THEORETICAL

Regular

POPULATION

BIOLOGY

and Chaotic

16, 172-190

Cycling

in Models

MARJORIE

A.

Department of Mathematics University of Georgia, Received

(1979)

of Ecological

Genetics*

ASMUSSEN

and Department Athens, Georgia

February

of Zoology, 30602

12, 1979

A model of density-dependent selection is investigated for the cyclical behavior associated with the analogous nonlinear models of population growth. If the population size regulating mechanism reacts too sharply to perturbations in population size, regular and chaotic limit cycles may result. It is established analytically that the population may converge to fixation or an invariant polymorphic gene frequency while the population size undergoes regufar or chaotic oscillations. The possibility of joint limit cycles in both the population size and gene frequency is demonstrated and investigated numerically. Such cycles may occur even though one or both fixation equilibria are locally stable. In the context of equilibrium cycles it is found that overdominance in carrying capacity is not necessary for the maintenance of genetic variation in the population. Furthermore, the genetic system appears able to exert a stabilizing influence on the overall system.

1.

INTRODUCTION

The potential for regular and chaotic cycling within nonlinear deterministic systems dramatically emphasizes the complicated behavior that is possible in even very simple models. Instead, or in addition to, isolated stable or unstable equilibrium states, there may exist stable limit cycles between several points. Even more surprising is the occurrence of completely unpredictable behavior with no discernible pattern. Such complex dynamics have now been reported for a number of systems of difference equations (see, e.g. May, 1974, 1975; Hassell, 1975; May and Oster, 1976; Raveh and Ritte, 1976; Guckenheimer et aZ., 1977; Auslander et al., 1978). The same general behavior can also be found in models with differential and integral equations (Oster and Guckenheimer, 1975). It is a surprising and significant observation that such complicated behavior can be produced by simple deterministic models with no stochastic elements. Two of the simplest models which have been found to produce cyclic behavior are given by the two equations of population growth with nonoverlapping generations N t+l = N, (1 + r - +), * Research

supported

in part

by NSF

Grant

172 0040-5809/79/050172-19$02.00/O Copyright All rights

0 1979 by Academic Press, Inc. of reproduction in any form reserved.

DEB78-09937.

(1)

REGULAR

AND

CHAOTIC

CYCLING

173

May (1974) first reported that as the intrinsic growth rate r is increased above 2, the equilibrium fi = K becomes unstable and the population enters cyclic oscillations between two population sizes, fil and Ra , then stable cycles between four points, eight points, and so on. Finally, for I sufficiently large (roughly 2.6 for Eq. (l), 2.7 for Eq. (2)) the population reaches what Li and Yorke (1975) term “chaos” (i.e., from an uncountable number of initial points, the system does not settle into any finite cycle. On the other hand, it is possible to find initial points that give rise to cycles of arbitrary finite period). The complicated behavior here appears to be due to the use of discrete time difference equations, for in the continuous time differential equation analogs, fi = K is a globally stable equilibrium for all positive birth rates r. A certain amount of theory has been developed to gredict the behavior of single difference equations of this sort (see, e.g., Chaundy and Phillips, 1936; Li and Yorke, 1975; May and Oster, 1976). Similar growth rate equations appear in models of ecological genetics. In particular, in models of density-dependent selection which have been developed to study the interaction between the genetic components of a population and the ecological pressures upon it, the genotypic response to density has often been assumed to be analogous to the logistic growth rate functions in (1) and (2) (Anderson, 1971; Charlesworth, 1971; Roughgarden, 1971; Leon, 1974; Asmussen and Feldman, 1977). It should be recalled that, although cast in discrete generations, the classical one-locus-two-allele selection model with constant viabilities does not admit cycling. Except in the neutral case, it can be shown that starting from any initial gene frequency, the population will converge monotonically to a unique stable equilibrium value. The incorporation of density-dependent selection can, however, dramatically alter the dynamics of the population genetic system. The anomalies that arise when linear logistic fitnesses become negative have been reported elsewhere (Asmussen and Feldman, 1977). The possibility of cyclic behavior has also been predicted (Anderson, 1971; Charlesworth, 1971; Asmussen and Feldman, 1977). The present paper extends previous work by investigating how regular and chaotic limit cycles arise in models of density-dependent selection. It will be shown that the interaction between an inherently stable genetic system and a potentially unstable population size regulating mechanism is very complex and can produce surprising behavior. Whether such behavior is actually found in natural or even laboratory populations remains to be demonstrated. The results here, in fact, provide an additional reason why cycling may not arise. The significant qualitative differences in the conclusions based on the earlier analyses of isolated equilibrium points, as opposed to those taking equilibrium limit cycles into account, are nonetheless important to model building itself, pointing out the shortcomings of a local stability analysis in understanding the behavior of a model.

174

MARJORIE

2.

