污染环境中的非自治种群模型的生存分析

研究了环境污染对种群的长期影响.考虑到新生个体的出生对种群体内毒素的影响,以及死亡的种群个体将体内毒素带回环境,建立了一个非自治数学模型.主要运用比较定理得到了种群一致持续生存、弱持续生存以及绝灭的判据.     

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.     

Effect of pollution tax on population in a polluted environment

This paper establishes a mathematical model to study the long behavior of a single‐species population living in a polluted environment. In this paper, we suppose that pollution tax is imposed on toxicant emitters if their emission crosses the permissible limit, limit up to which there is no harm to the population. Some sufficient conditions for the persistence of the population are obtained. Copyright © 2011 John Wiley & Sons, Ltd.     

EFFECTS OF A TOXICANT ON A SINGLE-SPECIES POPULATION WITH PARTIAL POLLUTION TOLERANCE IN A POLLUTED ENVIRONMENT

We study a model for the long-term behavior of a single-species population with some degree of pollution tolerance in a polluted environment. The model consists of three ordinary differential equations: one for the population density, one for the amount of toxicant inside the living organisms, and one for the amount of toxicant in the environment. We derive sufficient conditions for the persistence and the extinction of the population depending on the exogenous input rate of the toxicant into the environment and the level of pollution tolerance of the organisms. Numerical simulations are carried out to illustrate our main results.     

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．     

Dynamics of a two-species Lotka-Volterra competition system in a polluted environment with pulse toxicant input

In most models of population dynamics in a polluted environment, the emission of toxicant is generally considered to be continuous, but it is often the case that toxicant is emitted in regular pulses. This paper deals with the effects of pulse toxicant input with constant rate on two-species Lotka-Volterra competition system in a polluted environment. The thresholds between persistence and extinction of each population are obtained. Moreover, our results indicate that the release amount of toxicant and the pulse period will affect the fate of each population. Finally, the results are verified through computer simulations.     

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.     

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.     

Long term behaviors of stochastic single-species growth models in a polluted environment

Thresholds for extinction and persistence are important for assessing the risk of mortality in systems exposed to toxicant. In this paper, three single-species models with random perturbation in a polluted environment are proposed and investigated. One is the generalized logistic model and the other two are the stochastic resource–consumer models of Leslie and Gallopin. For each model, the survival threshold is obtained in some cases. In general, each threshold is determined by intensity of the random noise, the mean stress measure in organisms, the population intrinsic growth rate and the stress response rate.     

Persistence and extinction of a single-species population system in a polluted environment with random perturbations and impulsive toxicant input

Taking both white noises and colored noises into account, a stochastic single-species model with Markov switching and impulsive toxicant input in a polluted environment 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. Some simulation figures are introduced to illustrate the main results.     

A time-delay model for the effect of toxicant in a single species growth with stage-structure

In this paper, we investigate a single-species growth model with stage-structure consisting of immature and mature stages for the effects of toxicants with constant maturation time-delay. We study the dynamics of our model in three cases: an instantaneous emission of toxicant, a constant emission of toxicant, and a periodic emission of toxicant into the environment. We present results on positivity and boundedness of all solutions under appropriate conditions. The model equations are analyzed mathematically with regard to the nature of equilibria and their stabilities using the theory of nonlinear differential equations and computer simulations. It is shown that under suitable conditions, there exists a globally asymptotically stable positive equilibrium. It is concluded from the analysis that as the concentration of toxicant in the environment increases, equilibrium densities of both immature and mature populations decrease. It is also noted that the effects of toxicants are more on the equilibrium level of immature population in comparison to the mature population.     

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.     

污染环境下具有脉冲输入环境毒素的单种群模型的灭绝阈值研究

本文研究了污染环境下具脉冲输入环境毒素的单种群模型.利用乘子理论和小振幅扰动法,当脉冲周期小于一个临界值时,我们得到了种群灭绝周期解是全局渐近稳定的,同时我们还得到了种群持久的条件.从生物学的观点看,污染环境下保护物种的方法是控制环境毒素的排放周期或排放量.我们的结论为资源环境下的生物资源管理提供了策略基础.     

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.     

Dynamical model of a single-species system in a polluted environment

The effect of toxicants on ecological systems is an important issue from mathematical and experimental points of view. Here we have studied dynamical model of a single-species population-toxicant system. Two cases are studied: constant exogeneous input of toxicant and rapidly fluctuating random exogeneous input of toxicant into the environment. The dynamical behaviour of the system is analyzed by using deterministic linearized technique, Lyapunov method and stochastic linearization on the assumption that exogeneous input of toxicant into the environment behaves like ‘Coloured noise’.     

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.     

PERSISTENCE IN AN AGE-STRUCTURED POPULATION FOR A PATCH-TYPE ENVIRONMENT

A discrete-time model for an age-structured population in a patch-type environment is presented and analyzed. Comparison techniques for difference equations are used to find sufficient conditions for population persistence or extinction. The persistence and extinction theorem is used to define the critical patch number, the threshold for population persistence. Several examples are presented which illustrate the results of the theorems. The model is applied to a watersnake population.     

Qualitative analysis on a diffusive SIS epidemic system with logistic source and spontaneous infection in a heterogeneous environment

In this paper, we consider an SIS epidemic reaction–diffusion model with spontaneous infection and logistic source in a heterogeneous environment. The uniform bounds of solutions are established, and the global asymptotic stability of the constant endemic equilibrium is discussed in the case of homogeneous environment. This paper aims to analyze the asymptotic profile of endemic equilibria (when it exists) as the diffusion rate of the susceptible or infected population is small or large. Our results on this new model reveal that varying total population and spontaneous infection can enhance persistence of infectious disease, which may provide some implications on disease control and prediction.     

A resource-dependent competition model: Effects of toxicants emitted from external sources as well as formed by precursors of competing species

In this paper, a nonlinear mathematical model is proposed and analyzed to study the survival of resource-dependent competing species. It is assumed that competing species and its resource are affected simultaneously by a toxicant emitted into the environment from external sources as well as formed by precursors of competing species. Stabilities of all the equilibria are studied using the theory of differential equations and computer simulation. A condition which determines the persistence of the system is also obtained. It is concluded from the analysis that as the cumulative rates of emission and formation of toxicants into the environment increase, the densities of both competing species and its resource decrease. It is also concluded that the usual competitive outcomes for the resource biomass altered in the presence of precursors.     

MODELING EFFECTS OF PRIMARY AND SECONDARY TOXICANTS ON RENEWABLE RESOURCES

ABSTRACT. In this paper a nonlinear mathematical model to study effects of primary and secondary toxicants on the biomass of resources such as forestry, agricultural crops, etc., is proposed and analyzed. The primary toxicant is emitted into the environment with a constant prescribed rate by an external source and a part of which is transformed into a secondary toxicant, which is more toxic, both affecting the resource simultaneously. By using stability theory of differential equations, it is shown that the biomass density of resource attains an equilibrium level, the magnitude of which is smaller than its original (toxicant independent) carrying capacity and it decreases as the emission rate of primary toxicant increases. It is also shown that the decrease in biomass density of resource is more than the corresponding case of a single toxicant due to large transformation and uptake rates and high toxicity of secondary toxicant. It is pointed out that the resource may even become extinct if emission rate of primary toxicant and transformation rate of secondary toxicant are very large and their effects on resource are sufficiently harmful due to large uptake and high toxicity of secondary toxicant which is more toxic.