Bifurcation analysis and optimal harvesting of an intraguild predation three-level food web model with harvesting on top two levels

Abstract

The present paper discusses the dynamics and optimal harvesting of an intraguild predation three-level food web model incorporating nonlinear Michaelis-Menten type harvesting on the intermediate predator and proportional harvesting on the intraguild predator. The positivity and boundedness of solutions, as well as the existence and stability of equilibria, are established, and unconditional survival of the prey species is observed. The effect of harvesting is studied through a detailed bifurcation analysis, revealing rich dynamical behaviors and threshold harvesting levels that prevent predator extinction. The existence of saddle-node, transcritical, pitchfork, and Hopf bifurcations is shown. The qualitative dynamics are discussed through two-parameter bifurcation diagram. Parameter regions of extinction and coexistence are identified. At higher harvesting rates, Bogdanov-Takens and generalized Hopf bifurcations reveal parametric regions in which either both predator species will eventually be driven to extinction or all three species may coexist, depending on the initial values. At lower harvesting rates, Zero-Hopf and generalized Hopf bifurcations reveal parametric regions in which either intermediate predator eventually goes extinct or all three species may coexist, depending on the initial population densities. It is shown that the system can exhibit multistability and sensitivity to initial conditions, with bistability between coexistence attractors and predator-free attractors. From an economic perspective, an optimal harvesting policy is derived, maximizing the total economic return from harvesting while preventing overharvesting and ensuring ecological sustainability. A numerical example shows that both economic benefits and ecological balance can be achieved by controlling both predators harvesting rates.