Qualitative analysis of a two-species ecosystem with mutual interference: effect of additional food and complex habitat

Habitat complexity and additional food supplied to the predator are major drivers of temporal changes in prey–predator communities. The effects of these factors are well studied independently, but the researchers poorly understand the relative importance of these two phenomena. To assess their relative effects, we examine the influence of habitat complexity with additional food provided to the predator, which follows the functional response of Beddington–DeAngelis type that incorporates the mutual interference of predators. The existence of equilibrium states and their stability features together with the feasibility of solutions are discussed. By considering the habitat complexity and quantity of the supplementary food, we explore critical threshold conditions for saddle-node, Hopf, transcritical, and Bogdanov–Takens bifurcations. Using the signs of the first Lyapunov coefficients, the direction and stability of Hopf bifurcation are also investigated. It is observed that when there are two interior equilibrium states, one of them is always saddle, while the other exhibits instability behavior through the Hopf bifurcation. In this situation, the local periodic solution is initially stable but transitions to instability via generalized Hopf bifurcation. Finally, some numerical simulations are performed to validate the theoretical findings and illustrate the results.