Modelling effects and control of soil contaminants on the dynamics of plant–herbivore species system

In this paper, we have proposed and analysed a mathematical model to study the effects of impulsive input of contaminants in soil on the dynamics of interacting species system consisting of contaminants, soil-nutrients, plant biomass and herbivore population in order to investigate the criteria for the survival or extinction of plant biomass and herbivore population. The uptake function for herbivore population considered in the model of the paper shows saturation effect and this type of uptake function behave linearly with respect to plant biomass for low density. Also, the saturation for large plant biomass is a reflection of the limited herbivore capability or perseverance when the plant biomass is abundant. Therefore, the uptake function considered in this paper is more natural to lotka volterra and holling type functional responses. We have shown that the plant–herbivore extinction periodic solution exists and also the plant–herbivore extinction periodic solution is locally and globally stable if the impulsive depletion rate of nutrients due to contaminants is more than some critical value. Further, it is concluded that if the impulsive input period for contaminants is increased and impulsive depletion rate of nutrients due to contaminants is decreased then both the plant and herbivore population will survive otherwise they will tend to extinction.