Global dynamics in a generalized slow-fast predator-prey model

This paper studies a generalized slow-fast predator-prey model extended from the classic Gause model, where the predator reproduction rate is taken as a small singular perturbation parameter. First, by applying Dulac's criterion, we derive the sufficient conditions under which the local asymptotic stability of the system's unique positive equilibrium implies its global asymptotic stability in the first quadrant. Second, based on the geometric singular perturbation theory, we prove the existence of a unique relaxation oscillation surrounding the positive equilibrium, and show that this relaxation oscillation converges to the transcritical slow-fast cycle in the Hausdorff distance as the perturbation parameter approaches zero. Finally, with relaxation oscillation properties and Zhang Zhifen's limit cycle uniqueness theorem, we further derive sufficient conditions for the existence and uniqueness of stable limit cycles. The results enrich dynamical researches on slow-fast predator-prey systems and are applicable to related ecological dynamic analyses.