Journal of Theoretical Biology 354 (2014) 12–24

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Optimal allocation patterns and optimal seed mass of a perennial plant Andrii Mironchenko a,n, Jan Kozłowski b a b

Institute of Mathematics, University of Würzburg, Emil-Fischer Straße 40, 97074 Würzburg, Germany Institute of Environmental Sciences, Jagiellonian University, Gronostajowa 7, 30-387 Kraków, Poland

H I G H L I G H T S








We model a perennial plant as a self-controlled system. The periodization of a plant life is derived from the model. Model encompasses annual, monocarpic and polycarpic species. Sufficient condition for a plant to be monocarpic is provided. Concavity of photosynthetic rate function leads to production of small seeds.

art ic l e i nf o

a b s t r a c t

Article history: Received 30 September 2013 Received in revised form 24 February 2014 Accepted 12 March 2014 Available online 19 March 2014

We present a novel optimal allocation model for perennial plants, in which assimilates are not allocated directly to vegetative or reproductive parts but instead go first to a storage compartment from where they are then optimally redistributed. We do not restrict considerations purely to periods favourable for photosynthesis, as it was done in published models of perennial species, but analyse the whole life period of a perennial plant. As a result, we obtain the general scheme of perennial plant development, for which annual and monocarpic strategies are special cases. We not only re-derive predictions from several previous optimal allocation models, but also obtain more information about plants' strategies during transitions between favourable and unfavourable seasons. One of the model's predictions is that a plant can begin to re-establish vegetative tissues from storage some time before the beginning of favourable conditions, which in turn allows for better production potential when conditions become better. By means of numerical examples we show that annual plants with single or multiple reproduction periods, monocarps, evergreen perennials and polycarpic perennials can be studied successfully with the help of our unified model. Finally, we build a bridge between optimal allocation models and models describing trade-offs between size and the number of seeds: a modelled plant can control the distribution of not only allocated carbohydrates but also seed size. We provide sufficient conditions for the optimality of producing the smallest and largest seeds possible. & 2014 Published by Elsevier Ltd.

Keywords: Optimal phenology Size–number trade-off Biomass partitioning

1. Introduction The pioneering work (Cohen, 1971) gave rise to a new class of mathematical models of plants based on methods of optimal control theory. In these models it was assumed that a plant can control resource allocation in order to maximise its fitness, which is often identified with the mass of seeds produced by a plant during its lifetime.

n

Corresponding author. E-mail addresses: [email protected], [email protected] (A. Mironchenko), [email protected] (J. Kozłowski). http://dx.doi.org/10.1016/j.jtbi.2014.03.023 0022-5193/& 2014 Published by Elsevier Ltd.

In the early models, devoted to the development of annual plants, it was assumed that a plant consists of a number of compartments – at least of a vegetative compartment (leaves, roots, stems) and a reproductive compartment (seeds and auxiliary tissues), although storage and defensive tissues could also be included. This basic model, posited by Vincent and Pulliam (1980), results in a bang-bang transition from the allocation to vegetative tissues to the allocation to seeds. This annual plant model has been extended in many directions, in particular in Iwasa and Roughgarden (1984) a model with multiple vegetative compartments was analysed, and in Ziolko and Kozlowski (1995) and Ioslovich and Gutman (2005) a model with additional physiological constraints, resulting in periods of mixed growth (where both vegetative and reproductive parts of a plant grow simultaneously). Optimal allocation strategies in

A. Mironchenko, J. Kozłowski / Journal of Theoretical Biology 354 (2014) 12–24

stochastic environments have been investigated in particular in De Lara (2006), while allocation to defensive tissues was encountered in Yamamura et al. (2007) and Takahashi and Yamauchi (2010), to cite a few examples. The survey of early works in this field is provided in Fox (1992), and for a general overview of resource allocation in plants see monographs (Bazzaz and Grace, 1997; Reekie and Bazzaz, 2005). In contrast to annual plants, less attention has been devoted to the modelling of perennials' optimal phenology. Usually, the behaviour of a perennial plant is modelled in the following way: its lifetime is divided into discrete seasons during which environmental conditions are favourable for photosynthesis. The model of a plant within every season is continuous and is treated with the methods used in annual plant models (Schaffer, 1983; Iwasa and Cohen, 1989; Pugliese and Kozlowski, 1990). To model the behaviour of a plant between seasons (when the weather is unfavourable), some simple transition rules are used that show which parts of compartments are saved during the season and which are not. The solution to such problems is divided into two parts: first, the model on one season is solved using Pontryagin's Maximum Principle (PMP), see e.g. Alekseev et al. (1987), and then a solution to the whole model is sought by applying the dynamic programming method. Although these models provide quite interesting qualitative results regarding the behaviour of perennial plants, they have an important disadvantage, namely that the detailed description of plant strategies during a season contrasts with the simple jump from the end of one season to the beginning of the next one. In this paper we propose a continuous-time model of a perennial plant, which allows us to describe more precisely the dynamics of a plant during seasons with unfavourable environmental conditions for photosynthesis, and to avoid the introduction of additional parameters for describing jumps between seasons. With the help of PMP we derive a general scheme of perennial plant development, which contains models for annual as well as monocarpic plants as special cases. We also prove that within our model monocarpy is always optimal if there are no losses of storage parts and there is no mortality before the end of life. With the help of numerical examples we show that many developmental patterns from previous papers can be derived by using our model. In particular, one can use it to study annual plants with multiple reproduction periods (King and Roughgarden, 1982); perennial plants which grow to a certain size for a number of periods, and in the following periods they firstly regrow to this size and then produce reproductive tissues (Iwasa and Cohen, 1989); the evergreen polycarpic plants as well as monocarps. Necessary conditions for existence of singular controls (whose biological meaning is an existence of periods with mixed vegetative and reproductive growth) can be obtained on the basis of our model. Some of them are similar to those obtained in Yamamura et al. (2007) for annual plants which undergo losses of vegetative and generative tissues. Moreover, our model is much better suited for the study of transitions from favourable to unfavourable climate conditions, and one of its predictions is that plants begin to generate vegetative tissues not at a time when environmental conditions are favourable for photosynthesis, but slightly earlier in order to enter into the suitable period with developed vegetative tissues. Overall, the model describes the behaviour of plants from dormancy of seeds up to senile stage. Having established an optimal allocation model we will connect it to the theory of trade-offs between the size and the number of seeds. This topic occupies an important place in the study of plants. In particular the evolution of seed size and its relation to the seed number, plant size and type of seed dispersal have been investigated in Moles et al. (2005, 2007) and Eriksson (2008), to cite a few. The basic mathematical model of seed size–number

13

trade-offs has been proposed in a seminal work (Smith and Fretwell, 1974), where it was assumed that the fitness of a plant is equal to the sum of the fitnesses of its descendants. Afterwards this model has been generalised in a number of directions (for a review see Fenner, 2000). In this framework, optimal size is sought depending on the properties of the fitness function. This makes possible the quite general treatment of size–number trade-offs, but the question remains as to how to formalise the dependency of fitness on size and the number of seeds and how to find the properties of the function that characterises this dependency. Our next aim is to investigate the trade-offs between the number and the size of seeds in the context of optimal allocation models. Within this framework fitness is properly formalised, and we can investigate the optimal size of a seed depending on the properties of the photosynthetic rate function and other physiological parameters of a plant, which offer more distinct criteria than abstract fitness. We provide the analysis for the model developed in Section 2 of this paper, but the results are valid also for a number of other optimal allocation models. We prove that, according to our plant model, if the photosynthetic rate function is concave (that is, if the rate of photosynthesis per unit mass decays with an increase in the size of a plant), then the seeds have to be as small as possible. Such behaviour is particularly typical in colonising species (see Section 5). The outline of the article is as follows. In Section 2.1 we introduce a perennial plant model. In Section 2.2 we summarise the predictions of the model, provide a general plant development scheme and consider a number of special cases (annual and monocarpic plants). In Section 3 we show the results of numerical simulations of different plant development scenarios, while in Section 4 we consider trade-offs between the size and the number of seeds. The results of the paper are discussed in Section 5 and Section 6 draws conclusions from the results and outlines some directions for future work. In Appendices A and B we provide derivations of our theoretical results.

