The principal eigenvalue for periodic nonlocal dispersal systems with time delay

The theory of the principal eigenvalue is established for the eigenvalue problem associated with a linear time-periodic nonlocal dispersal cooperative system with time delay. Then we apply it to a Nicholson's blowflies population model and obtain a threshold type result on its global dynamics.     

The Effect of Dispersal on Population Growth with Stage-structure

The effect of dispersal on the permanence of population in a polluted patch is studied in this paper. The authors constructed a single-species dispersal model with stage-structure in two patches. The analysis focuses on the case that the toxicant input in the polluted patch has a limit value. The authors derived the conditions under which the population will be either permanent, or extinct.     

Influence of time delay and nonlinear diffusion on herbivore outbreak

Herbivore outbreaks, a major form of natural disturbance in many ecosystems, often have devastating impacts on their food plants. Understanding those factors permitting herbivore outbreaks to occur is a long-standing issue in conventional studies of plant-herbivore interactions. These studies are largely concerned with the relative importance of intrinsic biological factors and extrinsic environmental variations in determining the degree of herbivore outbreaks. In this paper, we illustrated that how the time delay associated with plant defense responses to herbivore attacks and the spatial diffusion of herbivore jointly promote outbreaks of herbivore population. Using a reaction-diffusion model, we showed that there exists a threshold of time delay in plant-herbivore interactions; when time delay is below the threshold value, there is no herbivore outbreak. However, when time delay is above the threshold value, periodic outbreak of herbivore emerges. Furthermore, the results confirm that during the outbreak period, plants display much lower density than its normal level but higher in the inter-outbreak periods. Our results are supported by empirical findings.     

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…     

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.     

Aggregation and heterogeneity from the nonlinear dynamic interaction of birth,maturation and spatial migration

We consider a single species structured population distributed in two identical patches connected by spatial dispersal. Assuming that the maturation time for each individual is a random variable with a gamma distribution and that the spatial dispersal rate is constant, we obtain from a hyperbolic differential equation a system of six ordinary differential equations for the matured populations and their moments. Our qualitative analysis and numerical simulations show that the nonlinear interaction of birth process, the maturation delay and the spatial dispersal can lead to a new mechanism for individual aggregation in the form of the existence of multiple stable heterogeneous equilibria, even though the spatial dispersal is assumed to be proportional to the population gradients with a constant rate.     

The survival analysis for a population in a polluted environment

This paper concentrates on studying the long-term behavior of a single species in a polluted closed environment. We improve the rudimentary population model of Hallam and the classical Gallopin resource–consumer model, assuming that a born organism takes with it a quantity of internal toxicant, and the amount of toxicant stored in each living organism which dies is drifted into the environment. Sufficient criteria for persistence or extinction of the consumer population are obtained. The threshold between persistence and extinction will be established in some cases.     

Wavefronts of a Nonlocal State-dependent Time Delay Model

This paper is concerned with the travelling wavefronts of a nonlocal dispersal cooperation model with harvesting and state-dependent delay,which is assumed to be an increasing function of the population density with lower and upper bound.Especially,state-dependent delay is introduced into a nonlocal reaction-diffusion model.The conditions of Schauder's fixed point theorem are proved by constructing a reasonable set of functionsΓ(see Section 2)and a pair of upper-lower solutions,so the existence of traveling wavefronts is established.The present study is continuation of a previous work that highlights the Laplacian diffusion.     

Weak average persistence and extinction of a predator–prey system in a polluted environment with impulsive toxicant input

In this paper, we have investigated a predator–prey system in a polluted environment with impulsive toxicant input at fixed moments. We have obtained two thresholds on the impulsive period by assuming the toxicant amount input is fixed to the environment at each pulse moment. If the impulsive period is greater than the big threshold, then both populations are weak average persistent. If the period lies between of the two thresholds, then the prey population will be weak average persistent while the predator population extinct. If the period is less than the small threshold, both populations tend to extinction. Finally, our theoretical results are confirmed by own numerical simulations.     

The effect of dispersal on the permanence of a predator–prey system with time delay

A predator–prey model with prey dispersal and time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of each of the nonnegative equilibria is discussed. By using an iteration technique, a threshold is derived for the permanence and extinction of the proposed model. Numerical simulations are carried out to illustrate the main results.     

