Wave of chaos and Turing patterns in Rabbit–Lynx dynamics: Impact of fear and its carryover effects

Abstract

An attempt has been made to understand the joint impact of predator induced fear and its carryover consequences with diffusion. The prey population such as European rabbit is captured and consumed by the predator, Iberian lynx. In the absence of diffusion, the system undergoes saddle–node and Hopf-bifurcation with respect to the carryover and fear parameters. Both the fear and carryover parameter affect the system dynamics in a contradictory manner, i.e., higher amount of fear level destabilizes the system dynamics whereas higher amount of carryover level stabilizes it. Additionally, the creation and destruction of interior equilibrium points have been observed under the variation of both these parameters independently. Furthermore, the temporal system undergoes Cusp bifurcation in two parametric plane such as $f_1-f_2$, $f_1-\delta$ and $f_2-\delta$ plane. The global stability of the temporal system has been analyzed both analytically and numerically. However, in the presence of diffusion, the system experiences Turing instability. Numerical simulation shows the occurrence of spatio-temporal pattern formation for the proposed system. Further, it exhibits wave of chaos phenomenon for lower level of fear and carryover parameter value which is very important phenomenon to understand the spread of disease dynamics. Furthermore, the effect of the predator induced fear on the system dynamics has been explored in non-local sense for the spatio-temporal system. Our research integrates the model dynamics with its analysis by a variety of figures and diagrams that visually represent and reinforce our results. By examining non-linear models, we reveal unique and noteworthy patterns that offer fresh perspectives. These discoveries are particularly useful for biologists aiming to deepen their understanding of eco-epidemiological system dynamics in a practical context. The graphical depictions throughout our study play a key role in delivering a thorough analysis, making the findings more approachable and relevant to both researchers and field practitioners.