A. ASMUSSEN

PRELIMINARIES

The potential for cyclic behavior is anticipated in simple models of densitydependent selection on the basis of the observation that there need not be a stable equilibrium, either on the boundary or in the interior, whenever the population size regulating mechanism reacts too sharply to perturbations from the equilibrium values. This is apparent from the equilibrium analysis of the basic model of density regulated selection which, for convenience, is briefly reviewed here (see, e.g., Asmussen and Feldman, 1977). In the general model with two alleles at a single autosomal locus, changes in the gene frequency of allele A,@,) and the total population size (Nt) are governed by the two-dimensional recursion system

Pt WY

Pt+1 = -- w(t)

Nt,,

= NPt)

= .&(Pt 3 Nt), = g,(p, ) Nt).

(3b)

In generation t, the marginal fitness of allele Ai is

WY = Wi(Pt , Nt) = PtW&)

+ (1 - Pt) W$

(i=

1,2)

(4

and the mean fitness of the population is

W’ = iv(pt, NJ= ptwy+ (1- pt)wy.

(5)

The fitness of genotype AiAj in generation t, W$ = Wij(Nt) is assumed to be a function of the changing population size. Associated with each genotype are one or more carrying capacity parameters Kij where W,,(K,,) = 1. In the context of this model, a population state (p, N) is called an interior point only if 0 < p < 1 and N > 0. Those corresponding to fixation, with = 0 or 1, or extinction, with N = 0, are called boundary points. A joint gene frequency-population size equilibrium (4, fi) is locally stable if and only if both

p

and -1 K,, , K,, . For such fitnesses an interior equilibrium exists if and only if there is either overdominance or underdominance in carrying capacity (see, e.g., Asmussen and Feldman, 1977). Z%US, as Edith co~tunt viabilities, there will always be at least ow equilibrium at which the genetic stability criterion is met if Jitnesses strictly decreasewith increasingpopulation size. To be stable in the joint system (3, condition (7) must also be satisfiedat the equilibrium. Extinction, corresponding to N = 0, can never be stable, since population size always increasesat low numbers in this classof fitnesses. For the linear logistic fitnessesanalogousto (l), W(f)= 23

1 .,..L& t3

(6-i = 1, 21,

KG.

(8)

the population size regulating component (7) of the stability criterion is simply 0 < F = pq1 + 2$(1 - $) Tr2+ (1 - 8)” raa < 2,

(9

where i is the meanbirth rate in the population at the equilibrium. Consequently, with birth rates rii above 2 it is possibleto have no stable equilibria, neither on the boundary nor in the interior. Sin.cethe population cannot converge to a single equilibrium state, more complicated behavior must result. For the exponential fitnessesanalogousto (2), W(f)

= 13

e+ij(l-NtlK*j) (i,j

=

1,

21,

(10)

condition (7) is again 0 < iii < 2 at the fixation equilibria, but at an interior equilibrium ($, fl) (7) becomes

The relation between (11) and the magnitude of the birth rate parametersis not immediate but has been explored numerically. Results show that it is possible to have no stable equilibria for (10) as well. The complicated behavior which ensuesin thesetwo modelsis examined in the following sections.

176

MARJORIE A. ASMUSSEN

3. FIXATION LIMIT

CYCLES

The possibleexistence of equilibrium cycles in the population size N with fixation for one of the allelesis an immediate consequenceof the results from single growth equations. With fixation for Ai , the joint recursion system (3) reducesto the single growth equation Nt,, = NP’ii(Ni).

(12)

For the linear logistic (8) and exponential (10) fitnesses,this is equivalent to Eqs. (1) and (2), respectively, with r = rit and K = Kii . As r is increased above 2, there will be cyclic solutions with period 2” which are ultimately replaced by a chaotic regime (May, 1974, 1975). The question here is under what conditions the population can converge to such a limit cycle in the two-dimensionalsystem(3). This issuewill be addressed by determining the local stability properties of the limit cycle. Consider, first, an equilibrium cycle between two distinct population points ($, , fir) and ($a, fia), where p, = $r and N, = Nr implies p,, = 8s and N,,, = fir, impliesp,,, = $I and N,,, = fir , and so on. The analysisof a two-point cycle hinges on viewing the two points, (jl , fir) and (4s , fis), as equilibria of the two-step recursion system

which is obtained by iterating (3) over two generations. For the linear logistic fitnesses(g), the system(13) possesses three equilibrium solutions for N correspondingto fixation for A, . The two solutions

fil , rcr, = 2

(2 + 12

Yjj

&

(Yfj

- 4)1/Z),

exist provided rii > 2. These determine an equilibrium two-point cycle in N with fixation for allele Ai . The equilibrium & = Kji of (3) is alsoa solution of (13), but representsa degeneratecycle of period 2, and hence will not be considered a valid limit cycle. For the exponential fitnesses(lo), the nondegeneratetwo-point cycles cannot be readily obtained explicitly, but the population sizesare given implicitly by the roots of 1 = W,,(N) @‘~,[NW~,(N)Iv (15) which reducesto

