Abstract
We study the qualitative behavior of a class of predator-prey models with Beddington-DeAngelis-type functional response, primarily from the viewpoint of permanence (uniform persistence). The Beddington-DeAngelis functional response is similar to the Holling type-II functional response but contains a term describing mutual interference by predators. We establish criteria under which we have boundedness of solutions, existence of an attracting set, and global stability of the coexisting interior equilibrium via Lyapunov function.
Highlights
Introduction and Mathematical ModelStandard Lotka-Volterra systems are known as the predator-prey systems, on which a large body of existing predator-prey theory is built by assuming that the per capita rate of predation depends on the prey numbers only 1
The Beddington-DeAngelis functional response admits a range of dynamics which include the possibilities of extinction, persistence, and stable or unstable equilibria
It will contain the numerical simulations to justify the obtained results
Summary
Standard Lotka-Volterra systems are known as the predator-prey systems, on which a large body of existing predator-prey theory is built by assuming that the per capita rate of predation depends on the prey numbers only 1. It is similar to the well-known Holling type-II functional response but has an extra term β1Y in the first right term equation modeling mutual interference among predators. This kind of type functional response given in 1.2 is affected by both predator and prey, that is, the so-called predator dependence by Arditi and Ginzburg 2. We have x > 0 and y > 0 in Int R2 ; every solution φ t x t , y t of system 1.3 , which starts in Int R2 , satisfies the differential inequality dx/dt ≤ 1 − x t x t This is obvious by considering the first equation of 1.3. This completes the proof; we conclude that system 1.3 is dissipative in R2
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