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.     

A mathematical model for chemical defense mechanism of two competing species

In this paper, a non-linear mathematical model is proposed and analyzed to study the phenomenon of a chemical defense mechanism involving two competing species, where each species produces a toxicant affecting the other. It is shown that if the emission rate coefficient of toxicant, produced by one species increases, the equilibrium density of the other species decreases and its magnitude is lower than its original carrying capacity. It is found that the usual principle of competitive exclusion (coexistence) in the absence of toxicant may change in the case under consideration. It is also observed that increases in the values of production rates of toxicants by the competing species and depletion rates of environmental toxicants due to its assimilation by the species has a destabilizing effect, and decrease in the washout rates of environmental toxicants has a destabilizing effect on the dynamics of the system. In the case of allelopathy, where only one species produces a toxicant affecting the other species, it is shown that the affected species is driven to extinction for large production rate of this toxicant.     

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.     

Global attractivity of nonautonomous stage-structured population models with dispersal and harvest

The nonautonomous stage-structured single-species dispersal model with harvesting of mature individuals in an N-patch environment is considered, in which the individual members of the population have a life history that takes them through two stages, immature and mature. By using the theory of monotone and concave operators to functional differential equations, we establish conditions under which this population dynamical system admits a positive periodic solution which attracts all positive solutions.     

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.     

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.     

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.     

Optimal Harvesting and Stability for a Predator-prey System with Stage Structure

The dynamics of a predator-prey system, where prey population has two stages, an immature stage and a mature stage with harvesting, the growth of predator population is of Lotka-Volterra nature, are modelled by a system of retarded functional differential equations. We obtain conditions for global asymptotic stability of three nonnegative equilibria and a threshold of harvesting for the mature prey population. The effect of delay on the population at positive equilibrium and the optimal harvesting of the mature prey population are also considered.     

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’.     

Global asymptotic stability of a two species competitive system with stage structure and harvesting

Introduction' There have recently appeared in the literature several mathematical models of stagestructured population growth, i. e., models which take into account the faCt that individuals in a population may belong to one of two classes, the immatures and the matureslllZI.Cannibalism has been observed in a great variety of species, including a number of fish species.Cannibalism models of various types have also been investigatedI3"l. In these models, the ageto maturity is represented by a…     

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.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

Global Attracting Behavior of Non-autonomous Stage-structured Population Dynamical System with Diffusion

A non-autonomous single species dispersal model is considered, in which individual member of the population has a life history that goes through two stages, immature and mature. By applying the theory of monotone and concave operators to functional differential equations, we establish conditions under which the system admits a positive periodic solution which attracts all other positive solutions.     

A two species competition model under the simultaneous effect of toxicant and disease

A mathematical model is proposed to study the simultaneous effects of toxicants and infectious diseases on a competing species system. It is assumed that the competing populations are adversely affected by the toxicant and one of them is vulnerable to an infectious disease. In this paper, two models are studied separately. The first model is developed to study the effect of only infectious diseases on the existence of a two competing species system in the absence of a toxicant, whereas in the second model the presence of a toxicant is also taken into account. In both the models, conditions for the existence of interior equilibria are derived. The models are analyzed using stability theory, and conditions for the nonlinear stability of the interior equilibria are obtained using Lyapunov’s direct method. Further, the models are studied numerically by taking two sets of numerical values for each model and the results are compared.     

The survival analysis for a single-species population model in a polluted environment

This paper concentrates on studying the long-term behavior of a single-species population living in a polluted environment. A new mathematical model is derived 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 uniform persistence, weak persistence in the mean or extinction of the population are obtained. Also we find some sufficient conditions, depending on the parameters of the model and the clean up rate, under which the population will be persistent.     

具有年龄结构的单种群模型的脉冲控制及其捕获策略

给出单种群阶段结构模型,利用脉冲微分方程的比较原理,通过状态反馈和输出反馈对模型变换后的系统进行了脉冲控制.对成年、幼年种群同时捕获,通过状态反馈,得到了单种群阶段结构模型在正平衡点渐近稳定的充分条件;通过输出反馈得到了相应的结论;并给出了脉冲控制时间间隔的上界估计值.分别对其幼年种群和成年种群捕获问题,给出以最大捕获可持续均衡收获(MSY)为目标的最优捕获策略.     

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.     

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.     

On global asymptotic stability of the equilibrium of “predator–prey” system in varying environment

This paper considers a predator–prey system of differential equations. This ecological system is a model of Lotka–Volterra type whose prey population receives time-variation of the environment. It is not assumed that the time-varying coefficient is weakly integrally positive. We obtain sufficient conditions of global asymptotic stability of the unique interior equilibrium if the time-variation is bounded.     

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.