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.

Caputo fractional derivative solves the fractional initial value problem in Riemann-Liouville (R-L) fractional calculus. The definition of a Caputo-type derivative is in the same form as the definition of an integral differential equation, including the restriction of the value of the integral derivative to the value of the unknown function at the endpoint $t=a$. Therefore, this paper introduced the Caputo fractional derivative (CFD) to establish the transmission model of leptospirosis. First, to ensure that the model had a particular significance, we proved the dynamic properties of the model, such as nonnegative, boundedness, and stability of the equilibrium point. Second, according to the existence mode and genetic characteristics of pathogenic bacteria of leptospirosis, and from the perspective of score optimal control, we put forward measures such as wearing protective clothing, hospitalization, and cleaning the environment to prevent and control the spread of the disease. According to the proposed control measures, a control model of leptospirosis was established, and a forward-backward scanning algorithm (FB algorithm) was introduced to optimize the control function. Three different disease control strategies were proposed. Finally, the numerical simulation of different fractional orders used the fde12 (based on Adams–Bashforth–Moulton scheme) solver. The three optimized strategies, A, B, and C, were compared and analyzed. The results showed that the optimized control strategy could shorten the transmission time of the disease by about 80 days. Therefore, the above methods contributed to the study of leptospirosis and the World Health Organization.