Lyapunov functions for a generalized Gause-type model

Lyapunov functions are given to prove the global asymptotic stability of a large class of predator-prey models, including the ones in which the intrinsic growth rate of the prey follows the Ricker-law or the Odell generalization of the logistic law, and the functional predator response is of Holling type.Work supported by M.U.R.S.T., 60%.     

Extinction of top-predator in a three-level food-chain model

 In this paper we extend the Lyapunov functions, constructed by A. Ardito and P. Ricciardi for predator–prey system [1], to the three level food chain models. We first consider a general three-level food-chain model. A criterion for the extinction of top predator will be given. Then we restrict our attentions to the case in which the prey is of logistic growth and predators have Holling’s type II functional responses. Received: 10 October 1997     

Lyapunov functions for Lotka–Volterra predator–prey models with optimal foraging behavior

 The theory of optimal foraging predicts abrupt changes in consumer behavior which lead to discontinuities in the functional response. Therefore population dynamical models with optimal foraging behavior can be appropriately described by differential equations with discontinuous right-hand sides. In this paper we analyze the behavior of three different Lotka–Volterra predator–prey systems with optimal foraging behavior. We examine a predator–prey model with alternative food, a two-patch model with mobile predators and resident prey, and a two-patch model with both predators and prey mobile. We show that in the studied examples, optimal foraging behavior changes the neutral stability intrinsic to Lotka–Volterra systems to the existence of a bounded global attractor. The analysis is based on the construction and use of appropriate Lyapunov functions for models described by discontinuous differential equations. Received: 23 March 1999     

Bifurcation Analysis of Models with Uncertain Function Specification: How Should We Proceed?

When we investigate the bifurcation structure of models of natural phenomena, we usually assume that all model functions are mathematically specified and that the only existing uncertainty is with respect to the parameters of these functions. In this case, we can split the parameter space into domains corresponding to qualitatively similar dynamics, separated by bifurcation hypersurfaces. On the other hand, in the biological sciences, the exact shape of the model functions is often unknown, and only some qualitative properties of the functions can be specified: mathematically, we can consider that the unknown functions belong to a specific class of functions. However, the use of two different functions belonging to the same class can result in qualitatively different dynamical behaviour in the model and different types of bifurcation. In the literature, the conventional way to avoid such ambiguity is to narrow the class of unknown functions, which allows us to keep patterns of dynamical behaviour consistent for varying functions. The main shortcoming of this approach is that the restrictions on the model functions are often given by cumbersome expressions and are strictly model-dependent: biologically, they are meaningless. In this paper, we suggest a new framework (based on the ODE paradigm) which allows us to investigate deterministic biological models in which the mathematical formulation of some functions is unspecified except for some generic qualitative properties. We demonstrate that in such models, the conventional idea of revealing a concrete bifurcation structure becomes irrelevant: we can only describe bifurcations with a certain probability. We then propose a method to define the probability of a bifurcation taking place when there is uncertainty in the parameterisation in our model. As an illustrative example, we consider a generic predator–prey model where the use of different parameterisations of the logistic-type prey growth function can result in different dynamics in terms of the type of the Hopf bifurcation through which the coexistence equilibrium loses stability. Using this system, we demonstrate a framework for evaluating the probability of having a supercritical or subcritical Hopf bifurcation.     

Egg-eating age-structured predators in interaction with age-structured prey

In this paper, we consider a system of integrodifferential equations which models a predator-prey system with both species (predator and prey) age-structured and predators living only on the eggs of prey. The present model is a generalization of the model given in [20]. The existence, stability, and instability of nonnegative equilibria is studied assuming a general fecundity rate function for the prey. With a special choice of fecundity rate function for the predator it is shown here that a large maturation period m of the predator leads to stability. This seems to be contrary to the usual rule of thumb that increasing delays in growth rate responses cause instabilities.     

A comparison of two predator-prey models with Holling's type I functional response

In this paper, we analyze a laissez-faire predator-prey model and a Leslie-type predator-prey model with type I functional responses. We study the stability of the equilibrium where the predator and prey coexist by both performing a linearized stability analysis and by constructing a Lyapunov function. For the Leslie-type model, we use a generalized Jacobian to determine how eigenvalues jump at the corner of the functional response. We show, numerically, that our two models can both possess two limit cycles that surround a stable equilibrium and that these cycles arise through global cyclic-fold bifurcations. The Leslie-type model may also exhibit super-critical and discontinuous Hopf bifurcations. We then present and analyze a new functional response, built around the arctangent, that smoothes the sharp corner in a type I functional response. For this new functional response, both models undergo Hopf, cyclic-fold, and Bautin bifurcations. We use our analyses to characterize predator-prey systems that may exhibit bistability.     