2. Optimal allocation model 2.1. Model description In optimal allocation models it is usually assumed that all allocated photosynthate is immediately used for the construction of tissues. In models taking into account the presence of a storage compartment, a plant can also allocate resources from storage, depending on the mass of the storage. Such a method ignores the fact that a photosynthate is not immediately allocated to certain structures but instead exists for some time in a free state. We shall take this effect into account and assume that an intermediate stage exists whereby carbohydrates have already been photosynthesised but have not yet been permanently allocated to a given structure. Let a plant consists of three parts: a vegetative compartment, a reproductive compartment and non-structural carbohydrates (free glucose travelling from leaves to other organs, including speciesspecific storage organs, and starch in storing organs) hereinafter called ‘storage’. Let


x1 ðtÞ be the mass of the vegetative compartment at time t,
x2 ðtÞ be the mass of the reproductive compartment at time t,
x3 ðtÞ be the mass of storage at time t. We model the dynamics of a plant via the following equations: x_ 1 ¼ v1 ðtÞgðx3 Þ  μðtÞx1 ; x_ 2 ¼ ðvðtÞ  v1 ðtÞÞgðx3 Þ; x_ 3 ¼ ζ ðtÞf ðx1 Þ  vðtÞgðx3 Þ  ωðtÞx3 :

ð1Þ

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Here f ðx1 Þ describes the rate of photosynthesis of the plant with vegetative mass x1 in optimal environmental conditions and gðx3 Þ means the maximal rate of allocation of non-structural carbohydrates for a plant with a mass of non-structural carbohydrates x3. It is natural to assume that f and g increase monotonically and f ð0Þ ¼ gð0Þ ¼ 0. We assume that the time-interval on which the plant is living is fixed and equals to ½t 0 ; T, where T is a maximum longevity. The initial mass of the seed and all its compartments is given as follows: xi ðt 0 Þ ¼ x0i ;

i ¼ 1; 2; 3:

ð2Þ

Problem of optimisation of a seed mass will be considered in Section 4. Climate influence is modelled by three functions: ζ : ½t 0 ; T½0; 1 and μ; ω : ½t 0 ; T-½0; 1Þ,


ζ ðtÞ models the dependence of the rate of photosynthesis on the climate (ζ ðtÞ ¼ 0 if at time t no photosynthesis is possible);
μðtÞ is the loss rate of vegetative tissues per unit mass at time t;
ωðtÞ is the loss rate of the storage parts per unit mass due to external factors (decaying, grazing by animals, etc.) at time t.

2.2. Model predictions We perform an analysis of (1) and (3) with the help of PMP, with the aim of determining in what order the periods of a plant's life follow each other. The full analysis is presented in Appendix A, but for now we summarise the results derived therein. First, we construct a Hamiltonian H, corresponding to the problem (1) and (3), which is equal to the scalar product of the right-hand side of (1) times the functional coefficients p1 ; L; p3 , see (9). The coefficients p1 and p3 are the so-called ‘adjoint functions’ to the equations, as they govern the dynamics of x1 and x3. Their dynamics is described by Eq. (11). The development of a plant according to the model (1) involves three main periods:

(V) Vegetative period, during which holds p1 ðtÞ 4 maxfLðtÞ; p3 ðtÞg: (R) Reproductive period, characterised by LðtÞ 4maxfp1 ðtÞ; p3 ðtÞg: (S) Storage period, during which

Note that photosynthesised carbohydrates firstly enlarge the mass of storage before they can be allocated to other compartments. We assume that a plant can control the total allocation rate with the control vðtÞ A ½0; 1 and the allocation rate to vegetative tissues with the control v1 ðtÞ A ½0; vðtÞ; consequently, the allocation rate to reproductive tissues at time t is controlled by v2 ðtÞ ¼ vðtÞ  v1 ðtÞ, and vðtÞ ¼ 0 means that resources are not being relocated from storage at time t. To model the mortality of a parental plant, we introduce the function L : ½t 0 ; T-½0; 1. L(t) is the probability of survival of a parental plant to age t. We assume in this paper that mortality is only age-dependent and does not depend on the size of a plant. It is natural to assume that L is a non-increasing function and that LðtÞ 4 0 for all t A ½t 0 ; TÞ. In fact, if LðtÞ  0 on ½T  ε; T for some ε 4 0, then this means that at moment T  ε a plant will already be dead, and we can therefore consider the optimal control problem on time-period ½t 0 ; T  ε. We choose the maximisation of the expected total yield of seeds over a lifespan as the fitness measure. Thus Z

T t0

LðsÞx_ 2 ðsÞ ds ¼

Z

T

LðsÞðvðsÞ  v1 ðsÞÞgðx3 ðsÞÞ ds-max:

ð3Þ

p3 ðtÞ 4 maxfp1 ðtÞ; LðtÞg:

Thus, the choice of a compartment to which a plant should allocate available resources depends on the relation between functions p1 ; L; p3 : all resources should be allocated to vegetative parts if p1 is the largest of the trio, to the storage if p3 is the largest and to reproduction if L is the largest. If the maximum maxfp1 ðtÞ; p3 ðtÞ; LðtÞg is achieved by two or three functions from the set fp1 ; p3 ; Lg on an interval of non-zero length, a singular control is possible, which means either a simultaneous allocation to both vegetative and reproductive tissues or allocation from storage with non-maximal speed. In this section we will consider the case when singular controls do not appear. Some basic results concerning necessary conditions for emergence of these controls can be found in Appendix 1.1. The periods V, R, S can be further subdivided into sub-periods which follow each other, as depicted in Fig. 1:


D – seed dormancy.
S:2 – preparing for unfavourable climate conditions.
S:1 – life in unfavourable climate conditions.

t0

It is implicitly assumed in the above definition of fitness that reproductive tissues produced before the death of the maternal plant also contribute to fitness. This is a common simplifying assumption in optimal allocation models, which does not change solutions qualitatively and has only minor quantitative effect unless within-season mortality is heavy, which is unlikely for plants. A plant can maximise fitness by choosing controls v and v1 defined on ½t 0 ; T. We assume that the functions on the right-hand side of Eq. (1) are smooth enough to guarantee existence and uniqueness of solutions for (1). We also assume that the system (1) is forward complete, i.e. for each initial condition and each admissible control, the solution of (1) exists for all time. From the biological viewpoint this means that it is impossible to achieve an infinite yield over a finite amount of time, which in turn ensures that the solution to the problem (1)–(3) exists.

Fig. 1. Stages of development of a perennial (a), annual (b) and monocarpic (c) plant. D stays for seed dormancy, V – vegetative growth after germination, S.2 – preparing for unfavourable climate conditions, S.1 – life in unfavourable climatic conditions, V.2 – vegetative period, V.1 – allocation to vegetative tissues before reproduction and R – reproductive allocation.

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V:2 – vegetative period that starts close to the end of the period




with unfavourable conditions. At the beginning of this period, a plant starts allocating to vegetative tissues as preparation for climate conditions which are favourable for photosynthesis. V:1 – Allocation to vegetative tissues before reproduction.

