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.     

Global stability of population models

Local stability seems to imply global stability for population models. To investigate this claim, we formally define apopulation model. This definition seems to include the one-dimensional discrete models now in use. We derive a necessary and sufficient condition for the global stability of our defined class of models. We derive an easily testable sufficient condition for local stability to imply global stability. We also show that if a discrete model is majorized by one of these stable population models, then the discrete model is globally stable. We demonstrate the utility of these theorems by using them to prove that the regions of local and global stability coincide for six models from the literature. We close by arguing that these theorems give a method for demonstrating global stability that is simpler and easier to apply than the usual method of Liapunov functions.     

Stability of an age specific population with density dependent fertility

A nonlinear version of the Lotka-Sharpe model of population growth is considered in which the age specific fertility is a function of the population size. The stability of an equilibrium population distribution is investigated with respect to both global and local perturbations. Sufficient conditions for such stability are presented, as are estimates for the rate of return of the population to the equilibrium configuration. Particular attention is paid to those situations in which the age dependent stability criteria coincide with those of age independent models.     

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.     

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.     

THE WELL-POSEDNESS OF SIZE STRUCTURED POPULATION MODELS WITH DENSITY DEPENDENT PARAMETERS

Thewell-posednessofnonlinearsizestructuredpopulationmodelsisstudied.Thenonlinearitiesareintroducedbyassumingthevitalparameters(thebirthrate,thedeathrate,andthegrowthrate)tobedensitydependent.TheidealadoptedhereisbasedonthemethodofGurtinandMacCamy[4]usedfornonlinearage-dependentpopulationmodels.Thenetreproductivenumberisintroducedandusedtodeterminethelocalandglobalstabilityoftrivialequilibrium.Thestabilityconditionsoftrivialequilibriumareobtained.     

Behavioral choices based on patch selection: a model using aggregation methods

The aim of this work is to study the influence of patch selection on the dynamics of a system describing the interactions between two populations, generically called 'population N' and 'population P'. Our model may be applied to prey-predator systems as well as to certain host-parasite or parasitoid systems. A situation in which population P affects the spatial distribution of population N is considered. We deal with a heterogeneous environment composed of two spatial patches: population P lives only in patch 1, while individuals belonging to population N migrate between patch 1 and patch 2, which may be a refuge. Therefore they are divided into two patch sub-populations and can migrate according to different migration laws. We make the assumption that the patch change is fast, whereas the growth and interaction processes are slower. We take advantage of the two time scales to perform aggregation methods in order to obtain a global model describing the time evolution of the total populations, at a slow time scale. At first, a migration law which is independent on population P density is considered. In this case the global model is equivalent to the local one, and under certain conditions, population P always gets extinct. Then, the same model, but in which individuals belonging to population N leave patch 1 proportionally to population P density, is studied. This particular behavioral choice leads to a dynamically richer global system, which favors stability and population coexistence. Finally, we study a third example corresponding to the addition of an aggregative behavior of population N on patch 1. This leads to a more complicated situation in which, according to initial conditions, the global system is described by two different aggregated models. Under certain conditions on parameters a stable limit cycle occurs, leading to periodic variations of the total population densities, as well as of the local densities on the spatial patches.     

The effects of large and small-scale random events on the synchrony of metapopulation dynamics: a theoretical analysis

We present an analysis of the conditions under which migration and global random factors may determine large scale synchrony in the dynamics of spatially structured populations. We derive an analytic approximation which describes how the desynchronizing influence of local environmental stochasticity combines with the synchronizing influences of larger scale environmental stochastic variation and migration to determine population cross correlation coefficients. Despite the simplifications made by this analysis, computer simulations show that the behaviour of more complicated models is well described by our approximation over considerable regions of parameter space. We conclude that population synchrony is largely determined by the coefficients of variation (CVs) of the local and larger scale stochastic processes, and that migration alone is only likely to maintain population synchrony when the CV of the local stochastic process is very small.     

Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.     

Species' borders and dispersal barriers

Range limits of species are determined by combined effects of physical, historical, ecological, and evolutionary forces. We consider a subset of these factors by using spatial models of competition, hybridization, and local adaptation to examine the effects of partial dispersal barriers on the locations of borders between similar species. Prompted by results from population genetic models and biogeographic observations, we investigate the conditions under which species' borders are attracted to regions of reduced dispersal. For borders maintained by competition or hybridization, we find that dispersal barriers can attract borders whose positions would otherwise be either neutrally stable or moving across space. Borders affected strongly by local adaptation and gene flow, however, are repelled from dispersal barriers. These models illustrate how particular biotic and abiotic factors may combine to limit species' ranges, and they help to elucidate mechanisms by which range limits of many species may coincide.     

