A Simple Food Chain Model with Delay

Abstract

In this paper we consider a chemostat-like model for a simple food chain of microorganisms. This consists of a well stirred nutrient substance that is the food for a prey population of microorganism. At the bottom of the food chain there is a predator population of miroorganisms that grows up on the prey. The nutrient-uptake of each population is of Holling type I (or Lotka-Volterra) form. We show the existence of the global attractor for the solutions of the model and also we show that the positive globally asymptotically stable equilibrium point of the system undergoes a Hopf bifurcation when we suppose that the dynamic of the microorganisms at the bottom of the chain depend on the past history of the prey population by mean a distributed delay that take an average of the microorganisms in the middle of the chain.