The precise definitions of the above periods as well as the proof that the periods can follow each other only in the order depicted in Fig. 1 you can find in Appendix A. Important special cases of the general scheme are: 1. Annual plant with the possibility of multiple reproduction periods (Fig. 1b). Multiple reproduction periods may appear if losses of vegetative mass due to external factors modelled by the function μ are severe. This particular case was analysed in the early work (King and Roughgarden, 1982). We show a numerical example of this scenario in Section 3.1. If μ  0, then multiple reproductive periods for annual plants are not possible. 2. Monocarpic plants: A sufficient (but not necessary) condition for a plant without mortality (L  const) to be monocarpic is the negligibility of ω (in particular, if ω  0); in other words, the mass of storage cannot decrease due to external factors. In this case, transitions R -S:2 and V:1-V:2 are not possible, as shown in Fig. 1c. We show a numerical example of this scenario in Section 3.2. 3. Numerical examples In this section we present examples of numerical solutions for the model (1), which represent various patterns of plant life histories. The examples are not intended to mimic any specific plant species but to show that the model is adequate enough to include a broad range of strategies previously modelled separately. Additionally, we show that parameter changes may lead to qualitatively different results. We start with annual plants with single or multiple periods of reproductive allocation, following which we present the cases of monocarpic and evergreen polycarpic plants as well as a polycarpic plant losing almost all of its vegetative parts but retaining its storage during unfavourable seasons. Finally, we show that annual strategy can evolve under certain conditions, even when lifespan is not predefined. In all examples time is measured in months, with the beginning and the end of the simulation placed in the middle of a winter. The mass of all three compartments (vegetative mass, storage and reproductive mass) is measured in energy units, say MJ, to avoid using coefficients in order to take into account water content and differences in energy density between compartments. Seeds of the size 0.3 units contain 95 per cent of storage and 5 per cent of vegetative mass. The photosynthetic rate is described via the following saturation function: f ðx1 Þ ¼

ax1 ; bx1 þk

x1 Z 0;

ð4Þ

where a; b; k 4 0 (Vincent and Pulliam, 1980, p. 224). Another reasonable choice could be an allometric function axb1, used in particular within Metabolic Theory of Ecology (MTE), Dynamic Energy Budgets (DEBs), etc. (see Vandermeer, 2006; Lavigne, 1982). We model the dependence of the photosynthetic rate on climate as follows: 

ζ ðtÞ≔0:2 þ 0:8 sin

 π   t ; 12

15

where jxj denotes an absolute value of x. The maximal release rate of storage tissues is linear, i.e. gðx3 Þ ¼ cx3 , x3 Z 0 for some c 4 0 – while the actual release rate depends on the control variable v(t). Storage and vegetative mass losses are defined separately for particular cases. If the contrary is not mentioned explicitly, we assume that there is no mortality ðL  1Þ. Computations are made in Matlab, with the help of the optimal control solver GPOPS. 3.1. Annual plants We start with the simple scenario of an annual plant with a single reproduction period at the end of the season. The results of the simulation are depicted in the left-hand column of Fig. 2. A seed stays dormant for several weeks and then germinates using resources from storage. Germination takes place when environmental conditions are still harsh, to prepare a plant for vegetative growth when these conditions improve. Next, there is a phase of pure vegetative growth, when all assimilated resources are allocated to vegetative mass. After an instantaneous switch, all resources are allocated to reproductive mass and vegetative mass decays, reaching a very small value at the end of life equal to 12 months. If vegetative mass losses are heavier, for example through repeated grazing, after the vegetative growth phase there are multiple switches between vegetative and reproductive allocation, as shown in the right-hand column of Fig. 2. This scenario is a reminiscent of an early model (King and Roughgarden, 1982). We want to stress that the switches between reproductive and vegetative stages are due to heavy losses of vegetative mass and not due to periodicity by itself. Because of periodicity of μ in this numerical simulation, the switches between reproductive and vegetative periods are close to periodic. In any case in the final stage all resources are allocated to reproductive mass. Fig. 2e and f shows the dynamics of costate variables (see Appendix A) as well as of a survivability function L (which is constant in these examples) for annual plants with one reproductive period and for annuals with multiple reproductive periods. Taking into account that L replaces the costate variable in the case of the reproductive compartment, we can say that resources should be devoted to the compartment with the highest costate variable. In Fig. 2e we see that if p3 is not maximal over some time span, then it is only slightly smaller than the maximal costate variable on this interval. To explain this phenomenon, let us assume that at a given time t, allocation to reproductive tissues is optimal. If a plant does not behave optimally and does not allocate to reproductive tissues at t, resources are retained in storage and can be allocated from storage to reproductive tissues a bit later. Such a small delay decreases fitness minimally, which means that the ‘usefulness’ of allocation to storage is only slightly smaller than the ‘usefulness’ of allocating to vegetative tissues. One can see the numerical consequences of this fact in Fig. 2c. The strategy of a plant changes twice. The first switch is instantaneous, but the second switch is not, and one can see on the graph two points with the mixed control (0 o v1 ðtÞ o 1). The transition from v1 ¼ 1 to v1 ¼ 0 is not instantaneous because at the moment of a second switch the values of p1, L and p3 are almost the same, and p3 is only a bit lower than L for the following several months, see Fig. 2e. Therefore the numerical solver of the optimal control problem cannot distinguish the optimal and suboptimal solutions, which leads to an interval of mixed controls on a small time-interval. To enhance the difference between the costate variables for storage and reproduction (Fig. 2f), we assumed in the annual plant with multiple reproductive periods model that storage losses increase over time.

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Fig. 2. Annual plant with single (left column) or multiple (right column) reproduction periods. a and b are the states, c and d are the corresponding controls and e and f are the costate variables. The maximum photosynthesis rate is given by f ðx1 Þ ¼ 1:5x1 =ð1 þ 0:3x1 Þ, x1 Z 0 and the maximum storage release rate is defined as gðx3 Þ ¼ 5x3 , x3 Z 0. The rate of destruction of vegetative tissues μðtÞ ¼ 0:8 j cos ððπ=12ÞtÞj for the left-hand column and μðtÞ ¼ 1:8 j cos ðπtÞj for the right-hand column. The rate of storage losses ω  0:05 for the left column; although it was possible to obtain multiple switches for the same ω, this function has been changed in the right-hand column to ωðtÞ ¼ t=ð1 þ 0:5tÞ to better visualise the difference in costates between storage and reproductive output. (a) Annual with one reproduction period: states. (b) Annual with multiple reproduction periods: states. (c) Annual with one reproduction period: controls. (d) Annual with multiple reproduction periods: controls. (e) Annual with one reproduction period: costates. (f) Annual with multiple reproduction periods: costates.

Life span was set to 12 months for the cases illustrated in Fig. 2. To show why annual strategy could evolve, we set the lifespan to several years and take the parameters as in the caption of Fig. 3. If the losses of a storage mass are not large (Fig. 3b), a plant is a typical polycarpic plant. But if the losses are becoming larger (Fig. 3a), the entire amount of storage is relocated to reproductive mass, while only remnants of vegetative mass survive through to the next season and there is no vegetative growth in that season. This is seemingly only virtually annual strategy, but we can easily imagine that a real plant would relocate all movable hydrocarbons

to reproductive mass before winter and then die, because remnants of vegetative mass are practically useless. Such relocation, considered by Kozłowski and Wiegert (1986), was not allowed in the model. 3.2. Monocarpic plants Our next scenario involves a monocarpic plant with a five-year lifecycle. The results are depicted in Fig. 4. Since a plant starts its development during winter, it does not grow because the loss of

A. Mironchenko, J. Kozłowski / Journal of Theoretical Biology 354 (2014) 12–24

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Fig. 3. High losses of storage in winter can change optimal strategy from annual (a) to perennial (b). Storage losses omega equal to zero for 6 month (seed stage) in both cases, then follows the function ωðtÞ≔aj cos ððπ=12ÞtÞj, for t Z 6 with a ¼0.8 for perennials and a¼ 1.6 for annuals. Other parameters are identical for both cases: the maximum photosynthesis rate f ðx1 Þ ¼ 2:5x1 =ð1 þ 0:01x1 Þ and the maximum storage release rate gðx3 Þ ¼ 5x3 . The rate of destruction of the vegetative tissues is μðtÞ ¼ 2:4j cos ððπ=12ÞtÞj. (a) Non-forced annual. (b) Smaller losses lead to perennial strategy.

