Steady states in population models with monotone,stochastic dynamics

Existence, uniqueness and asymptotic stability of stochastic equilibrium are established in multi-dimensional population models with monotone dynamics.     

Rapidly converging numerical algorithms for models of population dynamics

We propose algorithms for the approximation of the age distributions of populations modeled by the McKendrick-von Foerster and the Gurtin-MacCamy systems both in one- and two-sex versions. For the one-sex model methods of second and fourth order are proposed. For the two-sex model a second order method is described. In each case the convergence is demonstrated. Several numerical examples are given.     

Bottom-up derivation of discrete-time population models with the Allee effect

This paper presents a framework in which various single-species discrete-time population models exhibiting the Allee effect are derived from first principles. Here, the Allee effect means a reduction in individual fitness at low population sizes. The derivation is based on the distribution of female and male individuals among discrete resource sites, in addition to competitive and cooperative interaction among individuals. These derivations show how the derived population models depend on the type and the intensity of competition, and the degree of clustering of individuals. Along with these models exhibiting the Allee effect, this paper also presents first-principles derivation of population models without the Allee effect which include a parameter relating to the intensity of competition.     

Blowing-up of deterministic fixed points in stochastic population dynamics

We discuss the stochastic dynamics of biological (and other) populations presenting a limit behaviour for large environments (called deterministic limit) and its relation with the dynamics in the limit. The discussion is circumscribed to linearly stable fixed points of the deterministic dynamics, and it is shown that the cases of extinction and non-extinction equilibriums present different features. Mainly, non-extinction equilibria have associated a region of stochastic instability surrounded by a region of stochastic stability. The instability region does not exist in the case of extinction fixed points, and a linear Lyapunov function can be associated with them. Stochastically sustained oscillations of two subpopulations are also discussed in the case of complex eigenvalues of the stability matrix of the deterministic system.     

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450-463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173-205].     

Separable models in age-dependent population dynamics

A class of population models is considered in which the parameters such as fecundity, mortality and interaction coefficients are assumed to be age-dependent. Conditions for the existence, stability and global attractivity of steady-state and periodic solutions are derived. The dependence of these solutions on the maturation periods is analyzed. These results are applied to specific single and multiple population models. It is shown that periodic solutions cannot occur in a general class of single population age-dependent models. Conditions are derived that determine whether increasing the maturation period has a stabilizing effect. In specific cases, it is shown that any number of switches in stability can occur as the maturation period is increased. An example is given of predator-prey model where each one of these stability switches corresponds to a stable steady state losing its stability via a Hopf bifurcation to a periodic solution and regaining its stability upon further increase of the maturation period.     

Stochastic von Bertalanffy models, with applications to fish recruitment

We consider three individual-based models describing growth in stochastic environments. Stochastic differential equations (SDEs) with identical von Bertalanffy deterministic parts are formulated, with a stochastic term which decreases, remains constant, or increases with organism size, respectively. Probability density functions for hitting times are evaluated in the context of fish growth and mortality. Solving the hitting time problem analytically or numerically shows that stochasticity can have a large positive impact on fish recruitment probability. It is also demonstrated that the observed mean growth rate of surviving individuals always exceeds the mean population growth rate, which itself exceeds the growth rate of the equivalent deterministic model. The consequences of these results in more general biological situations are discussed.     

Stability results for delayed-recruitment models in population dynamics

This paper relates the stability properties of a class of delay-difference equations to those of an associated scalar difference equation. Simple but powerful conditions for testing global stability are presented which are independent of the length of the time delay involved. For models which do not have globally stable equilibria, estimates of stability regions are obtained. Some well known baleen whale models are used to illustrate the results.     

Linking the coevolutionary and population dynamics of host–parasitoid interactions

The interplay between coevolutionary and population or community dynamics is currently the focus of much empirical and theoretical consideration. Here, we develop a simulation model to study the coevolutionary and population dynamics of a hypothetical host–parasitoid interaction. In the model, host resistance and parasitoid virulence are allowed to coevolve. We investigate how trade-offs associated with these traits modify the system's coevolutionary and population dynamics. The most important influence on these dynamics comes from the incorporation of density-dependent costs of resistance ability. We find three main outcomes. First, if the costs of resistance are high, then one or both of the players go extinct. Second, when the costs of resistance are intermediate to low, cycling population and coevolutionary dynamics are found, with slower evolutionary changes observed when the costs of virulence are also low. Third, when the costs associated with resistance and virulence are both high, the hosts trade-off resistance against fecundity and invest little in resistance. However, the parasitoids continue to invest in virulence, leading to stable host and parasitoid population sizes. These results support the hypothesis that costs associated with resistance and virulence will maintain the heritable variation in these traits found in natural populations and that the nature of these trade-offs will greatly influence the population dynamics of the interacting species. Received: December 20, 1999 / Accepted: July 17, 2000     

The Galilean turn in population ecology

The standard mathematical models in population ecology assume that a population's growth rate is a function of its environment. In this paper we investigate an alternative proposal according to which the rate of change of the growth rate is a function of the environment and of environmental change. We focus on the philosophical issues involved in such a fundamental shift in theoretical assumptions, as well as on the explanations the two theories offer for some of the key data such as cyclic populations. We also discuss the relationship between this move in population ecology and a similar move from first-order to second-order differential equations championed by Galileo and Newton in celestial mechanics.     