REGULAR

AND

CHAOTIC

CYCLING

Using (15) it can be shown that the points on a two-point satisfy

177 cycle, fi, and 20, ,

and hence lvii = (& + #,)/2. In other words, the average of the population sizes along a two-point cycle with fixation for Ai is equa1 to the carrying capacity of the &A, homozygote. One point on the cycle must therefore lie above, the other below Kii . The latter is also true for the linear logistic fitnesses, except that the values along the cycle are centered about N = Kii(Q + I/rii), which is less than Kit . An equilibrium cycle between the two points, (jl , fil) and (& , NJ, is locally stable if and only if the individual population points ($3 , Ri) are locally stable as equilibria of the two-step system (13). L inearizing (13) in the neighborhood of (ji , flJ p ro d uces the local stability matrix

where all partial derivatives are evaluated at (hi , fiJ. Since 8fl/i3V vanishes when jt = 0 or 1, the local stability eigenvalues for a two-point fixation cycle are

(174

The two-point cycle in N with fixation for Ai is 1ocaIly stable if and only if both eigenvalues A1 and A, in (17) are less than one in magnitude. Note that A, , the eigenvalue associated with the change in population size, coincides with that for the two-step equations for simple population growth in (12). Consequently, the conditions for A, to be less than unity in absolute value for the linear logistic (8) and exponential (10) fitnesses are precisely those for the local stability of a two-point cycle for (1) and (2), respectively. May (1974, 1975) has found these to be 2 < yii < 2.449 (1) and 2 < r 0 (e.g., W,,(N) < W,,(N) < W,,(N) all N > 0), as when all three have the same birthrate parameter rij z Y. Then 1hi / < 1 if and only if K,, < Kii , as in the exponential model, provided no negativity problems are encountered. In general,

4 = (1 + ,zY - +&

(l +

rd2

+ rii) + (s)’

(2

+ yii)*

(18)

Restricting attention to those parameter values {iii , Kij , i, j = 1, 2) which ensure all three fitnesses are positive along the cycle fir and fiZ , one can find examples where with 2 < rii < 2.449: (1) (2) (3)

the two-point cycle may be unstable even though Kii > K12 ; the two-point cycle is locally stable even though Kii < K,, ; or stability of the two-point cycle requires Kii < K,, .

The analysis of equilibrium cycles of period d = 2” in N with fixation for Ai proceeds in a similar fashion. The existence of such limit cycles of period 2,4, 8, and so on for the fitnesses (8) and (10) follows immediately from the analysis of the growth equations (1) and (2) (May, 1974, 1975). The points ($i , fli), ($2 , fis),..., (hd , &z, f orm an equilibrium cycle of period d if p, = jr and Nt = fii imply pt+, = #a and N,,, = fia , and,..., p,,, = $i and Nttd = fli where (ji , IGi,) # (j$ , fij) f or i # j. These are obtained as equilibria of the d-step recursion system produced by iterating (3) over d generations. It can then be shown by mathematical induction that the limit cycle between the population sizes Nl, Ri, ,..., fid , with fixation for Ai , is locally stable if and only if the two local stability eigenvalues

are both less than unity in absolute value. ha again corresponds to the local stability eigenvalue of the d = 2” point cycle of the single equation (12). For the exponential fitnesses (lo), the average value of the d population sizes along a d-point cycle with fixation for A, can again be shown to equal Kji , the carrying capacity of the A,A, homozygote. Thus from (19), 1A1 1 < 1 if and only if K12 < Kti . A stable limit cycle in N with fixation for A, will therefore exist for the exponential fitnesses whenever Kl, < Kii , provided rii > 2 lies below a critical level, the period of the cycle depending on the magnitude of rii . This is also true of the linear logistic fitnesses when the three genotypic fitnesses

REGULAR

AND

CHAOTIC

179

CYCLING

remain positive and are in the same relative order for all N > 0. Otherwise, there can be a very complicated relationship between the magnitude of /\1 and the ratio of Kii/Kl, , as seen in the case d = 2 above. A final observation may be made. Should W,,(N) < W,,(N) < Ws,(N) all N > 0 or W,,(N) < W,,(N) < W,,(N) all N > 0, the gene frequency under (3) will monotonically approach fixation for the allele with the maximum fitness. This will occur whatever the behavior of the population size. Consequently, it should also be possible for the gene frequency to become fixed, while the population size exhibits chaotic oscillations. 4.

INTERIOR

LIMIT

CYCLES

The next step in the analysis of the cyclic behavior of the density-dependent model (3) is to investigate the existence of limit cycles in the interior, with a polymorphic gene frequency and positive population size at each point in the cycle. From (13), all interior two-point cycles are given by the solutions (p, N) of the complicated pair of simultaneous equations

for which 0 < p < 1 and N > 0. This is an extremely complicated set of equations, even for the linear logistic fitnesses. Certain special cases afford some insight. Consider a gene frequency p * that is invariant under the transformation (3) in the sense that p, = p* implies pttl = p*; in other words, once the gene frequency equals p* it remains there, independent of the population size. The gene frequencies p = 0 and p = 1 corresponding to fixation lie in this category. A polymorphic value p* is invariant under (3) if and only if W&J*, N) = W,(p*, N) for all N > 0. This is equivalent to the condition that p*(N)

=

2W,,(N)

- W,,(N)

- W,,(N)

= ‘*

be independent of N. This holds, for instance, if the two homoaygotes have identical fitnesses, i.e., W,(N) = W,,(N), for which p* = Q, or as has been shown for the linear logistic fitnesses (8) when rij = r (Asmussen and Feldman, 1977) with

when there is overdominance or underdominance in carrying capacity. Once p, = p* the joint system (3) reduces to the single equation N t+1

--

NP(p*,

NJ-

(22)

180

MARJORIE

A.

