations [56],

(l-4) is relatively more tractable; see Section 4.5. A second class of models is increasingly being discussed in biological applications, partly because it is becoming clear that a fuller understanding of, for example, community ecology requires more sophsiti-

4

VIVIAN

HUTSON

AND KLAUS

SCHMITT

cated models, and partly because mathematical techniques for their treatment are more readily available. These are models that lead to essentially infinite dimensional problems, and it is to treat these that the preparations in Sections 2 and 3 need to be fairly extensive. We treat two representatives of this class, where spatial and delay effects, respectively, are included. When spatial variation of the species is important, diffusion effects must be included. The background is discussed in [84], [30], and [12]. We shall consider here only the simplest model of diffusion, when its effects may be modeled by the Laplacian A, and when there is no migration across the boundary of the spatial region 1R.The governing problem is then of initial/boundary value type: 6%.

-$

= u;fi( u) +

dUi -0

(x

dv

u(x,O) =&J(x)

/_L;

E

(xEO,i=l,...,

AZ+

n),

(1.5)

aa), (XEfi),

where u,(x,t) represents the species density at the point x at time t. The definition of permanence must now be refined to take the spatial variation into account. It is noteworthy that for a nontrivial class of models the permanence question may be fully resolved, as is shown in Sections 4.4 and 5.3. A model that takes into account the “history” of the system is a set of autonomous functional differential equations; see [25] or [79]. For example, the basic equations might be of the form (i= l,...,n),

tii=gi(u(t-T1),...,U(t-Tm))

(1.6)

or there may be integral terms on the right-hand side modeling the total history of the past states of the system. An example is treated in Section 5.2; for further information concerning a system with “infinite delay,” see [17]. Finally, we consider a class of models that has received little attention in the literature, but which we suggest shows some promise in the context of biology. The motivation for considering this model is that in practice experiments are extremely hard to perform, and usually the interaction terms are known only very loosely. Suppose that for each state u of the system it is known only that the interactions lie in some set I;;(u). Then it is appropriate to consider the differential inclusion, see [91,

ziiE E;;:(24)

(i=l

,-**, a),

(l-7)

PERMANENCE

5

where a solution is any “smooth” trajectory satisfying (1.7). The technicalities are now more difficult, but some progress can be made with permanence; see Section 2.6. At the present time the theory is far from complete, but we believe it worthwhile to include a brief discussion here in the hope that this might stimulate further investigation. There are other models that arise in biological applications that we have not space to consider here. Perhaps the most important are systems based on nonautonomous differential equations. It is shown in [18] and [42] that a dynamical system approach is applicable, and another approach is reported in [27-291 and [78]. However, a great deal more work is needed in this area. Also, it is possible to argue (see [77]) that it is essential to include stochastic effects in biological models, but so far as we are aware, questions of the type addressed here have not been much studied for such models; however, see [41]. 1.3.HISTORICAL BACKGROUND OF PERMANENCE

The development of stability concepts in biology is likely to have important implications for major areas of the subject. Certainly the history of this development is of interest, and its study may be useful in highlighting the issues that are currently under discussion. In this section we shall thus make some remarks on this history, and for simplicity of presentation emphasize here largely systems (1.1) of autonomous ordinary differential equations. It is convenient first to introduce some loose definitions for the general reader, more precision being adopted later. The first is the idea that a set is invariant for (1.1) if a solution with initial values in that set remains in the set for all time. Of course, biological considerations imply that (1.1) is only considered in the positive cone R:, and we note that this set, its interior int R:, and its boundary GJIR: are all invariant. Clearly points in JR: correspond to situations where are at least one of the species is absent. An equilibrium E is a point where the right-hand side of (1.1) is zero for eachi, and an interior equilibrium is one in int IFI:. If E is an attracting point, its basin of attraction is the set of points attracted to it. From a biological point of view, long-term survival of all species in a community seems at first sight to present a problem that is readily formulated. One imagines a community that is disturbed by some external event, perhaps a flood, and asks whether every species will survive; this is naturally thought of as a sort of stability. However, on closer examination the question is rather loose. For example, it is not clear whether all types of disturbances should be allowed (even granted that no species is completely annihilated). Indeed, should the disturbance be temporary? Furthermore, very small population levels should

6

VIVIAN

HUTSON AND KLAUS SCHMITT

perhaps be excluded. It is in fact rather obvious that no single criterion is enough to provide useful answers to such a general question. For it is certainly relevant to ask how close species densities come to zero and how long they remain near zero after the disturbance has terminated. Unfortunately practicalities impose severe restrictions on the adoption of criteria, for current techniques do not allow answers to either of the preceding questions except in the simplest cases. Hence, in devising criteria, a compromise between tractability and finding a test that yields useful practical information must clearly be sought. The history of the early development of stability notions in biology is a subject of considerable interest and not a little complexity. However, it is appropriate to concentrate here on later events and to recommend the interesting account [69] for the early period. Whatever may have been the thinking among ecologists themselves, the mathematical treatment developed into a rather clear pattern that we describe next. For the system (1.1) it was required that there should be a unique interior equilibrium point E and that this should be globally asymptotically stable (for initial values in int 153:1. Hence all perturbations would eventually be compensated for, and the system would return to E. Volterra invented a clever Liapunov function that enabled the question of stability in this sense to be settled for a nontrivial range of cases. Unfortunately, it soon became clear that, although the function worked very well for Lotka-Volterra equations for two species, for three or more species when the equations are Lotka-Volterra, or even for two species when they are not, unless some rather stringent condition is imposed, the Liapunov function was not adequate. Extensive further investigation over the years has failed to achieve significant advances in settling this question. Another criterion that has something of the same flavor is that of asymptotic stability (also known in ecological circles as local or neighborhood stability), where it is only required that all orbits with initial values sufficiently close to the equilibrium point should return to E. A relatively simple and very well known test dating back to Liapunov exists for asymptotic stability: that all eigenvalues of the linearization at E should have negative real parts. In many nontrivial cases theoretical attacks have been successful with this test, and it is particularly simple from a computational point of view. The practicability of the test has no doubt had much to do with its widespread adoption, and it permeates much of theoretical biology since Volterra, as a perusal of the literature soon reveals; see [SO], [891, 1861to quote but a few texts. One drawback with asymptotic stability, and it is a very major one, is that “sufficiently close” cannot be made more specific; the basin of attraction could be the whole of int lF4: or an extremely small neighbor-

PERMANENCE

7

hood of E. There are other difficulties, also applying to global asymptotic stability, that are less obvious but perhaps just as important. Why should only one interior equilibrium be allowed (of course, if there are two or more, one must be unstable)? Perhaps several rest states may exist in real communities. Further, why should the system have to settle down to rest at all? Indeed, well-known and much-quoted examples (for example the lynx-hare cycle) exist that seem to suggest that communities that oscillate violently could survive perfectly well. Certainly one reason for continuing to use these criteria was that mathematical techniques for dealing with alternative criteria did not appear to exist. There is perhaps though another reason, and that is that the “spirit of the time” seemed to require that a system that was to persist should have a single nontrivial equilibrium and that this should be globally asymptotically stable, or a poor second, asymptotically stable at the very least. In support of this view we quote [8], itself quoting not a biologist but an economist, who wrote in 1954: “Multiple equilibria are not necessarily useless, but from the standpoint of any exact science the existence of a uniquely determined equilibrium is, of course, of the utmost importance, even if proof has to be purchased at the price of very restrictive assumptions; without any possibility of proving the existence of a uniquely determined equilibrium or, at all events, of a small number of possible equilibria at however high a level of abstraction, a field of phenomena is really a chaos that is not under analytical control.” Current thinking has no difficulty with imagining the persistence of systems that exhibit quite wild oscillations, multiple equilibria and so on. Indeed the study of chaotic systems that allow a form of persistence is now extremely fashionable; examples come from engineering and the physical sciences on the one hand, and biology and economics on the other. It is difficult to apportion with certainty credit for initiating a change in the point of view. However, we are aware of two early papers, published in the sixties, that presage the approach reported here. The first is by the eminent geneticist R. C. Lewontin [77]. This contains the following assertion: “The presence or absence of species is sometimes the point of interest regardless of some variation in their numbers or relative abundance from time to time.” It also contains a definition of “dynamic boundedness,” which is a concept loosely allied to permanence, although no mathematical techniques for dealing with this were given. The paper contains other interesting ideas and may still be read with profit. The second paper [82] is by J. Maynard-Smith, and actually includes the word “permanent” in allowing communities that “. . . survive, either in a static equilibrium or a limit cycle.” Again the idea of allowing nonconstant asymptotics is clearly present. That biologists later

8

VIVIAN HUTSON AND KLAUS SCHMITT

were searching for more satisfactory concepts may be seen in several papers from a conference in 1974 [76]. However, although many criteria were proposed, little progress with devising mathematical techniques for dealing with them is apparent. In summary, by the sixties there were doubts among some biologists that either the idea of asymptotic stability or of global asymptotic stability reflected well the broad concept of long-term survival. However, there was no agreement as to what concept or concepts should replace these ideas. Furthermore, no group of mathematical ideas that seemed helpful in treating these concepts was readily available. The approach that we shall describe has as its movitation the idea that any asymptotic (that is, large-time) behavior of orbits should be allowed so long as the orbits do not remain too close to the boundary. Or to put it another way, the boundary should be a repeller in some sense. As stated the idea is not precise, but it certainly fits broadly with the intuitive view previously discussed of Lewontin and Maynard-Smith. To meet these requirements, the idea of weak persistence, that is limsup ui( t) > 0

(1.8)

t-z

for all i, was introduced in [34]. A disadvantage of this concept is that orbits of a weakly persistent system may approach c?IR: by spiraling out towards a heteroclynic cycle; see [81], [93]. The stronger condition of permanence that avoids this difficulty was introduced in [94] and is based on dR: being repelling in a strong sense. The system (1.1) is said to be permanent if there are numbers m, M with 0 < m Q M t,,i=l,.,.,

n).

(1.9)

The broadly analogous terms cooperativeness [51], permanent coexistence [67], and uniform persistence [19] have also been used in the literature. A condition intermediate between weak persistence and permanence, the notion of strong persistence was formulated in [35], its definition being obtained by replacing “limsup” by “liminf’ in (1.8). It is frequently the case that if it can be verified that a system is strongly persistent, it may also be shown to be permanent, and we hence concentrate on the latter here. An investigation of the connections between the various concepts noted here and others is given in [33]. A preliminary examination of the concept of permanence reveals some obvious advantages. First, it is global, the quantities m, M being independent of the initial values. Secondly, no solution can approach

PERMANENCE

9

the boundary. Thirdly, only the behavior near all%: is relevant. Lastly, any asymptotic behavior consistent with (1.1) is allowed; indeed even the existence of a strange attractor is not ruled out, a possibility that can occur in relatively simple-looking systems; see [7] and [92]. There is one clear disadvantage. The definition says nothing about how large m need be, the possibility existing that it could be extremely small, which would allow orbits to approach asymptotically extremely close to the boundary. This is certainly not ideal from a biological point of view. Our comment here is that the great contribution of [94] is that a reasonable definition was devised and a test for it given that was practical in a useful range of examples. Up to the present time, as can be seen from [43], attempts to specify m except in the simplest cases have not been successful. After seeing the criterion in action we shall return in Section 7 to make some further remarks on its value and applicability, but we now turn to the mathematical methods for dealing with it. The key article [94] that initiated this area of study is by Schuster, Sigmund, and Wolff in 1979. Interestingly, the idea arose not in the traditional areas of ecology and genetics, but in the relatively new field of prebiotic evolution due to Eigen. In this article the concept of permanence was introduced, and some analysis presented that showed that the concept could be tackled for a nontrivial example. To put the approach used into context, recall that if a function I/ defined on a neighborhood N of c?R: can be found that is zero on JR: and increases along orbits, then orbits must eventually escape from N and not return, so permanence must hold; such a function would be naturally thought of as a Liapunov function, developing the original idea of a Liapunov function for a globally asymptotically stable equilibria to a repelling set (with a sign change). In the present context there has been little success in finding conditions ensuring the existence of such a function even in low dimensions. The approach in [94] is based on showing that much weakened requirements on I/ (see Section 2.4 for the details) still lead to the conclusion that permanence holds, although no estimate for the repelling neighborhood N would be available. This function has come to be known as an average Liapunov function. Inherent in the technique is the idea that only the situation in a neighborhood of cCJR: is relevant. A second key reference is [51], where the essential feature of the technique were extracted and made explicit in the context of ordinary differential equations, and a dynamic systems approach is put forward in [59]. Subsequently a considerable body of literature has been devoted to this technique. One direction of development is to increase the range of applicability for ordinary differential equation models (1.1). There has been some success with models for four or even more species (see [70-731) and an approximate method for

10

VIVIAN


higher dimensional problems has been developed in [40]. Another direction that has been taken is to attempt to cover some of the more difficult models discussed in Section 1.2, and for example a reaction-diffusion model is discussed in [64-651 and a differentialdelay system in [17]. Many references will be given when we discuss the theory in Section 2 and a range of examples in Section 4; see also the survey article [102]. A second approach to permanence can be traced to a paper [35] by Freedman and Waltman in 1984 where it was observed that a careful analysis of the flow in the boundary led to convenient conditions for strong persistence. Subsequent work reported in 1986 1191showed how ndeed permanence (or uniform persistence in the terminology of the authors) held under somewhat strengthened conditions; see also [33] for further discussion and references. The first application of this method to a reaction-diffusion model is given in [26], but the most general version of the method appears in [48], and it is this that we shall describe. A large number of applications have been tackled, and further references are given in Section 5. The great majority of examples that have been successfully tackled by one method can be done by the other. There are just a few known examples where this is not the case and we illustrate the contrasting advantages by a pair of examples. Usually, though, the choice of method is best determined by convenience. In some cases a unification of the methods may be necessary; see Section 2.7 for comments. 1.4.

