Bifurcations, multistability, and chaos in a three-level intraguild predation model with Michaelis–Menten-type harvesting on both predators

Abstract

A three-level intraguild predation model is proposed by incorporating nonlinear Michaelis–Menten-type harvesting in both the intermediate and the intraguild (IG) predator populations. Positivity and boundedness of solutions are established, the existence and stability of equilibria are analyzed, and the unconditional survival of the prey species is demonstrated. The impact of harvesting on dynamics is studied in the parameter plane by varying the intermediate and IG predator harvesting rates, revealing complex dynamics characterized by both codimension-one bifurcations (saddle-node, transcritical, pitchfork, Hopf) and codimension-two (Bogdanov–Takens, Cusp, Cusp bifurcation of limit cycles, Zero-Hopf, generalized Hopf) bifurcations. Moreover, the complexity of the system’s dynamics is also reflected in the emergence of invariant sets that are more complicated than equilibrium points or limit cycles, including invariant tori and chaotic attractors. The coexistence of all three species via quasiperiodic orbits with chaotic motion is numerically shown, and the transition to chaos is confirmed through a period-doubling cascade and the computation the maximum Lyapunov exponent. The results indicate that moderate and balanced harvesting promotes biodiversity, whereas excessive or imbalanced harvesting may lead to the extinction of one or both predator populations. It is observed that, in certain parameter regions, the system exhibits bistability, tristability, and quadristability, resulting in strong sensitivity to initial conditions. Unlike previous models, the present system exhibits multistable buffer regions in which coexistence is not immediately lost but depends on initial population levels.