Bifurcation Analysis In A Predator-prey System With Nonmonotonic Functional Response And Prey Harvesting

In this paper we study the dynamical properties of a Predator-Prey system here,x(t) and y(t) are functions of time representing population densities of prey and predator, respectively. is the simplified Holling type-IV functional response, which is nonmonotonic in the first quadrant, h. d, μ and α are all positive constants and stand for the constant rate harvesting, the death rate of the predator, the conversion rate of prey to predator and the half-saturation constant, respectively.It is shown that numerous kinds of bifurcation phenomena occur for the model, such as the saddle-node bifurcation, the Hopf bifurcation of codimension1and2, the cusp bifurcation of codimension2(i.e. Bogdanov-Takens bifurcation), as the values of parameters of the model vary. Hence, there are different parameter values for which the model has a limit cycle, or a homoclinic loop, or two limit cycles, or exhibits the so-called paradox of enrichment phenomenon. These results reveal far richer dynamics compared to the model with no harvesting. Numerical simulations for the existence of one limit cycle or two limit cycles bifurcated from the multiple focus are also given in support of the Hopf bifurcation of codimension1and2.