Fig. 4. Monocarp plant with a lifespan set to 60 months. State variables and controls are depicted in (a) and (b), respectively. The maximum photosynthesis rate is f ðx1 Þ ¼ 0:5x1 =ð1 þ 0:01x1 Þ and the storage release rate equals gðx3 Þ ¼ 2:5x3 . The rate of destruction of vegetative tissues is μðtÞ ¼ 0:4j cos ððπ=12ÞtÞj and the rate of storage losses ω  0:1.

storage is much lower than the loss of vegetative tissues. At the beginning of spring we see the regrowth of the plant from storage, and then vegetative growth due to photosynthesis, which is followed by allocation to storage as preparation for winter. This pattern of development continues until the last season, which ends with reproduction. The larger are the losses of storage ω, the earlier is a regrowth of a plant from storage. In the limit it leads to the evergreen perennial, as shown in the next subsection. 3.3. Perennial polycarpic plants Fig. 5 presents numerical examples of perennial polycarpic plants. In the case shown in the upper row, plant mortality is neglected, but storage losses are moderate. After a few years of pure vegetative growth, with the peak size of a plant increasing, there are years with both vegetative and reproductive growth. Peak size is constant over several years and it decreases close to the end of life, which is arbitrarily set at 96 months. Without such limitation, yearly cycles would be repeated infinitely. Each cycle starts with regrowth from storage during the end phase of an unfavourable season, superseded by the growth of vegetative parts from the current photosynthesis, followed by an instantaneous and complete switch to reproductive allocation (if it appears),

followed by the instantaneous and complete switch to building storage. In parallel, vegetative mass decreases during both reproductive allocation and building storage, and this decline continues during an unfavourable season. Only a small amount of vegetative mass persists over an unfavourable season, but stored resources allow for quick regrowth in the next season. Note that the size of storage which is optimal for flowering is reached in the winter before the first reproduction phase. The middle row in Fig. 5 represents an evergreen plant. Here, storage losses are so high and losses of vegetative mass are low enough so that a plant can survive during an unfavourable season in the form of vegetative parts and storage is not being built. Because the variable x3 represents not only storage but also free sugars still not allocated to another compartment, x3 is not exactly equal to zero. Note that essentially the pattern of growth of an evergreen perennial plant resembles the life history of an annual with multiple reproductive periods, except for longer multi-year life. The lowest row in Fig. 5 represents a perennial plant that does not lose storage but is subject to a constant mortality rate (i.e. exponentially decreasing survival probability L(t)). To show that mortality has a qualitative effect similar to storage losses, we choose the case with ω ¼ 0. A plant subject to both mortality and storage losses will achieve smaller size, but the general

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Fig. 5. Perennial plant, with mainly storage persisting over an unfavourable season (a, b, e, f), and an evergreen with permanent vegetative parts and no storage (c and d). In a–d mortality is neglected, but storage losses are moderate in a and b (ω  0:15) or high in c and d (ω  1). In e and f storage losses are neglected, but a constant mortality rate of 0.03 is assumed (probability of survival to age t, LðtÞ ¼ e  0:03t ); note that the results in a and b and e and f are qualitatively similar. a, c and e are the states and b, d and f are the corresponding controls. The maximum photosynthesis rate f ðx1 Þ ¼ 0:5x1 =ð1 þ 0:1x1 Þ and the maximum storage release rate gðx3 Þ ¼ 2:5x3 . The rate of destruction of vegetative tissues μðtÞ ¼ 0:4j cos ððπ=12ÞtÞj in A, B, E and F and μðtÞ ¼ 0:1j cos ððπ=12ÞtÞj in c and d. (a) Perennial plant: states. (b) Polycarpic plant: controls. (c) Evergreen polycarp: states. (d) Evergreen polycarp: controls. (e) Polycarp with mortality: states. (f) Polycarp with mortality: controls.

pattern will be the same. Reproductive output is very high in the illustrated case, because it represents plants that survive to the final time T. For fitness calculation, x2 is weighted by L.

4. Optimisation of seed mass In the previous sections we defined the fitness of a plant as an expectation of the mass of reproductive tissues produced by the

plant during its lifetime. To maximise fitness, a plant controls the allocation of photosynthate. However, it is well-known that for plants that propagate exclusively through seeds, fitness depends crucially on the quantity and size of the seeds a plant produces. Current models of optimal allocation do not provide this information, and so the mass of a seed is treated as an external parameter. Essentially, though, choice of the mass of a seed is an additional control which a plant can use in order to allocate the photosynthate efficiently. Therefore in this section we extend the model

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from the previous one by giving the plant an additional control over the mass of a seed. Assume that the whole reproductive yield of a plant is given and assume that the proportion in which this mass will be subdivided between vegetative, generative and storage compartments of seeds, equals ðy01 ; y02 ; y03 Þ≔y0 and cannot be controlled by a plant (the whole mass of a yield is then y01 þ y02 þ y03 ). Let a plant can divide its yield between a seeds, while a A ½1; 1Þ and a can be either natural or real number, and thus the mass of compartments of a seed is given as 1 xðt 0 Þ≔ðx1 ðt 0 Þ; x2 ðt 0 Þ; x3 ðt 0 ÞÞ ¼ y0 : a

ð5Þ

We model the dynamics of all individual plants as before by Eq. (1). We look for the values of control variables that maximise the total mass of reproductive tissues produced by all direct descendants: Z T max Q a ¼ aξ LðsÞx_ 2 ðsÞ ds; ð6Þ 0 r vðtÞ r 1; 0 r v1 ðtÞ r vðtÞ; a A ½1;1Þ

t0

where a constant ξ is the fraction of germinating seeds. We assume that ξ does not depend on the size of a seed and thus the optimal solution of the problem (1), (5) and (6) does not depend on ξ. In contrast to the problem (1) and (3) with the fixed mass of the seed, the problem (1), (5) and (6) may have no solution, in the sense that every finite amount of seeds may not be optimal, which means (with a slight abuse of mathematical rigorousness) that the size of the seeds may be infinitely small (a-1), i.e. no admissible controls v, v1 and parameter a generate the optimal value of Qa. Recall that a function f is called concave on the set M, if 8 z1 ; z2 A M, 8 α A ½0; 1 an inequality f ðαz1 þ ð1  αÞz2 Þ Z αf ðz1 Þ þ ð1  αÞf ðz2 Þ

ð7Þ

holds. If the inequality (7) holds with r instead of Z , then the function f is called convex on set M. From biological viewpoint important is the case when f and g are concave functions, i.e. the rate of photosynthesis and the maximal speed of chemical reactions in a plant are saturated as a result of the growth of the mass of a plant (due to self-shading of leaves, nutrient depletion in the soil, etc.). For concave f ; g satisfying f ð0Þ ¼ gð0Þ ¼ 0, one can prove (see Appendix B) that Qa, a A ½1; 1Þ increases when a increases. It follows from this result that when f and g are concave, the best strategy for a plant is to produce as many seeds as possible, which means that the seeds should be as small as possible. A similar argument shows that for convex functions f, g, the optimal mass of the seed has to be as large as possible (without additional restrictions on the quantity of seeds a¼ 1). The boundary case occurs when both f and g are linear functions, that is they are concave and convex at the same time. In this case it follows from (22) that the yield of a plant does not depend on the mass of the seeds. Another observation is that if f and g are concave and continuously differentiable in the neighbourhood of 0, then the solution of the linearised model (1), (5) and (6) provides the ‘theoretical’ upper bound for the fitness of the plant under consideration.

5. Discussion Cohen (1971) produced the first explicit model for optimising the allocation of limited resources to growth or reproduction, and this model was applied to annual plants, with the photosynthesis rate dependent linearly on vegetative mass. Denholm (1975)

