It is well known that the classical Lotka–Volterra diffusive predator–prey model (1.1) has no any spatiotemporal patterns. In this paper, we show that disease transmission in prey in model (1.1) can be regarded as a new mechanism to induce spatiotemporal dynamical patterns. We first establish an eco-epidemic Lotka–Volterra predator–prey model (1.6) with disease transmission in prey. Then we investigate the spatiotemporal dynamics of the model (1.6), and corresponding reaction-diffusion models (1.7) and (1.8), respectively. For the models (1.6) and (1.7) with the special case \((\alpha _1,\,\alpha _2)=(\beta _1,\,\beta _2)\) , we study the local and global asymptotical stabilities of nonnegative constant steady-state solutions. The findings show that the kinetic model (1.6) has no periodic solutions and the reaction-diffusion model (1.7) has no any spatiotemporal patterns. Whereas, the dynamics of models (1.6) and (1.7) with the general case \((\alpha _1,\,\alpha _2)\ne (\beta _1,\,\beta _2)\) change rapidly. We find that Hopf bifurcation occurs in kinetic model (1.6). Moreover, the positive equilibrium point \(E^*\) exhibits multiple stability switches phenomenon. For the reaction-diffusion model (1.7), we derive sufficient conditions of Turing instability of both the Hopf bifurcating periodic solutions and the endemic constant steady-state solution \(E^*\) . When Turing instability of the Hopf bifurcating periodic solutions occurs, we find numerically that model (1.7) can create new spatiotemporal patterns, i.e., model (1.7) turns from temporal periodic oscillatory pattern to spatial periodic oscillatory pattern. When Turing instability of \(E^*\) occurs, numerical simulations show that (1.7) can undergo stripes pattern, stripe-spot mixtures pattern and sparse patches pattern. For the reaction-diffusion model (1.8) with special case \((\alpha _1,\,\alpha _2)=(\beta _1,\,\beta _2)\) , we first study the existence and uniqueness of semi-trivial nonnegative steady-state solutions and positive steady-state solution. Then we prove the global asymptotic stabilities of trivial and semi-trivial nonnegative steady-state solutions.
