Co-dimension 2 bifurcation analysis of a tri-trophic food chain model with strong Allee effect and Crowley–Martin functional response

This paper focuses on the species interaction within a tri-trophic food chain model involving species’ mutual interference, characterized by Crowley–Martin functional response with strong Allee effect in prey species. First, the evaluation of the solutions’ positivity and boundedness is conducted. Subsequently, the local stability of equilibrium points is analysed, and the Lyapunov function is employed to assess the global stability of interior equilibrium. We have also established adequate criteria for detecting local bifurcations. Sotomayor’s theorem infers the presence of saddle–node and transcritical bifurcations. Utilizing the centre manifold theory and the Hopf bifurcation theorem, the emergence and stability of Hopf bifurcation are investigated. Following this, numerical simulations are performed so as to validate the theoretical findings and analyse the impacts of Allee effect in greater details. The analysis reveals that the system undergoes a range of local bifurcations in co-dimension one, including saddle–node, subcritical and supercritical Hopf bifurcations. Furthermore, the existence of global dynamics can be observed by the appearance of Generalized-Hopf bifurcation in the co-dimension two bifurcation diagram. The findings suggest that strong Allee effect could trigger instability and induce a bi-stability phenomenon in the system’s dynamical behaviours.