ASMUSSRN

For the linerar logistic fitnesses with ru = r and p* given by (21), (22) is equivalent to (1) with

which is the equilibrium population size corresponding to the equilibrium gene frequency p* in (3). Equation (22) therefore has a two-point equilibrium cycle analogous to (14) w h en r > 2. The full two-dimensional system (3) with linear logistic fitnesses and rij = r has equilibrium limit cycles of period 2, 4, 8, 16, and so on in the population size N, with the gene frequency fixed at the invariant value p* as r is increased above 2 if Krs > Kll , K,, or I(,, < Kll , I(,, . The same is true when W,,(N) = W,,(N) and p* = 4, since (22) is then equivalent to (1) with r = (rll + rJ2 and

K = (rll + r12)(E + j$-)-‘. The exponential fitnesses are less readily analyzed. The existence of limit cycles in N with the gene frequency fixed at an invariant value p* depends on the behavior of the population size as governed by Eq. (22). The local stability criteria for such an interior limit cycle are determined by linearizing the d = 2” step system, obtained by iterating (3) for 2n generations, in the neighborhood of one of the points (p*, RI),..., (p*, Icjd) along the equilibrium cycle. This produces the two local stability eigenvalues

A, = [ W(p*, Nl) + fil g

(p*, R,]

--. [W(p*,

R)

+ RI g

(p*, Icid)]. (23b

For the linear logistic fitnesses with rij 3 r andp* given by (21) there is a locally stable d = 2” point cycle between the points (p*, I$), (p*, &a),..., (p*, @&) if and only if IS,, > K,, , K,, and r is in the range for which Eq. (1) has a stable limit cycle of period d = 2”. Similar results hold when W,,(N) = Was(N) and p* = l/2. If the three genotypes do not have the same birth rate, the magnitude of hr has a more complicated dependence upon the parameters, analogous to that in (18), with rl, replaced by rrr , rii by r, K,, by K,, , and Kti by K. Again in the special cases for which the three genotypic fitnesses are in the same relative order for all N > 0, much stronger results hold. In this case, if there is an invariant polymorphic gene frequency, the fitnesses must satisfy K,(N) -C ~&V, W,,(N) all N > 0 or W12(N) > W,dN), W&Y all

REGULAR AND CHAOTIC CYCLING

181

N > 0. For the classof fitnesseswhich strictly decreasewith increasingpopulation size, theseconditions are equivalent to K,, < K,, , K,, or K,, > K,, , K,, respectively. By examining the sign of dp = p,,, - pt from (3), it can be shown that for any initial polymorphic gene frequency, the population will approach the invariant value p* monotonically in the overdominant case,but will proceed monotonically to fixation in the underdominant case. These gene frequency changes will occur whatever the corresponding changes in population size. Once genetic equilibrium is reached,changesin population sizewill be governed by (12) or (22). In fact, it is possiblefor the gene frequency to converge to a polymorphic value p*, while the population size undergoeschaotic oscillations characteristic of Eq. (22). In general there will not be an invariant polymorphic gene frequency for the two-dimensional system(3). Due to the complicatedinterdependenceof the two variablesin (3) it is very difficult to analyze the full cyclic behavior of the system. For instance,to determinewhether there is an interior two-point cycle, one would need to solve the complicated simultaneousequations given in (20). To gain further insight, numerical studieshave been made.

5. NUMERICAL

RESULTS

Numerical examplesconfirm the behavior predicted from the analytic study of (3) in the previous sections. For both the linear logistic and exponential fitnessesit is possiblefor the population to converge to an equilibrium limit cycle where the gene frequency is fixed at 0 or 1 or an invariant polymorphic value, while the population size exhibits the cyclic oscillations characterizing Eqs. (12) or (22) ( seeTables I and IV). In fact, the gene frequency may reach an equilibrium value while the population size undergoeschaotic cycling with no discernible pattern (Table I). Numerical investigations have also provided new insight into the behavior of the model. In particular, the density-dependent system (3) may produce +int limit cycles in p and N for suitable parameter values. The population may ultimately alternate betweentwo points (A , fii) and ($a , fla) with 4X # $a and & # fia . For other parameter values limit cycles between four points, eight, and so on, may be found (see Tables II-IV). A seeminglychaotic state in the interior with neither p nor N exhibiting a discemible pattern is alsopossible(Tables II-IV). Such complicated behavior can arise for a wide range of parameter values from which the examplesin Tables I-IV have been drawn. The results in Tables I-IV by no meansrepresent exhaustive studies of the outcomesunder all possibleinitial conditions. Rather, for eachset of parameter values, trajectories were followed for a relatively small number (usually lessthan 10) of different initial points (pO , NO)to determinethe possibleultimate behavior of the population. All outcomesobservedare recordedin the tabIes.For instance,

