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Harvesting of the Vegetation Biomass And Grazer Population with Its Effects on Predator Population: A Mathematical Model

Manju Agarwal, Sapna Devi

Abstract


In this paper, a nonlinear mathematical model is proposed and analyzed to see the effect of harvesting of vegetation biomass and grazer population (like deer) on the predator population (like tiger). The model is formulated as a food chain model by considering the hyperbolic interaction processes. It is assumed that the dynamics of vegetation biomass follows regrowth equation. Mathematical analyses of the model equations with regard to the boundedness of solutions, existence of equilibria and their stabilities in both local and global manner are carried out. Butler – McGehee lemma is used to identify the conditions which influence the persistence of the system as well as impermanence. The role of delay in the above system is also discussed and the critical value of delay for which stability change occurs is obtained. These analytical findings are substantiated numerically using fourth order Runge – Kutta method. It is concluded that increase in catchability coefficients of harvesting of vegetation biomass and grazer population, without any limit, will make predator population to go towards extinction. Maximum values of catchability coefficients are obtained to keep away predator population from extinction.

Keywords


Vegetation biomass; grazer population; predator population; harvesting; delay; simulation

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