Marrakech 2002

# Stability and Bifurcations in Delayed Predator-Prey Systems

### S. Ruan

Department of Mathematics and Statistics

Dalhousie University

Halifax, Nova Scotia

Canada B3H 3J5

## Absolute Stability, Conditional Stability and Bifurcation in Kolmogorov-Type Predator-Prey Systems with Discrete Delays

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.

## Predator-Prey Models with Delay and Prey Harvesting

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.

## Multiple Bifurcations in a Delayed Predator-Prey System with Nonmonotonic Functional Response

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.

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