Local and global stability for population models

In general, local stability does not imply global stability. We show that this is true even if one only considers population models.We show that a population model is globally stable if and only if it has no cycle of period 2. We also derive easy to test sufficient conditions for global stability. We demonstrate that these sufficient conditions are useful by showing that for a number of population models from the literature, local and global stability coincide.We suggest that the models from the literature are in some sense simple, and that this simplicity causes local and global stability to coincide.     

Graph theoretical criteria for stability and boundedness of predator-prey systems

We introduce a graphical approach in the study of the qualitative behavior ofm species predator-prey systems. We prove that tree graphs imply global stability for Volterra models and local stability for general models; furthermore, we derive sufficient conditions so that loop graphs imply stability and boundedness of the solutions.     

Stability of discrete one-dimensional population models

We give conditions for local and global stability of discrete one-dimensional population models. We give a new test for local stability when the derivative is −1. We give several sufficient conditions for global stability. We use these conditions to show that local and global stability coincide for the usual models from the literature and even for slightly more complicated models. We give population models, which are in some sense the simplest models, for which local and global stability do not coincide.     

A unifying framework for chaos and stochastic stability in discrete population models

 In this paper we propose a general framework for discrete time one-dimensional Markov population models which is based on two fundamental premises in population dynamics. We show that this framework incorporates both earlier population models, like the Ricker and Hassell models, and experimental observations concerning the structure of density dependence. The two fundamental premises of population dynamics are sufficient to guarantee that the model will exhibit chaotic behaviour for high values of the natural growth and the density-dependent feedback, and this observation is independent of the particular structure of the model. We also study these models when the environment of the population varies stochastically and address the question under what conditions we can find an invariant probability distribution for the population under consideration. The sufficient conditions for this stochastic stability that we derive are of some interest, since studying certain statistical characteristics of these stochastic population processes may only be possible if the process converges to such an invariant distribution. Received 15 May 1995; received in revised form 17 April 1996     

Global stability in discrete population models with delayed-density dependence

We address the global stability issue for some discrete population models with delayed-density dependence. Applying a new approach based on the concept of the generalized Yorke conditions, we establish several criteria for the convergence of all solutions to the unique positive steady state. Our results support the conjecture stated by Levin and May in 1976 affirming that the local asymptotic stability of the equilibrium of some delay difference equations (including Ricker's and Pielou's equations) implies its global stability. We also discuss the robustness of the obtained results with respect to perturbations of the model.     

On global stability of discrete population models

A necessary and sufficient condition for the global stability of a large class of discrete population models is provided which does not require the construction of a Liapunov function. The general result is applied to difference equations defined in terms of “two hump” functions and to an example of frequency dependent selection.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet–Kolmogorov L ¹-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

Differential susceptibility epidemic models

We formulate compartmental differential susceptibility (DS) susceptible-infective-removed (SIR) models by dividing the susceptible population into multiple subgroups according to the susceptibility of individuals in each group. We analyze the impact of disease-induced mortality in the situations where the number of contacts per individual is either constant or proportional to the total population. We derive an explicit formula for the reproductive number of infection for each model by investigating the local stability of the infection-free equilibrium. We further prove that the infection-free equilibrium of each model is globally asymptotically stable by qualitative analysis of the dynamics of the model system and by utilizing an appropriately chosen Liapunov function. We show that if the reproductive number is greater than one, then there exists a unique endemic equilibrium for all of the DS models studied in this paper. We prove that the endemic equilibrium is locally asymptotically stable for the models with no disease-induced mortality and the models with contact numbers proportional to the total population. We also provide sufficient conditions for the stability of the endemic equilibrium for other situations. We briefly discuss applications of the DS models to optimal vaccine strategies and the connections between the DS models and predator-prey models with multiple prey populations or host-parasitic interaction models with multiple hosts are also given.This research was partially supported by the Department of Energy under contracts W-7405-ENG-36 and the Applied Mathematical Sciences Program KC-07-01-01.     

Global dynamics of a discrete two-species Lottery-Ricker competition model

In this article, we study the global dynamics of a discrete two-dimensional competition model. We give sufficient conditions on the persistence of one species and the existence of local asymptotically stable interior period-2 orbit for this system. Moreover, we show that for a certain parameter range, there exists a compact interior attractor that attracts all interior points except Lebesgue measure zero set. This result gives a weaker form of coexistence which is referred to as relative permanence. This new concept of coexistence combined with numerical simulations strongly suggests that the basin of attraction of the locally asymptotically stable interior period-2 orbit is an infinite union of connected components. This idea may apply to many other ecological models. Finally, we discuss the generic dynamical structure that gives relative permanence.     

Traveling waves in the discrete fast buffered bistable system

We study the existence and uniqueness of traveling wave solutions of the discrete buffered bistable equation. Buffered excitable systems are used to model, among other things, the propagation of waves of increased calcium concentration, and discrete models are often used to describe the propagation of such waves across multiple cells. We derive necessary conditions for the existence of waves, and, under some restrictive technical assumptions, we derive sufficient conditions. When the wave exists it is unique and stable.      

