Dynamic Analysis of Prey-Predator Model with Infection in the Prey Species Under the Effect of Primary and Secondary Toxicants

Abstract

This paper depicts a predator-prey model to study the synchronous impact of two toxicants with various toxicities, discharged into nature, affecting the biological species. In the model, we consider simultaneous effect of two toxicants, in which primary toxicant is being
converted into secondary toxicant and where secondary toxicant is more harmful. Also, transformation rate from primary toxicant to secondary toxicant is directly proportional to concentration of primary toxicant in the environment. Properties of two of these toxicants are different and hence the uptake rate of these toxicants by different species vary. Feasible equilibrium points have been obtained for the model. Using Routh Hurwitz method, conditions for local stability of non-trivial equilibrium points have been established. Also, using a suitable Lyapunov function, global stability of a non linear system at interior equilibrium point has been established. It has been observed in this paper that when conversion rate from primary to secondary toxicant is more, equilibrium is attained at a lower level and survival of population is less. We observe that although there is lesser infection in species in presence of pollution, survival of species becomes difficult with increase in pollution, therefore, either pollution has to be controlled or certain measures should be taken to reduce the conversion rate so that infection decreases and species also survives.