Closing the larval loop: linking larval ecology to the population dynamics of marine benthic invertebrates

The majority of marine benthic invertebrates exhibit a complex life cycle that includes separate planktonic larval, and bottom-dwelling juvenile and adult phases. To understand and predict changes in the spatial and temporal distributions, abundances, population growth rate, and population structure of a species with such a complex life cycle, it is necessary to understand the relative importance of the physical, chemical and biological properties and processes that affect individuals within both the planktonic and benthic phases. To accomplish this goal, it is necessary to study both phases within a common, quantitative framework defined in terms of some common currency. This can be done efficiently through construction and evaluation of a population dynamics model that describes the complete life cycle.

Two forms that such a model might assume are reviewed: a stage-based, population matrix model, and a model that specifies discrete stages of the population, on the bottom and in the water column, in terms of simultaneous differential equations that may be solved in both space and time. Terms to be incorporated in each type of model can be formulated to describe the critical properties and processes that can affect populations within each stage of the life cycle. For both types of model it is shown how this might be accomplished using an idealized balanomorph barnacle as an example species. The critical properties and processes that affect the planktonic and benthic phases are reviewed. For larvae, these include benthic adult fecundity and fertilization success, growth and larval stage duration, mortality, larval behavior, dispersal by currents and turbulence, and larval settlement. It is possible to predict or estimate empirically all of the key terms that should be built into the larval and benthic components of the model. Thus, the challenge of formulating and evaluating a full life cycle model is achievable. Development and evaluation of such a model will be challenging because of the diverse processes which must be considered, and because of the disparities in the spatial and temporal scales appropriate to the benthic and planktonic larval phases. In evaluating model predictions it is critical that sampling schemes be matched to the spatial and temporal scales of model resolution.     

Mathematical analysis of an integral equation arising from population dynamics

In this paper, we establish the existence of travelling wave solution to an intrinsically non-linear differential–integral equation formed as a result of mathematical modelling of the evolution of an asexual population in a changing environment. This equation is first converted to a non-linear integral equation. The discretization and manipulation of the corresponding eigenvalue problem allows us to use the theory of positive matrices to get some very useful estimates and then to confirm the existence of solution. We also exhibit numerical simulation results and explain the biological meaning of the results.     

The effects of spatial heterogeneity in population dynamics

The dynamics of a population inhabiting a heterogeneous environment are modelled by a diffusive logistic equation with spatially varying growth rate. The overall suitability of an environment is characterized by the principal eigenvalue of the corresponding linearized equation. The dependence of the eigenvalue on the spatial arrangement of regions of favorable and unfavorable habitat and on boundary conditions is analyzed in a number of cases.Research supported by National Science Foundation grant #DMS 88-02346     

Equilibrium solutions of age-specific population dynamics of several species

Summary A mathematical model describing the dynamics of a population consisting of several species is studied. The interactions in the population are assumed to be age-specific. Using an evolution equation approach, sufficient conditions for well-posedness in L ¹ of the dynamics and for existence as well as for stability of equilibrium solutions are given.     

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R₀, is calculated and it is shown that if R₀<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R₀>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.     

Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.     

From arctic lemmings to adaptive dynamics: Charles Elton's legacy in population ecology

We shall examine the impact of Charles S. Elton's 1924 article on periodic fluctuations in animal populations on the development of modern population ecology. We argue that his impact has been substantial and that during the past 75 years of research on multi-annual periodic fluctuations in numbers of voles, lemmings, hares, lynx and game animals he has contributed much to the contemporary understanding of the causes and consequences of population regulation. Elton was convinced that the cause of the regular fluctuations was climatic variation. To support this conclusion, he examined long-term population data then available. Despite his firm belief in a climatic cause of the self-repeating periodic dynamics which many species display, Elton was insightful and far-sighted enough to outline many of the other hypotheses since put forward as an explanation for the enigmatic long-term dynamics of some animal populations. An interesting, but largely neglected aspect in Elton's paper is that it ends with speculation regarding the evolutionary consequences of periodic population fluctuations. The modern understanding of these issues will also be scrutinised here. In population ecology, Elton's 1924 paper has spawned a whole industry of research on populations displaying multi-annual periodicity. Despite the efforts of numerous research teams and individuals focusing on the origins of multi-annual population cycles, and despite the early availability of different explanatory hypotheses, we are still lacking rigorous tests of some of these hypotheses and, consequently, a consensus of the causes of periodic fluctuations in animal populations. Although Elton would have been happy to see so much effort spent on cyclic populations, we also argue that it is unfortunate if this focus on a special case of population dynamics should distract our attention from more general problems in population and community dynamics.     

From discrete to continuous evolution models: A unifying approach to drift-diffusion and replicator dynamics

We study the large population limit of the Moran process, under the assumption of weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.