Existence of limit cycles in a three level trophic chain with Lotka-Volterra and Holling type II functional responses

In this paper we analyze a three level trophic chain model, considering a logistic growth for the lowest trophic level, a Lotka-Volterra and Holling type II functional responses for predators in the middle and in the cusp in the chain, respectively. The differential system is based on the Leslie-Gower scheme. We establish conditions on the parameters that guarantee the coexistence of populations in the habitat. We find that an Andronov-Hopf bifurcation takes place. The first Lyapunov coefficient is computed explicitly and we show the existence of a stable limit cycle. Numerically, we observe a strange attractor and there exist evidence of the model to exhibit chaotic dynamics.