Ecological and epidemiological thresholds in a predator–prey system with cross-infection: stability, bifurcations, and reaction-diffusion dynamics

This paper investigates a four–dimensional predator–prey model with cross–species infection and Holling type Ⅱ functional response. The model incorporates logistic growth for susceptible prey, susceptible–infected (SI)–type disease transmission with mass–action incidence, and a biologically realistic mechanism by which predators become infected through consumption of infected prey. We derive five key ecological and epidemiological threshold parameters governing predator persistence and disease invasion in different ecological scenarios. Analytical results establish positivity, boundedness, and conditions for the existence, feasibility, and stability of all equilibria. These include disease–free coexistence, endemic prey–only, and full endemic coexistence states. Global stability of the disease–free coexistence equilibrium is obtained using the Lyapunov method. Bifurcation analyses reveal transcritical, Hopf, and saddle–node bifurcations, explaining the transitions between extinction, stable coexistence, oscillatory dynamics, and bistability. Co–dimension–two analysis identifies organizing centers that structure the parameter space and clarifies mechanisms underlying regime shifts. Numerical simulations using MATLAB and MatCont confirm theoretical findings and illustrate diverse dynamical behaviors. These behaviors include predator extinction driven by highly infectious diseases, predation–mediated disease control, sustained oscillations, and multiplicity. Spatial extension via a reaction–diffusion framework demonstrates diffusion–driven Turing instability and pattern formation. The results provide integrated ecological and epidemiological insights into cross–infection dynamics and predator–prey coexistence.