ROLE OF TIME-DELAY IN AN ECOTOXICOLOGICAL PROBLEM

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 6, Number 1, Winter 1997 ROLE OF TIME-DELAY IN AN ECOTOXICOLOGICAL PROBLEM J. CHATTOPADHYAY, E. BERETTA AND F. SOLIMANO ABSTRACT. The present paper deals with the problem of a two species Lotka-Volterra type ecotoxicological model proposed by Maynard Smith and recently modified by Chattopadhyay. We have introduced the delay terms on the growth rate of toxic substances produced by each of the species in the model equations of Chattopadhyay and have observed the local dynamical behavior of this model around the partially feasible equilibria and positive equilibrium by characteristic equations. Moreover, we have established a basin of asymptotic stability of positive equilibrium by constructing a proper Liapunov functional. 1. Introduction. Lotka-Volterra models were introduced by Volterra [20]for the case where the species are competing with each other for a common pool of resources or where the species are the predators of the others, and by Lotka [13]for the case of symbiosis and parasitism. These models employ nonlinear differential equations describing linear 'per capita' growth and quadratic interactions between the variables. This is an important branch of mathematical biology to which many scientists in recent years have contributed. Several investigations have been made to study the effect of time lag on the stability of Lotka-Volterra population models. Relevant references are so many which are placed in the books and also are scattered in various journals ([14, 19, 15, 10, 7, 11, 1 to mention a few). Apart from those studies the ecotoxicological problems are also an important area of research. Many experimental studies have been conducted to study the effect of a single toxicant, and two or more toxicants (by itself or in combination) on both terrestrial and aquatic ecosystems [2, 6, 16, 17, 181. The point is that though several experimental studies on the effect of toxicants on biological species have been conducted, little attention has been paid to model such a phenomenon. Recently, in a series of works [4, 8, 91 Hallam and his co-workers have observed the effect of toxicant on various Accepted for publication by the editors on June 5, Copyright Rocky Mountain Mathematica Consortium

2 20 J. CHATTOPADHYAY. E. BERETTA AND F. SOLIMANO ecological scenarios by utilizing mathematical models. In particular, Hallam et al. in [9] have modelled the interaction of toxicant in the environment with the population by assuming the growth rate of population density linearly depends upon the toxicant concentration in the populations. The ecotoxicological problem would be more visible if one could incorporate the effect of environmental toxicant in the carrying capacity of the system. Reedman and Shukla [5] have considered and observed the effect of toxicant on equilibrium levels of single species dynamics and predator-prey interactions in a closed homogeneous environment, the carrying capacity of which is also affected by the exogeneous introduction of toxicant. It is to be observed that the ecotoxicological problem in a two species Lotka-Volterra system is also an interesting and challenging part of research both from mathematical and biological view points. Maynard Smith [19] studied the effect of toxic substance in a two species Lotka-Volterra competitive system by considering that each species produces a substance toxic to the other at a constant rate. Recently, Chattopadhyay [3] has modified Maynard Smith's model by considering that each species produces a substance toxic to the other, but only when the other is present. But in reality one species requires some time for producing a substance toxic to the other, which we may call the 'maturity time' for the species. As the ecotoxicological problems might be more visible if one could incorporate the time delay effect into the model equations, the main motivation of this paper is to introduce time delay terms in the model proposed by Chattopadhyay [3] and to see their effects. Before writing down our basic differential equations we shall present a brief history of the development of mathematical models in this particular issue starting from the Lotka-Volterra two species competitive system. The Lotka-Volterra two species competitive system can be written as: (1.1) -- dn1 - c1~l dt ((1-2) - B ~~N~} -- dn2 - E2~2{(1-2) - P ~~N~}, dt where Nl(t) and N2(t) are the population densities of two competing species for a common pool of resources, ~ i, i = 1,2, is the intrinsic bi-

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