Numerical Equilibrium Analysis for Structured Consumer Resource Models

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Bulletin of Mathematical Biology : DOI /s ORIGINAL ARTICLE Numerical Equilibrium Analysis for Structured Consumer Resource Models A.M. de Roos a,o.diekmann b, P. Getto
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Bulletin of Mathematical Biology : DOI /s ORIGINAL ARTICLE Numerical Equilibrium Analysis for Structured Consumer Resource Models A.M. de Roos a,o.diekmann b, P. Getto c,, M.A. Kirkilionis d a Institute for Biodiversity and Ecosystem Dynamics IBED, University of Amsterdam, P.O. Box 94084, 1090GB Amsterdam, The Netherlands b Department of Mathematics, University of Utrecht, Budapestlaan 6, P.O. Box 80010, 3508 TA Utrecht, The Netherlands c BCAM Basque Center for Applied Mathematics, Bizkaia Technology Park, Derio, Bizkaia, Spain d Department of Mathematics, University of Warwick, CV4 7AL Coventry, UK Received: 3 March 2009 / Accepted: 16 July 2009 / Published online: 31 July 2009 The Authors This article is published with open access at Springerlink.com Abstract In this paper, we present methods for a numerical equilibrium and stability analysis for models of a size structured population competing for an unstructured resource. We concentrate on cases where two model parameters are free, and thus existence boundaries for equilibria and stability boundaries can be defined in the two-parameter plane. We numerically trace these implicitly defined curves using alternatingly tangent prediction and Newton correction. Evaluation of the maps defining the curves involves integration over individual size and individual survival probability and their derivatives as functions of individual age. Such ingredients are often defined as solutions of ODE, i.e., in general only implicitly. In our case, the right-hand sides of these ODE feature discontinuities that are caused by an abrupt change of behavior at the size where juveniles are assumed to turn adult. So, we combine the numerical solution of these ODE with curve tracing methods. We have implemented the algorithms for Daphnia consuming algae models in C-code. The results obtained by way of this implementation are shown in the form of graphs. Keywords Numerical equilibrium analysis Structured populations Stability boundaries Hopf bifurcation Consumer resource models Delay equations Renewal equations Delay differential equations Daphnia models 1. Introduction In this paper, we present methods for a numerical equilibrium and stability analysis of a class of models of the interaction between a structured consumer population and an unstructured resource population. Corresponding author. address: P. Getto. 260 de Roos et al. In unstructured consumer resource models and in predator prey models, one typically finds a Hopf-bifurcation marking the destabilization of an interior equilibrium and the emergence of a stable limit cycle Rosenzweig, In de Roos et al. 1990, one finds a model of a size structured population of Daphnia magna consuming algae, which is parameterized on the basis of experimental data. Based on extensive numerical calculations, it is concluded in de Roos et al that incorporating size structure of the consumer induces additional features, namely the possibility of the coexistence of a stable and an unstable limit cycle and even the coexistence of two stable limit cycles. In Diekmann et al. 2007, a theoretical framework is described that establishes existence and uniqueness, the principle of linearized stability and the Hopf-bifurcation theorem for a class of abstract integral equations. For given resource, the dynamics of a structured population can be described by a renewal equation; see Metz and Diekmann This equation can also be classified as a Volterra integral equation or a functional equation of delay type with continuously distributed delay; see Diekmann et al Incorporating competition then amounts to adding an equation for the resource dynamics Diekmann et al., This equation has on the right-hand side a similar structure as the renewal equation, but the left-hand side consists of the time derivative of the unknown resource concentration, whereas in the case of the renewal equation, it consists of the unknown function itself. It hence should be called an integro-differential equation or a delay differential equation again with continuously distributed delay. Hence, in summary, we shall speak of a system of delay equations, a renewal equation coupled to a delay differential equation. Such systems are equivalent to special cases of the mentioned abstract integral equations; see again Diekmann et al., In de Roos et al. 2009, the mathematical setting of Diekmann et al is used to establish an analytical stability and bifurcation theory for structured consumer resource models with special attention for the Daphnia model in de Roos et al Our aim here is to complement the analytical theory in de Roos et al with tools for the numerical equilibrium and stability analysis of such models. Even though we use Daphnia models