OUTLINE OF CONTENTS

The treatments of the models described in Section 1.2 are based on technicalities of widely differing difficulty. For example, the basics for the system (1.1) of ordinary differential equations are rather widely known, whereas the treatment of the reaction-diffusion system (1.5) requires a great deal more mathematical sophistication as does that of the system (1.6) of differential-delay equations, and the techniques for these may not be among the armory of some workers in this area. We do believe though that it is not sufficient today to restrict attention in mathematical biology to models governed by ordinary differential equations, and that the study of “harder” systems is necessary if understanding of such questions as permanence of communities, and indeed of many other key issues, is to advance. We have thus thought it essential to present an outline of enough of the basic technicalities for infinite dimensional systems to cover all our models. This is intended to give a flavor of the method rather than to be a comprehensive account, for which the reader must refer to standard texts, some of which are referenced at the appropriate place.

PERMANENCE

11

In fact, from a certain point of view the use of the basic theory of partial differential equations is not really very much harder than that of ordinary differential equations. The key word here is “use.” To set up the theory from scratch is a great deal more difficult, but as we hope we can convince the reader, if one is willing to accept certain “hard” technical basics, by exploiting an elegant abstract setting, the use of the theory can be made quite tractable. We try to bring out this point of view in the next two Sections. The key idea is to abstract the notion of “solution,” say of a system of differential equations, to the setting of a “dynamical system.” In this abstract setting we then derive the key theorems yielding permanence. They are Theorem 2.17 for the average Liapunov function method and Theorem 2.21 for the acyclicity method. These ideas are developed in Section 2, which also contains a brief section where we tackle an issue that is much less well known-the area of set-valued dynamical systems. This is done so that the reader may see that a dynamical systems approach is possible even for differential inclusions (1.7). It is not necessary to follow the proofs of the basic theorems to apply them to practical examples. We give two Sections on the applications of the two techniques (Sections 4 and 5, respectively), intended to illustrate the fundamental ideas. Examples showing their relative strengths and weaknesses are included. Some of the examples include cases where there may be very complex asymptotics. We also give a brief description of the theory as applied to Lotka-Volterra systems. This is a very restrictive class of systems, but the results are so elegant and powerful that we believe they are worthy of note. Finally, in Section 6 we treat first the question of the existence of stationary interior equilibria, corresponding biologically to a constant coexistence state, when permanence holds. This provides a necessary condition for permanence under a wide variety of conditions. The second topic is the question of the “robustness” of permanence. Ideally we should like permanence to be invariant under small perturbations of the governing equations. Essentially this remains an open question. However, we describe briefly results that show that the somewhat weaker condition of stability under perturbations holds. In Section 7 some remarks are made concerning the general question of the applicability and utility of the permanence criterion. There is now a considerable body of literature in the area of permanence. Our broad strategy is to include as many references as we know of up to the middle of 1991 that are relevant to the applicability of the theory. Although no attempt is made to list every problem that has been tackled from the permanence point of view, we have included a fairly extensive and we hope representative range of applications and references.

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2. 2.1.


DYNAMICAL PRELIMINARY

SYSTEMS AND PERMANENCE REMARKS

The material in this section forms the theoretical background to the discussion of permanence in applications. There are two basic results. These are applicable to a wide class of specific biologic examples, and the applications are discussed in Sections 4 and 5. As we have seen in the introduction, there are many types of models in which the concept of permanence is important, so to unify the treatment the analysis will be carried out in the abstract setting of a semidynamical system on a metric space Y. For example, in the case of ordinary differential equations (1.0, Y will simply be the positive cone IW:={xEW: Xi>O, i=l,..., n}, whereas for a system of reactiondiffusion equations (1.61, a part of a Banach space E will be used. For situations modeled by difference equations the time variable is discrete and its domain will be the positive integers Z, rather than [w,. Permanence is of course a qualitative property of the asymptotic behavior of the system. Very little progress can be made with questions concerning this unless there is some compactness associated with the system. For ordinary differential equations this causes little difficulty because the phase space rW: is locally compact. This is not the case, for example, with partial differential equations, but in most applications initial data are “smoothed” by the flow, which leads to enough compactness for the theory to be carried through. The task of setting up the dynamical system and proving compactness properties in specific models is tackled in Section 2.3. The next section gives the dynamical system terminology; for the general background, [ll], [91], [46], or [471 may be consulted. Section 2.3 summarizes some results on the structure of the asymptotics for permanent flows. Sections 2.4 and 2.5 review the central results of this survey, which are used for establishing permanence. Finally, in Section 2.6 we outline a generalization of the average Liapunov function method to differential inclusions as described in the introduction. 2.2. SEMIDWAMICAL

SYSTEMS

Let (Y,d) be a metric space. Points in Y will usually be denoted by l&u,... and subsets of Y by U,V,.... For c-neighborhoods of U we use B(U,E)={uEY:3uEUs.t.d(u,u) is called a semidynamical system or semiflow if T: Y X [[D, --, Y satisfies for all u E Y, 6) 7440) = u; (ii) ~(~(u,f),s)=~(~,f+~),(~,t (iii> 7r is continuous.

ED+);

For convenience we often write &, t) = u.t. If D, = R,, the semidynamical system is referred to as continuous; if it equals Z +, it will be called discrete. For subsets II of Y and 2 of D,, U.I=

u u tcluGU

u.t.

u is said to be an equilibrium point if u.t. = u,(t E D, ). It is periodic if there is a T > 0 such that UJ. = u.(t + T) for t E ED,; [note that in view

of (ii) of Definition 2.1 it is enough for this, if u.T = u]. The set y+(u)=(u:u=u.t.

forsometED+}

is called the semiorbit through u and y+ (U> is defined by taking unions. Semiorbits may or may not have a continuation backwards in time. We shall be interested sometimes in situations where, at least for some U, there is a continuation back to -m. Definition 2.2 A solution 4 through u is a continuous map C#J:D *Y such that +(O) = u and T(&(T), t) = c#& + T) for t E D,, T E D. The range of 4 is denoted by y(u) and is called an orbit through U. Definition 2.3. A set U c Y is said to be forward invariant if U.t = U,(t E ID, 1.

invariant

if yf (U> c U, and

14


Invariance has been defined in this fashion to accommodate the possible backward “splitting” of orbits, as may be the case for systems of differential equations that are uniquely solvable in forward time but not in backward time. In such cases there may be more than one such orbit, and it may even happen that there are orbits through u not contained in U. In the case of dynamical systems defined by ordinary differential equations with Lipschitz continuous nonlinear terms, backward invariance may be similarly defined and then a set will be invariant in case it is both forward and backward invariant. It is easy to show that through each point u of an invariant set U there is an orbit y(u)cU. We next introduce some terminology for studying the asymptotic behavior of semidynamical systems and give some results under dissipativity conditions. A set U is said to be an attracting set for I/ if lim, _r ti(u.t, U) = 0, (vE V); it is a globally attracting set if I/ = Y. The semidynamical system is said to be dissipative if there is a bounded globally attracting set. U is a global attractor if it is compact, invariant and lim f-t 3c dW.t,17)= 0 for all bounded V. The set U is said to be absorbing for V if it is forward invariant, and y+ (u) r? U # d for all u E V. Definition 2.4

The elimit

set w(u) of u is defined to be

oJ( u) =

{?I: 3( tJ

with t, -+ 03such that u .t, -+ u},

or equivalently w(u)=

n 720

uz4.t.

(2.1)

f>T

For a subset U c Y o(U)=

u

w(u).

UEU

The following concept based on (2.1) will also occasionally tioned; see [47, p. 431, [48]: G(U) =

n u T>O

be men-

U.t.

f,T

h(U) is generally a larger but better behaved set than o(U). For proofs of the following standard results see [ill.

15

PERMANENCE

LEMMA

2.5.

Suppose that y+(u) is relatively compact. Then w(u) is nonempty, compact, and invariant. If D, = R,, then w(u) is connected. Zf U is closed and forward invariant, o(U) c U for u E (I. Definition 2.6 Let U be a compact invariant set. Suppose for u E U, y(u) is an orbit through u with y(u) c U. The a-limit set of y is defined to be

a,(u) LEMN4

=

{u:3{t,)

with t, + -CO such that c$( t,) --) u}.

2.7

Under the assumptions of these definitions, a,(u) and invariant. Also CY.,(U> c U.

is nonempty, compact

The stable and unstable manifolds are well known and important concepts in describing the flow in a neighborhood of a hyperbolic equilibrium point for a smooth dynamical system defined by ordinary differential equations or systems of difference equations. The following definitions generalize these concepts and in addition describe a weaker notion of attraction. Definition 2.8 Let U be compact and invariant. Then the stable, unstable, weakly stable, and weakly unstable set of U are defined, respectively, as follows.

W”(U)

={u:o(u)

W”(U)

= {u: 3 an orbit y(u)

W,s(U) ={u:

z&3, o(u) cu) such that a,(u)

cU)

~(u)nu#8}

W,U(U)={u:3anorbit

y(u)

suchthat

cu,(u)nU+d}.

Definition 2.9

A compact set U is said to be stable if every neighborhood of U contains a forward invariant neighborhood of U. U is asymptotically stable if it is stable and Ws(U) contains a neighborhood of U. The existence of a compact set that absorbs or attracts orbits may be regarded as a broad stability property of the flow in that it ensures that asymptotically orbits are forced into a relatively restricted region. Theorems 2.11 and 2.12 below show how dissipativity together with certain


16

compactness conditions yield this property. A technical lemma that will often be useful in the sequel is first given. LEMMA 2.10

Let N c Y be open and R compact. Suppose that V, where N c V c Y, is forward invariant, and assume that y+ (u)n N # fi,(u E V). Then the following hold. (i) y+ CR> is a forward invariant compact subset of V. (ii) If K c V is compact, yf (K) is contained in a compact subset of V, and further, there is a t(K) E D+ such that u.t. E y+(N) for u E K and t > t(K). Proof Let A c V be compact. Since y+(u) intersects N for each u E A, given u E A there exists T(u) E (0,~) such that u.T(u) E N. As N is open there is an open neighborhood U(U) of u in A such that U(u).T(u) c N. Now U, E /, U(u) is an open cover, so as A is compact there is a finite subcover U(u,>, . . . , U(u,), say, of A. From the continuity of r, the set T(A) = ;

a(ui).[O,T(ui)]

i=l

is compact. (i) Take A = N. r(N)

includes all orbits leaving #, so y+ (fl> c = T(N), which has

U&9. The opposite inclusion is obvious, so y+(N)

been shown to be compdct. (ii) Take A = K, t( K) = maxI d i ~ n T&j. Then every orbit reaches N before t(K), and (ii) follows as y+ (N) is forward invariant. THEOREM 2. I I

Let Y be local& compact, and assume that there is a compact globally attracting set K. Then there is a compact absorbing set X for Y. Proof B(K, E) is open, B(K, E) compact, and all orbits intersect B(K, E). The result follows from Lemma 2.10 on taking N = LICK,E). THEOREM 2.12 [48]

Let Y be complete, and suppose that the semidynamical system (Y, IT,R, ) is dissipative. Assume that there is a t, > 0 such that rr(., t) is compact for t > t,. Then there is a nonempty global attractor. 2.3.

ABSTRACT PERMANENCE

Guided by the models described in the introduction, we suppose that Y = Y, u 8 Y,, where Y0 is open, and assume that Y, and aY, are forward invariant.

