Bifurcations and Homoclinic Orbits of a Model Consisting of Vegetation-Prey-Predator Populations

This study provides a comprehensive analysis of the dynamics of a three-level vertical food chain model, specifically focusing on the interactions between vegetation, herbivores, and predators in a Snowshoe hare-Canadian lynx system. By simplifying the model through dimensional analysis, we determine conditions for equilibrium existence and identify various types of bifurcations, including Saddle-Node and Hopf bifurcations. Additionally, the study explores codimension-two bifurcations such as Bogdanov-Takens (BT) and zero-Hopf bifurcations. Coefficient formulas of normal forms are derived through the use of center manifold reduction and normal form theory. The study also presents an approximation of homoclinic orbits near a BT bifurcation of the system by computing explicit asymptotics based on regular perturbation methods. Utilizing the MATLAB package MATCONT, a family of limit cycles and their associated bifurcations are computed, including limit point cycles, period-doubling bifurcations, cusp points of cycles, fold-flip bifurcations, and various resonance bifurcations (R1, R2, R3, and R4). The biological implications of the findings are discussed in detail, highlighting how the identified bifurcations and dynamics can impact the population dynamics of vegetation, herbivores, and predators in real-world ecosystems. Numerical experiments validate the theoretical results and provide further support for the conclusions.