The effect of harvesting with threshold on the dynamics of prey predator model

Abstract

Abstract This article deals with the dynamics of Lotka-Volterra prey predator population. The populations are considered as economically valuable stocks and then exploited. There is no harvesting when the densities of population are still low and the populations are harvested when the threshold value is achieved. The rate of harvesting is assumed to be an increased function and bounded. Phase portrait and linearization approach are used to analyze the behavior of the populations. There exists one equilibrium point for system without harvesting and it is a centre. The trajectories of the population oscillate around the stable equilibrium point. It is possible to find one, two, three, or none equilibrium points for model with harvesting. From the analysis we found that when the populations are not harvested then the equilibrium point becomes a centre. But when the populations are harvested with a smaller value, the equilibrium point becomes unstable spiral. When the value of harvesting rate is increased, the equilibrium point becomes either stable spiral or stable node. When the equilibrium points are unstable, the populations will meet a condition where their sizes are smaller than the threshold value and then the populations must stop being harvested.

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