The transition between two ecological limit cycles in one predator-two competitive prey model

In this paper we analyze an ecological model with a single predator and two competitive prey species. This model incorporates several key elements, including logistic growth dynamics for the prey populations, a Holling type II functional response governing predator-prey interactions, and the inclusion of intraspecific competition among predators. Our study establishes stringent conditions on the model parameters to ensure the existence of two coexistence equilibrium points (CEPs). Of particular interest is one CEP that undergoes a Hopf bifurcation, resulting in a continuous transition between two limit cycles residing in different dimensions. More precisely, we observe a periodic solution within a three-dimensional phase space, alongside another periodic solution confined to an invariant phase plane. The bialternate sum matrix criterion serves as a vital tool in demonstrating the existence of this Hopf bifurcation. Furthermore, employing Lyapunov exponents, we provide numerical evidence showcasing the emergence of chaotic dynamics within the model. This comprehensive analysis sheds light on the intricate behavior of the ecological system under consideration, offering valuable insights into the complex interplay of ecological factors and nonlinear phenomena within the predator-prey dynamics that has not been previously detected.