PERMANENCE

17

Definition 2.13

The semidynamical system on Y is said to be permanent exists a bounded attracting set U for Y, with &U, aY,J > 0.

if there

A variation on this definition is sometimes used, for example in [481 where the semidynamical system is said to be uniformly persistent if there is an n > 0 such that liminf, -tz d(u.t, aY,) > n for u E Y,,. Clearly for a dissipative system uniform persistence is equivalent to permanence. Dissipativity will be required in all cases to be treated. Together with the available compactness this will often enable us to show that there is a compact absorbing set X for the semidynamical system; one technique for doing this is described in Theorem 2.11. Then we will regard X itself as a metric space with the induced topology and consider the restriction (X, m, D, 1 of the original semidynamical system to X. S = aY,, fl X will be a compact subset of X with empty interior, and S and X\S will be forward invariant. As an example, in the ordinary differential equations case Y = rW: and JY, is the union of the “faces” ui = 0. The set X would then be a compact neighborhood of the origin, and S the intersection of this with aY,,, and so will consist of that part of the faces which lies in X (but will not be the whole of aXI. The semidynamical system will thus be permanent if and only if there exists a compact attracting set MO for X\S with M, c X\S. It is then easy to prove the following. THEOREM2.14 Let X be compact, and let S c X be compact with empty interior. Assume that S and X \S are forward invariant. Then if the system is permanent, there exists a compact asymptotically stable set M c X\S which is absorbing for X \ S.

A set U c Y,, is said to be strongly bounded if it is bounded and #J, JYJ > 0. A, is said to be a global attractor relative to strongly bounded_ sets if it is a compact invariant subset of Y0 and lim , ~ md(U.t, A,) = 0 for all strongly bounded U. THEOREM2.15 Assume that Y is complete and (Y, IT, R + ) is dissipative. Let Y. and JY, be forward invariant. Suppose that there is a t, > 0 such that T(., t) is compact for t > t,. Then if permanence holds, there are global attractors A, A, for rJ (that is T restricted to JY,), and a global attractorA, relative to strongly bounded sets.

The compactness and invariance of the attractors A and A,, are consequences of the smoothing action of the flow and are basic to the

18

VIVIAN

HUTSON

AND

KLAUS

SCHMITT

approach in Section 2.5. For the general background to applications using the existence of global attractors the reader may refer to l4.51;for a proof and further discussion of Theorem 2.15, see [48]. The following heuristic remarks may help the reader in visualizing the tactics for establishing permanence. First, clearly it is enough to examine the flow in any neighborhood of S. The most obvious way in which permanence might fail is if for some u E X\S, w(u) n S #d. If this happens we might intuitively expect w(u)nS to be “carried along” by the flow in S towards w(S), and indeed it is easy to show that then du)n o(S) f d. Permanence may also fail if there is a sequence {u,} in S with lirnn+% d( w(u,,), S) = 0, but then under suitable compactness assumptions, the limit w of w(u,) will intersect S, and a similar conclusion may be expected. This suggests that the flow in a neighborhood of w(S) will be of special importance. The condition just described is not easy to check so it is worthwhile to search for a condition that is more easily verified. Clearly the system is not permanent if o(u) c w(S) for some u E X \S. One might then ask whether, if this is ruled out by requiring that W(w(S))nS=S,

(2.2)

permanence will follow. The simple semidynamical system described in Figure 1 on a triangle with w(S) = {A,B,C} shows that this is too optimistic. However, if (a generalization of) the above “spiralling out” behavior is ruled out, (2.2) does ensure permanence (see Section 2.5). These remarks suggest that the flow near o(S) plays a central role, but another relatively weak condition on the rest of S is required. In both approaches to be taken this outline is followed. 2.4.

THE AVERAGE LIAPUNOVFUNCTIONAPPROACH

TO PERMANENCE

Consider the semidynamical system (X, rr, ID, > with compact X as described in the previous section, and take S = ~JY, I? X. An obvious approach to permanence is to attempt to construct a non-negative Liapunov function that vanishes on S only, and which is increasing along orbits; this is the time-reversed version of the more usual situation when attracting sets are being considered. Thus if there exists a neighborhood U of S and V E C(U, R, > such that

I(i)

V-'(O)

= S,

1

w4

>

(ii>

qu)

1

(t > 0,u E U\S),

(2.3)

19

PERMANENCE

permanence might be expected to hold. However, in anything but the simplest situations it turns out to be extremely difficult to find such a I/. The approach here is to weaken the conditions on I/, and in outline the basic theorem 2.17 will prove that it is enough if the second part of (2.3) holds: (a) on w(S) only, after taking a suitable limit; (b) for some t (rather than all t > 0). These conditions are significantly easier to manage; for example o(S) in (a) is usually a much smaller set than U \ S. We shall thus be able to show in the sequel that an average Liapunov function may be found in a wide range of models. A representative range of applications of Theorem 2.17 is discussed in Section 4. In the main theorem of this section, essentially in the version from 1591 and [65], our aim is to be as general as practicable to encompass infinite dimensional models such as partial differential equations, equations with delay, and difference equations. For a related approach, see [31]. For ordinary differential equations a compact globally attracting set K may often easily be found, and then the existence of a compact absorbing set X follows from Theorem 2.11. For more general models locating such a set is more difficult and in addition Theorem 2.11 is no longer applicable. However, it is often possible to find a bounded absorbing set X,,, say, for (Y, 7~,D, 1. Take S, = X0 n S. Then, since in many models the flow has a smoothing action, m(X,, [ 1, ~1) = X,, will be relatively compact, and we may take X = x,. We shall follow this strategy. Of course, in the case of ordinary differential equations this

FIG. 1. A section of the flow generated see [81] and [93].

by a simple competing

species problem;

20

VIVIAN


will be unnecessary for the reasons just given, and in Theorem 2.17 we may simply replace S,, X0 by S, X, respectively. Let X,, be a bounded absorbing set for (Y, T, D + 1 and consider the semidynamical system ( X,, , T, D + 1. Put X, = ~(XO,[Lq),

S, = +J,

[Lm)),

and take finally X = x,, S = S,, the topology always being that inherited from X0. LEMMA 2.16

Let S,, X,,\S, be forward invariant, and suppose that S, is closed with int S, = 8. Then the following hold: (i) X,, S,, X,\ S, are forward invariant; (ii) X,S, X\S are forward invariant and rr(X,[l,m))c (iii) X, /S, is dense in X.

X,;

Proof 6) follows immediately from the semigroup property definition 2.1 (ii). (ii) X and S are forward invariant since the closure of forward invariant sets are forward invariant. We remark next that if u.t E S, for some t > 0, then u E S,, as otherwise the forward invariances of X,,\S, is contradicted. We show that S, = X, f~ S,. Clearly X f~ S, c S,. To prove the opposite inclusion, take any u E X, f~ S,. Then there exist u E Xa,t > 1 with u = u.t, and u E S, implies by the preceding remark that u E S,, whence u E S,. Thus S, = X, f~ S,, and as S, is closed, it follows that S = X n S,. To complete the proof we argue by contradiction, and suppose that ZJE X \S but u.t E S for some t > 0. Then by the preceding remark, u E S,. Hence u E X n S, = S, yielding a contradiction. (iii) S, has empty interior, and so also does S,. Thus X,\S, is dense in X, andsoin X=X,. THEOREM 2.17 Let S,, X,,\ S, be forward invariant subsets of the semidynamical system (X,,, m, D, >, and suppose that S, is closed with int S, = 8. Assume that X, is relative& compact. Let P: X,\S, + R, be continuous, strict& positive, and bounded. Define

o(t,u)

P( u.t) = liminf ___ (UEXr), V’U P(u) WEx,\s,

21

PERMANENCE

and suppose that

supa(t,u) t>o Then there is a compact

>

1

(UE +q),

0

(UES).

l

(2.4)

absorbing set M for X0\ S, with SE< M, S) > 0.

Proof: It is clearly enough to confine the analysis to the compact subspace X = x, of X0 and to prove that M is absorbing for X \S. Observe next that a(t,.) is lower semicontinuous. For t, t’ > 0 and z4E x,

a(t + t’,U) = liminf U’U

P( u.( t + t’)) P( WC)

LJEXl\SI

P( u.t) . P(u)

(2.5)

> a(t,u)a(t’,u.t).

Therefore

for any ti> 0 (i = 0,l,...,k),

The first step is to prove that the first part of (2.4) holds for all u E S, and as a preliminary we claim that given u E S and T > 0, there is a to a T such that a(t,,u) > 0. For if not, for some u E S, 7:=inf{t,:

a(t,u)

= 0,

t 2 to}

is finite, which in combination with the second part of (2.4) implies that T E (0,~). Hence there is a nondecreasing sequence {t,} with t, 7 T such that a(tn, u) > 0; the possibility t, = T for all n is allowed. By (2.4) there is a t > 0 such that cx(t, u.7) > 0, and by the lower semicontinuity of a(t;) there is a neighborhood lJ of LLT such that a(t,u> > 0 for u E U. Now there is an n, such that for n > no, both u.t, E U (by continuity of W) and t, > T - t. But by (2.5), a(t,

+ t,u)

2

a(t,,u)LY(t,u.t,)

Since t, + t > 7 for n 2 n,, this contradicts establishes the claim. Take any u E S, and define the sets U(h,t)={uEX:

a(t,u)>l+h}

> 0.

the definition

(h,t

> 0).

of r, and

22


By the lower semicontinuity of a-(t, -), each U(h,t) is open, and by (2.4) they form an open cover of o(u). However, since S, is relatively compact, w(u) is compact, so there is a finite subcover. It follows that there are a finite set F c (0, m>and an ?r > 0 such that

u U(h,t):=W.

o(u)c

t6F

By the definition of *limit set and the conclusion of the previous paragraph, there is a t, such that u.t E W for t 2 t, and T:= cw(t,, U) > 0. Choose n such that

(2.7) Define inductively the sequence_{tJ with ti E F for each i by arranging that u.0, + C,+ *a-+ tj_ ,) E UN, ti) for each i a 1. Then by (2.6) and (2.7) a(t0 + .‘.+r,,U)~l).(l+h)n>l. It follows that as asserted supa(t,u)

> l(” E S).

t>o

Since S is compact, we may conclude by repeating the covering argument of the last paragraph that there is a finite set G c (O,m) and an h* > 0 such that SC

U U(h*,t):= tse

W,.

Since WI is open, there is a closed neighborhood V of S with V c W,. With N = X\V, it will be shown that A4 = y+(N) is the required absorbing set. In fact this will follow from Lemma 2.10 if it can be shown that given u E X\S there is a t > 0 such that ~.t E N. If this is false, u.t E V for some u and c 2 0. As previously, we may define an infinite sequence (ti) with ti E G by requiring that u E U(h*,&),

u.&, E U(h*,t,),

Hence from (2.6), for each II, P(u.(r,+--+t,))>(l+h*)“+‘P(u).

u.t,~lJ(h*,t,+tZ);--

23

PERMANENCE

This contradicts the boundedness of P on letting II +CC and completes the proof of the theorem. The following theorem, which is a generalization of a result in [3], may be regarded as a partial converse of Theorem 2.17. Its main use here is in establishing conditions for a system nor to be permanent and thus in obtaining necessary and sufficient conditions for permanence. Its proofs may be supplied by making minor modifications in the proof of [56, Theorem 2.71. THEOREM 2.18 With the dynamical system as in Theorem 2.17 define p(t,u)

P( u.t) = limsup ~

“‘U

P(u)

*

u E x,\s, Let U be a subset of S such that 0.t c int UCt > 01, where the interior is taken in the relative topology of S. Suppose

inf p(t,u)

f>O

z.4E WA9, (ii) there exists an orbit y(u) with a,(u)

c M.

A4l,...,Mk is called a chain if M, -+i%f2 --f .** +M,(M, k = 1). The chain is a cycle if Mk = M,.

Definition

-+M,

if

2.20

w(a Y,) is said to be acyclic if there exists an isolated covering U t= ,M,, such that no subset of the M, form a cycle, the covering then being called acyclic. THEOREM

2.21 [48]

Let Y = YOu JY, be a complete metric space with YO open, and assume that YO and c?Y, are forward invariant for the semidynamical Jystem

FIG. 2. M is chained to N since w(u)c need not be contained in M.

N and (Y,,~(u)c M. Note that

aTz(u)

PERMANENCE

25

(Y,n,lR+ ). Suppose that: (8 There is a t, > 0 such that m(*, t) is compact for t > t,; (ii) m is dissipative; (iii) o(aY,,) is isolated and acyclic (see Figure 3).