19

confirmed Cohen's result through the PMP method. The model was generalised to a non-linear photosynthesis rate by Vincent and Pulliam (1980) and Ziółko and Kozłowski (1983). Both of these papers used the PMP method to analyse the problem, and in the last one mortality was introduced to the model. King and Roughgarden (1982) also considered vegetative mass losses and showed that multiple switches from vegetative to reproductive allocation may be optimal if these losses are heavy – the result that we were able to reproduce in our general model. The first papers on optimal allocation in perennial plants appeared later in Pugliese (1987, 1988) and Iwasa and Cohen (1989). Our model combines several specific models within the same framework, which is a great advantage. This goal was achieved through a crucial modification in comparison to the previous models: photosynthates are not allocated directly to vegetative and reproductive tissues, but firstly to storage, from which they could be relocated to other compartments, if optimal at a given time. Such a redefinition of a storage compartment, also including sugars just produced, is biologically reasonable as well as fruitful for mathematical modelling. Analytical analysis of the model leads to the development of general schemes of life history phases, illustrated in Fig. 1. Such schemes will be very useful in building more advanced models, as well as in planning field studies. The analysis of the model confirmed that optimal switches are instantaneous, in other words resources should be allocated to only one compartment at any given time t. Resources should go to the vegetative compartment if p1 is larger than L and p3, to storage if p3 is larger than L and p1 and to reproduction if L is the largest of the three. Simultaneous allocation is optimal only if maxfp1 ; L; p3 g is achieved by 2 or 3 functions from the set fp1 ; L; p3 g on an interval of nonzero length. In this case the optimal control is called singular. Singular controls play an important role for transitions between periods. In Yamamura et al. (2007) singular controls for the model of annual plant which undergoes grazing pressure have been studied. Since in our model of a perennial plant there are more state and control variables, even more possibilities for emergence of singular controls exist. The necessary conditions for its existence have been studied in Appendix 1.1. In particular, on this way some results from Yamamura et al. (2007) have been obtained. Moreover, we showed in numerical examples that the difference between costate variable for storage p3 and a maximum of L and p1 can be fairly small even if the controls are non-singular, which indicates that the price for suboptimal strategies can be very low. Thus, we should not expect strictly bang-bang switches in nature. Some constraints can also force the optimality of nonbang-bang solutions, as discussed later in this paper. Apart from the most important novelty, i.e treating storage as the primary sink for photosynthates, we added several other novelties to the model. Seasonality was modelled by changing the photosynthesis rate, vegetative tissue losses and storage losses over time. Although we used simple periodic functions in our numerical examples, functions extracted from real data could be applied as well. Such functions could take into account a common in some geographical regions mid-summer depression, i.e. decreasing the rate of photosynthesis in summer months, usually caused by water limitation (e.g. Rosef and Bonesmo, 2005), as well as early decreases in light penetration down to the forest floor, which forces some perennials to bloom and to set seeds early in spring. In other papers on optimal allocation, seasonal changes have not been gradual: after a favourable season, the onset of winter is rather abrupt, and then the next favourable season appears instantaneously. Such an approach precludes discovering of an interesting phenomenon: under some storage losses and vegetative parts, germination or spring regrowth may still appear in an unfavourable season, which allows for using the

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photosynthetic potential of a favourable part of the season in full. If losses of vegetative parts in winter are low, an evergreen strategy without producing storage is optimal, as shown in Fig. 5c and d. Such perennials are common, but earlier models were not able to show this strategy because of the non-realistic treatment of seasonality. Note that the transition from the strategy of persisting over winter in the form of almost an exclusive storage organ to an evergreen strategy with virtually no storage is gradual – if we decrease the losses of vegetative tissue with respect to storage losses, regrowth from storage becomes gradually earlier. Storage losses were not considered earlier in allocation models, except in the context of storage as a backup for an unpredictable and basically a-seasonal environment (Iwasa and Kubo, 1997). Although we do not show such a result, annuals starting to grow in autumn and continuing the process through to spring, as in the case of winter cereals, would be optimal under some specific loss functions for vegetative parts. The interpretation of some of the terms used in this paper could be more general. As stated previously, ‘storage’ (x3 in the model) contains real storage in the form of starch or fat, as well as free carbohydrates just produced. Similarly, losses of vegetative parts may include not only grazing or the decaying of leaves, but also a decrease in photosynthetic potential typical of ageing leaves. Storage losses may mean a usage of a part of storage for metabolic processes, but also partial consumption of storage organs by animals. Our model, as each model, has its limitations. Assuming that f depends only on vegetative mass (x1 in the model), we neglect the possibility of photosynthetic activity in reproductive tissues. Such a possibility was considered by King and Roughgarden (1982), albeit only for annual plants. Since we suppose that g is a function of x3, the rate of allocation from storage to vegetative and/or reproductive tissues is limited only by the mass of storage. This leads to the rapid regrowth of a plant from storage, because the allocation rate can be very high, even if the mass of existing vegetative tissues is extremely small. To exclude this effect one could consider more general functions g, depending also on x1 ; x2 . Another possibility, investigated in Ziolko and Kozlowski (1995) and Ioslovich and Gutman (2005), is to introduce additional constraints to the allocation rate from one type of tissue to another. However, this issue is outside of the scope of this paper. We also do not allow for the relocation of resources from vegetative mass to reproduction or storage, which could be optimal when the rate of photosynthesis drops to a very low level (Kozłowski and Wiegert, 1986). An important and special case for perennial plants are monocarps. At present, no unified theory explains the origins of monocarpy (Klinkhamer et al., 1997; Iwasa, 2000). In this paper we provide several possibilities for the emergence of a monocarpic strategy. The first sufficient condition for the genesis of a monocarpy, shown in Appendix A, is the negligibility of mortality and storage losses. In this case the strategy is not forced by the choice of the length of life T, and qualitatively it does not depend on the choice of functions f and g. In this case for a plant there is no sense in reproducing earlier because it can save energy in the storage and, since storage cannot be lost, use it later for producing either vegetative or reproductive tissues. One may argue that the mortality of a parental plant usually cannot be neglected for perennial species, but if we also add losses of reproductive tissues, then the negligibility of storage losses may still lead to the development of a monocarpic lifecycle. Monocarps (or annuals) may also arise from our model when storage losses are moderate, exemplified in Figs. 2 and 4. However, these strategies are in a sense forced by limited life span, as the plants have no choice but to reproduce at the end of their life. Such a scenario is believable if certain constraints are not allowing the

plant to live longer; for example, if a plant species has no morphological or physiological mechanisms allowing it to survive during the winter, it must be annual. However, we can find many plant genera in which annual, monocarpic and polycarpic species coexist. Even more so, some species represent different strategies in different regions, which means that constraining to one strategy is infrequent. A third possible scenario which leads to monocarpy is a rapid increase in storage losses over time. As we show in examples illustrated in Fig. 3, even without a limited lifespan an annual strategy can evolve if storage losses are very high in the unfavourable season. We call such plants non-forced annuals. In annuals producing resistance to frozen storage may have high overhead costs, and necessary resources may be more efficiently used for additional seed production. Monocarps must be equipped with mechanisms allowing for winter survival, so we therefore have to seek another explanation. de Jong and Klinkhamer (2005) suggest that mass blooming of monocarps attracts enemies. For example, flowering Senecio jacobaea plants have twice as many Tyria jacobaeae butterfly egg batches than non-flowering plants. Losses of vegetative and storage organs would be so high that producing additional seeds would increase fitness far more than storing resources for further life. Similarly, Cygnoglossum officinale setting seed plants are almost always attacked by the root weevil Ceutorhynchus cruciger, whereas non-reproducing plants are almost never attacked. To explain the development of monocarpy for such species, one may consider storage losses, depending on a mass of reproductive tissues, as done in Klinkhamer et al. (1997). It is possible to include such behaviour in our model by taking ω ¼ ωðt; x2 Þ. We hope that an analysis of this generalisation of our model will provide new insights into the genesis of monocarpy in perennial plants. We have shown that the optimal size of a seed depends crucially on the form of the functions f and g. For concave functions that are often used to take into account self-shading, the boundedness of resources, etc. (see e.g. Iwasa and Cohen, 1989), we have proved that according to our model seeds have to be as small as possible. For plants living in open environments and for species occupying early phases in succession (colonising species), the assumption of concavity is not an oversimplification. The behaviour that our model predicts, namely that the optimal strategy is to produce a vast amount of small seeds, is typical for these species (Harper et al., 1970). However, in closed and shady environments, under mineral shortage, or if there is strong competition from established vegetation, the rate of photosynthesis per unit of mass can increase with the increasing mass of the plant, i.e. function f is convex on some ½0; p, p 4 0 and the seeds cannot be too small. These predictions are, in general, in accordance with experiments of Jakobsson and Eriksson (2000) and Westoby et al. (1996), but see Metcalfe et al. (1998). Note that a similar result for offspring size in animals was obtained by Taylor and Williams (1984) and Kozlowski (1996): in a-seasonal environments, offspring should be as small as possible if the dependence of production rate/mortality rate is concave, and they should have some optimal size if dependence is convex. Because in their models the necessary condition for the existence of optimal adult size is the concavity of the ratio, optimal size for both adults and offspring can appear only if the ratio has an inflection point(s). Another important question is assessing how realistic are fitness measures (3) and (6). Such measures of fitness, known also as ‘lifetime offspring production’, are reasonable only in stable populations. In populations changing their size, the timing of reproduction is also important. Descendants which have been produced earlier can in turn produce their own offspring earlier,