Effective competition determines the global stability of model ecosystems

We investigate the stability of Lotka-Volterra (LV) models constituted by two groups of species such as plants and animals in terms of the intragroup effective competition matrix, which allows separating the equilibrium equations of the two groups. In matrix analysis, the effective competition matrix represents the Schur complement of the species interaction matrix. It has been previously shown that the main eigenvalue of this effective competition matrix strongly influences the structural stability of the model ecosystem. Here, we show that the spectral properties of the effective competition matrix also strongly influence the dynamical stability of the model ecosystem. In particular, a necessary condition for diagonal stability of the full system, which guarantees global stability, is that the effective competition matrix is diagonally stable, which means that intergroup interactions must be weaker than intra-group competition in appropriate units. For mutualistic or competitive interactions, diagonal stability of the effective competition is a sufficient condition for global stability if the inter-group interactions are suitably correlated, in the sense that the biomass that each species provides to (removes from) the other group must be proportional to the biomass that it receives from (is removed by) it. For a non-LV mutualistic system with saturating interactions, we show that the diagonal stability of the corresponding LV system close to the fixed point is a sufficient condition for global stability.     

一类含间隙分布时滞的种群增长模型的稳定性

本文首先利用间隙分布时滞函数来建立更为符合实际的种群增长模型,然后运用两种不同的方法,对其平衡位置的局部稳定性进行了全面的讨论,得出了局部渐近稳定的充分必要条件,在参数平面上划分出了稳定和不稳地的区域。     

Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE)

One goal of this paper is to give an algorithm for computing a threshold condition for epidemiological systems arising from compartmental deterministic modeling. We calculate a threshold condition T(0) of the parameters of the system such that if T(0)<1 the disease-free equilibrium (DFE) is locally asymptotically stable (LAS), and if T(0)>1, the DFE is unstable. The second objective, by adding some reasonable assumptions, is to give, depending on the model, necessary and sufficient conditions for global asymptotic stability (GAS) of the DFE. In many cases, we can prove that a necessary and sufficient condition for the global asymptotic stability of the DFE is R(0)< or =1, where R(0) is the basic reproduction number [O. Diekmann, J.A. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation, Wiley, New York, 2000]. To illustrate our results, we apply our techniques to examples taken from the literature. In these examples we improve the results already obtained for the GAS of the DFE. We show that our algorithm is relevant for high dimensional epidemiological models.     

A note on persistence about structured population models

In this paper, we report some results on persistence in two structured population models: a chronic- age-structured epidemic model and an age-duration-structured epidemic model. Regarding these models, we observe that the system is uniformly strongly persistent, which means, roughly speaking, that the proportion of infected subpopulation is bounded away from 0 and the bound does not depend on the initial data after a sufficient long time, if the basic reproduction ratio is larger than one. We derive this by adopting Thieme's technique, which requires some conditions about positivity and compactness. Although the compactness condition is rather difficult to show in general infinite-dimensional function spaces, we can apply Fréchet-Kolmogorov L(1)-compactness criteria to our models. The two examples that we study illuminate a useful method to show persistence in structured population models.     

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k>2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.     

Oscillatory viral dynamics in a delayed HIV pathogenesis model

We consider an HIV pathogenesis model incorporating antiretroviral therapy and HIV replication time. We investigate the existence and stability of equilibria, as well as Hopf bifurcations to sustained oscillations when drug efficacy is less than 100%. We derive sufficient conditions for the global asymptotic stability of the uninfected steady state. We show that time delay has no effect on the local asymptotic stability of the uninfected steady state, but can destabilize the infected steady state, leading to a Hopf bifurcation to periodic solutions in the realistic parameter ranges.     

A review on spatial aggregation methods involving several time scales

This article is a review of spatial aggregation of variables for time continuous models. Two cases are considered. The first case corresponds to a discrete space, i.e. a set of discrete patches connected by migrations, which are assumed to be fast with respect to local interactions. The mathematical model is a set of coupled ordinary differential equations (O.D.E.). The spatial aggregation allows one to derive a global model governing the time variation of the total numbers of individuals of all patches in the long term. The second case considers a continuous space and is a set of partial differential equations (P.D.E.). In that case, we also assume that diffusion is fast in comparison with local interactions. The spatial aggregation allows us again to obtain an O.D.E. governing the total population density, which is obtained by integration all over the spatial domain, at the slow time scale. These aggregations of variables are based on time scales separation methods which have been presented largely elsewhere and we recall the main results. We illustrate the methods by examples in population dynamics and prey–predator models.     

具有时滞的细胞神经网络模型的全局指数稳定性

利用拓扑度理论、推广的Halanaly矩阵时滞微分不等式、Lyapunov原理以及Dini导数,研究了具有时滞的细胞神经网络模型的全局指数稳定性．去掉了有关文献中要求输出函数fj在实数集R上有界、可微的条件,给出了更弱的判定平衡点的存在唯一性以及全局指数稳定性的判据,推广和改进了前人的相关结论,最后的数值例子说明本文结果不仅保守性小,而且计算简单．