182

MARJORIE

A. ASMUSSEN

TABLE Limiting

Behavior

with

Exponential Is Indicated

I Fitnesses Where by Arrows

Cyclical

Behavior

Yll

T22 Kn &a %a

KI, f&z Kaz

&I &s &s

1.0

1.0

100

loo

loo

r12

120

loo

(P, N)

150

loo

200

loo

(P, NJ

(P, N)

0.5

(0.5,

106.1)

(0.5,113.7)

(0.5,

123.6)

1.5

(0.5,

111.7)

(0.5,

128.1)

(0.5,

153.7)

2.0

(0.5,

113.1)

(0.5,

131.9)

(0.5,

162.0)

3.0 $k:: 4.0 c;;:::

1;::g

33

4.5

5.0

(0.5, N “chaotic”)

(E:

2:g

(0.5, N in eight-point

rI:::: cycle)

2%;)~

(0.5, N “chaotic”)

(0.5, N “chaotic”)

(0.5, N “chaotic”)

(0.5, N “chaotic”)

(0.5, N “chaotic”)

in the first column of Table II, with r,, 2 3.5, the population either converged to fixation or to an interior limit cycle, depending on the initial condition. Indeed, all interior cycles listed in Tables I-IV were reached from many different initial points, suggesting that these interior cycles were locally stable or perhaps, in some cases, globally stable. Due to the complexities of the mathematical system (13), the stability properties associated with joint interior limit cycles in the subsequent discussion are based on such empirical results. The ultimate behavior was classified as chaotic if no discernible pattern had appeared by 1000 generations. With the linear logistic fitnesses (8), the onset of cycling is directly correlated to the value of the birth rates in the population, as suggested by the local stability criterion (9). With birth rates above 2, there will be no stable equilibria. Unfortunately, negativity problems (Asmussen and Feldman, 1977) are also soon encountered for birth rates in this range. Consequently, a more detailed numerical study has been based upon the exponential fitnesses (10). For very high birth rates (8 or lo), these fitnesses also break down, in that the population size may fall so low as to be judged extinct, even though extinction is not a stable state. This behavior is also found (May and Oster, 1976) with the single growth rate equation (2). Numerical investigations with more moderate birth rates have revealed a rich array of very complicated and often unexpected behavior from exponential fitnesses in the density-dependent selection model (3). The most significant of these have been summarized below.

REGULAR

AND

CHAOTIC

TABLE Limiting

y11 1.1

rae 1.0 TlZ

Behavior

with

Kn 110

Exponential Is Indicated

Ku 100

Km 105

(P, N)

183

CYCLING

II Fitnesses by Arrows’ KI, 130

Where

Ku 100

Cyclical

&a 125

Behavior

&I 150

(P> N)

K,e 100

Km 140

(P, N

0.5

Fixation

Fixation

Fixation

2.0

Fixation

Fixation

Fixation

3.0

Fixation

Fixation

Fixation

3.5

Fixation

Fixation

Fixation

Fixation

Fixation

(0.51;;6 45 1) $0.5228:194:9)) 4.0

Fixation (0.4;6 $0.5009:

4.6

5.0

a Fixation and (1, K,,)

26 2) 27917))

(0.5:8 $0.5274:

Fixation or +(0.4745, 10.4)) (0.4989, 332.7) (0.4832, 18.5)) -(0.4994,414.5)3

Fixation

Fixation or “chaos”

(0.4;2 $0.47341

234 2) 49:3)) Fixation 417.3) 19.3))

(0.5:s ((0.51451

Fixation

Fixation

in interior

below indicates that both of the Iocally stable were reached for certain initial values (p, , NJ.

408 2) 30:4+

(0.5k3 $0.5009:

fixation

equilibria

15 0) 545:4))

(0,

K,,)

5.1. Overdominancein Carrying Capacity is not Necessaryfor the Maintenance of Genetic Variation For the class of density-dependent fitnesseskhich strictly decreasewith increasing population size, heterozygote superiority in carrying capacity is necessaryfor the existence of a locally stable interior equilibrium in gene frequency and population size for the model in (3). Numerical examples have shown, however, that there may be stable limit cycles in the interior thereby maintaining both allelesin the population, even though the AlA heterozygote is intermediate (Tables III and IV) or inferior (Table II) in carrying capacity.

184

MARJORIE

A.

TABLE Limiting

rll

ha

1.0

1.0 fl,

Behavior

with

&I

Exponential 18 Indicated

Ku Km

95

100

105

(P, N)

ASMUSSEN

III Fitnesses Where by Arrows*

Ku KIZ K,, 95

100

Cyclical

Behavior

K,,

K,,

Km

95

100

150

125

(P, N)

(P> N)

0.1

Fixation

Fixation

Fixation

2.0

Fixation

Fixation

Fixation

3.5

Fixation

Fixation

Fixation

4.0

Fixation

Fixation

Fixation

Fixation

Fixation

(0.308196 25 7) ((0.4770: 267:2)) 4.6

Fixation

(0.2;3 c(0.4468:

5.0

Fixation

27 9) 310:9)3

Fixation

Fixation

Or

“chaos”

a Fixation below reached for certain

5.2. Limit C’cles Points

in interior

indicates that the locally initial values (PO , No).

