EFFECTS OF INDIVIDUAL HETEROGENEITY IN ESTIMATING THE PERSISTENCE OF SMALL POPULATIONS

ABSTRACT. Population viability models are commonly used to estimate the probability of persistence of small, threatened, or endangered populations. Demographic, temporal, spatial, and individual heterogeneity are important factors affecting the probability of persistence of small populations. Because stochastic process are intractable analytically (Lud-wig [1996]), computer simulation models are often used for estimating population viability via numerical techniques. Although demographic, spatial, and temporal stochasticity have been incorporated into some population viability models, individual heterogeneity has not been included. In this paper we include individual heterogeneity in a simulation model and examine probabilities of population persistence at different levels of heterogeneity and population size. Individual heterogeneity may increase the probability of persistence of small populations. The mechanism for the extension in persistence may be explained by natural selection. Genotypes persisting through a decline may be those that survive better under the conditions causing the decline. These individuals that survive and reproduce in the face of adverse conditions may extend the probability that a small population persists.     

DYNAMICS OF AN AGE‐STRUCTURED METAPOPULATION MODEL

ABSTRACT. We introduce a metapopulation model that includes both landscape changes (patch destruction and recreation) and age‐dependent metapopulation dynamics. A threshold quantity is derived and related to the existence of an ecologically nontrivial equilibrium, to the stability of the species‐free equilibrium, and to weak and strong persistence of the species. We provide examples to illustrate how age‐related changes in patch colonization and extinction rates can alter metapopulation persistence. Future field studies may need to address the temporal dynamics that characterize local populations in fragmented landscapes.     

EFFECTS OF DOMAIN SIZE ON THE PERSISTENCE OF POPULATIONS IN A DIFFUSIVE FOOD-CHAIN MODEL WITH BEDDINGTON-DeANGELIS FUNCTIONAL RESPONSE

ABSTRACT. A food chain consisting of species at three trophic levels is modeled using Beddington-DeAngelis functional responses as the links between trophic levels. The dispersal of the species is modeled by diffusion, so the resulting model is a three component reaction-diffusion system. The behavior of the system is described in terms of predictions of extinction or persistence of the species. Persistence is characterized via permanence, i.e., uniform persistence plus dissi-pativity. The way that the predictions of extinction or persistence depend on domain size is studied by examining how they vary as the size (but not the shape) of the underlying spatial domain is changed.     

Dynamical Behavior of a Stochastic Nutrient-Plankton Food Chain Model with L\''{e}vy Jumps

In aquatic ecosystem, plankton populations are easily affected by environmental fluctuations due to the unpredictability of many physical factors. To better understand how environmental fluctuations influence plankton populations, in this paper, we propose and investigate a stochastic nutrient-plankton food chain model with L$\acute{\rm e}$vy jumps. Firstly, by constructing a suitable Lyapunov function, we prove that the stochastic model has a unique global positive solution for any given positive initial value. Then, we establish sufficient conditions for the persistence and extinction of plankton. Finally, we provide some numerical simulations to illustrate the analytical results.     

A METAPOPULATION MODEL WITH DISCRETE SIZE STRUCTURE

ABSTRACT. We consider a discrete size‐structured meta‐population model with the proportions of patches occupied by n individuals as dependent variables. Adults are territorial and stay on a certain patch. The juveniles may emigrate to enter a dispersers' pool from which they can settle on another patch and become adults. Absence of colonization and absence of emigration lead to extinction of the metapopula‐tion. We define the basic reproduction number R₀ of the metapopulation as a measure for its strength of persistence. The metapopulation is uniformly weakly persistent if R₀> 1. We identify subcritical bifurcation of persistence equilibria from the extinction equilibrium as a source of multiple persistence equilibria: it occurs, e.g., when the immigration rate, into occupied patches, exceeds the colonization rate (of empty patches). We determine that the persistence‐optimal dispersal strategy which maximizes the basic reproduction number is of bang‐bang type: If the number of adults on a patch is below carrying capacity all the juveniles should stay, if it is above the carrying capacity all the juveniles should leave.     