Attractivity of coherent manifolds in metapopulation models

The likelihood that coupled dynamical systems will completely synchronize, or become “coherent”, is often of great applied interest. Previous work has established conditions for local stability of coherent solutions and global attractivity of coherent manifolds in a variety of spatially explicit models. We consider models of communities coupled by dispersal and explore intermediate regimes in which it can be shown that states in phase space regions of positive measure are attracted to coherent solutions. Our methods yield rigorous and practically useful coherence criteria that facilitate useful analyses of ecological and epidemiological problems.     

Neutral evolution in spatially continuous populations

We introduce a general recursion for the probability of identity in state of two individuals sampled from a population subject to mutation, migration, and random drift in a two-dimensional continuum. The recursion allows for the interactions induced by density-dependent regulation of the population, which are inevitable in a continuous population. We give explicit series expansions for large neighbourhood size and for low mutation rates respectively and investigate the accuracy of the classical Malécot formula for these general models. When neighbourhood size is small, this formula does not give the identity even over large scales. However, for large neighbourhood size, it is an accurate approximation which summarises the local population structure in terms of three quantities: the effective dispersal rate, sigma(e); the effective population density, rho(e); and a local scale, kappa, at which local interactions become significant. The results are illustrated by simulations.     

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.     

具有扩散的n-斑块单种群系统的全局稳定性

研究了具有扩散的ｎ－斑块多种环境下单种群非自治模型,在假定该模型所有系数连续有界的情况下,得到了系统全局稳定的充分条件。     

Persistence and convergence of ecosystems: An analysis of some second order difference equations

Summary Three second order difference equation models are analyzed and numerical solutions computed. It is shown that two concepts of ecosystem stability, the local property of convergence and the global property of persistence, do not coincide, and that the existence of either need not imply the other. Conditions for the existence of either form of stability are obtained and shown as parameter space diagrams. Examples of solution trajectories representative of different regions of this space are computed and discussed. A wide range of oscillatory behavior, as noted in recent papers by several authors, results. In addition, the erratic nature of regions of convergence to stable solutions is discussed.     

Models for spatially distributed populations: The effect of within-patch variability

This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are nonlinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average population density and this effect does not disappear in large populations. This result is an outcome of the interaction of the three ingredients in the models and it says that stochastic models of systems of patches can be expected to give results for average population density that differ systematically from those of deterministic models. The magnitude of these differences is related to the degree of nonlinearity of local dynamics and the magnitude of local variability. These results explain those obtained from a number of previously published models which give conclusions that differ from those of deterministic models. Results are also obtained that show how stochastic models of systems of patches may be simplified to facilitate their study.     

Plankton blooms and patchiness generated by heterogeneous physical environments

Microscopic turbulent motions of water have been shown to influence the dynamics of microscopic species living in that habitat. The number, stability, and excitability of stationary states in a predator–prey model of plankton species can therefore change when the strength of turbulent motions varies. In a spatial system these microscopic turbulent motions are naturally of different strength and form a heterogeneous physical environment. Spatially neighboring plankton communities with different physical conditions can impact each other due to diffusive coupling. We show that local variations in the physical conditions can influence the global system in form of propagating pulses of high population densities. For this we consider three different local predator–prey models with different local responses to variations in the physical environment. The degree of spatial heterogeneity can, depending on the model, promote or reduce the number of propagating pulses, which can be interpreted as patchy plankton distributions and recurrent blooms.     

The interacting effects of food,spring temperature,and global climate cycles on population dynamics of a migratory songbird

Although long‐distance migratory songbirds are widely believed to be at risk from warming temperature trends, species capable of attempting more than one brood in a breeding season could benefit from extended breeding seasons in warmer springs. To evaluate local and global factors affecting population dynamics of the black‐throated blue warbler (Setophaga caerulescens), a double‐brooded long‐distance migrant, we used Pradel models to analyze 25 years of mark–recapture data collected in New Hampshire, USA. We assessed the effects of spring temperature (local weather) and the El Niño Southern Oscillation index (a global climate cycle), as well as predator abundance, insect biomass, and local conspecific density on population growth in the subsequent year. Local and global climatic conditions affected warbler populations in different ways. We found that warbler population growth was lower following El Niño years (which have been linked to poor survival in the wintering grounds and low fledging weights in the breeding grounds) than La Niña years. At a local scale, populations increased following years with warm springs and abundant late‐season food, but were unaffected by spring temperature following years when food was scarce. These results indicate that the warming temperature trends might have a positive effect on recruitment and population growth of black‐throated blue warblers if food abundance is sustained in breeding areas. In contrast, potential intensification of future El Niño events could negatively impact vital rates and populations of this species.