具有时滞和避难所的捕食者-两共存食饵模型

研究了一类具有时滞和避难所的捕食-被捕食模型的一致持久性和全局稳定性.利用比较原理讨论了模型的一致持久性,运用Lyapunov函数方法得到了模型全局渐近稳定的充分条件.     

Incorporating prey refuge in a prey–predator model with a Holling type I functional response: random dynamics and population outbreaks

A prey–predator discrete-time model with a Holling type I functional response is investigated by incorporating a prey refuge. It is shown that a refuge does not always stabilize prey–predator interactions. A prey refuge in some cases produces even more chaotic, random-like dynamics than without a refuge and prey population outbreaks appear. Stability analysis was performed in order to investigate the local stability of fixed points as well as the several local bifurcations they undergo. Numerical simulations such as parametric basins of attraction, bifurcation diagrams, phase plots and largest Lyapunov exponent diagrams are executed in order to illustrate the complex dynamical behavior of the system.     

Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects

Recently, the population dynamic systems with impulsive controls have been researched by many authors. However, most of them are reluctant to study the seasonal effects on prey. Thus, in this paper, an impulsively controlled two-prey one-predator system with the Beddington–DeAngelis type functional response and seasonal effects is investigated. By using the Floquet theory, the sufficient conditions for the existence of a globally asymptotically stable two-prey-free periodic solution are established. Further, it is proven that this system is permanent under some conditions via a comparison method involving multiple Lyapunov functions and meanwhile the conditions for extinction of one of the two prey and permanence of the remaining two species are given.     

Global dynamics of two-compartment models for cell production systems with regulatory mechanisms

We present a global stability analysis of two-compartment models of a hierarchical cell production system with a nonlinear regulatory feedback loop. The models describe cell differentiation processes with the stem cell division rate or the self-renewal fraction regulated by the number of mature cells. The two-compartment systems constitute a basic version of the multicompartment models proposed recently by Marciniak-Czochra and collaborators [25] to investigate the dynamics of the hematopoietic system. Using global stability analysis, we compare different regulatory mechanisms. For both models, we show that there exists a unique positive equilibrium that is globally asymptotically stable if and only if the respective reproduction numbers exceed one. The proof is based on constructing Lyapunov functions, which are appropriate to handle the specific nonlinearities of the model. Additionally, we propose a new model to test biological hypothesis on the regulation of the fraction of differentiating cells. We show that such regulatory mechanism is incapable of maintaining homeostasis and leads to unbounded cell growth. Potential biological implications are discussed.     

基于比率的具有反馈控制的非自治捕食系统的动力学行为

讨论了一类基于比率的具有反馈控制的非自治捕食系统,所有的参数都是时滞的.先研究了该系统的一致持久性和全局渐近稳定性,并通过构造适当的Lyapunov函数,得到了系统存在惟一渐近稳定的正概周期解的充分性条件.最后,通过一个例子说明了结论的可行性.     

Metapopulation dynamics for spatially extended predator–prey interactions

Traditional metapopulation theory classifies a metapopulation as a spatially homogeneous population that persists on neighboring habitat patches. The fate of each population on a habitat patch is a function of a balance between births and deaths via establishment of new populations through migration to neighboring patches. In this study, we expand upon traditional metapopulation models by incorporating spatial heterogeneity into a previously studied two-patch nonlinear ordinary differential equation metapopulation model, in which the growth of a general prey species is logistic and growth of a general predator species displays a Holling type II functional response. The model described in this work assumes that migration by generalist predator and prey populations between habitat patches occurs via a migratory corridor. Thus, persistence of species is a function of local population dynamics and migration between spatially heterogeneous habitat patches. Numerical results generated by our model demonstrate that population densities exhibit periodic plane-wave phenomena, which appear to be functions of differences in migration rates between generalist predator and prey populations. We compare results generated from our model to results generated by similar, but less ecologically realistic work, and to observed population dynamics in natural metapopulations.     

生态系统中Lyapunov函数的构造

本文讨论生态模型中Liapunov函数的构造方法．典型的Liapunov函数的构造步骤和某些应用被综述．     

Global stability of pathogen-immune dynamics with absorption

In this paper, we consider the global stability of the models which incorporate humoural immunity or cell-mediated immunity. We consider the effect of loss of a pathogen, which is called the absorption effect when it infects an uninfected cells. We construct Lyapunov functions for these models under some conditions of parameters, and prove the global stability of the interior equilibria. It is impossible to remove the condition of parameters for the model incorporating humoural immunity.     

