Complex Dynamics in a Tritrophic Chain Model With Prey Stage Structure

A tritrophic chain model with prey stage structure (reproductive and nonreproductive classes) is analyzed. This model is provided by an ODE system of dimension four and the main premise consists in assuming that the interaction between reproductive population and predator is governed either by a functional response Holling type II or IV (where a defense mechanism is considered). The aim on the present paper is to show the dynamic richness of the model, which ranges from stable sets to chaotic behaviors. In fact, on each case, we show the parameter conditions for proving the validity of the analytic nondegeneracy hypotheses for having a nonresonant Hopf–Hopf bifurcation (HHB), and whose unfolding will have a normal form determined by the cubic normal coefficients associated to the corresponding truncated amplitude system. Derived from our analytic results, it is also shown by using numerical simulations that the coexistence of the species takes place by means of invariant sets with a more complex dynamics than the one given by a stable limit cycle. In fact, we show the existence of quasiperiodic orbits, of ω-limit sets coming from an invariant two dimensional torus and of orbits with a chaotic motion. Finally, the contribution of our theoretical and numerical findings is appreciated in two directions: We provide a continuous dynamical system with chaotic behavior, and the coexistence of all the populations is obtained by means of interesting invariant sets that complement those obtained in previous published works by using stable equilibrium points and periodic orbits.