THE EFFECTS OF REDUCING MATING LIKELIHOOD ON POPULATION VIABILITY

Abstract The success a species may have invading a patch previously unoccupied is of considerable interest for pest managers and conservation ecologists. The purpose here is to present a mechanistic approach to analyze reproductive Allee effects appearing through the failure in the process of fertilization in a two‐sex population and observe how the survival in an invaded patch is affected. This is in contrast to the usually employed stochastic models with a deterministic skeleton that describe the presence of Allee effects. A Poisson–Ricker model, which includes stochastic demography and sex determination with females classified as successfully fertilized or not fertilized, is used. Numerical approximations to the probabilities of extinction and the mean time to extinction are presented, for fixed parameter values, suggesting how stochasticity in the mating process combined with random fluctuations in the male and female densities, at each generation, contribute to the risk of extinction of a population which started an invasion at a low density.     

THE EFFECT OF DECREASING SYSTEM SIZE ON BIRTH AND DEATH MODELS OF OPEN-ACCESS FISHERIES AND PREDATOR-PREY ECOSYSTEMS

Birth and death simulation, developed by Pielou, is a form of Markov stochastic process for describing the time evolution of populations. Applied to modelling the human element of a fishery, it expresses two features of fishing effort dynamics absent in systems of differential equations: (1) discreteness of events, such as a fishing trip or entry of a vessel into the fleet, and (2) demography stochasticity, expressed as randomness in the time occurrence of successive events. Birth and death simulation is based on randomly selecting the waiting time between events from a negative exponential distribution, derived under the assumption of Markov. Histograms from commercial landings data of waiting times between events of boats returning to port in a Nova Scotia fishery yielded good agreement with the predicted negative exponential. Algorithms are presented for stochastically modelling two processes: (1) catch and (2) the open-access hypothesis for changes in fleet size in response to changing levels of profit. The solutions qualitatively diverge from that predicted by differential equations: As the numbers of vessels and fish schools decline (i.e., as the system size scale shrinks), a birth and death formulation predicts increasing instability of the predator-prey cycle solution about the deterministically stable open-access equilibrium. Open-access models are a form of predator-prey model. In choosing the minimum wilderness preserve area needed to sustain a population of top predators, numbered in the low hundreds, a predator-prey model formulated with differential equations could underestimate instability and thus the risk of extinction, when the discreteness and randomness of predator-prey birth, death, and capture events is significant.     

The threshold between permanence and extinction for a stochastic Logistic model with regime switching

A stochastic logistic model under regime switching is proposed and investigated. Sufficient conditions for extinction, non-persistence in the mean, weak persistence and stochastic permanence are established. The threshold between weak persistence and extinction is obtained. Then we show that this threshold also is the threshold between stochastic permanence and extinction under a simple additional condition. The results show that firstly, the stationary probability distribution of the Markov chain plays a key role in determining the permanence and extinction of the population. Secondly, different types of environmental noises have different effects on the permanence and extinction of the population. Thirdly, the more the stochastic noises, the easier the population goes to extinction.     

Permanence and extinction of a stochastic hybrid model for tumor growth

A tumor growth model perturbed by both white noise and Markov switching is formulated and explored. The threshold between permanence and extinction is obtained. Some effects of environmental stochasticity on permanence and extinction of the model are revealed.     

ALLEE EFFECTS: POPULATION GROWTH,CRITICAL DENSITY,AND THE CHANCE OF EXTINCTION

This paper develops mathematical models to describe the growth, critical density, and extinction probability in sparse populations experiencing Allee effects. An Allee effect (or depensation) is a situation at low population densities where the per-individual growth rate is an increasing function of population density. A potentially important mechanism causing Allee effects is a shortage of mating encounters in sparse populations. Stochastic models are proposed for predicting the probability of encounter or the frequency of encounter as a function of population density. A negative exponential function is derived as such an encounter function under very general biological assumptions, including random, regular, or aggregated spatial patterns. A rectangular hyperbola function, heretofore used in ecology as the functional response of predator feeding rate to prey density, arises from the negative exponential function when encounter probabilities are assumed heterogeneous among individuals. These encounter functions produce Allee effects when incorporated into population growth models as birth rates. Three types of population models with encounter-limited birth rates are compared: (1) deterministic differential equations, (2) stochastic discrete birth-death processes, and (3) stochastic continuous diffusion processes. The phenomenon of a critical density, a major consequence of Allee effects, manifests itself differently in the different types of models. The critical density is a lower unstable equilibrium in the deterministic differential equation models. For the stochastic discrete birth-death processes considered here, the critical density is an inflection point in the probability of extinction plotted as a function of initial population density. In the continuous diffusion processes, the critical density becomes a local minimum (antimode) in the stationary probability distribution for population density. For both types of stochastic models, a critical density appears as an inflection point in the probability of attaining a small population density (extinction) before attaining a large one. Multiplicative (“environmental”) stochastic noise amplifies Allee effects. Harvesting also amplifies those effects. Though Allee effects are difficult to detect or measure in natural populations, their presence would seriously impact exploitation, management, and preservation of biological resources.     