Effects of deterministic and random refuge in a prey-predator model with parasite infection

Most natural ecosystem populations suffer from various infectious diseases and the resulting host-pathogen dynamics is dependent on host's characteristics. On the other hand, empirical evidences show that for most host pathogen systems, a part of the host population always forms a refuge. To study the role of refuge on the host-pathogen interaction, we study a predator-prey-pathogen model where the susceptible and the infected prey can undergo refugia of constant size to evade predator attack. The stability aspects of the model system is investigated from a local and global perspective. The study reveals that the refuge sizes for the susceptible and the infected prey are the key parameters that control possible predator extinction as well as species co-existence. Next we perform a global study of the model system using Lyapunov functions and show the existence of a global attractor. Finally we perform a stochastic extension of the basic model to study the phenomenon of random refuge arising from various intrinsic, habitat-related and environmental factors. The stochastic model is analyzed for exponential mean square stability. Numerical study of the stochastic model shows that increasing the refuge rates has a stabilizing effect on the stochastic dynamics.     

带饱和项的互惠交错扩散模型整体解的存在性和稳定性

应用能量估计方法和Gagliardo-Nirenberg型不等式证明了带饱和项的Shigesada-Kawasaki-Teramoto两种群互惠模型在齐次Neumann边值条件下整体解的存在唯一性和一致有界性.通过构造Lyapunov函数给出了该模型正平衡点全局渐近稳定的条件.     

A Markov chain analysis of predator strategy in a model-mimic system

A predator which is preying on a model-mimic system can choose either the single-trial strategy or a multi-trial strategy as its behavior in learning to prudently harvest such a prey system. In this learning behavior, an important and often-posed problem is to determine which among these two strategies is better suited for the predator and why one is preferable over the other. We present in this article, using Markov chain methods, an extensive analysis of these strategies (and also of eat-everything, strategy). We conclude that the multi-trial strategy is the one that the predator should adopt (but we will also describe the situations when the single-trial strategy seems to be better). Our conclusions are based on the comparisons of quantities such as the mean benefit to the predator, energy derived by a predator from the model-mimic system and (a newly introduced notion of) contagion in eating mimics and models (these quantities are computed for different strategies). The first two quantities are functions of the abundancep and noxiousnessb of models. The contagion is a function of onlyp; and, though independent ofb, it is also in support of multi-trial, strategy. We introduce, in the present context, a biological analog of the d'Alembert principle and also derive functions describing the influences of eating a specified type of prey at a given time on eating any type of prey at a later time. Various results of Estabrook-Jespersen (single-trial strategy) and Bobisud-Potratz (multi-trial strategy) are re-derived as special cases of our more general results. A central limit theorem under the eat-everything strategy is given.     

Qualitative dynamics of three species predator-prey systems

Summary A qualitative analysis of some two and three species predator-prey models is achieved by application of the method of averaging in conjunction with a Lyapunov function constructed from the appropriate Volterra-Lotka model. We calculate the limit cycle solution for a two-species model with a Holling type functional response of the predator to its prey by means of a time-scaled transformation. The existence of a bifurcation of steady states for a community of three species is discussed and the periodic solution around one of the unstable steady states is calculated to the lowest approximation. Several comments are made regarding the behavior of these systems under changes of some control parameters.This work was supported in parts by USERDA, Contract number E(11-1)-3001.     

Evolutionary stability of optimal foraging: Partial preferences in the diet and patch models

In this article the patch and diet choice models of the optimal foraging theory are reanalyzed with respect to evolutionary stability of the optimal foraging strategies. In their original setting these fundamental models consider a single consumer only and the resulting fitness functions are both frequency and density independent. Such fitness functions do not allow us to apply the classical game theoretical methods to study an evolutionary stability of optimal foraging strategies for competing animals. In this article frequency and density dependent fitness functions of optimal foraging are derived by separation of time scales in an underlying population dynamical model and corresponding evolutionarily stable strategies are calculated. Contrary to the classical foraging models the results of the present article predict that partial preferences occur in optimal foraging strategies as a consequence of the ecological feedback of consumer preferences on consumer fitness. In the case of the patch occupation model these partial preferences correspond to the ideal free distribution concept while in the case of the diet choice model they correspond to the partial inclusion of the less profitable prey type in predators diet.     

Should "handled" prey be considered? Some consequences for functional response, predator-prey dynamics and optimal foraging theory

Predator-prey models consider those prey that are free. They assume that once a prey is captured by a predator it leaves the system. A question arises whether in predator-prey population models the variable describing prey population shall consider only those prey which are free, or both free and handled prey together. In the latter case prey leave the system after they have been handled. The classical Holling type II functional response was derived with respect to free prey. In this article we derive a functional response with respect to prey density which considers also handled prey. This functional response depends on predator density, i.e., it accounts naturally for interference. We study consequences of this functional response for stability of a simple predator-prey model and for optimal foraging theory. We show that, qualitatively, the population dynamics are similar regardless of whether we consider only free or free and handled prey. However, the latter case may change predictions in some other cases. We document this for optimal foraging theory where the functional response which considers both free and handled prey leads to partial preferences which are not observed when only free prey are considered.