Abstract
In this paper, a predator-prey system with Holling type function response incorporating prey refuge is presented. By applying the analytical approaches, the dynamics behavior of the considered system is investigated, including stability, limit cycle and bifurcation. The results show that the shape of the functional response plays an important role in determining the dynamics of the system. Especially, the interesting conclusion is that the prey refuge has a destabilizing effect under some certain conditions.
Highlights
Researches on predation systems are always a popular issue in contemporary theoretical ecology and applied mathematics [ – ]
Results based on non-spatial systems have shown that the effect of prey refuge played an important role in determining the dynamical consequences of predator-prey systems [, – ]
We have considered a predator-prey system with a general functional response incorporating a prey refuge
Summary
Researches on predation systems are always a popular issue in contemporary theoretical ecology and applied mathematics [ – ]. X(t) and y(t) are the density of prey and predator populations at time t, respectively, and they are all positive numbers. For the positive equilibrium point E (x, y), the Jacobian matrix is as follows:. The interior equilibrium point E (x, y) is locally asymptotically stable. The interior equilibrium point E (x, y) is always asymptotically stable whenever the proportion of prey refuge is. ]n, the system has a unique globally stable limit cycle surrounding the interior equilibrium point E (x, y) which is unstable. ], the prey and predator populations stably oscillate around the unique interior equilibrium point. ] /n , the two populations tend to reach a globally asymptotically stable equilibrium point at the first quadrant. The system ( . ) undergoes a Hopf bifurcation at E (x, y) when the effect of prey refuge β passes the threshold value
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