Analysis of a stochastic tumor-immune model with regime switching and impulsive perturbations

In this article, an impulsive stochastic tumor-immune model with regime switching is formulated and explored. Firstly, it is proven that the model has a unique global positive solution. Then sufficient criteria for extinction, non-persistence in the mean, weak persistence and stochastic permanence are provided. The threshold value between extinction and weak persistence is gained. In addition, the lower- and the upper-growth rates of tumor cells are estimated. The results demonstrate that the dynamics of the model are intimately associated with the random perturbations and impulsive perturbations. Finally, biological implications of the results are addressed with the help of real data and numerical simulations.     

PERSISTENCE OF POPULATIONS FOR A PREDATOR-PREY SYSTEM IN A POLLUTED ENVIRONMENT

1IntroductionTheinfluenceoftoxicantsonthepersistenceandextinctionofpopulationsinapollutedenvironmentisveryimportant.In1983,T.C.Hallametal.firstproposedadeterministicmodeltostudythetoxiceffectsonpopulationgrowth[fi.In1986,T.G.HallamandZ.Maproposedtheconceptofpersistenceinthemeanofpopulations[2],anewmethodtodeterminethethresholdsbetweenpersistenceandextinctionofpopulationsofnonautonomousmodelshasbeenprovidedintheirpaper.Thethresholdsbetweenpersistenceandextinctionforsomemodelshavebeenobtaine…     

CONSIDERATION OF STOCHASTIC DEMOGRAPHY IN THE DESIGN AND MANAGEMENT OF BIOLOGICAL RESERVES

A generalized birth–and–death process serves as a simple, flexible model for computing the expected persistence time of a small population in a random world. We may reparametrize the model in ways that allow explicit incorporation of density dependence, random differences in events experienced by different individuals, and random environmental variation experienced by all individuals in concert. This model seems to capture the important features of real population dynamics for purposes of computing the mean persistence time, even though the underlying mechanisms presumed in the mathematics of the model are decidedly unrealistic. The lack of isomorphism between birth and death rates, as they feature in the model, and vital rates of real biological populations can lead to extremely misleading results, if the classic formulation, rather than the reparametrization is applied without due circumspection. Using the reparametrized model, we find that environmental variation poses a greater problem for population persistence than does individual variation. In particular, with purely individual variation, the expected persistence time increases approximately with the power of the ceiling on population size; but with purely environmental variation, the expected persistence time increases somewhat less than linearly with the size of the population ceiling. The birth–and–death process model can also be applied to calculating the persistence time of a population on an ensemble of reserves which are linked by natural migration or by deliberate reintroduction programs. Results of this model, for an idealized ensemble, show that multiple independent reserves with a sufficient recolonization rate (natural or otherwise) will confer a longer persistence time than a single reserve with the same total carrying capacity, but in the absence of recolonization, the system of smaller separate reserves confers a shorter persistence time than the single large reserve.     