>(0.4:1, 342.3)~ (0.2551, 22.3) (0.4717,443.7)) I-(0.2063, 12.2))

stable

fixation

equilibrium

(0, Kss)

was

May Occur S~~ultaneo~~ with Locally Stable Equilibrium

Cyclic behavior was predicted for the joint system (3) because parameter values exist for which there are no locally stable equilibria. Furthermore, under such circumstances regular and chaotic cycling occurs for the simple growth equations (1) and (2). One does find fixation or interior cycles in these cases for the joint system (3) (see, e.g., Tables I and IV). There may, however, be stable limit cycles in the interior, even though one or more of the fixation equilibria, (p = 0, N = K,,) and ( p = 1, N = Ki,), is locally stable (Tables II and III).

REGULAR

AND

CHAOTIC

TABLE Limiting

y11 1.1

Behavior

with

Exponential Is Indicated

Pa%

KI,

2.2

95

185

CYCLING

IV Fitnesses Where by Arrows

Cyclical

Behavior

K,, Km loo 105

0.5

2.3

3.0

(0.6039, 43.2) f(0.5074,165.5)J

4.0

(0.5780,200.8) $0.8014, 28.6))

4.1

4.5

Eight-point

5.0

“Chaos”

cycle in interior

in interior

5.3. Interior Limit Cycles and Fixation Equilibria or Limit Cycles May Be Simultaneously Stable For the density-dependent system (3) with the class of decreasing fitnesses, a stable polymorphic equilibrium only exists if K,, > K,, , K,, , in which case both fixation equilibria are unstable. Whether a population ultimately reaches a fixation or interior equilibrium (4, fi) thus depends upon the order of the carrying capacities and not upon the initial value (p,, , N,,). Which of the two outcomes occurs with constant viability selection also depends only on the parameter values. (In the underdominant case, fixation will result, although which allele will ultimately be eliminated does depend on the initial conditions.) This is not true for density regulated selection when the possibility of equilibrium cycles is taken into account, for it is possible to have one or both fixation equilibria or fixation cycles simultaneously stable with an interior limit cycle (Tables II-IV). 6.53/16/z-6

186

MARJORIE

A.

ASMUSSEN

5.4. There Can Be Apparent Chaos in the Presence of Stable Equilibria It is surprising to find that the population may persist in an apparent chaotic regime despite the presence of a locally stable equilibrium. For example, with Yll = 4, r12 = 1.1, rz2 = 1.0; K,, = 110, K,, = 100, Ks, = 105, the fixation state (p = 0, N = 105) is locally stable. Nonetheless, trajectories were found which converged to p = 1, while the population size exhibited the chaotic behavior expected for (2) with r = 4. The population may also exhibit seemingly chaotic oscillations in the interior even though there is a stable fixation equilibrium. When r12 = 5 in Table II, irregular oscillations in the interior persist even after 2000 generations, despite the fact that both (p = 0, N = 105) and ( p = I, N = 110) are locally stable. Chaotic behavior can also occur in the presence of stable limit cycles.

5.5. The Effect of Birth RatesuponInterior CyclesIs Very Complicated Regular and chaotic cycling automatically arises in the simple models of population growth (1) and (2) when the population’s birth rate exceeds2. The exponential fitnessesin the joint system (3) allow regular and chaotic fixation cycles for the samebirth rates rid > 2, provided Krs < Kii . The onset of interior limit cycles is lessstraightforward. In Tables I-IV, increasingthe heterozygote’s birth rate gives rise to interior cycles of increasingperiod. Increasing a homozygote’s birth rate can have the sameeffect aswhen rll is increasedfrom 0.1 to 4.0 with rrz = 4.5, rz2 = 2.2, K,, = 95, K12 = 100, KS2 = 105. On the other hand, it may lead to interior cycles of decreasingperiod as when rrr is increasedfrom 1.0 to 4.0 with rra = 5, rz2 = 1, K,, = 95, K,, = 100, K,, = 105. Similarly, how the three birth rates are assignedto the genotypescan lead to profoundly different outcomes. For example, depending upon which of the six possible ways the birth rates 1.0, 2.2, and 4.5 are assignedto the three genotypeswith carrying capacitiesK,, = 95, K,, = 100, K,, = 105, there may be a stabletwo- or eight-point interior cycle, chaosin the interior or in N alone with fixation, or convergenceto a fixation equilibrium. 5.6. Altering the Carrying CapacitiesCan Have a Stabilizing l@ect The onset of cycling in the interior also dependsstrongly upon the relative size of the three carrying capacity parameters.Consider, for example, the case where all three genotypes have a common birth rate rij 5 r. As K1,/Kii was increasedfrom I to 5, the value of r at which interior cycling aroseincreased from 2.0 to 2.6. The transitions from cycles of period 2” to period 2n+f also occurred at correspondingly higher r valuesasK,,/K,, wasincreased.The critical vaIue for r at which an apparentchaotic regime is reachedcan alsobe appreciably delayedby increasingthe ratio K,,/K,, . For Eq. (2) this occurs near Y = 2.692