to test our algorithms, our approach works for more general structured consumer resource models. Moreover, we use formulations that should be easy to generalize to other structured population models. As key results of the analysis of qualitative behavior, in de Roos et al curves in two-parameter space marking stability boundaries where Hopf-bifurcations occur were presented. Integrals, which arise naturally when modeling structured populations, could in the case of the Daphnia model be evaluated analytically, and thus the curves could be approximated by coding established predictor- corrector methods Allgower and Georg, 1990 without it being necessary to refer to numerical integration. We present a method here to compute stability boundaries for models specified in terms of general vital rates. Integrals involving these general vital rates can be evaluated by integration of a system of coupled ODE, which is generally not possible by hand. The idea, which can be found already in Kirkilionis et al. 2001, is to combine the curve tracing methods with numerical integration of the ODE. We use this idea here to establish methods to compute existence and stability boundaries for structured consumer resource models formulated as systems of delay equations. The Daphnia models in de Roos et al and2009 exhibit a discontinuity of the consumers vital rates at the size at which juveniles turn adult, caused by an abrupt change of behavior, notably an abrupt onset of reproduction, upon reaching this size. We therefore allow also here for such discontinuities. Numerical Equilibrium Analysis for Structured Consumer Resource 261 Much of the length of this paper is caused by the complexity of the characteristic equation. This complexity is due to the fact that the following issues are involved in its computation: differentiation of solutions of ODE with respect to infinite dimensional parameters, discontinuities at the size where adulthood is reached, the rewriting of ingredients of the characteristic equation as solutions of real ODE. In de Roos et al. 2009, an expression for the ingredients of the characteristic equation for our class of structured consumer resource models is given. Our present derivation of the characteristic equation is much inspired by this paper, but we found that for the numerical computation of the elements a different representation of the characteristic equation is more convenient. In Section 2, we introduce the analytic setting for a class of models describing the interaction of a structured consumer population with an unstructured resource population. We formulate the population equations and discuss the existence of a unique interior equilibrium. Then we concentrate on the case where two model parameters are free and define existence boundaries for the equilibrium in the two-parameter plane. In this plane, we then define a curve marking Hopf-bifurcation points. The main ingredient here is the characteristic equation obtained via linearization of the population equations. In Section 3, we present methods to numerically approximate the curves defined in Section 2. We start with an algorithm for the numerical integration involved in the approximation of equilibria. In a further algorithm, we show how to achieve this approximation by a combination of the numerical integration method and a Newton-method. In the case of two free parameters, we show how the curves can be continued by combining first tangent prediction and then correction with a Newton method with numerical integration. Here, the map defining stability boundaries is much more involved than the one defining existence boundaries, due to the already mentioned complexity of the characteristic equation. In Section 4, we present the results obtained via the implementation of the various algorithms using as examples models of a structured Daphnia population interacting with an unstructured algal population. We use specifications of vital rates in terms of parameters and parameter values as in de Roos et al and de Roos We have implemented the algorithms to compute existence and stability boundaries by numerical integration combined with curve tracing for the Daphnia model in a C-code. The parts of this code that deal with numerical integration and curve tracing are problem independent, whereas in other parts of the code the ingredients specific for the Daphnia model are used. We present results of the computations for the Daphnia model in the form of numerical examples and graphs of traced curves in two-parameter planes. We tested our algorithms by comparing our results with stability curves computed following the approaches in de Roos et al. 1990and de Roos 1997, in which curves were traced after deriving analytic expressions for the integrals over the vital rates for the Daphnia model. The curves we compute in this paper match with the curves computed in de Roos et al and de Roos 1997. 