A. Mironchenko, J. Kozłowski / Journal of Theoretical Biology 354 (2014) 12–24

and thus they are more valuable than descendants produced at a later stage. This indicates that the right measure of fitness of the plant is not the production of offspring over a lifetime, but lifetime offspring production by this plant and its descendants. This leads to a considerably more complicated (and more challenging) optimal control model, which to our knowledge has not been addressed in the literature for continuous-time models of plants. Much more frequent is a simpler version of fitness, defined as a number of descendants discounted to the present (Mylius and Diekmann, 1995): Z

T

WðsÞðvðsÞ  v1 ðsÞÞgðx3 ðsÞÞ ds-max;

ð8Þ

t0

where WðsÞ ¼ LðsÞe  rðs  t 0 Þ , for all s Z t 0 , r is the population growth rate (Houston and McNamara, 1992; Kawecki and Stearns, 1993; Kozłowski, 1993). If we assume that r Z0, then W is a decreasing function of time with Wðt 0 Þ ¼ Lðt 0 Þ, and an analysis of life-histories – as performed in this paper – is still valid for fitness measure (8) if we change L to W. Despite some limitations, our model, which unifies previous specific allocation models, provides deep insights into a broad range of plant strategies. Restrictions on the functions f and g as well as on other functions describing seasonality in (1) are very mild and allow for right parametrisation of the model for a particular species or group of species. This makes the model helpful for planning experiments and field measurements.

21

Another interesting topic involves generalising the results in Section 4 to the case when the survivability of a seed depends on its size, which will be more realistic from the biological point of view. It seems that methods different from those used in our paper will be required to address this problem accordingly. We have derived only very basic results related to emergence of singular controls in the model of a perennial plant. A deeper study of singular controls as well as their dependence on the climate conditions will lead to a better understanding of the transitions between different periods of plant life. We have assumed in the paper that mortality is age-dependent, but size-independent. Introducing size-dependent mortality into the model (1) makes analysis with the aid of PMP considerably more complex. Due to a number of results achieved with the help of this model, as well as due to a variety of possible directions for future investigations, we believe that this paper is a good starting point for a fruitful research program in this area.

Acknowledgments We thank Filip Kapustka and Volodymyr Nemertsalov for fruitful discussions and constructive suggestions. The computations have been made in Matlab, with the help of the optimal control solver GPOPS. JK was funded by Jagiellonian University (DS/WBiNoZ/INoS 757/13).

6. Conclusions and outlook

Appendix A. Model analysis

In the present paper we have developed a model that describes the optimal allocation strategies of a perennial (as well as an annual) plant during all stages of its life. The model was analysed with the help of PMP, and as a consequence we have derived a description of the life of a perennial plant (Fig. 1) from dormancy until its death. This model encompasses the models of annual as well as monocarpic plants. The sufficient condition for monocarpicity has also been presented. The necessary conditions for existence of periods of mixed growth of vegetative and reproductive tissues as well as of a storage (singular controls) have been considered in Appendix 1.1. In Section 3 we have shown by means of numerical simulations that the patterns of development of annual plants with one or multiple reproduction periods, monocarpic, evergreen and polycarpic perennial plants, can be obtained through different choices of model parameters. In Section 4 we analysed the trade-off between size and the number of seeds for optimal allocation problems. We have provided sufficient conditions to ensure that an optimal strategy will produce as many (or as few) seeds as possible. The applicability of our results has been discussed in Section 5. Although our model encompasses a wide variety of different patterns of plant development, it is inapplicable to certain types of plants. In particular, it is hardly possible to explain the life histories of plants which at the beginning of the season produce flowers and only then start to develop vegetative tissues. It seems that competition for pollinators should be introduced to the model in order to explain such a strategy. Plants with vegetative reproduction are entirely beyond the scope of this model. In spite of the commonness of such plants in nature, only a few papers are devoted to investigation of their life strategies (Olejniczak, 2001, 2003). This makes the development of a unifying model for plants with vegetative and sexual reproduction a challenging direction for research.

For analysis of (1) we exploit Pontryagin's Maximum Principle. Note that we can drop the equation for x2, since the other equations of (1) as well as the cost functional (3) after substitution of x_ 2 do not depend on x2. The Hamiltonian of (1) and (3) is defined by     H ¼ p1 ðtÞ v1 ðtÞg x3 ðtÞ  μðtÞx1 ðtÞ     þ λ0 LðtÞ vðtÞ v1 ðtÞ g x3 ðtÞ       þ p3 ðtÞ ζ ðtÞf x1 ðtÞ  vðtÞg x3 ðtÞ  ωðtÞx3 ðtÞ :

ð9Þ

Here λ0 Z 0 and p1 ; p3 are the so-called adjoint functions. The equations determining their dynamics will be given later. To simplify the notation, we will frequently write in equations simply p1, x2, etc. instead of p1 ðtÞ, x2 ðtÞ, if there arises no ambiguity. We rewrite expression (9) in a more suitable form H ¼ p3 ζ ðtÞf ðx1 Þ p1 μðtÞx1 þ gðx3 Þðv1 ðtÞðp1  λ0 LðtÞÞ þ vðtÞðλ0 LðtÞ p3 ÞÞ p3 ωðtÞx3 :

ð10Þ

Equations for the adjoint function p are as follows: ∂f p_ 1 ¼ p1 μðtÞ  p3 ζ ðtÞ ðx1 Þ; ∂x1 p_ 3 ¼ 

∂g ðx3 Þðv1 ðp1  λ0 LðtÞÞ þ vðλ0 LðtÞ p3 ÞÞ þ p3 ωðtÞ: ∂x3

ð11Þ

The corresponding boundary conditions are p1 ðTÞ ¼ p3 ðTÞ ¼ 0:

ð12Þ

If λ0 ¼ 0, then from (12) and (11) we obtain that pi  0 on ½t 0 ; T, from which it follows that all the controls are possible. Let λ0 4 0. We can take in this case λ0 ¼ 1.

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Controls v and v1 maximize the value of v1 ðp1  LðtÞÞ þ v ðLðtÞ  p3 Þ, therefore for optimal v; v1 it holds that

To obtain the values of v; v1 , we solve the problem H-max;

0 rv r1;

0 rv1 r v:

v1 ðp1  LðtÞÞ þ vðLðtÞ  p3 Þ Z 0:

It is not hard to check that its solution is given by

1. If LðtÞ  p3 ðtÞ 4 0, 8 > < vðtÞ v1 ðtÞ ¼ 0 > :  EAC

then vðtÞ ¼ 1, and if p1 ðtÞ  LðtÞ 40; if p1 ðtÞ  LðtÞ o0;

ð13Þ

if p1 ðtÞ  LðtÞ ¼ 0:

2. If LðtÞ  p3 ðtÞ ¼ 0, then p1 ðtÞ  LðtÞ 40 ) vðtÞ ¼ 1; v1 ðtÞ ¼ vðtÞ p1 ðtÞ  LðtÞ ¼ 0 ) vðtÞ; v1 ðtÞ  EAC p1 ðtÞ  LðtÞ o0 ) vðtÞ  EAC; v1 ðtÞ ¼ 0

ð14Þ

3. If LðtÞ  p3 ðtÞ o 0, then
if p1 ðtÞ  LðtÞ r 0 then vðtÞ ¼ v1 ðtÞ ¼ 0.
if p1 ðtÞ  LðtÞ 4 0 then p1 ðtÞ  p3 ðtÞ o 0 ) vðtÞ ¼ v1 ðtÞ ¼ 0 p1 ðtÞ  p3 ðtÞ ¼ 0 ) vðtÞ  EAC; v1 ðtÞ ¼ vðtÞ p1 ðtÞ  p3 ðtÞ 4 0 ) vðtÞ ¼ v1 ðtÞ ¼ 1

ð15Þ

Here the abbreviation EAC stands for “every admissible control”. We introduce three main periods characterized by different values of controls:

