Abstract
A delayed Lotka-Volterra two-species predator-prey system of population allelopathy with discrete delay is considered. By linearizing the system at the positive equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the positive equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs). Finally, some numerical simulations are carried out for illustrating the theoretical results.
Highlights
In recent years, the Lotka-Volterra predator-prey models modeled by ordinary differential equations (ODEs) have been proposed and studied extensively since the pioneering theoretical works by Lotka [1] and Volterra [2]
The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations (FDEs)
According to the Hopf bifurcation theorem for FDEs, we find that the system can undergo a Hopf bifurcation of nonconstant periodic solution at the positive equilibrium when the delay crosses through a sequence of critical values
Summary
The Lotka-Volterra predator-prey models modeled by ordinary differential equations (ODEs) have been proposed and studied extensively since the pioneering theoretical works by Lotka [1] and Volterra [2]. Rice [19] has suggested that “all meaningful, functional ecological models will eventually have to include a category on allelopathic and other allelochemic effects” To our knowledge, such viewpoint haven’t been investigated in predator-prey model so far. The production of toxic substance by the predator species will not be instantaneous, but mediated by some time lag, see [7,19,20,21,22] From this viewpoint and combining the factors appeared above of different type of time delay and allelopathic effect in predator-prey model, we have modified the model of (3).
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