Predator-Prey Models with Discrete Time Delay

Abstract

Our goal in this thesis is to study the dynamics of the classical predator-prey model and the predator-prey model in the chemostat when a discrete delay is introduced to model the time between the capture of the prey and its conversion to biomass. In both models we use Holling type I response functions so that no oscillatory behavior is possible in the associated system when there is no delay. In both models, we prove that as the parameter modelling the delay is varied Hopf bifurcation can occur. However, we show that there seem to be differences in the possible sequences of bifurcations. Numerical simulations demonstrate that in the classical predator-prey model period doubling bifurcation can occur, possibly leading to chaos while that is not observed in the chemostat model for the parameters we use. For a delay differential equation, a prerequisite for Hopf bifurcation is the existence of a pair of pure imaginary eigenvalues for the characteristic equation associated with the linerization of the system. In this case, the characteristic equation is a transcendental equation with delay dependent coefficients. For our models, we develop two different methods to show how to find values of the bifurcation parameter at which pure imaginary eigenvalues occur. The method used for the classical predator-prey model was developed first. However, it was necessary to develop a more robust, less complicated method to analyze the predator-prey model in the chemostat with a discrete delay. The latter method was then generalized so that it could be applied to any second order transcendental equation with delay dependent coefficients.