The effect of diffusion on the permanence of Smith population model in a polluted patch

IntroductionThe effect of diffusion on the permanence of population has been studied in some refer-ences. LevinI1] set up the followiIlg model to study the effect of diffusion on the permanence ofpopulation:: f \' \ =.tvhere ur(t) defines the number of population i in patch p, uu = (ut,'. u:). f,u(uu) isthe int!.i11sic growth rate fOr population t, and D:' is the (1iffosive rate of population l frompatch 7 to patch U. Hastingsi2J proved that the positive equilibrium state is stab1e for suf…     

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.     

EDGE PERMEABILITY AND POPULATION PERSISTENCE IN ISOLATED HABITAT PATCHES

ABSTRACT. Population persistence in isolated habitat fragments is investigated using integrodifference equations. The propensity of individual dispersers encountering the boundary of the patch to emigrate is defined by edge permeability. A dispersal model incorporating movement, settlement and edge permeability defines dispersal success as a function of a disperser's starting location. This dispersal model is used to generate dispersal kernels for integrodifference equation models, analysis of which gives a condition for population persistence in terms of edge permeability, patch size and average dispersal distance. An approximation reduces the spatial problem to a simple nonspatial model that can be easily analyzed.     

How Do the Spatial Structure and Time Delay Affect the Persistence of A Polluted Species

In this article, we consider the effects of diffusion and time delay on the species in a polluted environment. Persistence-extinction thresholds are given for population in the toxicant stressed logistic growth model with discrete diffusion or time delay. It is proved that dispersal allows a larger threshold, that is, dispersal can increase the endurance effectiveness of the population subjected to toxicant, and time delay has no effect on the threshold result.     

The effect of population dispersal on the spread of a disease

The effect of population dispersal among n patches on the spread of a disease is investigated. Population dispersal does not destroy the uniqueness of a disease free equilibrium and its attractivity when the basic reproduction number of a disease R₀<1. When R₀>1, the uniqueness and global attractivity of the endemic equilibrium can be obtained if dispersal rates of susceptible individuals and infective individuals are the same or very close in each patch. However, numerical calculations show that population dispersal may result in multiple endemic equilibria and even multi-stable equilibria among patches, and also may result in the extinction of a disease, even though it cannot be eradicated in each isolated patch, provided the basic reproduction numbers of isolated patches are not very large.     

The diffusive Lotka–Volterra competition model in fragmented patches I: Coexistence

It is an ecological imperative that we understand how changes in landscape heterogeneity affect population dynamics and coexistence among species residing in increasingly fragmented landscapes. Decades of research have shown the dispersal process to have major implications for individual fitness, species’ distributions, interactions with other species, population dynamics, and stability. Although theoretical models have played a crucial role in predicting population level effects of dispersal, these models have largely ignored the conditional dependency of dispersal (e.g., responses to patch boundaries, matrix hostility, competitors, and predators). This work is the first in a series where we explore dynamics of the diffusive Lotka–Volterra (L–V) competition model in such a fragmented landscape. This model has been extensively studied in isolated patches, and to a lesser extent, in patches surrounded by an immediately hostile matrix. However, little attention has been focused on studying the model in a more realistic setting considering organismal behavior at the patch/matrix interface. Here, we provide a mechanistic connection between the model and its biological underpinnings and study its dynamics via exploration of nonexistence, existence, and uniqueness of the model’s steady states. We employ several tools from nonlinear analysis, including sub-supersolutions, certain eigenvalue problems, and a numerical shooting method. In the case of weak, neutral, and strong competition, our results mostly match those of the isolated patch or immediately hostile matrix cases. However, in the case where competition is weak towards one species and strong towards the other, we find existence of a maximum patch size, and thus an intermediate range of patch sizes where coexistence is possible, in a patch surrounded by an intermediate hostile matrix when the weaker competitor has a dispersal advantage. These results support what ecologists have long theorized, i.e., a key mechanism promoting coexistence among competing species is a tradeoff between dispersal and competitive ability.     

Discrete-time Metapopulation Dynamics and Unidirectional Dispersal

The effects of unidirectional dispersal on single pioneer species discrete-time metapopulations where the pre-dispersal local patch dynamics are of the same (compensatory or overcompensatory) or mixed (compensatory and overcompensatory) types are studied. Single-species unidirectional metapopulation models behave as single-species single-patch models whenever all pre-dispersal local patch dynamics are compensatory and the dispersal rate is low. The pioneer species goes extinct in at least one patch when the dispersal rate is high, while it persists when the rate is low. Unidirectional dispersal can generate multiple attractors with fractal basin boundaries whenever the pre-dispersal local patch dynamics are overcompensatory, and is capable of altering the local patch dynamics in mixed systems from compensatory to overcompensatory dynamics and vice versa.     

Analysis of a nonautonomous Gompertz population growth model with dispersal in a polluted environment

In this paper, the dynamic behavior of a nonautonomous system with mixed functional response is studied. The population has a history that takes them through two stages, immature and mature. The effects of diffusion on population growth in a polluted patch environment are discussed. Some sufficient conditions on permanence and extinction of population are obtained. Under some appropriate conditions, the asymptotically stability of the periodic solution is obtained. Moreover, a stochastic model is proposed and the conditions for the existence of a global positive solution are discussed.     