Then the flow is permanent

if and only if for all M, W”(M,)nY,=&

(2.8)

The necessity of (2.8) is obvious. For sufficiency we use a contradiction argument and show that if permanence does not hold, every isolated covering is cyclic. Throughout the proof repeated use is made implicitly of Theorem 2.12 guaranteeing the existence of a global attractor A, which is compact and invariant by definition and which contains O(U) for u E Y. In the following lemma we envisage the failure of permanence as discussed at the end of Section 3. LEMMA

2.22

Let M be an isolated invariant set, and let y,, = y+ (u,,) be a sequence of relatively compact semiorbits with w-limit sets w,:= w(u,,), the possibility being allowed that the u, are identical. Suppose there is a sequence (p,} with p,, E w,,\ M such that p, + M as n + 03. Then given any neighborhood V of M, there exist subsequences, indexed still by n, {q,), {r,,}, {w,,} with q,, r,, E w,, and q, + q, r,, +r, ~,,+wasn-+m,and qEW(M)\M)n(wnv),

rE(lP(M)\M)n(wnV). Proof We prove the result for the *limit sets, the remainder being similar. Let K be the set of nonempty compact subsets of w(Y) with the metric d. Since A and w, are compact, then on E K and there is an o E K such that lim, em d(w,, w) = 0 for some subsequence {w,}; w is evidently invariant. Take U to be an isolating neighborhood of M, and choose an open V such that M c V c v c 17.Since p, + M, p, E V for large enough n. Since pn is in the invariant set o,, there is an orbit y(p,Jc o,, and it follows that there are sequences {q,} with q, E dV and {r,J with r,, > 0 such that q,.t E V(t E (0, r,)) and qn.Tn = pn (for otherwise o,(p,) c V c U, which since p,, E M contradicts the maximality of MI. As q,, E w, c A, there is a convergent subsequence still denoted by {q,) and qEaVnowithq,+qasn+m.FurtheruGM. If (7,) is bounded, there is a convergent subsequence, still denoted by (7,) with lim 7, =r, n-m

26

VIVIAN

HUTSON

AND KLAUS

SCHMITT

say, and 4.7 = lim qn .T, = lim p, E M.

Hence

u(q)

c U so o(q)

= w(q.7) c A4 c M,

and q E WS(M)\M. If (7,) +w, y+(q) C v as otherwise the maximality of M is contradicted. In

both cases u(q) c M, q E M. Proof

of Theorem

2.21.

If permanence

does not hold, either

(a> There is a u E Y, with w(y) n aY,, f d, or (b) There is a sequence {u,} with u, E Y, such that there sequence {v,} with u,, E CO,,and un + CRY,.

is a

In case (a), by compactness of o(u), there is a v E JY, n w(u). In case (b), there are subsequences {v,,l,{w,) with u,, E w,,, u, + u, wn + o such that u E aY, n o. In either case w(u) c w(a Y,>, so there is an M,, say M,, with W(U)n M, # d and w(u) c o (as w is forward invariant). Thus there is a sequence {p,) with p,, E w,,, pn --f M, but p,, 4 M,. AS the M, are disjoint, it follows from Lemma 2.22 that there is a q1 E W”( M, )\ M, but q, CCu M,. Since by (2.8) W”(M,)c dY,, then q, E G’Y,. Now q, E w, so there is an orbit y(q,)c w, and by the forward invariance of Y,, y(q,)c c?Y,. Thus ay(ql) is contained in both a,(q,)

FIG. 3. The situation described by lemma 2.22.

27

PERMANENCE

and

UM,,so

It follows that there is an M,,, say M,, such that a,,(q,)fl M, # d, or to put it another way, q, E W,“(M,). We now need a result, roughly a “backward” version of Lemma 2.22 with all o, = w; for details of the proof see [20]. LEMMA 2.23

If u E W,“(M)\W”(M)

then a,(u)

n W”(M)\M#kf,

CX,(U)~W~(M)\M#~?.

To continue with the proof of the theorem, we must consider two possibilities: 6) ay(q,> is not contained in any M,,, and (ii) ay(q, > c M2. Case (il. Then q, E W,“ c w, so a,,(q,) c w as o is closed. Therefore, by Lemma 2.22 there is a q2 E Ws(M2)\ M2 with q2 6i U M,. Repeating

28

VIVIAN


the argument that starts the proof of the theorem (with q, replaced by return to case (3 or remain in case (ii). Whichever possibility holds, a cycle is eventually achieved. This contradiction completes the proof of the theorem.

q2), we either

2.6.

PERMANENCE FOR SET-VALUED

DmAMICAL

SYSTEMS

An outline is now presented of a dynamical systems approach for tackling permanence for the system of differential inclusions (1.7); for the details see [661. The basic idea is that solutions of systems of differential inclusions form a “funnel” in phase space, so the corresponding semidynamical system must be set valued to capture this behavior. With this modification an average Liapunov function result analogous to Theorem 2.17 may be proved. Let (X,d) be a metric space, and let A(X) denote the set of nonempty subsets of X. Definition 2.24

The map f: X + A(X) is upper semicontinuous if given any u0 E X and any open set U of there is a neighborhood V of u,, such that f(U)CU for UE V. Consider the map .+zr:X X [w, * AX),

and for U c X, I c [w, put

It will be assumed that the set valued semidynamical system (X, r, R, ) satisfies the following axioms analogous to those of Definition 2.1. (i) 7r(u,O> = u. (ii> 4lr(u,t,),t,> = 7&t, + t2) (t,,t, E R, >. (iii) 7r is upper semicontinuous and compact valued. The semiorbit through u is defined to be y+(u) = {v: u E T(U, t> for some c > O}, and for U a set, y+ (U> is defined by taking unions. U is forward invariant if y+ W> c U. In the following theorem X is thought of as a part of rW: and S as X n alR=. As usual, dissipativity is required for the flow, and it will in fact be assumed simply that X is compact. THEOREM 2.25

Assume that X is compact, and let S c X be compact interior. Let S, X \ S be forward invariant.

with empty

29

PERMANENCE

Suppose that P: X --, R + is continuous and P- ’ (0) = S. Assume finally that the following holds.

sup liminf v+u wJ$“,,) I>0 UE x\s

P( w> P(u)

Then there exists a compact set M c X/S is a t, such that ~T(u, t) c M for t & t,.

> I(” E s).

(2.10)

such that given u E X \S there

The technique for proving this result is broadly similar to that used in proving theorem 2.17, but rather more careful analysis is needed. Condition (2.10) may be thought of as ensuring that for the “worst possibility” orbits are repelled by S. A simple predator-prey example has been treated in [66] using this theorem; however there remains much scope for further investigation in this direction. 2.7.

CONCLUDING

REMARKS

The two techniques described here produce identical results for a wide class of models based on ordinary differential equations. However, in some cases one but not the other technique works. The best illustration of their contrasting merits is a comparison of the models described in Sections 4.2 and 5.1. When the phase space is not locally compact, the situation is not as yet clear, since only a few applications have been successfully treated. Recent work ([381, [39], [53], [57]) places the study of permanence in a more general setting and shows how the approaches may be unified; an application is given in [73].

3.

SOME SPECIAL DYNAMICAL

SYSTEMS

3.1. INTRODUCTION

To apply the abstract results of Section 2 to specific models of the type described in Section 1.2, it is necessary first to show that solutions generate a semidynamical system, and second to establish the required compactness. For models governed by ordinary differential equations (1.0, the first objective is readily achieved via an elementary standard existence theorem mentioned in Section 3.2. The compactness is automatic since the phase space is finite dimensional. However, the system (1.5) of reaction-diffusion equations is essentially infinite dimensional, and matters are not at first sight quite so simple. In fact, there is now a well established theory for dealing with such systems, and the main aim of this chapter is to give an outline of this. This is contained in Sections 3.3 to 3.5; Section 3.6 gives a brief outline of results for another infinite

30

VIVIAN


dimensional system-of differential delay equations. The basic idea used here in tackling reaction diffusion equations is to regard the solution u(x,t) for each t as a point in a certain Banach space E, say. Thus we write u(x, t) as u(t) where U: R, --f E, that is u is a Banach space valued function of t. The Laplacian A cannot be defined sensibly on the whole of E, but a careful choice of domain yields a closed, but not continuous, operator A corresponding to A. We may then write (1.6) in the form

$+Ad=f(u), a system of ordinary differential equations in the Banach space E. The procedure, covered in Section 3, is to show how a linear semigroup theory paralleling the elementary theory for ordinary differential equations may be developed if A is “sectorial,” a condition satisfied by the Laplacian and indeed a wide class of elliptic operators. The hard work comes in proving the sectoriality, but the results are standard and we merely quote them here. The remainder of the analysis is then not difficult. In Section 4 the abstract results of Section 3 are applied to reaction-diffusion systems. It is finally shown in Section 5 how a priori bounds may be established and thus global existence proved. 3.2.

ORDINARY DIFFERENTUL

EQUATIONS

Let f: U c IL!” --, R” be a mapping that is locally Lipschitz continuous

on the open set U and consider the system of ordinary equations l.i=f(u),

u(0) = then it is [24]) that value u,, condition.

u() E

differential

.=-g

u;

well known (the theorem of Picard Lindeliif; see, e.g., 1491or this problem has a unique solution u(t,u,J for each initial and that this solution depends continuously upon the initial Hence if we define

r(qj,t)

= u(t,uo),

then, if global existence holds, the triple (U, 72,R, > will be a dynamical system, where we regard U as a metric space with metric induced by the norm in R”.

31

PERMANENCE 3.3.

SEMIGROUPS OF OPERATORS.

In this section we wish to demonstrate how certain systems of partial differential equations (systems of reaction diffusion equations) yield semidynamical systems in appropriate function spaces and hence are amenable to the application of the abstract results developed previously. Thus we shall be able to give permanence results for systems of predator-prey equations for populations that are allowed to diffuse throughout their habitat. To set up the dynamical systems we shall need some results about semigroups of operators and their infinitesimal generators from linear functional analysis. We shall briefly review the necessary topics here to make this paper partly self contained. For many of the omitted details we refer the reader to the semigroup treatments in [37], [50], [85], [95], and [99]. Let E be a Banach space with norm )I~11 and let A be a closed linear operator on E with the domain of A, B(A), dense in E. Such an operator is called a sectorial operator, if for some 4 E (0, 7~), for some M > 1, and some a E R, the sector of complex numbers S o,g={h:

4Glarg(h-a)lGrr,

h#u)

is in the resolvent set p(A), and the resolvent satisfies

A family {T(t): t > 0) of bounded linear operators from E to E is called an analytic semigroup provided the following conditions hold: (1) (21 (3) (4)

T(0) = I, the identity mapping, T(t)T(s) = 7-0 + s), t, s > 0, T(t)u-,u as t-,O+,all UEE, The mapping t - T(t)u is real analytic on (0,~) for each u E E.

The infinitesimal

generator Lwli:+

of such a semigroup is defined by: f(T(r)u-u),

with L8( L) = {u E E: above limit exists in E) . We write T(t) = eL’.

32

VIVIAN

HUTSON AND KLAUS SCHMI-IT

THEOREM 3.1.

If A is a sectorial operator, then -A is the infinitesimal generator of an anabtic semigroup {e-A’: t z 01, where e-A~

&lg(A+ A)-‘eA’dh,

_ -

and %Tis a contour in p( -A)

with arg A -+ f 0 as I Al -+ ~0for some 8

may be analytically continued into a sector

{t ZO: largtl
a,(i.e.,Reh>a,

VAE~(A)),

then for t > 0

for some constant C. Furthermore d

-AI

=

_

A~-A’

ze

3

t >

0.

If A is a sectorial operator and Re (T(A) > 0, one defines fractional powers of the operator A as follows: For any (Y> 0 A-*

= ,_(L)

the

Oxt”le-A’dt, 1

where r is the Gamma function. THEOREM 3.2

Zf A is a sectorial operator on E with Re a(A) > 0, then for any (Y > 0, the operator A- a is a bounded injectiue operator that satisfies: /“A-P

=

A-“-P 3

It follows from Theorem 3.2 that for any such operator A the operator A-” has an inverse that we denote by A”, with domain

33

PERMANENCE

given by the range S’CA Pa 1. The next result collects some needed properties of fractional powers of operators.

