Electron. J. Diff. Eqns., Vol. 2009(2009), No. 76, pp. 1-10.

Hopf bifurcation for simple food chain model with delay

Mario Cavani, Teodoro Lara, Sael Romero

Abstract:
In this article we consider a chemostat-like model for a simple food chain where there is a well stirred nutrient substance that serves as food for a prey population of microorganisms, which in turn, is the food for a predator population of microorganisms. The nutrient-uptake of each microorganism is of Holling type I (or Lotka-Volterra) form. We show the existence of a global attractor for solutions of this system. Also we show that the positive globally asymptotically stable equilibrium point of the system undergoes a Hopf bifurcation when the dynamics of the microorganisms at the bottom of the chain depends on the history of the prey population by means of a distributed delay that takes an average of the microorganism in the middle of the chain.

Submitted May 11, 2009. Published June 16, 2009.
Math Subject Classifications: 34D99
Key Words: Simple food chain model; Hopf bifurcation; Holling type I; attractor.

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Mario Cavani
Departamento de Matemáticas, Núcleo de Sucre
Universidad de Oriente, Cumaná 6101, Venezuela
email: mcavani@sucre.udo.edu.ve
Teodoro Lara
Departamento de Física y Matemáticas, Universidad de los Andes
Núcleo Univeristario Rafael Rangel, Trujillo, Venezuela
email: tlara@ula.ve
Sael Romero
Departamento de Matemáticas, Núcleo de Sucre
Universidad de Oriente, Cumaná 6101, Venezuela
email: sromero@sucre.udo.edu.ve

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