Bifurcation Analysis for a General Gause-Type Cross-Diffusion Prey-Predator Model in Spatially Heterogeneous Environment

This work analyzes a quasilinear elliptic system modeling a general Gausetype cross-diffusion prey-predator model in a spatially heterogeneous environment. Treating d as a bifurcation parameter, we establish the global bifurcation structure of positive solutions emanating from the semi-trivial solution branch, utilizing bifurcation theory coupled with the method of upper and lower solutions. We prove that when the prey birth rate r is relatively high (i.e., r > μ 1 –K 1 (0)Δ; σ(x)) and d lies within a moderate interval (i.e., between B(r) and μ 1 –K 2 (0)Δ), coexistence of prey and predator occurs, that is, the system admits positive solutions. Furthermore, we derive explicit conditions determining the direction of this bifurcation. Crucially, we demonstrate that the spatial distribution of prey resources, σ(x), significantly influences the existence of positive solutions for the system. Our results obtained in this paper can be applied to prey-predator models with Holling-type II functional response and non-monotonic functional response.