HA”)

THEOREM 3.3

Let A be as in Theorem 3.2, then the following properties are valid:

(1) If a > 0, A” is a closed and densely defined operator. (2) (3) (4) (5)

Zf a > B, then g_(Aa)c9(AP). Powers of A commute on their common ae-AI _ - eWA’A”, on B(Aa). A Zf Rev(A)> 6 > 0, then

IIAaemAfl( < C,teaemS’, where C, is a constant. (6) A-’ is compact if and only if A-” and only if e -A’ is compact for t > 0.

domain of definition.

t > 0,

is compact,

for any a > 0, if

If A is a sectorial operator on the space E, one defines for each a >, 0 the fractional power spaces E” as follows: E” = “((A

+ u)~),

with llulla = lI( A + a) uuII,

where a has been so chosen that Re a(A + a) > 0. Different choices of a will generate equivalent norms on E*. THEOREM 3.4

If A is a sectorial operator on the Banach space E, then E* is a Banach space when endowed with the norm II*11 a, for any a z 0, (here, of course E” = E) and for a > B 2 0, E a is a dense subspace of E p with continuous inclusion. If A has compact resolvent, then the inclusion is compact, whenever a > B z 0. Let A be a sectorial operator on the Banach space E, and let u. E E. Then the function u(t) = e-Afuo solves the differential equation

II

$+Au=O,t>O

u(O) =uo,

(3.1)

34

VIVIAN


in the following sense: (1) u E c([oA,E>n cWo,4,E). (2) vt E (O,@J),u(t) E 53(A). (3) Vt E (O,w), 9 = - Au(t). (4) lim, ~ 0+ u(t) = uo. Furthermore emA’u,, is the unique solution of this problem. On the other hand, if f: (0, a 1 + E is a function that is locally Holder continuous and is locally integrable near 0, that is,

then in a similar sense, the function

u(t)

solves the differential

= epA’uo +

L’e-nc’ps)f( s)ds

(3.4

equation

(

$+Au=f(t),t>0

u(0) = uo

(3.3)

uniquely; that is, a variation of constants formula holds. If the forcing term f satisfies additional smoothness conditions, then more can be asserted about solutions, namely we have the following theorem, whose proof we give for the sake of later calculations. THEOREM 3.5

Suppose thatf: [O,T]-, E” with supsEtO,,,llfII 6 > 0. Assume that u. E E”. Then, for any /3 < (Y + 1, we have that u(t) E E p, t > 0. Furthermore,

given t, > 0

where M,, M2 are constants depending only on t.

PERMANENCE Proo$

II+)

35

It follows from (3.3) that IIP= IIA %r) II

4lA De-Al, 11 llAaf(s)

$Cpt-"e-bfllu,ll+~fllA"-ae~A'f-s)

IIds

Since /3 < cr + 1, the integral converges, and we obtain (3.4). If A- ’ is compact, then using Theorem 3.4, we immediately obtain the following compactness result. COROLLARY3.6 Let the conditions of Theorem 3.5 hold and assume that (Y < p. Then if A has compact resolvent and u0 E E e, the set (u(t): t E [O,a)} is relatively compact in Ea.

We next consider initial value problems containing nonlinear forcing terms; in fact we shall only consider autonomous forcing terms, i.e., we shall be interested in the initial value problem $+Au=f(u),t>O

(3.5)

u(0) = ug.

The following theorem may be proved by the contraction mapping principle; the proof is rather straightforward once one accepts the results just stated. THEOREM3.7 Let A be a sectorial operator with Re o(A) > 6 > 0. Let (Y E [0,1) and let f: U c ELI -+ E be locally Lipschitz continuous and map bounded sets into bounded sets. Then for any u0 E U, there exists a maximal T = T(u,) such that the initial value problem (3.5) has a unique solution u: (0, T) + E =. That is, there exists a unique u E C([O,T), satisfjkg T is a semiaynamical system, where we regard U as a metric space with metric induced by the norm of E”. 3.4. PARTIAL DIFFERENTIAL

EQUATIONS

In this section we shall apply the results of the previous section to formulate systems of reaction-diffusion equations as semidynamical

PERMANENCE

31

systems. Let Sz c [w" be a bounded domain with smooth boundary, an, of class C2*y, 0 < y G 1. Let f: IR” + 58” be a function that is locally Lipschitz continuous and let D = diag( p,, . . . , p,,) be a constant n X n matrix with lLi > 0, i = 1,. . . , n. We consider the system of reaction diffusion equations

$=P~Au;+J(u),

XE:R,

t>O (3.6)

“i -

dv

XEdfl,

0,

t >

0,

i=l ,*.*>n.

This system we may write for short in vector notation as follows:

s=DAu+f(~), -= d” du

XEfl,

t>O (3.7)

0

’

XEdR,

t > 0,

where -& is the normal derivative operator with respect to alR. To use the development in the section preceding, to rewrite (3.7) in abstract form, one employs the LJ'theory of such equations. Solutions are sought among those functions all of whose components have the property that they, together with their generalized derivatives up to order two, belong to LP(fl). (That is, they are elements from the Sobolev space W**P(0).) Let p>m anddefinefor i=l,...,n

9(Ai)=

{lJEW*.p(iq: $0,

xEm),

where normal derivatives are understood in the sense of traces. The Laplace operator is defined in a distributional sense on a(AJ. Let Ai: 9(Ai)+ LP(fi)be defined as Ai = - pi A. Letting E =(LJ'(fl)Y = LP(fl,lFt"), and A = lly=,Ai: ll;==,NAi)+ E, then A is a sectorial operator (since each A, is; see [50]) and thus defines a family of fractional power spaces (E"},a a 0.If f G (YG 1, then it follows from an embedding result in [50, Theorem 1.6.11 or [371 that

38

VIVIAN


that is, the space Ea is continuously embedded in the Holder space Cp, where 0 Q p < 2 cx - 2. We may therefore define the Nemitskii operaP tor f: E”-,E bY

f(u)(x) = f(u(x)),

u E E",

x E fi

(we use the same letter f for both the function and the Nemitskii operator it generates, since no confusion will arise) and conclude that f satisfies the conditions of Theorem 3.7. Now consider the evolution equation

$+Au=f(u), which is the abstract form of (3.7). We have thus arrived in the setting of Theorems 3.7 and 3.8 and may apply the theory of semidynamical systems to initial boundary value problems of the type (3.7), and, in fact, because of the continuous embedding obtain classical, that is, smooth, solutions of the problem. We emphasize here, as follows from Corollary 3.6, that if {&~~,t): t > 0) is a semiorbit of Equation (3.8) that remains in a bounded subset of Ea, then this semiorbit is precompact in E”. We consider also the case that E = C0(6,Rn)=(Co(fiV. In this case the Nemitskii operator determined by f is much easier to handle and the existence theory for the nonlinear initial boundary value problem has also been developed in detail (see e.g., 1833 and [98]). In this case one takes as the domain of A the domain we have chosen here, subject to the restriction that Au E E, in which case A will be a sectorial operator and it is the case that for arbitrary initial conditions chosen in E the initial value problem for (3.8) will have a unique classical solution. Furthermore, bounded semiorbits will be precompact in E and as was done in the foregoing discussion, one may define a semidynamical system on a compact metric space provided suitable a priori bounds are available. 3.5.

COMPARISON RESULTS-

A PRIORI BOUNDS

In this section we shall collect some results that will be of help in identifying invariant regions for systems of semilinear parabolic equations. In particular we shall consider again system (3.6) written in vector

39

PERMANENCE

form as (3.7) and we shall assume the requirements imposed there. In establishing such invariant regions results the following maximum principle (see e.g. [87], [971, or [loll) for scalar equations will be of use. LEMMA

3.9

Let D, = 0 X(0, T) and let c: D, + R be a bounded function. Suppose u satisfies g u(x,O) 1

< dAu + c(X,t)u, d 0,

in D,,

d>O,

in a, on JRx(O,T).

s=O,

continuous

Then u(x, t) < 0 in D, and ifu(x,O) in Dr.

(3.9)

< 0 for some x E a, then u(x, t) < 0,

This result has an immediate consequence all diffusion coefficients di = d.

for systems (3.7) provided

THEOREM3.10 Consider the system (3.71, the vector form of system (3.61, and assume that all diffusion coeficients di, 1 < i < n are equal to a constant d. Let there exist a closed convex set K c R." such that for every v E dK there exists an outer normal vector n(v) with the property that n(v)*f(u)

G 0.

(3.10)

Furthermore assume that uO: h+

K.

Then if u is the solution of (3.7) with initial condition uO, it follows that u(* t): fi --, K, for each t in its internal of existence. Proof Let u E JK and let n(v) be the corresponding outer normal vector such that (3.10) holds. Since K is convex, it follows that Kc{u:

n(u)*(u-u)

GO}

and further K=n

vEdK{~: n(u)(u-u)GO}.

Hence, we merely need to show that n(u).(u(X,t)-u)

GO

40

VIVIAN

for each such u. Let w(x, t) = n(v)Mx,


t> - v>. Then w satisfies

~~dAw+n(v).(f(u)-f(v)), w(x,O) GO,

in fI,

dW

dy=O,

on dCIx(O,T).

One now treats first the case where int K #d and the initial condition belongs to int K. The more general case is treated by an approximation argument. We refer the interested reader to [4], [lo] and [103]. If it is the case that the diffusion coefficients are not necessarily equal, we may obtain a result of the type just discussed, provided we restrict the set K to be (a not necessarily bounded) closed parallelpiped whose faces are parallel to the coordinate planes. THEOREM 3.11

Let the conditions of Theorem 3.10 hold, except that we do not assume that the diffusion coefficients are all equal. Let K be a closed parallelpiped with faces parallel to the coordinate planes. Then the conclusion of Theorem 3.10 holds. Proof Let {uE[W”: ui = /I} contain one of the faces of K and assume that K lies to the left of this face (in case K lies to the right a similar argument may be used). Then condition (3.10) is simply fi(u I,...,“j-I,P,ui+*,...,u~)

Go7

for all C=(u, ,..., u~_~,~,u~+~,..., u,,) E K. Hence, if let wi = ui - p, we obtain that wi satisfies (3.9) for an appropriate bounded function c and d = di, hence yielding that wi < 0. A further easy consequence of Lemma 3.9 is the lemma of Nagumo-Westphal, which we state here for the sake of completeness. LEMUA 3.12 Let g: IF&! + R

be locally Lipschitz continuous and let u, v be functions

that satis& XEfi,

$-dAu-g(u)+dAu-g(u), u(x,O) Q u(x,O), /

d7J -=dv

t > 0,

XEf2,

au dv

’

XEdfl,

t > 0.

(3.11)

41

PERMANENCE

7’hen u(x, t) < u(x, t), for x E CI, t > 0, with strict inequality, whenever 74x, 0) < 4x, 01, for some x.

There is an extensive literature on invariance results of this type; we refer to the papers [4], [lo] and [lo31 and the books [97], [loll. We end this section by proving an extension of a lemma of [l] that will be needed to obtain Lx bounds on solutions (see Section 5.3). The proof we give here is due to Treibergs [loo]. LEMMA 3.13

Let u be a solution of the scalar equation f

0,

XEdR, XEfl,

(3.12)

where c is a locally Lipschitz continuous function that is bounded above by a constant a. Then there exist a constant K, depending on sup, r ,,\I& t)iiL,I and Ilu(.,O)ll~~, and a constant K~ depending only on the first of these quantities such that

that is an L” bound for u follows from an L’ bound. Proof

constant satisfies

It follows from Lemma 3.9 that u(x, t) > 0. Let a be a so that c(x,t) < a, for all (x,t), and let u = ePnru. Then u

$=dAu+(c-a)u,

and hence by Lemma 3.9 it follows that

and hence (3.13)

42


On the other hand, if we denote by w the unique solution of g=dAw,

x~fi,

x E 0,

w(x,O) = e-“u(x,s), dW

x=0,

i

t>o,

XEdfi,

s > 0,

t>o,

(3.14)

then it again follows from Lemma 3.9 that u(x,t+s)o

this estimate with (3.13) we obtain the desired result. like the preceding may also be generated by using general analytic semigroups on L’(R) that have nice regularity Such results may be found in [5].

EQUATIONS

WITH FINITE DELAY

In this section we shall briefly discuss how differential delay equations involving a finite delay generate semidynamical systems. This theory is again well developed and we refer to [E-161, [441, and [47l for the details.

43

PERMANENCE

Let T > 0 (the delay) and let E = C”([ - T,O], Rm) equipped with the maximum norm. If U: [- r,T) + R”,T > 0 is a continuous function we denote by u,, 0 G t < T, that element in E defined by

ut( 0) = u(t + fl),

-T 0, and the system (4.1) is permanent if

;>u:@u.s)ds>O(uE

w(S)).

(4.4)

An application of this result to a three-species problem in ecology illustrates many of the key features of the method. Realistic versions of such problems may not fall naturally into any of the simple classifications, for example two prey-one predator, usually used in the literature. For example the interaction between two species could be positive (mutualistic) at certain densities and negative (competitive) at others. In fact, quite a considerable amount of progress on the permanence question can be made under minimal assumptions on the fi, see [62]. The representative example here may be thought of as a generalization of the three-competing-species system of [81], but the usual strong assumptions, such as dfi /duj < o(i f j), are not required. Consider then the system (4.1) for i = 1, 2, 3, and suppose the following hold: (Cl) There is a K such that h(x) < 0 (i = 1, 2, 3) if xi > K. (C2) There exist k,, k,, k, E (0,~) such that

with similar requirements for k,, k,. The points K, = (k,,O,O) etc. are equilibrium points, and the ki are the carrying capacity of species i. CC3 P12, &,

Pjl < 0, 021, P32, PI3 > 0, where P;j = fj(Ki)

(C4) There are no equilibria in the interior of a “face.” (Cl) yields dissipativity. (C4) implies that the zero isoclines in each two-dimensional subsystem do not intersect. The conditions taken together ensure that

With P(U) = up1u;*ua3 3 , from (4.2)

47

PERMANENCE

From Theorem 4.1, it is sufficient for permanence if there exist nonnegative (Y~such that $(u) > 0 for u E o(S), and this yields the conditions ff2 62

+

a3 P,3

al

P2,

+

‘y3 P23 > 0,

a1 P3,

+

ff2 P32

C

aifi(o)

> 09

> 07 >

O*

A brief calculation leads to the inequality 62

P23 P31 +

P21 P32 PI3

> ‘.

