Period-doubling cascade to chaos and optimal quadratic harvesting in a prey-predator-scavenger model using Crowley-Martin functional response.

In the present article, a prey-predator-scavenger model is proposed and investigated with quadratic harvesting of predator and scavenger populations. The system is assumed to follow the Crowley-Martin functional response to describe the interaction between prey and predator populations. The positivity and boundedness of the system with respect to positive initial conditions are established. The analysis included determining all feasible equilibrium points and assessing their local stability under appropriate conditions. The system exhibits limit cycles around the interior equilibrium point. It is also observed that the solution of the system undergoes a period-doubling route to chaos. The existence of local bifurcation around the equilibrium points is investigated. It is shown that the system admits a transcritical bifurcation and a Hopf point for certain parameter values. The system also undergoes a global bifurcation, i.e., a generalized Hopf bifurcation, with respect to different parametric planes. The uniform persistence of the system is derived under specific conditions. Furthermore, an optimal harvesting problem is proposed and analyzed to determine the optimal harvesting pathways that not only maximize net revenue but also effectively manage harvesting efforts. The existence and characterization of optimal controls are discussed using Pontryagin's maximum principle to balance the implementation of harvesting efforts. Extensive numerical simulations, including time series, phase portraits, and bifurcation diagrams, are performed to illustrate the theoretical results.