Global dynamics behaviors for a nonautonomous Lotka–Volterra almost periodic dispersal system with delays

A nonautonomous Lotka–Volterra dispersal system with continuous delays and discrete delays is considered. By using a comparison theorem and delay differential equation basic theory, we obtain sufficient conditions for the permanence of the population in every patch. By constructing a suitable Lyapunov functional, we prove that the system is globally asymptotically stable under some appropriate conditions. Using almost periodic functional hull theory, we get sufficient conditions for the existence, uniqueness and globally asymptotical stability for an almost periodic solution. This implies that the population in every patch exhibits stable almost periodic fluctuation. Furthermore, the results show that the permanence and global stability of system, and the existence and uniqueness of a positive almost periodic solution, depend on the delay; then we call it “profitless”.     

Permanence of dispersal population model with time delays

Permanence of a dispersal single-species population model is considered where environment is partitioned into several patches and the species requires some time to disperse between the patches. The model is described by delay differential equations. The existence of food-rich patches and small dispersions among the patches are proved to be sufficient to ensure partial permanence of the model. It is also shown that partial permanence ensures permanence if each food-poor patch is connected to at least one food-rich patch and if each pair in food-rich patches is connected. Furthermore, it is proved that partial persistence is ensured even under large dispersion among food-rich patches if the dispersion time is relatively small.     

Permanence of a single-species dispersal system and predator survival

This paper considers permanence of a single-species dispersal periodic system with the possibility of the loss for the species during their dispersion among patches. The condition obtained for permanence generalizes the known condition on the system without loss for the species in the process of movement. Next, we add predators into every patch and consider the survival possibility of the predator. It is shown that the total amount of the predators can remain positive, if the single-species (prey) dispersal system has a positive periodic solution and the quantity of prey in each patch is enough for survival of the predator.     

Evolution of dispersal in a spatially periodic integrodifference model

An integrodifference model describing the reproduction and dispersal of a population is introduced to investigate the evolution of dispersal in a spatially periodic habitat. The dispersal is determined by a kernel function, and the dispersal strategy is defined as the probability of population individuals’ moving to a different habitat. Both conditional and unconditional dispersal strategies are investigated, the distinction being whether dispersal depends on local environmental conditions. For competing unconditional dispersers, we prove that the population with the smaller dispersal probability always prevails. Alternatively, for conditional dispersers, it is shown that the strategy known as ideal free dispersal is both sufficient and necessary for evolutionary stability. These results extend those in the literature for discrete diffusion models in finite patchy landscapes and from reaction–diffusion models.     

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.     

脆弱斑块中植物种子的脉冲扩散

考虑了一类脆弱斑块中植物种子的脉冲漂移模型,得到了系统永久持续生存性.在此基础上,利用单调凸算子理论,得到了系统唯一全局渐进稳定的周期解.数值模拟也表明脉冲扩散能挽救脆弱斑块中的植物种群.     

The permanence and extinction of a nonlinear growth rate single-species non-autonomous dispersal models with time delays

In this paper, we consider the effect of diffusion on the permanence and extinction of a non-autonomous nonlinear growth rate single-species dispersal model with time delays. Firstly, the sufficient conditions of the permanence and extinction of the species are established, which shows if the growth rate and dispersal coefficients is suitable, the species is permanent, on the contrary, it is extinction. Secondly, an interesting result is established, that is, if only the species in some patches even in one patch is permanent, then it is also permanent in other patches. Finally, some examples together with their numerical simulations show the feasibility of our main results.     

Permanence and extinction of periodic predator–prey systems in a patchy environment with delay

This paper studies two species predator–prey Lotka–Volterra type dispersal systems with periodic coefficients and infinite delays, in which the prey species can disperse among n-patches, but the predator species is confined to one patch and cannot disperse. Sufficient and necessary conditions of integrable form for the permanence, extinction and the existence of positive periodic solutions are established, respectively. Some well-known results on the nondelayed periodic predator–prey Lotka–Volterra type dispersal systems are improved and extended to the delayed case.     

Effects of population dispersal and impulsive control tactics on pest management

In this paper, we firstly consider a Lotka–Volterra predator–prey model with impulsive constant releasing for natural enemies and a proportion of killing or catching pests at fixed moments, and we have proved that there exists a pest-eradication periodic solution which is globally asymptotically stable. Further, we extend the model for the population to move in a two-patch environment. The effects of population dispersal and impulsive control tactics are investigated, i.e. we chiefly address the question of whether population dispersal is beneficial or detrimental for pest persistence. To do this, some special cases are theoretically investigated and numerical investigations are done for general case. The results indicate that for some ranges of dispersal rates, population dispersal is beneficial to pest control, but for other ranges, it is harmful. These clarify that we can get some new effective pest control strategies by controlling the dispersal rates of pests and natural enemies.     

Multiple Attractors in Juvenile-adult Single Species Models

The role of age-structure and the Allee effect in generating multiple attractors in juvenile-adult single species single patch discrete-time models without dispersal are studied. In the presence of the Allee effect juvenile-adult single patch models support multiple attractors. However, in the absence of the Allee effect single attractors are supported when the dynamics are compensatory while multiple attractors are supported under overcompensatory dynamics. When the governing dynamics are compensatory, the boundaries of the basins of attraction have simple structure while complicated fractal basin boundaries are supported under overcompensatory dynamics.     

带有扩散、脉冲和时滞的非自治捕食系统的正周期解

考虑了一类食饵在斑块环境中扩散具有脉冲和时滞的捕食系统，通过灵活地运用Gaines和Mawhin的连续拓扑度定理，获得了一系列易验证的正周期解存在的充分条件．     

Coexistence in a discrete competition model with dispersal

We propose a discrete-time competition model between two populations to study the effects of dispersal upon population interactions. It is assumed that dispersal occurs after reproduction and in synchrony. We first analyse a two-patch single species population model with no interspecific competition. Based on these results, we derive sufficient conditions for population coexistence. It is proved that the system is uniformly persistent and possesses a unique coexisting equilibrium.