(4.5)

On the other hand, using the fact that 0 is a source, we may readily apply the converse theorem 2.18 to establish the following. Z’ (Cl)-(C4) hold, the system is permanent if (4.5) holds, and there is an attractor in c?R: if the inequality in (4.5) is reversed.

This is a rather strong result since it yields a necessary and sufficient condition [excluding equality in (4.511 for permanence. The result may be extended to include cases where there is not more than one equilibrium in each face, see [62]. If a face contains two or more equilibria, then as shown in Section 5.1 the necessity no longer holds. This example provides a good illustration of the contrasting merits of the two methods described in Sections 2.4 and 2.5 for establishing permanence; see Section 5.1 for further discussion. 4.3.

DIFFERENCE EQUATION MODELS

It is now well known that even in one dimension the u-limit sets of difference equations may be extremely complicated. Since the resolution of the permanence question in n dimensions requires in general a knowledge of the *limit sets of all the (n - 1) dimensional subsystems, this means that even in two dimensions only limited progress can be made. For the special class of Lotka-Volterra systems, a great deal more may be said; see Section 4.5. Here we examine a pair of predator-prey equations that do not belong to this special class to see how far the analysis may be carried; the discussion is based on [63]. For another approach, see [32]. Consider the system U’ = Au in R’, where (Au),=u,exp{r(l-xl/K)-x2/(1+x,)),

(Au),=u,exp{-1+2x,/(1+x,)},

48

VIVIAN


and r, K > 0. This represents a system with a “saturating” predator. Put cAu)i = Ui exp{fiW for i = 1, 2. It may be shown that the system is dissipative. Further, the *limit set of the x,-axis is just (0,O). For r < 2, the o-limit set of the xi-axis consists of the two points (0,O) and (K,O), but as r increases beyond 2 up to r. = 2.692.. ., cycles of order 2” successively appear. For r > rO, the behavior is extremely complicated, and it appears difficult to make any progress with the permanence question, so we shall consider here only the range 0 < r < r,,. Assume then that for some r < r0 there is an orbit of the restriction A, of A to the x,-axis with period m = 2”. Let this orbit consist of the points with u1 coordinates h,,l,. . . , h,,,. Then choosing P(u) = UFIU,“Z with LYEsufficiently large, it is easy to show that the system is permanent if for each m = 1,2l, . . . . 2”,

It does not seem easy to obtain an analytical criterion for this system of inequalities to be satisfied, but it is not hard to obtain the h,,, numerically (since they are of course the periodic points of the one dimensional map A,), and a diagram of the region of the r - K plane where permanence holds may be constructed. This has been done in [63] for 0 < r < 2.656.e. where there are at most period four orbits. From a biological point of view, it is interesting to note that this region is quite different from that for which asymptotic stability holds. 4.4.

REACTION- DIFFUSION SYSTEMS

Permanence will be considered here in the setting of Co@). In fact, as will be pointed out later, this is not completely satisfactory from the point of view of the biological applications, but the deficiencies in the result will be repaired by an additional argument. To illustrate the method without getting involved in excessive algebraic complications, we shall consider a two prey-one predator model with Lotka-Volterra reaction terms. It is pointed out later that the argument is valid for a considerably more general class of systems. The discussion is based on [64-651. An alternative treatment of this problem is given in Section 5.3. For a treatment of the more difficult case, when zero Dirichlet conditions are imposed, see [21]. Let R c Iw” be a smooth bounded domain in the sense of Sections 3.4 and 3.5. Consider the system (4.6a)

PERMANENCE

49

du,_

dt

du,_ at

- da2 - 4

- EZIU~-c

+ Plu, +

~22~2

132~2

-

(~2~3)

-

~3)

+

+

~2

~3

Au2

Au3

(4.6b) (4.6~)

with homogeneous Neumann conditions. Assume that all the coefficients are constant and strictly positive, and suppose also that all equilibria in alR: of the associated reaction system are hyperbolic. Of course only non-negative solutions will be considered, and it is obvious from the maximum principle and the form of the equations that non-negative initial values lead to solutions that are non-negative for all time. Boundedness of orbits and dissipativity are proved in Lemma 5.1 in Section 5.3, and this result together with Theorem 3.8 ensures that solutions exist for t > 0 and eventually reach and remain within a cube in Co(G) with sides p, say. Thus if P denotes the cube in rW: with sides /3 and one vertex at the origin, the sets

are forward invariant and X0 is absorbing. Observe next that the forward invariance of So and X,\S, are consequences of the form of Equations (4.6) and the maximum principle, respectively. As noted in Section 3.4, the abstract Laplacian A generates an analytic semigroup in C’(fi) and from [83], [98], (A + al)- ’ is compact for some a. Since there is an a priori bound in Co(a) on solutions, each orbit of (4.6) satisfies a linear homogeneous system

for some bounded and continuous F, and it then follows from Theorems 3.4 and 3.5 that X, = 7r(Xo,[l,a)) is relatively compact. The conditions of Theorem 2.17 will thus be satisfied if a suitable average Liapunov functional P can be found. Let 4: %nintlR:+R+ be a strictly positive C’ function. Noting that by the maximum principle if u E X,\S, each of its components is strictly positive on fi, we define the average Liapunov functional

Let “dot” denote differentiation along an orbit. From the partial differential equations (4.6) and the divergence theorem it is easy to

50


show that

where

Finally, define a lower semicontinuous extension of I) to ‘i!?.It is then straightforward to obtain the following analog of Theorem 4.1 for ordinary differential equations-for the details see [65]. THEOREM 4.2

Let 4: ~~iintlR~-+lR+ be a strictly positive bounded C’ function, and assume that (1, is bounded below on F. Suppose that the following conditions hold. 6) sup,,,/,‘dx~~~l(u(x,s))~Jc (ii) Q a 0 (u E X,\S,>.

0 (~4E(u E w(S,)>.

The system (4.6) is permanent in C’(a). Condition (i) is the analog of (4.4). A condition of the general character of (ii> is common in any Liapunov functional approach; see 121. Further results concerning Liapunov theory for reaction diffusion systems have been obtained in [SS], and it may be possible to base a modified approach to the average Liapunov functional technique on these. To apply Theorem 4.2 to the system (4.6), it is enough to remark first that with 4(u) = up’u;*u~ condition (ii) is satisfied, and secondly that the *limit sets of each two-dimensional subsystem consist of the constant equilibria of the reaction system; see [26], [12-131 and 1751. Therefore permanence for the reaction system implies permanence for (4.6). On the other hand, if the reaction system has an attractor in the boundary, clearly so does (4.6). Thus a complete characterization of permanence may be deduced from [67]; see [65] for the details. Note that this argument may be extended to many three-species reaction-diffusion systems of the Lotka-Volterra type since complete results for the corresponding reaction systems are known; see 1541.

51

PERMANENCE

We have considered here permanence in the sense of C’(a). This means that there is an E > 0 such that for initial values, none of which is identically zero, for large enough t maxui(x,t)>E(i=l xe3i

,..., n).

(4.7)

This is not a completely satisfactory condition since it allows species densities to be very small over parts of the domain. However, we can use Theorem 2.15 to remove this deficiency, and show that there is an n > 0 such that for large enough t, min u,(x,t)>n(i=l,..., XGTi

n).

(4.8)

To do this define Y={~~C~(~):u~(x)~OforxER,i=l,...,n}, Y, = {u E Y: for each i, 3~; such that ui( x0 > 0)) dYo={uEY:forsomei,~i(~)=O(_xE~)}. Theorem 2.15 asserts that there is an (invariant) global attractor A, relative to strongly bounded sets. If (4.8) does not hold, for some component ui, there are sequences {ul”‘}c A, and {x,} c n with lim,, ~ x uj”)(x,) = 0. From compactness there are convergent subsequences, still denoted by {ul”‘} and lx,) with ujn) -+ ui E A, and X, 4 x E fin, where UJX) = 0. But A, is invariant, so it is legitimate to go backwards in time. The maximum principle then leads to a contradiction, for ui(x,O) = 0 only if ui(x, - t) = 0 for all x E a. This establishes (4.8). 4.5.

LOTKA-VOLTERR

SYSTEMS

The application of the techniques described requires a knowledge of the wlimit sets of the boundary flow. However, even for ordinary differential equations in dimensions as low as three, locating the w-limit sets can present a taxing problem, and for dimensions of four or more will usually be extremely hard. This is a major obstacle to the study of permanence in biological systems of high dimension, which are clearly extremely important in applications. It is therefore of quite some interest that a great deal more progress is possible with an important, though rather special, class of systems, that of the Lotka-Volterra type. The key to the analysis is a simple observation of Volterra: that orbits enjoy a certain averaging property; see Lemma 4.3. For an n-dimensional system, an application of this result in the (n - l)-dimensional

52

VIVIAN

HUTSON

AND KLAUS SCHMIl-l-

faces reduces the problem to that of considering a system of inequalities &y at the boundary equilibria, a purely algebraic problem. This is the case however complicated the asymptotic behavior may be. To illustrate the outline of the argument we consider the system (1.2) of Lotka-Volterra ordinary differential equations and point out later other systems to which a similar argument has been applied. Define the time averages Ei(t) by setting i.i,( t) =

t-l@(s)

ds.

Take the average Liapunov function P(u) =

fJup.

i=l

Then by Theorem czi> 0 such that

4.1 permanence

ai-

will hold if there exists a choice of

ibijuj(s) j=l

ds>O I

for each u E o(S). In terms of the averages this inequality becomes SUP

~

r>oi=1

ai -

(Yi

[

~

bij~j(t)

j=l

I

> 0.

(4.9)

Consider next the Volterra time average result for a k-dimensional subsystem, which may be written with a reordering of indices as follows: (i=l,...,k)

ai=*i[ai-~lb.juj]

(4.10)

Furthermore, assume for simplicity that there is precisely one equilibrium U* in int Rk+. LEMMA 4.3

For some u = u(0) suppose there exist m, A4 E (0,~) and t, such that mgui(t)GM(i=l

,..., n,t2tt,).

Then lim Z(t) = u*.

t-+m

PERMANENCE

53

Proof Divide each equation of (4.10) by ui respectively, integrate over [O,tl, and divide by t, obtaining

t-’ log[u,(t)/u,(O)]

=ai - 5 bijuj(t). j=l

The result follows on letting I -00. If the conditions of the lemma are met in each subsystem of dimension 1,2,..., (n -0, it follows by taking t large in (4.8) that it is sufficient for permanence if (4.11) However, it may not be the case that the lower bound on the yj hold, and further analysis is needed to prove that the result is still valid. The following theorem is from [68]. THEOREM 4.4

Assume that dissipatiuity holds for the Lotka-Volterra system (1.3). Suppose also that there are real numbers (r,, . . . , a,,, > 0 such that (4.11) holds for every equilibtium u* in dR y. Then the system is permanent. This theorem provides a sufficient condition for permanence of a strictly algebraic nature. It has formed the basis for much attractive analysis [54] and several difficult higher dimensional results have been obtained in [70]-[73]. In fact the averaging property also holds for a class of delay-differential equations [17], and for the Lotka-Volterra system of difference equations (1.4), see [56]. This is a particularly useful observation, for even in low dimensions the w-limit sets of these systems may be very complex. We shall not attempt to survey here the extensive literature on this theme. Rather we close this section with the statement of a lemma from [55] which gives a flavor of the elegant results that may be proved. Let C be the convex hull of the boundary equilibria. Let D be the set where no species increases: D=

UER~: a,~

2 bijuj

for

.

j=l

LEMMA 4.5

If dissipativity holds and C and D are dkjoint, the Lotku- Volterra differential Equations (1.3) and difference Equations (1.5) exhibit permanence.

54

VIVIAN

HUTSON AND KLAUS SCHMIT-I-

This result gives a geometrical condition for permanence; remarkably, in dimensions up to three it effectively provides a characterization of permanence. For further discussion see [.54], [52]. 5. 5.1.

