Codimension‐Two Bifurcation Analysis of the Tritrophic Food Chain Model With the Increasing Functional Response

ABSTRACT

In this research article, we consider a tritrophic food chain model consisting of one prey, one intermediate predator (mesopredator), and one top predator (superpredator). The novelty of this paper lies in the analysis of a tritrophic food chain model that combines a Holling type III functional response for the prey‐mesopredator interaction with both prey‐ and predator‐dependent functional response (also known as increasing functional response) proposed by Cosner et al. (Theor Popul Biol 56:65‐75, 1999) describing the predation of the superpredator on the mesopredator. The positivity and boundedness of solutions of the proposed model and the stability conditions of different equilibrium points are discussed here. Among the possible bifurcations that a system could have are those of codimension one as transcritical, saddle‐node, and Hopf bifurcations, and of codimension two as Bogdanov‐Takens bifurcation, a Zero‐Hopf bifurcation, and a generalized Hopf bifurcation. The results of numerical simulation are provided to illustrate and justify our theoretical results of the detailed bifurcation analysis. Through bifurcation analysis, the coexistence of the three species by means of quasiperiodic orbits with chaotic motion is numerically shown, and the transition to chaotic behavior is established via period‐doubling bifurcation and shown by calculating the Maximum Lyapunov Exponent. Relevant results from previous known food chain models are compared with the current findings. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper. Key findings and novelty of the paper include the analysis of a tritrophic food chain model in a two‐parameter space, which reveals that multiple codimension‐two bifurcations may coexist within the same parameter region, leading to significantly richer dynamical behavior and showing that different dynamical mechanisms can determine the long‐term outcome of the system, affecting the coexistence of all three species or potentially leading to the extinction of one or both predator populations. Since combining a Holling type III functional response at the lower trophic level with a Cosner‐type functional response at the higher level provides a biologically credible framework for modeling trophic interactions across a wide range of ecological systems, the results of our study may offer useful insights for the conservation of biological species.