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(May, 1974). With K,,/K,, = 1.2, it did not occur until r = 2.8, whereas for K,,/K,, = 5 the population exhibited stable cycles until the common birth rate exceeded 3.6. It is noteworthy that overdominance in carrying capacity is the genetic component of the local stability criteria for an interior equilibrium (Section 2). These examples suggest that the degree of geneticstability can exert a stabilizing in&ence on thejoint system(3), allowing a stableequilibriumand stable limit cyclesto persistfor higher birth rates than possiblewithin the purely ecological system(2). Similar resultsare obtained by exaggeratingthe relative differencesin carrying capacity when the heterozygote is inferior (Table II) or intermediate (Table III) in carrying capacity. Recall that Kii > K12 is the genetic component of the local stability criteria for fixation of Ai . As I&,/K,, is increasedfrom left to right in Tables II and III, ever higher Y,, values are needed to produce interior cycles. Stable limit cycles are possiblefor ever higher rra valuesas K,,/K,, is increased, and chaotic oscillationsare postponed. Unfortunately, the opposite effect is alsopossible. For example, in Table I, exaggerating the degree of overdominance leadsto interior cycles or chaos at lower birth rates.

6.

DISCUSSION

The interaction between an inherently stable genetic systemand a potentially unstable population size regulating mechanismcan be very complex. A wide range of complicated behavior may be exhibited by comparatively simple deterministic models of density regulated selection. If the population size regulating mechanismreacts too sharply to perturbations in population size, regular and chaotic cycles may result. The population may converge to fixation or an invariant polymorphic gene frequency while the population size oscillates between two values, four values, eight values, and so on. There may also be joint limit cycles in both the population size and gene frequency. This type of behavior appears with the linear logistic and exponential fitnesseswhen the genotypic birth rates are sufficiently high and in the right configuration. Cycling is by no meansrestricted to growth ratesor fitnesseswhich are strictly decreasing functions of the population size. The sametype of behavior is found in models incorporating the Allee effect of low numbers, where both high and low population sizesare detrimental (Asmussen,1979). Such complicated behavior is in sharp contrast to the classicalmodels of constant viability selection, in which there is always convergenceto an equilibrium gene frequency. On the basis of these models, there is a temptation to think that there must always be at leastone stableequilibrium and to conclude that an equilibrium is locally or globally stable if there are no other stable points. The linear logistic, exponential, and two Allee fitness forms,

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however, show that this is not always the case. They all allow situations in which there are no locally stable equilibria. For the linear logistic and exponential fitnesses, an interior state ($, fi) may be the only locally stable equilibrium and yet not be globally stable due to the presence of locally stable fixation cycles. Similarly, an extinction state may be the only locally stable equilibrium with the Allee fitnesses, but not be globally stable due to the existence of a locally stable interior cycle. Numerical investigations based on the exponential fitnesses have produced a number of important and unexpected results, which contrast sharply with conclusions based solely upon the standard analysis of the existence and stability of isolated equilibrium points. One of the most surprising discoveries was that although cycling was anticipated for the various density-dependent fitnesses because there may be no locally stable equilibrium points, this turns out not to be a necessary condition for the presence of limit cycles. There may, for instance, be apparently stable interior limit cycles even though one or both of the fixation equilibria are locally stable. Even more striking is the occurrence of chaotic oscillations in the presence of locally stable equilibria. One is led to wonder if cycling is a latent property of many other models. Another numerical finding is that overdominance in carrying capacity is not necessary for the maintenance of genetic variation in the population. It is needed for a stable interior equilibrium; however, there may be apparently locally stable interior limit cycles in p and N which maintain both alleles in the population, even though the heterozygote is inferior or intermediate in carrying capacity. Fixation equilibria or cycles may also be simultaneously stable with interior limit cycles. Whether or not genetic variation is maintained in such situations depends upon the initial values (pa , N,) as well as the parameter values. In contrast, whether a fixation or interior equilibrium point is reached, depends only upon the parameter values and not on the initial conditions. In general, the occurrence of interior limit cycles depends in a complex fashion upon the birth rate and carrying capacity parameters. It is widely known that many organisms can exhibit regular fluctuations in population size (see, e.g., Lack, 1954; Keith, 1963; Macfadyen, 1963; Williamson, 1972). Examples of oscillations in gene frequencies are not as easily found. Dobzhansky (1943), h owever, observed pronounced cycling in inversion frequencies of D. pseudoobscura within breeding seasons.In the absenceof more demographicdata on the populations, it seemsdifficult to exclude a contribution to this behavior from a suitable birth rate configuration in the above densitydependent sense.On the basisof the present work, it would be very interesting to determine in the caseswhere fluctuations are observed,whether there are ever, in fact, concurrent oscillations in population size and gene frequencies. The complicated behavior possible in the simple density-dependent models also emphasizesthat regular or random fluctuations need not be due to stochastic forces or even to fluctuating parameters.