Low dimensional homeochaos in coevolving host–parasitoid dimorphic populations: Extinction thresholds under local noise

A discrete time model describing the population dynamics of coevolution between host and parasitoid haploid populations with a dimorphic matching allele coupling is investigated under both determinism and stochastic population disturbances. The role of the properties of the attractors governing the survival of both populations is analyzed considering equal mutation rates and focusing on host and parasitoid growth rates involving chaos. The purely deterministic model reveals a wide range of ordered and chaotic Red Queen dynamics causing cyclic and aperiodic fluctuations of haplotypes within each species. A Ruelle–Takens–Newhouse route to chaos is identified by increasing both host and parasitoid growth rates. From the bifurcation diagram structure and from numerical stability analysis, two different types of chaotic sets are roughly differentiated according to their size in phase space and to their largest Lyapunov exponent: the Confined and Expanded attractors. Under the presence of local population noise, these two types of attractors have a crucial role in the survival of both coevolving populations. The chaotic confined attractors, which have a low largest positive Lyapunov exponent, are shown to involve a very low extinction probability under the influence of local population noise. On the contrary, the expanded chaotic sets (with a higher largest positive Lyapunov exponent) involve higher host and parasitoid extinction probabilities under the presence of noise. The asynchronies between haplotypes in the chaotic regime combined with low dimensional homeochaos tied to the confined attractors is suggested to reinforce the long-term persistence of these coevolving populations under the influence of stochastic disturbances. These ideas are also discussed in the framework of spatially-distributed host–parasitoid populations.     

Threshold of disease transmission in a patch environment

An epidemic model is proposed to describe the dynamics of disease spread between two patches due to population dispersal. It is proved that reproduction number is a threshold of the uniform persistence and disappearance of the disease. It is found that the dispersal rates of susceptible individuals do not influence the persistence and extinction of the disease. Furthermore, if the disease becomes extinct in each patch when the patches are isolated, the disease remains extinct when the population dispersal occurs; if the disease spreads in each patch when the patches are isolated, the disease remains persistent in two patches when the population dispersal occurs; if the disease disappears in one patch and spreads in the other patch when they are isolated, the disease can spread in all the patches or disappear in all the patches if dispersal rates of infectious individuals are suitably chosen. It is shown that an endemic equilibrium is locally stable if susceptible dispersal occurs and infectious dispersal turns off. If susceptible individuals and infectious individuals have the same dispersal rate in each patch, it is shown that the fractions of infectious individuals converge to a unique endemic equilibrium.     

一类捕食与被捕食LV模型的扩散性质

本文证明了一类带有扩散的捕食与被捕食Ｌｏｔｋａ－Ｖｏｌｔｅｒｒａ模型的如下性质：当该模型存在正平衡点时,它的一切正解是强持续生存的;当扩散率较小时,该系统的正平衡点是稳定的;当扩散率增大且位于某一开区间内变化时,该系统的正平衡点是不稳定的,而且分支出唯一的小振幅空间周期解;当扩散率继续增大时,该系统的正平衡点又变为稳定的．     

污染环境中的Leslie系统

研究了环境污染对Leslie资源-消费者系统中消费者种群的长期影响．考虑到种群数量的变化对种群体内毒素浓度和环境毒素浓度的影响,建立了一个新的数学模型, 给出了消费者种群弱持续生存和绝灭的判据,并在一定条件下得到了弱持续生存与绝灭的阈值．     

THE SURVIVAL ANALYSIS FOR SINGLE-SPECIES SYSTEM IN A POLLUTED ENVIRONMENT

This article concentrates on the study of a mathematical model for the effect of toxicant levels on a single-species ecosystem in the case where there is a constant emission of a toxicant. Some sufficient conditions for weak persistence and extinction are found. The threshold between persistence and extinction can be established in some cases.     

污染环境中三维竞争系统的生存阈值

本文用积分均值法，首次讨论了带有毒素影响的三维竞争系统的Lotka－Volterra模型，得到了各种群平均持续生存与绝灭的阈值．文中所得结论对环境污染以及生物种群影响的理论研究和实际应用有重要意义．文[5,6]的主要结论包含在本文的结果中．     

Dynamics Behavior of a Stochastic Predator-Prey Model with Stage Structure for Predator and L\''{e}vy Jumps

In order to study the effects of external environmental noise on the interaction dynamics between predator and prey populations, in this paper, we develop a predator-prey model with the stage structure for predator and L$\acute{\rm e}$vy noise. By constructing an appropriate Lyapunov function, we first prove that the proposed model exists the uniqueness of global positive solution. Then, we analyze the persistence and extinction of the proposed model. Finally, we perform some numerical simulations to verify the correctness of the theoretical results.