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.     

Global asymptotic stability of density dependent integral population projection models

Many stage-structured density dependent populations with a continuum of stages can be naturally modeled using nonlinear integral projection models. In this paper, we study a trichotomy of global stability result for a class of density dependent systems which include a Platte thistle model. Specifically, we identify those systems parameters for which zero is globally asymptotically stable, parameters for which there is a positive asymptotically stable equilibrium, and parameters for which there is no asymptotically stable equilibrium.     

Global stability for cholera epidemic models

Cholera is a water and food borne infectious disease caused by the gram-negative bacterium, Vibrio cholerae. Its dynamics are highly complex owing to the coupling among multiple transmission pathways and different factors in pathogen ecology. Although various mathematical models and clinical studies published in recent years have made important contribution to cholera epidemiology, our knowledge of the disease mechanism remains incomplete at present, largely due to the limited understanding of the dynamics of cholera. In this paper, we conduct global stability analysis for several deterministic cholera epidemic models. These models, incorporating both human population and pathogen V. cholerae concentration, constitute four-dimensional non-linear autonomous systems where the classical Poincaré-Bendixson theory is not applicable. We employ three different techniques, including the monotone dynamical systems, the geometric approach, and Lyapunov functions, to investigate the endemic global stability for several biologically important cases. The analysis and results presented in this paper make building blocks towards a comprehensive study and deeper understanding of the fundamental mechanism in cholera dynamics.     

Global asymptotic stability of plant-seed bank models

Many plant populations have persistent seed banks, which consist of viable seeds that remain dormant in the soil for many years. Seed banks are important for plant population dynamics because they buffer against environmental perturbations and reduce the probability of extinction. Viability of the seeds in the seed bank can depend on the seed’s age, hence it is important to keep track of the age distribution of seeds in the seed bank. In this paper we construct a general density-dependent plant-seed bank model where the seed bank is age-structured. We consider density dependence in both seedling establishment and seed production, since previous work has highlighted that overcrowding can suppress both of these processes. Under certain assumptions on the density dependence, we prove that there is a globally stable equilibrium population vector which is independent of the initial state. We derive an analytical formula for the equilibrium population using methods from feedback control theory. We apply these results to a model for the plant species Cirsium palustre and its seed bank.     

Global behaviour of age-dependent logistic population models

We study the large time behaviour of a nonlinear population model with a general logistic term. It is proved that every solution must have a limit when time becomes infinite. We present conditions that guarantee the boundedness of the solution. Furthermore, we prove that in general no oscillation is possible for the total number of population. This is in sharp contrast to the linear case.This work was carried out with the aid of a grant from the International Development Centre, Ottawa, Canada     

Global asymptotic stability in Volterra's population systems

Sufficient conditions which can be verified easily are obtained for the global asymptotic stability of the positive steady state in Volterra's population system incorporating hereditary effects.     

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.     

Global stability in a class of prey-predator models

Global stability is established in a class of prey-predator models. This includes a prey-predator model in which the predator has Type 2 functional response and no intraspecific interactions. Two simple examples demonstrate that Kolmogoroff’s theorem does not apply to some members of this class of models.     

Complexity and demographic stability in population models

This article is concerned with relating the stability of a population, as defined by the rate of decay of fluctuations induced by demographic stochasticity, with its heterogeneity in age-specific birth and death rates. We invoke the theory of large deviations to establish a fluctuation theorem: The demographic stability of a population is positively correlated with evolutionary entropy, a measure of the variability in the age of reproducing individuals in the population. This theorem is exploited to predict certain correlations between ecological constraints and evolutionary trends in demographic stability, namely, (i) bounded growth constraints--a uni-directional increase in stability, (ii) unbounded growth constraints (large population size)--a uni-directional decrease in stability, (iii) unbounded growth constraints (small population size)--random, non-directional change in stability. These principles relating ecological constraints with trends in demographic stability are shown to be far reaching generalizations of the tenets derived from classical studies of stability in an evolutionary context. These results thus provide a new conceptual framework for explaining patterns of variation in population numbers observed in natural populations.     

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.     

A note on stability of discrete population models

P. Cull (1981) and G. Rosencranz (1983) studied a discrete population model described by the first-order difference equation xt+1 = g(xt) and obtained an important result on the global stability of the equilibrium point means when g(x) has only one extreme point (a maximum) in (0, means). Motivated by work of M. Kot and W. M. Schaffer (1984), a more general case is considered in which g(x) can have more then one maximum point in (0, means), and results on global stability are obtained. These results are applied to develop tests for global stability of the equilibrium point that imply other results in the literature on global stability.     