PERMANENCE

BY THE ACYCLICITY

AN ORDINARY DIFFERENToiL

THEOREM

EQUATION MODEL

As a first illustration we take a special case of an example discussed in [36] of a three-competing-species system. Some remarks are made at the end of this section contrasting the model and the results with those for the related model discussed in Section 4.2. Consider then the system li, = z.$f.( u)

(i = 1,2,3)

(5.1)

0 denotes the origin, and “face” will be used to describe the open positive cones in each of the two-dimensional subsystems. The following are assumed: (Hl) All boundary equilibria are hyperbolic. (H2) dfi/duj < 0 for all i,j= 1,2,3. Also f, 0 for i = 1,2,3. (H4) The stable manifold of no equilibrium intersects int RI. (H5) The zero isoclines in the l-3 and 2-3 planes do not intersect, and there are exactly 2 face equilibria P,, P2 in the l-2 plane. It follows from these conditions that there are exactly four equilibria on the axes 0, !?, =(ii,,O,O), 0, =(O,U,,O) and I!$ =(O,O,U,). From standard results for competing species systems, the phase portrait for each two-dimensional subsystem may easily be constructed. In the notation of Section 2.5, take

Each equilibrium is hyperbolic by (Hl) and so an isolated invariant set. M={M,,...,M,) is thus an isolated covering of ti(J!X: 1. Since (H4) holds, condition (2.8) is satisfied. Thus permanence will follow from Theorem 2.21 if it can be shown that M is acyclic. It is a simple matter to deduce this from the phase portraits in the three boundary planes. 0 is a source and so cannot form a part of a cycle. It is obvious that no Mi can be chained to

55

PERMANENCE

itself. M, and M3 are chained, and so are M, and M3. There could thus be a chain M, + M3 --) M2, say. However, it is clear from the phase portrait in the l-2 plane that the cycle cannot be closed, for certainly M, and M2 are not chained, and neither can a chain be formed by including P, and P,. This completes the proof of permanence. If the zero isoclines in Figure 4 do not intersect (so that P,, P2 disappear), then the chain can be closed to yield a cycle M, + M,, + M, + M,. This is precisely what happens in the May-Leonard example (see 1811 and [93]) and the additional condition (4.6) is needed for permanence. A comparison of the two approaches, that using an average Liapunov function, and that based on Theorem 2.21, is well illustrated by this example. For the model discussed in this section where there are face equilibria, it has not so far proved possible to invent an average Liapunov function to establish permanence. On the other hand, if as in Section 4.2 there are no face equilibria, Theorem 2.21 will not yield the result because a cycle appears. We finally recall that in more complicated models a combination of the two approaches may be fruitful; see Section 2.6. 5.2.

DIFFERENTDIL-

DELAY EQUATIONS

We come next to the first of two examples illustrating the application of Theorem 2.21 when the phase space is not locally compact. This

“2

(O,i,)

J

"1

FIG. 4. The u,-u2

phase plane.

56

VIVIAN


example, based on a two-competing-species model with delay, is taken from [48], to which the reader is referred for further details and discussion. Consider two populations ul, u2 for which the governing equations are li,(t)=T,Ul(f)[l-Ul(t-l)-~*UZ(I)], &(t)

= V+(t)]1

- +(t

- 1) - /M(t)]

7

(5.2)

with p,, p-L2E (O,l>. Assume also that r, and r2 are so small that each population grows to the carrying capacity 1 if it is initially not identically zero. With continuous initial data given on [ - l,O], take Y to be the positive cone of (two copies of) C”([ - l,O]). Dissipativity is proved by a comparison argument. Then, (see section 3.6) the equations generate a semidynamical system, ~(a t), which is compact for t > 1. These assumptions ensure that w(JY,) consists of just three constant solutions i&=(0,0), M,=(l,O) and M,=(O,l), say, so M,UM,UM, form a cover of o(J Y,). The next step is to carry out a local analysis of the flow in neighborhoods of the Mi. MO is clearly unstable. The characteristic equation for M, is [A-r,(l-CLI)][h+Tle-*]

=o,

and since pu, < 1, this has a positive eigenvalue with associated eigenvector (0, 1). It may be shown that the other eigenvalues have negative real parts and correspond to eigenvectors in JY,. A similar argument holds for (0,l). It follows that W”(o(JY,)) does not intersect Yo, and it may be readily checked that the Mj are isolated. Finally the instability of MO rules out the possibility of a cycle, so the cover is acyclic, and permanence follows from Theorem 2.21. 5.3. A REACTION- DIFFUSION MODEL The system (4.6) was considered in [26] by a generalization of the method of 1341 and [35] and a form of persistence somewhat weaker than permanence was proved; see also [22]. We show here how a permanence result may be obtained by the use of Theorem 2.21. The setting for the analysis is again C’(n).), and a dissipativity result is first proved. LEMMA 5.1

Assume that all coeficients in (4.6) are non-negative and E,, , lZ2> 0. Then all semiorbits are bounded and the vstem is dissipative if either y > 0 or c 7 0.

57

PERMANENCE Proof

From (4.6a) dU1 dt

f

~40, - E,,u~)+ ~1 Au,-

Let E, be the solution of the ordinary differential

equation

with E,(O) = Ilul(O>ll.Then by Lemma 3.12, Ilu,(t)ll G ii,(t) for t > 0, and it follows that for large enough t, Ilu,(t>ll G 2~2,/E],. Similarly Ilu,(tIl B 2~2,/Q for large enough t. If y > 0 the same argument may be applied to (4.6~). However, if y = 0 this procedure breaks down and we use a method due to [26] based on first obtaining a bound in L’(fl). By this argument, there is a constant 6, depending only on the equations such that for any initial value u(x,O) there is a t,, such that ]]ui(*t)ll G 6, for t 2 t, and i = 1,2, and Ilu,(* till is bounded for t < t,. Take t, as the new time origin, and let u = nzLu, + m2u2 + u-, where mi = pi /ai for i = 1,2. An integration over R yields, after a little algebra, the following differential inequality:

where S, depends only on the equations and the domain a. Since U, and u2 are non-negative, the following inequality is obtained by solving (5.2):

Therefore,

for large enough t,

The required bound in C’(fi> follows from Lemma 3.13. The assumptions on the system (4.6) are exactly as in Section 4.4. That is, we assume that all the coefficients in the equations are strictly positive, that all boundary equilibria of the associated reaction system are hyperbolic, and finally that permanence holds for the reaction system. This last assumption obviously implies that for the reaction system, the stable manifold of each equilibrium in JR: is contained in alR:. Note also that the u, - u2 reaction subsystem cannot have an interior saddle point, for if it does the system is not permanent; see [67].

58

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HUTSON

AND KLAUS SCHMI’IT

It follows from results in [261, [14], and 1751that the attractivity properties in the boundary JY, of the constant equilibria of the reaction-diffusion system in JY,, are effectively the same as those for the reaction system, with obvious minor modifications concerning the initial conditions. Thus for example, if there is an equilibrium (Z,, U,,O) with Ei,U, > 0 for the reaction system, and if u,(x,,O),u,(x,,O) > 0 for some x,, x2 E h, then lim t~z(UI(X,t),u2(X,1),0)=(U1,U2,0) in C”(fi). It follows that w(aY,) consists precisely of the constant equilibria on the reaction system. To establish permanence, with the terminology of Section 2.5 take each A4, to be a (constant) boundary equilibrium. We must show that each M, is isolated, that W”(M,) c aY, for each n, and finally that the covering of o( d Y,> by u M, is acyclic. It will first be shown that the zero solution M,,, say, is isolated; the proofs for the other A4, are similar. If not, there is an orbit different from M,, in every bounded neighborhood of M,. Thus, since a, > 0, there is an orbit and an E > 0 such that a1

-

EI,U,(X,f)

-

QU,(X,t)

for t E R and x E a. Therefore,

--du1

-

‘YI%(X,f)

2

6

from (4.6a), /_qAu, - EU~> 0.

dt

Letting o be the solution of the comparison equation o’ = EW with w(O) = inf, E fi ui(x,O), we deduce from Lemma 3.12 that u,(x, t> 2 w(t) for t z 0, whence u,(x, t) -+ 03 as t + ~0 for each x. This is a contradiction. The same argument holds for u2. The only remaining possibility is that there is an orbit with U, = u2 = 0, and this is ruled out by the observation that uJ = 0 is globally attracting for the u,-equation with u1 = uz = 0. This proves that M,, is an isolated invariant set. We next show that W”(M,,) c JY, for each n. Clearly a minor modification of the preceeding argument shows that W”(M,,l c JY,. Consider next a possible equilibrium, say (0, E,, i?,) = M,, in the interior of the “face” ui = 0, and assume on the contrary that W”(M,)I? Y0 #d. From the assumption of permanence for the reaction system, in a sufficiently small neighborhood of M,, for some E > 0, JUl -

at

-

p,

Au, - lul > 0,

and a similar comparison argument based on Lemma 3.12 shows that u&x, t) is increasing in t. This is a contradiction and shows that W”(M,) c aY,. The argument for an equilibrium on an axis is similar.

PERMANENCE

59

The final step is to show that uM, is acyclic. In view of the preceding remarks concerning the attractivity of equilibria, the argument yielding acyclicity is in effect the same as for the reaction system and proceeds as follows. If there is an equilibrium in the interior of the ur - u2 plane, this is attracting for the interior and so cannot form part of a cycle. This conclusion also holds for interior equilibria in the u, - u3 and u2 - u3 planes. Since there is no equilibrium on the u,-axis, it is then clear that a cycle, if it exists, must lie in the ur - u2 plane and must involve only the origin and the “axial” equilibria P,= (E,,O,O) and P2 = (0, i&,0). However, the origin is a source in this plane and so cannot form part of a cycle. Further, neither P, or P2 is chained to itself. The final remaining possibility P,- P2 -+P, is impossible also. For if there is an interior equilibrium it is attracting for the interior, whereas if no such equilibrium exists, one of P, or P2 is attracting for the interior. Thus the cover is acyclic. This completes the proof of permanence in C’(fii), and the stronger conclusion (4.8) holds by the argument given in Section 4.4.

6. 6.1.

MISCELLANEA EXISTENCE OF A STATIONARY INTERIOR EQUILIBRIUM

If permanence holds, orbits that do not start on the boundary are eventually trapped in an attractor at a nonzero minimum distance from the boundary. A question of some biological interest is whether then there exists a stationary equilibrium state in this attractor where, necessarily, coexistence of all species holds; in mathematical terms “does there exist a fixed point in the attractor”? It has been noted ([63], [54]) that in a variety of contexts permanence does indeed imply the existence of an interior equilibrium. The methods used to prove this have usually been based on appeals to quite deep results concerning the existence of fixed points of abstract dynamical systems; see for example 111, Ch. 51, [44, Ch. 41, and [45]. Here we follow a slightly different approach based on [61] and present two techniques that exploit directly the structure of the phase space. The first approach based on the asymptotic Schauder fixed point theorem is broadly applicable when the flow immediately “smooths” orbits, that is, when m(. t) is completely continuous for t> 0.The second approach based on Horn’s asymptotic fixed point theorem is used when this condition only holds for t a T where r > 0. A few preliminary remarks may help to clarify the broad strategy for those readers unfamiliar with this general area. Consider first a continuous map A: R” --, R“ and suppose that there is a set M, say which

60

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absorbs orbits. If M is compact and convex (it is certainly forward invariant since it is absorbing), a straight application of the Brouwer fixed point theorem yields a fixed point. However, in the present context the difficulty is that while it is easy to construct an absorbing set, this may not be convex; on the other hand it is easy to construct a convex set that is eventually reached by all orbits, but this need not be forward invariant. It is to get around this difficulty that an asymptotic fixed point theorem is used. For continuous flows this argument applied to r(. t> for fixed t > 0 yields the existence of a t-periodic orbit for every t > 0, and the existence of a fixed point follows from a standard result [ll, p. 811. The relevant theorems, those of Schauder [104] and Horn [58], respectively, are as follows, the second being a weakened version that is sufficient for the present purposes. THEOREM6.1 Let U be a nonempty bounded open convex subset of the Banach space E, and suppose that A: E + E is completely continuous. Assume that for some fixed prime p > 2, Aku c U for k = p,p + 1. Then A has a fuced point. THEOREM6.2 Let U, c U, c U, be convex subsets of the Banach space E with U, and U, compact and U, open in U,. Let A: U, + E be continuous and assume that AiU, c U, (j E Zi ). Suppose also that there exists an integer m > 0 such that AiU, c U, for j > m. Then A has a fuced point in U,. THEOREM6.3 Suppose that permanence holds for the set of difference equations (1.3) or the set of differential equations (1.1). Then there exists an equilibrium point in int Rn+. Proof Let A be the (completely continuous) map associated with the difference equations. From permanence there is a compact absorbing set M c int R:. This may obviously be enclosed in an open rectangle U with u c int rW: compact. From Lemma 2.10 there exists a k, such that Ako c U(k 2 k,). The result follows from Theorem 6.1 on taking any prime p 2 k,. The proof for the continuous flow generated by the system of differential equations follows from the preceding remark. In the context of permanence, it is clear that for continuous flows the key feature of the proof of Theorem 6.3 is that the map &a t) should be completely continuous for each t > 0, in which case the proof may be extended to infinite dimensional systems. As observed in Section 3, this holds for a wide class of systems of reaction-diffusion equations, in which case permanence will imply the existence of an equilibrium state.

