Abstract
A predator-prey model with simplified Holling type III response function incorporating a prey refuge under sparse effect is considered. Through qualitative analysis of the model, at least two limit cycles exist around the positive equilibrium point with the result of focus value, the Hopf bifurcation under a prey refuge is obtained. We also show the influence of prey refuge. Numerical simulations are carried out to illustrate the feasibility of the obtained results and the dependence of the dynamic behavior on the prey refuge. Through the results of computer simulation, it is further shown that under certain conditions the model has three limit cycles surrounding the positive equilibrium point.
Highlights
The dynamical relationship between predators and their preys is one of the dominant subjects in ecology and mathematical ecology due to its universal importance, see [ – ]
Some of the empirical and theoretical work have investigated the effect of prey refuges, the refuges used by prey have a stabilizing effect on the considered interactions, and prey extinction can be prevented by the addition of refuges [ – ]
Our results indicate that refuge had a stabilizing effect on prey-predator interactions and the dynamic behavior very much depends on the prey refuge parameter m, point that increasing the amount of refuge could increase prey densities and lead to population outbreaks
Summary
The dynamical relationship between predators and their preys is one of the dominant subjects in ecology and mathematical ecology due to its universal importance, see [ – ]. Motivated by the study of Huang et al [ ] and Ji and Wu [ ], we consider the following predator-prey model with Holling type III response function incorporating a prey refuge under sparse effect: rx If the initial value p which is in the first quadrant is not in the OABC, we can construct a curve Jp in the same way, denoting p ∈ Jp. the positive trajectory L+p which is pass through p goes through into the interior of Jp at the end. Considering the time change dτ = –A dt, we know that P (x , y ) is a stable fine focus with second-order.
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