(V) Vegetative period: p1 ðtÞ 4 maxfLðtÞ; p3 ðtÞg. In this case vðtÞ ¼ v1 ðtÞ ¼ 1, that is the vegetative parts are being constructed with the maximal rate. Eq. (11) in the vegetative period takes the form ∂f p_ 1 ¼ p1 ðtÞμðtÞ  p3 ðtÞζ ðtÞ ðx1 ðtÞÞ; ∂x1 ∂g p_ 3 ¼  ðx3 ðtÞÞðp1 ðtÞ  p3 ðtÞÞ þ ωðtÞp3 : ∂x3

ð16Þ

(R) Reproductive period: LðtÞ 4 maxfp1 ðtÞ; p3 ðtÞg. In this case vðtÞ ¼ 1; v1 ðtÞ ¼ 0 and reproductive tissues are being constructed with the maximal rate. Eq. (11) within this period takes the form ∂f p_ 1 ¼ p1 ðtÞμðtÞ  p3 ðtÞζ ðtÞ ðx1 ðtÞÞ; ∂x1 ∂g p_ 3 ¼  ðx3 ðtÞÞðLðtÞ  p3 ðtÞÞ þ ωðtÞp3 ðtÞ: ∂x3

ð17Þ

(S) Storage period: p3 ðtÞ 4 maxfp1 ðtÞ; LðtÞg. In this case vðtÞ ¼ v1 ðtÞ ¼ 0 and all allocated energy goes to storage. The corresponding Eq. (11) takes the form ∂f p_ 1 ¼ p1 ðtÞμðtÞ  p3 ðtÞζ ðtÞ ðx1 ðtÞÞ; ∂x1 p_ 3 ¼ ωðtÞp3 :

ð18Þ

We are going to analyse these periods more deeply and find out in what order these periods can arise in a life of a plant. To this end we investigate Eq. (11) from the end of the life of a plant.

ð19Þ

Note that in case, when ωðtÞ  0 (that is, if storage parts cannot be destructed due to external factors), this inequality and monotonicity of g imply that p3 is an non-increasing function on ½t 0 ; T. Let us analyse the behaviour of Lagrange multipliers pi and values of controls at the neighbourhood of time T. If the last period would be vegetative, then the equations governing the dynamics of p1 ; p3 would be (16). Due to wellposedness of (16), and since the conditions (12) hold, we obtain that p1 ðtÞ  0 and p3 ðtÞ  0 in the neighbourhood of time T. Since LðtÞ 40 for all t o T, we come to a contradiction with an assumption that the last period is vegetative. Analogously one can show that also the storage period cannot be the last period of a plant's life. This proves that the last period of a plant development is a reproductive period. From Eqs. (19) and (11) using monotonicity of g and inequality ω Z 0 we have that if for some τ A ½t 0 ; T p3 ðτÞ o 0, then p3 ðtÞ o 0 for all t A ½τ; T, which contradicts to (12). Thus, p3 Z 0 on ½t 0 ; T. Analogously one can prove that p1 Z0 on ½t 0 ; T. Now let us find out what period can precede to the reproductive period. According to Eq. (18) and due to ω Z0 we see that p3 cannot decrease during the storage period. Since L is a nonincreasing function, we see that starting in a storage period (p3 4L) we cannot obtain p3 oL at the end of this period. This tells us that before reproduction period the storage period is impossible. If the climate conditions (functions μ and ζ) are such that p1 ðtÞ ¼ LðtÞ for all t A ½t s ; t 1  for some t s o t 1 , then according to (13) a plant can have the period with singular control. The conditions, necessary for emergence of such controls will be studied in the next section. Within this section we will consider the case when these controls do not appear. If p1 ðtÞ  LðtÞ is increasing from the left at t ¼ t 1 , then one can distinguish one more reproductive period ½t 1 s; t 1 Þ for some s 4 0. Throughout this paper we follow the agreement to combine all such periods together with stages with mixed controls between these periods into one reproductive period. Let p1 ðtÞ  LðtÞ be decreasing. Then for some time interval preceding to the reproductive period we have p1 ðtÞ 4 LðtÞ 4 p3 ðtÞ and therefore on this time interval a plant has a vegetative period. We call it period V:1, in contrast to period V:2 characterized by relation p1 ðtÞ 4p3 ðtÞ 4 LðtÞ (this distinction will be useful for monocarpic plants). There are two possibilities for the plant behaviour before period V:1: either it will have one more R-period (if p1 decreases lower than L(t) while it remains true that p3 oLðtÞ), or it will exist t 2 o t 1 : p3 ðt 2 Þ ¼ Lðt 2 Þ. As mentioned before, we do not consider in this section the possibility of singular controls and consider the case p_ 3 ðt 2 Þ o 0. In this case period V:2 characterized by p1 ðtÞ 4 p3 ðtÞ 4 LðtÞ precedes the period V:1. Although the allocation pattern is the same in both periods V:1 and V:2, the distinction between these periods is useful for the study of phenology of monocarpic plants. To understand this difference let us consider the case, when the non-structural carbohydrates cannot be deconstructed due to external factors (i.e. ω  0, which implies, as was mentioned earlier, that p3 is non-increasing) and the probability of survival remains constant throughout the whole period (L  const). This implies that before period V:2 the reproduction periods are not possible (p3 4L) and consequently the plant exploits monocarpic strategy. In the general case, when ω≢0 both periods R and S can precede the V-period, or all the previous life of a plant can consist

A. Mironchenko, J. Kozłowski / Journal of Theoretical Biology 354 (2014) 12–24

of one vegetative period. In the first case a plant possesses one more reproduction period, which has been already analysed. If before vegetative period there is no other period, then the plant is annual. Let now the S-period precede to the V-period. Then there exist t 4 ; t 3 : t 4 o t 3 o t 2 , such that p1 increases on ½t 4 ; t 3  (due to the unfavourable climate conditions) and p1 ðt 4 Þ ¼ p3 ðt 4 Þ. We separate period between t4 and t3 in the season V:2:1 (p1 4 p3 4LðtÞ, but p1 increasing), whose distinctive feature is that although the climate conditions are not comfortable for photosynthesis a plant anyway allocates some part of stored resources to the construction of the vegetative tissues, so as to come into the better conditions with a certain amount of already developed vegetative mass. Now let there exist some r: p1 ðtÞ op3 ðtÞ for all t A ½r; t 4 Þ. Then a plant enters a storage period. If the climate conditions are unfavourable for all t ot 4 , that is, p1 ðtÞ op3 ðtÞ for all t A ½0; t 4 Þ, then the first period of time is only the storage of allocated photosynthate (this is hardly possible because a seed has a possibility to stay this period in dormancy). If it is not the case, then there exist some moments t 6 ; t 5 , t 6 o t 5 ot 4 , such that p1 is decreasing on ½t 6 ; t 5  and p1 ðt 6 Þ ¼ p3 ðt 6 Þ. We separate the period ðt 5 ; t 4 Þ, which we call period S:1 (when the climate conditions are disadvantageous and all the allocated materials are stored), and time-span ðt 6 ; t 5 Þ called period S.2 (when the climate conditions are kindly, but all the allocated materials are anyway stored for a preparation to the unfavourable climate conditions). Both reproductive and vegetative periods can precede to the storage period. It depends on the climate conditions and values of x0. Under reasonable biological assumptions the life of a plant starts with a seed dormancy period, followed by a vegetative period. If the climate conditions are chosen to be unfavourable for photosynthesis throughout all the time-interval ½t 0 ; T, then the model is inapplicable, because the strategy to stay in dormancy all the period is not allowed in the model.

