Abstract
Abstract A Lotka-Volterra type predator-prey system with Allee effect on the predator species and density dependent birth rate on the prey species is proposed and studied. For non-delay case, such topics as the persistent of the system, the local stability property of the equilibria, the global stability of the positive equilibrium are investigated. For the system with infinite delay, by using the iterative method, a set of sufficient conditions which ensure the global attractivity of the positive equilibrium is obtained. By introducing the density dependent birth rate, the dynamic behaviors of the system becomes complicated. The system maybe collapse in the sense that both the species will be driven to extinction, or the two species could be coexist in a stable state. Numeric simulations are carried out to show the feasibility of the main results.
Highlights
As was pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance
Huseyin Merdan [2] investigated the in uence of the Allee e ect on the Lotka-Volterra type predator-prey system
We mention here that to this day, though there are many scholars investigated the dynamic behaviors of the ecosystem with Allee e ect [1,2,3,4,5,6, 22,23,24,25,26], none of them considered the density dependent birth rate of the species
Summary
As was pointed out by Berryman [1], the dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. System (1.5) combines with the idea of Merdan [2] and Guan et al [17], will lead to the following Lotka-Volterra type predator-prey system with Allee e ect on the predator species and density dependent birth rate on the prey species dx dt. We mention here that to this day, though there are many scholars investigated the dynamic behaviors of the ecosystem with Allee e ect [1,2,3,4,5,6, 22,23,24,25,26], none of them considered the density dependent birth rate of the species. To the best of the authors knowledge, to this day, still no scholars propose a ecosystem with in nite delay and Allee e ect at the same time It seems that this is the rst time such kind of model are proposed and studied.
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