Nonlinear dynamics of a stage-structured interacting population model with honest signals and cues

Abstract

Recognizing predators is an important component in prey–predator interactions so that the prey species can modify their defence patterns. In this regard, biological signaling in such interactions is one of the most important phenomena. Recently, a model incorporating Holling type-II predation for two-species with two stages of prey including honest signals and cues has been proposed and analyzed by Al-Salman et al. (2021). The authors have proposed the model in detail and studied it for stability and Hopf-bifurcation. We have extended this study and observed numerically that the proposed system is capable of exhibiting chaotic dynamics, where solutions are highly sensitive to initial conditions. Therefore, the existence, uniqueness, and continuous dependence on the initial conditions of solutions are also investigated, together with positive invariance and dissipativeness. We have worked out the conditions for the existence, local asymptotic stability, and Hopf-bifurcation of steady-states of the proposed system. With the help of bifurcation diagrams, numerical simulations are performed to obtain the chaotic dynamics of the model. The sensitive dependence on initial conditions and the largest Lyapunov exponents are also verified to support these findings. Bifurcation diagrams for pairs of parameters are also plotted, which enables us to determine the combined effects. We also study the same model with Holling type-I functional response and compare the results obtained for this model with the earlier one. Our numerical simulations suggest that two Hopf-bifurcation thresholds are possible to ensure the stability switching, but there is no possibility of chaotic dynamics for this model.