A.1. Singular controls In the analysis made above we considered only bang-bang controls, defined according to Eqs. (13)–(15). These equations determine the optimal control in the case when maxfp1 ðtÞ; LðtÞ; p3 ðtÞg is achieved by only one of these functions for all t apart from a finite number of time-instants. If maxfp1 ðtÞ; LðtÞ; p3 ðtÞg is achieved by two or three functions from the set fp1 ; L; p3 g for all t from a certain interval ½a; b, Eqs. (13)–(15) do not provide enough information to solve an optimal control problem, and a further analysis is needed to find an optimal control, which does not have a “bang-bang nature” and is called a singular control. First consider the case when p1 ðtÞ ¼ p3 ðtÞ 4 LðtÞ for all t from some interval ½τ1 ; τ2 . PMP tells us that relations (15) should be satisfied, i.e. v1  v on ½τ1 ; τ2  and v is some unknown control. This implies that v1 ðp1  LðtÞÞ þ vðLðtÞ  p3 Þ ¼ 0 on ½τ1 ; τ2  and the equation for p3 for t A ½τ1 ; τ2  becomes p_ 3 ¼ ωðtÞp3 :

ð20Þ

Since p1 ðÞ ¼ p3 ðÞ on ½τ1 ; τ2  we get from (11)

and thus ∂f ðx1 ðtÞÞ ¼ μðtÞ  ωðtÞ ∂x1

is a necessary condition for existence of a singular control (of this type) on ½τ1 ; τ2 . Assume for now that ζ ; μ; ω are constant functions (that is a plant lives in constant climate conditions) and f is a convex increasing function. If ∂f μω ðx1 ðt n ÞÞ ¼ ∂x1 ζ

and

p1 ðt n Þ ¼ p3 ðt n Þ

at some tn and if there exist a control v ¼ v1 , which allows a plant to invest into the vegetative parts exactly as much carbohydrates so as to hold the level of x1 ðtÞ constant (i.e. ¼ x1 ðt n Þ), then this control will be a singular control, which allocates carbohydrates into the vegetative tissues with non-maximal rate, and does not allocate carbohydrates to the generative tissues. Another possibility for a singular control arises if LðtÞ ¼ p3 ðtÞ 4 p1 ðtÞ for all t from some interval ½τ1 ; τ2 . According to (14) it holds that v1  0 on ½τ1 ; τ2  and v is some unknown control. This leads to v1 ðp1  LðtÞÞ þ vðLðtÞ  p3 Þ ¼ 0 on ½τ1 ; τ2  and p3 for t A ½τ1 ; τ2  satisfies Eq. (20). However, in contrast to the previous case LðÞ ¼ p3 ðÞ on ½τ1 ; τ2  and since L is a non-increasing function, p_ 3 ðtÞ r0 for all t A ½τ1 ; τ2 . Since ωðtÞ Z 0, the necessary condition for singular control is ω  0 on ½τ1 ; τ2 , i.e. there are no storage losses. This implies due to (20) that p3 ðÞ ¼ LðÞ ¼ const on ½τ1 ; τ2 . The dynamics for p1 will be as in (11). If such a singular control exists, then the carbohydrates will be allocated to the generative tissues with non-maximal rate, and they will be not allocated to vegetative tissues at all. This pattern can appear during the preparation to winter conditions. In the case, when LðtÞ ¼ p3 ðtÞ ¼ p1 ðtÞ for all t from some interval ½τ1 ; τ2 , Eqs. (13)–(15) do not provide any information about the possible controls v; v1 . However, we can find out that necessary conditions for existence of a singular control are ω  0, p1 ðÞ ¼ p3 ðÞ ¼ LðÞ ¼ const on ½τ1 ; τ2  as well as fulfilment of a condition (21), which is a reminiscent of a necessary condition for existence of a singular control obtained in Yamamura et al. (2007) within the model of an annual plant with vegetative losses. Certainly, three above cases do not exhaust all possibilities for emergence of a singular control in the model (1). Moreover, in each of above cases one can proceed further in order to obtain not only necessary conditions for existence of a singular control, but also to find more information about this control by itself, which depends on the properties of the functions, involved in the description of the model (1). Necessary conditions for existence of a singular control, derived above, are sensitive to the parameters μ; ζ ; ω, describing an influence of a climate. Since exact values of these parameters are never known already due to errors in measurements, an interesting problem will be to investigate the dependence of a singular control on small perturbations of μ; ζ ; ω. A detailed analysis of possible singular controls is outside of the scope of this paper and is left for a future research.

Appendix B. Proof of the main proposition from Section 4

Proof. The problem (5) and (6) can be written in equivalent form, using new variables yi ðtÞ≔axi ðtÞ, i ¼1, 2, 3. Then we have y  3  μðtÞy1 ; a y  y_ 2 ¼ ðvðtÞ v1 ðtÞÞag 3 ; a y  y  y_ 3 ¼ ζ ðtÞaf 1  vðtÞag 3  ωðtÞy3 ; a a yðt 0 Þ ¼ y0 :

y_ 1 ¼ v1 ðtÞag

∂f ωðtÞp1 ¼ p_ 1 ¼ p1 μðtÞ  p1 ζ ðtÞ ðx1 Þ ∂x1

ζ ðtÞ

23

ð21Þ

ð22Þ

24

A. Mironchenko, J. Kozłowski / Journal of Theoretical Biology 354 (2014) 12–24

The corresponding maximum problem is Z T Qa ¼ ξ LðsÞy_ 2 ðsÞ ds: max

0 r vðtÞ r 1; 0 r v1 ðtÞ r vðtÞ; a A ½1;1Þ

ð23Þ

t0

Now the problem is similar to (1) and (3), but with af ðy1 =aÞ and agðy3 =aÞ instead of f ðx1 Þ and gðx3 Þ. Using concavity we have: f ðy1 =aÞ ¼ f ð1=ay1 þ a  1=a  0Þ Z 1=af ðy1 Þ þ a  1=af ð0Þ ¼ 1=af ðy1 Þ. Thus, for every y1 ðtÞ Z 0, a Z 1 it holds af ðy1 ðtÞ=aÞ Z f ðy1 ðtÞÞ and therefore af ðy1 ðtÞ=aÞ and agðy3 ðtÞ=aÞ are non-decreasing in a and supa A ½1;1Þ af ðy1 ðtÞ=aÞ and supa A ½1;1Þ agðy3 ðtÞ=aÞ yields, when a-1. Define the optimal trajectories of the problem (5) and (6) for a fixed a as yðÞ. Now take arbitrary n 4 a and consider a system y  y_ 1 ¼ v1 ðtÞag 3  μðtÞy1 ; a y  y_ 2 ¼ ðvðtÞ  v1 ðtÞÞag 3 ; a y  y  y_ 3 ¼ ζ ðtÞnf 1  vðtÞag 3  ωðtÞy3 ; n a yðt 0 Þ ¼ y0 : ð24Þ The solution of this system at time t subject to optimality ^ condition (22) we denote yðtÞ. If ζ ðt 0 Þ 4 0, then from nf ðy1 ðtÞ=nÞ 4 af ðy1 ðtÞ=aÞ we have that y^_ 3 ðt 0 Þ 4 y_ 3 ðt 0 ; aÞ and therefore there exists t n 4 t 0 : y^_ 3 ðtÞ 4 y_ 3 ðt; aÞ 8 t A ½t 0 ; t n Þ. Hence y^ 3 ðtÞ 4 y3 ðt; aÞ and agðy^ 3 ðtÞ=aÞ 4 agðy3 ðt; aÞ=aÞ for t A ðt 0 ; t n Þ. Let v and v1 be the optimal controls for the system (22). There exist controls 0 r v^ r v, 0 r v^ 1 r v1 for the system (24), ^ such that vðtÞagð y^ 3 ðtÞ=aÞ ¼ vðtÞagðy3 ðt; aÞ=aÞ and v^ 1 ðtÞagðy^ 3 ðtÞ=aÞ ¼ v1 ðtÞagðy3 ðt; aÞ=aÞ. Consequently, y^ i ðtÞ ¼ yi ðt; aÞ, t A ½t 0 ; t n Þ, i¼1, 2. Constructing ^ v^ 1 for all t A ½t 0 ; T, we obtain that y^ 2 ðTÞ ¼ y2 ðt; aÞ and thus for v; a given a and n 4a the optimal trajectory of a system (24) produces no less output than the best trajectory of (22). Analogously, the output of the following system is not less than that of the system (24): y  y_ 1 ¼ v1 ðtÞng 3  μðtÞy1 ; n y  y_ 2 ¼ ðvðtÞ  v1 ðtÞÞng 3 ; n y  y  y_ 3 ¼ ζ ðtÞnf 1  vðtÞng 3  ωðtÞy3 ; n n yðt 0 Þ ¼ y0 : ð25Þ Hence Qa is non-decreasing in a.

&

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