61

PERMANENCE

This is a trivial observation if the equations are spatially independent since it follows directly from the ordinary differential equation result just given, but it is much less obvious if the equations are dependent on the space variables since the equilibrium state may be nonconstant; see 1211.We shall not pursue this point here but shall turn to a model where the stated condition on 74. t) is not satisfied for all t > 0. Consider then the system of equations (1.6) with finite delay 7 discussed in Section 1.2. Let E be the Banach space C([ - 7,0],Rn) with norm 11.11. Take the phase space Y to be the positive cone (with respect to the usual ordering) of Y and let S be the subset of Y consisting of those u such that ~~(0) = 0 for some i where ui is the ith component of u. Let B(O,r) be the intersection of the open ball center 0 radius r of E with Y, and let Lip(L) be the set of functions in Y satisfying a Lipschitz condition with constant L. It is clear directly from the form of Equations (1.7) that under modest conditions on the right-hand sides, if for some r, T(U, t) E B(O,r) for t > 0, then there is an L(r) such that &A, t) E Lip( L(r)) for t a r. This observation implies the crucial compactness requirement. The permanence question may be treated for a wide variety of delay equation models by the average Liapunov function method even in the infinite delay case (see [17]) and also by the technique of Section 2.5; an example is presented in Section 5.2. Since some boundedness and dissipativity conditions are needed, the following represents a typical set of conditions for delay equations. (i) Ultimate uniform boundedness with bound b. There exists b such that given (Y> 0 there is a t, such that ~(u, t) E B(O,b) if u E &O,a) and tat,. (ii) Uniform boundedness. Given p > 0 there exists C( p) such that m(u,t)E@O,C(P))if u~B(O,P)and t2.O. (iii) Permanence. There exists m > 0 such that for any u E Y \S there is a t, such that for all i, [4u, t>],(O) > m(t > t,), where this denotes the value of the ith component at 0. (iv) Given (Y> 0 there exists L(a) such that if u E &O, (Y+ 1) n L@(L((Y)) then n-TT(u,t)e Lip(L(cu)) for t 2 0. THEOREM 6.4

Under conditions (iIthe delay system has an equilibrium consisting of a constant solution (x, , . . _, x,) E int I%!:.

state

Proof: By (ii) there exists c such that if Ilull < b + 1 then Ilu.t(J < c(t > 0). In (iv) take (Y= b + 1 and put L = L(b + 1). Define M,, U, c Y as

follows. U, = (u: Ilull < c,u E Lip(L)), M,={u: llullm/2

forall i]nu,.

VIVIAN

62

HUTSON

AND KLAUS

SCHMITT

Clearly U, is compact and convex, i&f,, is open in U,, and M,, (its closure in U,) is compact. By (i) if u E M,,, u.t E U2(t a 01, and by (i), (iii), and (iv) there is a t, such that u-t, E_M,. A minor modification of the proof of Lemma 2.10 shows that y+(M,) is compact and does not intersect S. Thus there is an Gr ~(0, m) such that if u E M, then {?T(u,t))i(o) > ci(t 2 0). Put U, = {u: /JuJJG b,u;(O) > fi U, ={u:

Ilull h/2

for all i}n&.

U, is open in U,, and its closure 0, in U, is compact. Again modifying slightly the proof of Lemma 2.10 we deduce that there is a T, such that if u E o,, then u.t E MO for some t a T,. It follows from the definition of Ei that r&, t>i(0) > fi /2 for t a T,. By (i) there is a T2 such that if UE&, JJu.tlJT,.Therefore u.tEU, for taT:=T,+T,. Take now any fixed t > 0 and put A = r(., t). Clearly AjU, c U, for all j, and for jt > T, AjU, CU,,. From Theorem 6.2, A has a fixed point in U,. Since t is arbitrary we may conclude that rr has a fixed point in u,* 6.2.

STABILITY UNDER PERTURBATION OF PERMANENCE

It could be argued that from the point of view of applications one requirement on a model if it is to be satisfactory is that the property of permanence should be robust, that is invariant under a suitably “small” perturbation of the governing equations. The question of robustness raises some interesting but difficult mathematical problems, with which so far as we are aware there has been no real progress. The following remarks concerning the system of ordinary differential equations (l.l), that is tii = Uif,( u),

(6.1)

are thus little more than preliminary observations. Tentatively, we might require that permanence be preserved under small C’ perturbations of the fi. For the examples considered in this account, permanence is robust in this sense. However, this is evidently not true for the one-dimensional equation ri = uf, where f(u) = ~(1 - u). This suggests that a hyperbolic@ condition for the boundary is required, and such a condition is indeed satisfied by the w-limit sets of the boundary in our examples. Some such condition is also a natural requirement on general grounds in view of the results from the theory of structural stability in dynamical systems. Althcugh we can at this juncture say no more concerning the robustness problem, there is a related but weaker concept that can be

63

PERMANENCE

tackled mathematically without undue difficulty, and a result on this from [60] is now outlined. This result concerns the perturbed system ti; = &fj(U)

+ g,(u,t)

(6.2)

and is broadly as follows. If (6.1) is permanent and g is small enough, then all solutions of (6.2) for t > 0 with initial values outside a thin “skin” around alK!‘J are pushed away from c?R:. Note that since it is not required that the gi should vanish on all%:, the condition that orbits should not start too near JR: can obviously not be relaxed. To state this result precisely some notation must be introduced. Let X c R’J be a compact absorbing set for (6.11, and take S = X f? d&Q:. Let A4 be a compact absorbing set for X \S. Define a function p: X + IR, measuring the distance of points in X from S, such that p-‘(O) = S, p(u) G 1 for u E X, and p has Lipschitz constant 1. Let B(a) be the open ball in R’J center 0 and radius (Y.Assume existence and uniqueness of solutions of (6.2) for t > 0, and let u*(t, t,, u,,) be the solution through u0 at t > t,. Suppose that solutions of (6.2) are uniformly bounded, that is, for each (Y> 0 there exists PC(Y)such that if t a t, and u,, E EC(Y) then u*(t,tO,u,,) E B( /3(a)). Define d, = max UE M d(u, O>, r, = max[ PC a), d, 1 and X, = Y’ @(r, )>. Put

The boundary c?R: will be said to be repulsive under perturbations if given CY> 0 there exist pO,r0 > 0, and given l > 0, functions T(E), -y(e) > 0, and T(E) & 0 such that if

then for any t, 2 0, u0 E B( (Y), (9 p(u*k t,,u,N 2 T(E) ( p&J > E, t > toI, (ii) p(u*(t, t,, 24,)) z r, ( p(u,) > E, t > to + 7(e)). THEOREM

6.5

Suppose that solutions of (6.2) are uniformly bounded. permanent, ~3IR: is repulsive under perturbations.

Then if (6.1) is

In outline the proof is based on first constructing a Lipschitz continuous Liapunov function on a neighborhood of S and then using differential inequalities; see [60] for the details.

64

7.

VIVIAN

CONCLUDING


REMARKS

The examples treated in Sections 4 and 5 give a representative sample of the applications of permanence. In this final brief section we may thus discuss more profitably some general questions affecting the broad view that might be taken on permanence. The first questions that come to mind are perhaps in the following area. Granted that permanence and asymptotic stability may be different in specific examples, is there a major difference in the sense that very many systems yield different criteria? And in particular, in this average sense it is easier for a system to be permanent or asymptotically stable? This is a question that might have relevance to the old problem of stability versus complexity of communities; see [SO]. It is indeed the case that for certain classes of systems permanence and asymptotic stability can give answers that are very different. To substantiate this statement, let us describe in outline two examples considered in [6]. The first is a pair of difference equations modeling a predator-prey interaction: xi = x1 exp{r( 1- x,) - x2}, x;=X2exp{-d+bx,}. The condition for permanence is simply b > d, but there are two conditions for asymptotic stability: b < d + 1 and rd(2 + d - b) < 4b. The reader may readily be convinced by drawing a diagram, say with d = 1, that the region of parameter space where asymptotic stability holds is a very small subset of that where permanence holds. The second example is a generalized food chain, where a basic resource is consumed by a predator, itself consumed by another predator that also consumes the resource. Again, asymptotic stability is a much more restrictive condition in this case. These examples may suggest that these two criteria may differ quite considerably for many systems, but it would be pushing the evidence much too far to conclude that one concept is “weaker” than the other in an average sense for all systems. Apart from the rather trivial remark that global asymptotic stability is stronger than both, we know of no theoretical result casting light on this issue, and apart from the article previously mentioned, we are not aware of any relevant calculations. It would certainly be interesting if more was known, and some numerical experiments would seem a good place to start. The next question to be considered is how widely applicable is the method? Clearly asymptotic stability is the easiest criterion to apply for systems of ordinary differential equations (1.1) since it requires only a knowledge of eigenvalues of a matrix. Even when a theoretical treatment is difficult, only a relatively easy computation is required. By

PERMANENCE

65

contrast, global asymptotic stability for the problems of interest here can in general only be done for two species; even for three-species Lotka-Volterra equations the question has not been completely resolved. Furthermore, a computational approach is extremely time-consuming since it requires in each example the calculation of orbits starting at a representative sample of initial conditions. At the conclusion of the calculations one may be left wondering whether indeed the sample is adequate and whether the orbits have been integrated over a long enough time interval. Permanence lies between these two extremes. Broadly, to resolve the permanence issue, it is necessary to have good information about the solutions in the boundary. Thus the technique allows a reduction of one dimension in the mathematical problem, a very considerable gain for many problems. Therefore a great range of three-species problems can be solved for ordinary differential equations and two-species problems are often tractable for difference equations. For systems with diffusion the picture is more complex. If Neumann boundary conditions are imposed as in (1.5), these remarks apply, but for zero Dirichlet conditions (not included in our discussion), it will be very difficult to determine even whether there is an interior equilibrium, far less its actual form. Here permanence is actually easier to apply than asymptotic stability and very much easier than global asymptotic stability; see [21]. Of course, in ecology one would like to tackle problems where many species are present, and here it is in general an intractable problem to determine whether permanence holds. There is, however, an important exception, and this the class of Lotka-Volterra equations. Here, based on the idea described in Section 4.5, there is a sufficient condition for permanence that can be tested by a straightforward computation even for a large number of species. For two or three species (but not more) this condition is also necessary, but it is not known by how much this condition is “too strong” for four or more species. Nonetheless, this idea seems to have interesting implications in ecology (see [741) and further investigation is desirable on the question just mentioned. There is room for a great deal of further research in the permanence area. Apart from the questions just raised, two further areas mentioned seem to require further investigation. The first concerns the question of how close orbits come to the boundary when permanence holds. Do they “often” stay very close to the boundary? This could perhaps be tackled numerically in the first place. But is there a useful theoretical approach? Another very interesting mathematical question is that of robustness; there seem to be some real obstacles to resolving this except in the rather weakened form discussed in Section 6.2. Another area that requires development is that concerning the range of models. It is desirable that more investigation is directed towards

66

VIVIAN


models such as those governed by nonautonomous equations that are clearly of great importance though seemingly little studied. It would be interesting to try and develop further a rather different type of model such as that governed by differential inclusions (1.7). Perhaps of particular importance would be an investigation of stochastic models from the point of view of an appropriate generalization of permanence. In conclusion we remark that a discussion of the broad area of stability in mathematical biology seriously needs reopening, with close interaction between biologists and mathematicians, and if this article leads to an increase in discussion in this area we should be amply rewarded for our efforts. The authors are grateful to Andrejs Treibergs and Andreas Stahel for many comments and suggestions during a seminar series given by the first author while a visiting Professor at the University of Utah, Salt Lake City. The ftrst author is grateful to the Leverhulme Trust for a fellowship in partial support of this project. He has profited much over the years from discussions with Josef Hofbaur and Karl Sigmund of the Institut fiir Mathematik Universitiit wien, and with Richard Law of the Department of Biology, The University of York The second author has profited much from collaboration with IGzysztof Rybakowski, Friebutg and Steve Dunbar, Lincoln, Nebraska, and discussions with Paul Waltman, Emory University APPENDIX B(U, E), &U, co d %A) D E E” y+ (u) l&3 %?A+) Int U aLJ u U\V X,Y Z,E+

1. NOTATION

l) Open, respectively closed e-neighborhoods of

U.

Space of continuous functions with supremum norm. Distance in a metric space. Domain of the operator A. Either R or Z as appropriate. Banach space Fractional space; see Section 3.3. Semiorbit through u. Flow; see Section 2.2. Reals, non-negative reals, respectively. Range of the operator A. Interior of U, Boundary of U Closure of U. Complement of V in U. Metric spaces. Integers, non-negative integers, respectively.

PERMANENCE

67

REFERENCES 1 2

3

4 5 6 7 8 9 10

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