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Cycling, nonetheless, is not often observed in natural populations. Indeed, it should be emphasized that cycling is not an inherent feature of densitydependent or discrete time models. It has been shown, for instance, that there is always a locally stable equilibrium for the hyperbolic fitnesses ,!!)2.3 =

1 + Tii 1 + (yij/Kj) Nt

(i,j = 1, 2).

Various numerical examples fail to produce any cyclic behavior. Similarly, in Eq. (12), fi = K is a globally stable equilibrium for all positive birth rates r. Second, even though a model which can lead to oscillations is appropriate, natural populations may not possess the parameter values which produce such behavior. For example, Hassell, Lawton, and May (1976) recently fit a variety of data from field and laboratory populations with approximately discrete generations to a simple model of density regulated population growth for which a full range of limiting behavior was possible, depending upon the parameter values. Examples of both regular and even apparently chaotic oscillations were found which agreed with the populations’ observed behavior. However, such cases were rare and mainly arose in laboratory populations. The natural populations selected did not have parameter values that produce cycling. Numerical results with the exponential fitnesses suggest another reason why cycling may not often be observed, even in models where it is possible.This is due to an apparent stabilizing influence from the genetic system.By exaggerating the relative differences in genotypic carrying capacity while preserving the samerelative order, stable equilibria and stablelimit cycles may persist for ever higher birth rates. In particular, for suitable carrying capacity configurations, stable equilibria and cycles are possible for birth rates which would result in chaosfor the analogousecologicalgrowth equation.

ACKNOWLEDGMENTS The reading

author wishes to thank of the manuscript.

Michael

Clegg

and

Wyatt

Anderson

for

their

critical

REFERRNCE~ ANDERSON, W. 1971. Genetic equilibrium and population growth under density-regulated selection, Amer. Natur. 105, 489498. ASMUSSEN, M. A. 1979. Density dependent selection 2: The allee effect, Amer. Natur., in press. ASMUSSEN, M. A. AND FELDMAN, M. W. 1977. Density dependent selection 1: A stable feasible equilibrium may not be attainable, J.Theoret. Biol. 64, 603-618. AUSLANDER, D., GUCKBNHEIMER, J., AND OSTW, G. 1978. Random evolutionarily stable strategies, Theor. Pop. Biol. 13, 276-293.

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CHARLESWORTH, B. 1971. Selection in density-regulated populations, Ecologvl 52,469-474. CHAUNDY, T. W. AND PHILLIPS, E. 1936. The convergence of sequences defmed by quadratic recurrence formulae, Quart. J. Math. O.xjord Ser. 7, 74-80. DOBZHANSKY, T. H. 1943. Genetics of natural populations. IX. Temporal changes in the composition of populations of Drosophila pseudoobscura, Genetics 28, 162-186. GUCKENHEIMER, J., OSTEX, G., AND IPAKTCHI, A. 1977. The dynamics of density-dependent population models, j. Math. Biol. 4, 101-147. HASSELL, M. P. 1975. Density dependence in single species populations, 1. An. Ecol. 44, 283-295. HASSELL, M. P., LAWTON, J. H., AND MAY, R. M. 1976. Patterns of dynamical behavior in single species populations, /. An. Ecol. 45, 471-487. KEITH, L. B. 1963. “Wildlife’s Ten-Year Cycle,” Univ. of Wisconsin Press, Madison. LACK, D. 1954. Cyclic mortality, J. Wildlife Mgmt. 18, 25-37. LEON, J. A. 1974. Selection in contexts of interspecific competition, Amer. Nutur. 108, 739-757. LI, T. Y., AND YORKE, J. A. 1975. Period three implies chaos, Amer. Math. Monthly 82, 985-992. MAY, R. M. 1974. Biological populations with nonoverlapping generations: Stable points, stable cycles, and chaos, Science 186, 645-647. MAY, R. M. 1975. Biological populations obeying difference equations: Stable points, stable cycles, and chaos, J. Theoret. Biol. 51, 511-524. MAY, R. M., AND OSTER, G. F. 1976. Bifurcations and dynamic complexity in simple ecological models, Amer. Natur. 110, 573-599. MACFADYEN, A. 1963. “Animal Ecology: Aims and Methods,” 2nd ed., Pitman, London. OSTER, G. AND GUCKENHEIMER, J. 1975. Bifurcation phenomena in population models. in “The Hopf Bifurcation” (J. Marsden and M. McCracken, Eds.) Lecture notes in Mathematics, Springer-Verlag, Berlin. OSTER, G., IPAKTCHI, A., AND ROCKLIN, S. 1976. Phenolypic structure and bifurcation behavior of population models, Thor. Pop. Biol. 10, 365-382. RAVEH, A., AND RITTE, U. 1976. Frequency dependence and stability, Math. Biosci. 30, 371-374. ROUGHGARDEN, J. 1971. Density-dependent natural selection, Ecology 52, 453-468. WILLIAMSON, M. 1972. “The Analysis of Biological Populations,” Arnold, London.