262 de Roos et al. 2. Consumer resource dynamics We present the analytic formulation of the class of models here that we will investigate. We first give the population equations and equilibrium conditions. Then we define existence and stability boundaries in two parameter planes. Much of Sections 2.1 and 2.2 overlap with de Roos et al and is presented here for the sake of completeness The model We first derive the population equations stepping from the individual level to the population level formally, i.e., we postpone smoothness discussions. We then give smoothness assumptions, discuss discontinuities induced by an abrupt onset of reproduction, and finally become more specific about the derivation of size and survival as functions of age The population equations formally Let us denote by St the available resource or food concentration at time t. Ourway of bookkeeping at the population level leads us to introduce histories, first as functions defined on, 0]. For the resource S, we introduce the notation S t σ := St + σ, σ, 0], 1 which is common in the theory of functional differential equations Hale, Then S t is a history for every t, the history of the resource at time t. Let us assume that there is only one possible size x b at which individuals are born. Next, we introduce Xa,Ψ as the size that an individual has at age a, given that it has experienced history Ψ in the time interval [ a,0]. ThenXa,S t is the size that an individual has at age a and time t, given that it has experienced resource concentration S in the time interval [t a,t]; seefig.1. Likewise, we introduce Fa, Ψ as the survival probability to age a of an individual, given that if it survives, it has experienced history Ψ in the time interval [ a,0].thenfa, S t is the probability for an individual to reach age a at time t given that it experiences resource concentration S in the time interval [t a,t]. Next, we denote by βx,y the rate of reproduction of an individual of size x under resource condition y and by γx,y the rate of food consumption of an individual of size x under resource condition y. So, for example, βxa,s t, St is the rate of reproduction of an individual of age a at time t. Finally, fy denotes the intrinsic rate of change of the resource, meaning the rate of change in absence of the consumer. Let us denote by bt the population birth rate, i.e., the number of individuals born in total or per unit of area or volume, depending on the context with size x b per unit of time at time t. Letus denote by h 0 the maximum lifetime of an individual under ideal food conditions. Then we can describe the population dynamics by the system of equations bt = S t = f St 0 β Xa,S t, St Fa, S t bt ada, 2 0 γ Xa,S t, St Fa, S t bt ada; 3 see again Fig. 1. TheEqs.2 3 form a system of a renewal equation 2 coupled to a delay differential equation 3. The qualitative behavior of such systems can be studied using the theory developed in Diekmann et al. 1995, 2007, and de Roos et al Numerical Equilibrium Analysis for Structured Consumer Resource 263 Fig. 1 Equation 2: reproduction at time t as induced by the history of resource and population birth rate Smoothness conditions and discontinuities caused by abrupt onset of reproduction Motivated by the Daphnia consuming algae model in de Roos et al. 1990, we suppose that at size x A x b juveniles mature. Keeping in mind that we would like to linearize later, we should formalize the assumption that rates are smooth, except for a possible jump in x A. We thus assume that β and γ can be defined on [x b, via functions that are C 1 on [x b,x A ] R + and on [x A, R +. This leads to unique definitions on [x b, \{x A } R + and to a double definition in x A,yfor all y R +. This double definition, however, is irrelevant, as we are ultimately interested in the integrated functions. Finally, we assume that βx,y = 0forallx [x b,x A and all y. So, neither for β, nor for γ there can be expected continuity, let alone differentiability, in x A. Next, suppose that f : R + R is C 1. We assume that S is continuous and restrict the domain of definition of histories, in particular of S t, from, 0] to [ h, 0]. To guarantee integrability in 2 3 andfor later differentiability in the S-component, we require that X and F are such that C [ h, 0], R L [0,h], R, ψ X,ψ, 4 ψ F,ψ are continuously differentiable maps. Next, if an individual matures exactly at the present time for a given food history ψ, wedenoteitsageatmaturationbya A, i.e., we define a A via the equation Xa A,ψ= x A, 5 264 de Roos et al. the solvability of which will be discussed in Section below. Finally, we assume that b is nonnegative and integrable. The population equations, reflecting the jump in x A are now given by bt = β Xa,S t, St Fa, S t bt ada, 6 a A S t S t = f St aa S t 0 γ Xa,S t, St Fa, S t bt ada γ Xa,S t, St Fa, S t bt ada, 7 a A S t where the integrals from a A S t to h should be interpreted as zero when a A S t h.in7, we have split the integration interval into two parts in order to highlight the discontinuity in the integrand. In the following, however, we shall simply write one integral from zero to h, as the jump discontinuity is harmless with respect to integration Computation of size and survival