Dynamic complexity of Holling-Tanner predator–prey system with predator cannibalism

Cannibalism is a common intraspecific interaction phenomenon and thus the elucidation of the mechanisms of cannibalism can enrich the ecological dynamics. In this paper, we investigate a Holling-Tanner system with predator cannibalism, which is rarely studied. For the non-spatial system, the local dynamics of the origin are fully characterized. The global dynamics of the constant positive steady state, including global stability, Hopf bifurcation and its directions, are examined. For the diffusion system, the Turing instability and global asymptotic stability for the constant steady state are derived, and the existence of Hopf bifurcation and Turing–Hopf bifurcation are studied. We found that predator cannibalism not only leads to complex dynamical behaviors around the origin in non-spatial system, but influences the global asymptotically stability and Turing instability of $E^{\ast }$, as well as results in Hopf bifurcation and Turing–Hopf bifurcation of diffusion system, which can reveal the reasons for the effects of predator cannibalism on biological systems. The numerical verification of the obtained results, the evaluation of the impact of predator cannibalism on the dynamics are also presented.