Stability and Bifurcations in Delayed Predator-Prey Systems

The dynamics of delayed systems depend not only on the parameters describing the models but also on the time delays from the feedback. A delay system is absolutely stable if it is asymptotically stable for all values of the delays and conditionally stable if it is asymptotically stable for the delays in some intervals. In the later case, the system could become unstable when the delays take some critical values and bifurcations may occur. We consider three classes of Kolmogorov-type predator-prey systems with discrete delays and study absolute stability, conditional stability and bifurcation of these systems from a global point of view on both the parameters and delays.

It is known that predator-prey systems with constant rate harvesting exhibit very rich dynamics. On the other hand, incorporating time delays into predator-prey models could induce instability and bifurcation. We will study the combined effects of the harvesting rate and the time delay on the dynamics of the generalized Gause-type predator-prey models and the Wangersky-Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities, while the harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular, one of these models loses stability when the delay varies and then regains its stability when the harvesting rate in increased. Computer simulations are carried to explain the mathematical conclusions.

A delayed predator-prey system with nonmonotonic functional response is studied by using the normal form theory of retarded functional differential equations developed by Faria and Magalhaes. The bifurcation analysis of the model indicates that there is a Bogdanov-Takens singularity for any time delay value. A versal unfolding of the model at the Bogdanov-Takens singularity is obtained. On the other hand, it is shown that small delay changes the stability of the equilibrium of the model for some parameters and the system can exhibit Hopf bifurcation as the time delay passes through some critical values.