Dynamics of a population in two patches with dispersal

A two-dimensional discrete system of a species in two patches proposed by Newman et al. is studied. It is shown that the unique interior steady state is globally asymptotically stable if the active population has a Beverton–Holt type growth rate. If the population is also subject to Allee effects, then the system has two interior steady states whenever the density-independent growth rate is large. In addition, the model has period-two solutions if the symmetric dispersal exceeds a critical threshold. For small dispersal, populations may either go extinct or eventually stabilize. However, populations are oscillating over time if dispersal is beyond the critical value and the initial populations are large.     

Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.     

Asymptotic solution of wave front of the telegraph model of dispersive variability

A number of nonlinear phenomena in physical, chemical, economical and biological processes are described by the interplay of reaction and diffusion or by the interaction between convection and diffusion. Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this paper we use the technique of asymptotic solution to find travelling wave solution for the telegraph model of dispersive variability which is a generalization of the Julian Cook model. Our model tackled special values of the time delay and the probability that an individual is disperser in a population of dispersers and nondispersers. The solution of our model is reduced to telegraph Fisher–Kolmogoroff invasion and Julian Cook models. Also, the effect of the time delay on the propagation speed is presented.     

一类带有种群迁移的SIS传染病模型的全局分析

通过假设同一地区内易感者和染病者具有相同的迁移率系数,建立了一类两地区间种群迁移的SIS传染病模型,得到了地方病平衡点存在的阈值条件,并借助比较定理和极限系统理论证明了无病平衡点和疾病不导致死亡时地方病平衡点的全局稳定性,最后讨论了种群迁移对传染病传播的影响.     

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．     

Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations

For abstract functional differential equations and reaction-diffusion equations with delay, an exponential ordering is introduced which takes into account the spatial diffusion. The induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics (extinction or persistence/convergence to positive equilibria) for a nonlocal and delayed reaction-diffusion population model.     

Modeling the survival of a resource-dependent population: Effects of toxicants (pollutants) emitted from external sources as well as formed by its precursors

In this paper, a nonlinear mathematical model is proposed and analyzed to study the survival of a resource-dependent population. It is assumed that this population and its resource are affected simultaneously by a toxicant (pollutant) emitted into the environment from external sources as well as formed by precursors of this population. It is shown that as the cumulative rates of emission and formation of the toxicant into the environment increase, the densities of population and its resource settle down to lower equilibria than their initial carrying capacities, and their magnitudes decrease as rates of emission and formation of the toxicant increase. On comparing different cases, it is noted that when population is not affected directly by the toxicant but only its resource is affected, the possibility of its survival is greater than the case when both are affected simultaneously. But for large emission rate of toxicant, the affected resource may be driven to extinction under certain conditions and the population which wholly depends on it may not survive for long even if it is not affected directly by the toxicant.     

Singular perturbation analysis of a model for the effect of toxicants in single-species systems

We consider a mathematical model for the effect of toxicant levels on a single-species ecosystem in the case where there is an initial instantaneous introduction of a toxicant into the environment. The population birthrate as well as the carrying capacity are assumed to be directly affected by the level of toxicant in the environment as it is absorbed by the population. The toxicant level in the population can be depleted at a constant specific rate, a part of which may return to the environment. Through a singular perturbation analysis, we are able to identify different dynamical behavior which may be possible to the system, including the existence of sustained oscillation in the levels of toxicant in the population and the environment.     

Periodic solution and almost periodic solution for a nonautonomous Lotka-Volterra dispersal system with infinite delay

This paper studies a nonautonomous Lotka-Volterra dispersal systems with infinite time delay which models the diffusion of a single species into n patches by discrete dispersal. Our results show that the system is uniformly persistent under an appropriate condition. The sufficient condition for the global asymptotical stability of the system is also given. By using Mawhin continuation theorem of coincidence degree, we prove that the periodic system has at least one positive periodic solution, further, obtain the uniqueness and globally asymptotical stability for periodic system. By using functional hull theory and directly analyzing the right functional of almost periodic system, we show that the almost periodic system has a unique globally asymptotical stable positive almost periodic solution. We also show that the delays have very important effects on the dynamic behaviors of the system.     

Stability properties for partial Volterra integrodifferential equations

Summary Stability properties of the solutions of a Volterra's population equation including infinite delay and diffusion terms are studied via a linearized stability argument, which leads to investigating on operational characteristic equation. For a specific class of delay kernels time periodic solutions are shown to appear as the delay is increased.