Spatiotemporal Tipping Induced by Turing Instability and Hopf Bifurcation in a Population Ecosystem Model with the Fear Factor

Population ecosystems can display the tipping points at which extinctions of species happens. To predict the appearance of tipping points and to understand their evolution mechanism are of uttermost importance for ecological balance. Using techniques from bifurcation theory, we can predict the emergence of tipping points based on a spatiotemporal predator-prey system having a fear effect before an instability is encountered. In the case of no time delay, the tipping induced by Turing instability is studied. The conditions for Turing instability and local asymptotic stability of coexisting equilibrium points are given. It is ascertained that the introduction of diffusion causes the ecosystem to change from stable to unstable, and then the tipping occurs. Then, we investigate another tipping due to Hopf bifurcation. The delay-dependent stability criterion and Hopf bifurcation condition are derived, and the onset of Hopf bifurcations (tipping point) is also determined. In order to further probe into the mechanism of tipping evolution, explicit formulae are derived to ascertain the stability of bifurcated oscillations and the direction of bifurcation via the center manifold reduction. It is revealed that many tipping points may exist in ecological competition systems, and the tipping occurs many times as the fear delay increases. Finally, several simulation examples are provided to substantiate the analytical results.