Global stability in chemostat-type plankton models with delayed nutrient recycling

 In this paper, we consider chemostat-type plankton models in which plankton feeds on a limiting nutrient and the nutrient is supplied at a constant rate and is partially recycled after the death of plankton by bacterial decomposition. We use a distributed delay to describe nutrient recycling and a discrete delay to model the planktonic growth response to nutrient uptake. When one or both delays occur, by constructing Liapunov functionals, we obtain some sufficient conditions for the global attractivity of the positive equilibrium, which can be regarded as estimates of the delays for persistence of global attractivity. Received: 9 January 1996 / Revised version: 1 October 1997     

Global stability and periodic orbits for two-patch predator-prey diffusion-delay models

We consider a predator-prey system where the prey can diffuse between one patch with a low level of food and without predation and one patch with a higher level of food but with predation. We assume a Volterra within-patch dynamics, and we assume further that the benefit for the predator comes also from predation in the past through an exponential-delay memory function. By homotopy techniques we prove that, if the prey diffusion is weak enough, then a nonzero globally stable equilibrium exists. This result essentially depends upon the self-regulating coefficient of the predator. If we put this coefficient equal to zero, assuming that the predator density is regulated only by predation, then we can prove the existence of a Hopf bifurcating orbit from the positive equilibrium. The main cause of periodic orbits is the time delay in the predator response functional. We prove that diffusion, lack of delay in the predator response, and increase in the rate of the exponential decay of the memory play stabilizing roles.     

Global stability of models of humoral immunity against multiple viral strains

We analyse, from a mathematical point of view, the global stability of equilibria for models describing the interaction between infectious agents and humoral immunity. We consider the models that contain the variables of pathogens explicitly. The first model considers the situation where only a single strain exists. For the single strain model, the disease steady state is globally asymptotically stable if the basic reproductive ratio is greater than one. The other models consider the situations where multiple strains exist. For the multi-strain models, the disease steady state is globally asymptotically stable. In the model that does not explicitly contain an immune variable, only one strain with the maximum basic reproductive ratio can survive at the steady state. However, in our models explicitly involving the immune system, multiple strains coexist at the steady state.     

Global stability of single-species diffusion volterra models with continuous time delays

In this paper we consider the stability property of single-species patches connected by diffusion with a within-patch dynamics of Volterra type and with continuous time delays. We prove that this system can only have two kinds of equilibria: the positive and the trivial one. By the assumption that the delay kernels are convex combinations of suitable non-negative and normalized functions, the linear chain trick gives an expanded system of O.D.E. with the same stability properties as the original integro-differential system. Homotopy function techniques provide sufficient conditions for the existence of the positive equilibrium and for its global stability. We also prove the local stability of any positive equilibrium and the local instability both of positive and trivial equilibria. The biological meanings of the results obtained are compared with known results from the literature. This work was performed under the auspices of G.N.F.M., C.N.R. (Italy) and within the activity of the Evolution Equations and Applications group, M.P.I. (Italy). I thank the Department of Applied Mathematics, Shizuoka University, Japan, which enabled me to visit Urbino.     

Linear stability criteria for population models with periodically perturbed delays

We examine some simple population models that incorporate a time delay which is not a constant but is instead a known periodic function of time. We examine what effect this periodic variation has on the linear stability of the equilibrium states of scalar population models and of a simple predator prey system. The case when the delay differs from a constant by a small amplitude periodic perturbation can be treated analytically by using two-timing methods. Of particular interest is the case when the system is initially marginally stable. The introduction of variation in the delay can then have either a stabilising effect or a destabilizing one, depending on the frequency of the periodic perturbation. The case when the periodic perturbation has large amplitude is studied numerically. If the fluctuation is large enough the effect can be stabilising.     

Age distribution and the stability of simple discrete time population models

Differences in age specific demographic characteristics can considerably alter the behaviour of the population dynamics of a species or community of species. In this analysis techniques are developed which enable the stability of the equilibria of a set of models involving age structure to be investigated. The underlying model in all cases is a simple matrix representation of first order difference equations. The analysis enables the results of computer investigations of various population models of this type to be explained.     

Conditions for global stability of two-species population models with discrete time delay

By constructing appropriate Liapunov functionals, asymptotic behaviour of the solutions of various delay differential systems describing prey-predator, competition and symbiosis models has been studied. It has been shown that equilibrium states of these models are globally stable, provided certain conditions in terms of instantaneous and delay interaction coefficients are satisfied.     

Global asymptotic stability of the size distribution in probabilistic models of the cell cycle

Probabilistic models of the cell cycle maintain that cell generation time is a random variable given by some distribution function, and that the probability of cell division per unit time is a function only of cell age (and not, for instance, of cell size). Given the probability density, f(t), for time spent in the random compartment of the cell cycle, we derive a recursion relation for _n(x), the probability density for cell size at birth in a sample of cells in generation n. For the case of exponential growth of cells, the recursion relation has no steady-state solution. For the case of linear cell growth, we show that there exists a unique, globally asymptotically stable, steady-state birth size distribution, _*(x). For the special case of the transition probability model, we display _*(x) explicitly.This work was supported by the National Science Foundation under grants MCS8301104 (to J.J.T.) and MCS8300559 (to K.B.H.), and by the National Institutes of Health under grant GM27629 (to J.J.T.).