Local dynamics of a predator–prey community in a moderate period of time

Abstract

In this work, we have introduced an ecological model of a prey–predator system. It is assumed that the prey species grows logistically, but the total number of predator is constant in the time interval. Positivity and boundedness of the solution ensure that the proposed model is well-posed. Local stability conditions of the equilibrium points have been analysed by the Routh–Hurwitz criterion. The persistence of the system has also been shown under a parametric restriction. Numerical analysis has indicated that both axial and interior steady states can exist only for moderate consumption rate (searching efficiency). But if this rate becomes high (or low), then only the prey-free equilibrium (or one of the interior equilibriums) exists as a steady state. Further, the equilibrium points can change their stability through transcritical and saddle-node bifurcations by varying the consumption rate of the predator. Analytical results provide an interesting phenomenon about this model: the system can never show any oscillating behaviour for any parametric values, i.e. no limit cycle can occur through Hopf bifurcation around an equilibrium point. The axial equilibrium becomes stable from an unstable situation when the consumption rate becomes high and the interior state which is stable remains stable as time goes by.