functions as solutions of ODE We show how X and F can be computed for the case that one has given rates gx,y and μx,y of individual growth and mortality. Just like for γ, we also assume that g,μ : R 2 + R + are C 1 on [x b,x A ] R + and on [x A, R +. We additionally assume the existence of positive lower bounds for g and μ that are uniform for all sizes and uniform for all resource conditions outside a neighborhood of zero which should be chosen sufficiently small not to contain positive equilibria which will be defined below. The definition of X and F in terms of the history of a time dependent resource via the rates g and μ leads to a certain notational complexity. We here use the same notation as in de Roos et al We denote by xα = xα; a,ψ the size of an individual at age α, given that at age a, if still alive, it has experienced resource history ψ. Likewise, we denote by Fα = Fα; a,ψ the probability that an individual survives up to age α, given that at age a, if still alive, it has experienced resource history ψ. Then we can define Xa,S t := xa; a,s t and Fa, S t := Fa; a,s t. Fα by solving the system of one-sidedly coupled nonau- We compute xα and tonomous ODE x α = g xα,ψ a + α, 0 α a, x0 = x b, F α = μ xα,ψ a + α Fα, 0 α a, F0 = Equations 8 9 are linearized with respect to ψ in Appendix A to compute expressions for the derivatives of X and F with respect to ψ. Numerical Equilibrium Analysis for Structured Consumer Resource Steady states For the system 6 7, an equilibrium is a pair of constants b,s, such that b, S := b,s fulfills 6 7. If b = 0, S should be such that fs = 0. In this case, we have a trivial equilibrium 0, S, which we disregard here. A nontrivial equilibrium is given by a pair of constants b,s fulfilling R 0 S 1 = 0, 10 f S Θ S b = 0, 11 h R 0 S := β X a,s, S F a,s da, Θ S := where we denote by := a A S 0 γ X a,s, S F a,s da, 12 the age at which individuals mature under steady state conditions. Note that R 0 S and ΘS are, respectively, the expected lifetime offspring production and the expected lifetime resource consumption of a consumer individual, which gives obvious interpretations of the steady state conditions. As, for ΘS 0, 11 can be solved explicitly with respect to b, we write b S := fs ΘS, 13 and reduce the steady state problem to finding an S satisfying 10. A typical case is that 10 has a unique solution; see, e.g., de Roos et al. 1990, and also here we assume that this holds. We give a method to approximate the solution S of 10 in Section Existence boundaries We suppose in the following that two model parameters, which we denote by α 1 and α 2, are free. We will or will not incorporate free parameter dependence of functions into the notation according to convenience and relevance in the context. We then rewrite as R 0 α1,α 2, S 1 = 0, 14 b := fα 1,α 2, S Θα 1,α 2, S, 15 where it will prove convenient in Section 3 to denote first parameters and then the equilibrium. The b component of the equilibrium becomes positive if the curve in the α 1 α 2 -plane defined by the two equations f α 1,α 2, S = 0 16 266 de Roos et al. and 14 is crossed in the appropriate sense. We hence call this curve the existence boundary for the nontrivial equilibrium Existence boundaries for Daphnia models and equilibrium curves In the models of Daphnia consuming algae, we assume in absence of Daphnia chemostat or logistic algal dynamics and choose α 1 as the mortality for Daphnia and α 2 as the carrying capacity for algae. Hence, typically, for given S, f is independent of α 1 and R 0 independent of α 2. Moreover, 16 is equivalent to S = α 2 for chemostat or to S = 0or S = α 2 for logistic dynamics; see Section 4 below. Finally, typically 14 has no solution for S = 0, as in the absence of food there is no reproduction. In summary, for Daphnia, the existence boundary defined by 14and16 can equivalently be defined by S = α 2, R 0 α1, S 1 = 0, 17 where we dropped the nonmanifesting α 2 -dependence in the notation of R 0. It is hence clear that in this special setting the existence boundary in the α 1 α 2 -plane and the curve describing how the equilibrium changes if one parameter varies are equal. We will give methods to compute existence boundaries for both the general setting and the setting for Daphnia in Section Steady state defining functions as ODE and stopping criteria for integration As in Kirkilionis et al. 2001, in view of the interdependence of functions and the discontinuity at x A we propose to compute several model ingredients as solutions of ODE. We introduce two alternative criteria for ending the integration of the ODE. The choice of which one to use should depend on the biological problem one considers. The first is the reaching of a maximum age A max, in which case we should choose h := A max. The second criterion is the one proposed in Kirkilionis et al. 2001: For every ε 0, 1, we define a ε = a ε S via the equation Fa ε, S = ε, i.e., as the age at which the probability that an individual survives up to it has decreased to ε, and assum
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