Spatiotemporal dynamics of the prey-taxis model with pathogen invasion into predator species

This paper primarily investigates the impact of pathogen invasion on the prey-taxis system, focusing on the spread of pathogens among predator species. In contrast to the classical prey-taxis system, the predators are split into two parts: susceptible predator and infected predator. The global boundedness of its classical solutions in two dimensions is demonstrated by combining space-time estimates and energy estimates. Furthermore, we propose the large time behavior of nonnegative constant steady states, such as prey-only, disease-free, susceptible predator-free and endemic. These results show that pathogen invasion may break the dynamic behavior of the original prey-taxis system. The stability of the discussed system remains consistent with the corresponding results by Jin and Wang (JDE, 262, 2017), regardless of whether there is weak predation or weak infectiousness. However, this system posses strong predation and strong infectiousness, the infectious disease becomes epidemic among predator species. The further enhanced predation and infectiousness can even lead to the extinction of the susceptible predators. Finally, to elucidate the impact of prey-taxis mechanisms on the dynamic behaviors of this discussed system, we prove that steady-state and Hopf bifurcations can be induced at the endemic equilibrium with large prey-tactic coefficients, then numerical investigations reveal the emergence of diverse spatial patterns.