Turing bifurcation and linear Hopf threshold in a diffusive predator–prey model with Smith growth and ratio-dependent Holling type Ⅲ functional response

This paper studies a diffusive predator–prey model with Smith growth, a ratio-dependent Holling type Ⅲ functional response, and Leslie–Gower predator dynamics. By analyzing the characteristic equations, we derive conditions for Turing instability and a linear stability threshold that signals the onset of Hopf-type oscillations. For the Turing branch, we prove the local existence of non-constant steady states bifurcating from the positive equilibrium, treating simple cases using the Crandall–Rabinowitz theorem. Numerical simulations are then carried out to illustrate these theoretical results and to explore the system's behavior near the intersection of the Turing and Hopf curves, where long-lived transient spatial patterns, spatially inhomogeneous periodic solutions, and complex spatio-temporal oscillatory patterns are observed. Overall, this work offers a mathematical account of how nonlinear growth, ratio-dependent predation, and spatial diffusion together shape the dynamics of this predator–prey system.