Stability Analysis and Numerical Simulation of Prey-Mesopredator-Apex Predator Dynamic Model with Supplementary Food for Apex Predator

This study formulates and mathematically analyzes a three-species dynamic model involving prey, mesopredator, and apex predator, considering the presence of supplementary food available only to the apex predator. The model is expressed as a three-dimensional nonlinear differential equation system and analyzed by proving the existence and uniqueness of solutions, positivity, and solution limitations to ensure mathematical validity in the biological domain. Furthermore, we study the local stability of the six equilibrium points of the system using the eigenvalue approach and the Routh-Hurwitz criterion. We perform numerical simulations and find that the stability of the system is highly sensitive to the parameters of predation efficiency and the capacity to utilize additional food. In addition, species extinction, dominance, or long-term coexistence also occur. The model shows how the relationships between different species and the support from external energy sources can change the community structure and affect whether predator species survive.