Stability, convergence, limit cycles and chaos in some models of population dynamics

In this paper, we study three models of population dynamics: (i) the classical delay logistic equation, (ii) its variant which incorporates a harvesting rate and (iii) the Perez-Malta-Coutinho (PMC) equation. For all these models, we first conduct a local stability analysis around their respective equilibria. In particular, we outline the necessary and sufficient condition for local stability. In the case of the PMC equation, we also outline a sufficient condition for local stability, which may help guide design considerations. We also characterise the rate of convergence, about the locally stable equilibria, for all these models. We then conduct a detailed local bifurcation analysis. We first show, by using a suitably motivated bifurcation parameter, that the models undergo a Hopf bifurcation when the necessary and sufficient condition gets violated. Then, we use Poincar, normal forms and centre manifold theory to study the dynamics of the systems just beyond the region of local stability. We outline an analytical basis to establish the type of the Hopf and determine the stability of the limit cycles. In some cases, we are able to derive explicit analytic expressions for the amplitude and period of the bifurcating limit cycles. We also highlight that in the PMC model, variations in one of the model parameters can readily induce chaotic dynamics. Finally, we construct an equivalent electronic circuit for the PMC model and demonstrate the existence of multiple bifurcations in a hardware-based circuit implementation.