Complex dynamics in a reaction-cross-diffusion model with refuge depending on predator–prey encounters

Abstract

The contemporary study copes with a generalist predator–prey model with nonlinear cross-diffusion embracing prey refuge in proportion to both the species along with the ratio-dependent functional response. The investigation commences with the well-posedness keeping in view of the existence of all feasible non-negative equilibria together with their global dynamics for the corresponding temporal model. The refuge parameter plays a key role in the dynamics of the model in general and mediates the uniform persistence, the stability of the boundary and coexistent equilibria and even the Turing instability space, in particular. Another important observation is that refuge possessing a constant proportion predator–prey encounters leads to a weaker stabilizing effect than refuge possessing a constant proportion prey under certain system parameters. Subsequent exploration on three eminent classes of mechanism for Turing instability around the positive spatially homogeneous steady state reveals dynamical complexity stimulated by generalist predator of the model system. Some comparisons are given between refuge depending on prey and refuge depending on both the species. Finally, numerical simulations expose to view the growth of spatiotemporal patterns controlled by prey refuge together with both self- and cross-diffusion following the sequence of spots, stripe–spot mixtures, stripes, labyrinth, stripe–hole mixtures and holes as well. Thus the proposed model appears to be rich and complex from the dynamical point of view having novelty to contribute much to the system in the realm of ecology.

Appendix A: The proof of Lemma 3.2

Proof

It follows from $s_4<0$ that $\delta -\gamma -\lambda <0$, which implies $s_1<0$. Since $s_0>0, s_1<0, s_4<0$, we get the total number of sign changes from one coefficient to the next in the sequence $s_0, s_1, s_2, s_3, s_4$ is either equal to three or one. By $Descartes'\,\,Rule\,\,of\,\,signs$, we obtain that the number of positive real roots of the polynomial Eq. (12) is either equal to one or three. Next we claim that Eq. (12) has only one positive real root under the assumptions $s_3>0$ and $s_1^2<\frac{8}{3}s_0s_2$. Since $f^{\prime \prime }(x)=12s_0x^2+6s_1x+2s_2$, it follows that $f^{\prime \prime }(x)>0$ according to $s_0>0$ and $\Delta :=36s_1^2-96s_0s_2<0$, which implies that $f^\prime (x)$ is strictly monotone increasing in the interval $[0,\infty )$. Since $f^\prime (0)=s_3>0$, we obtain $f^\prime (x)>0$ in the interval $[0,\infty )$, which implies that f(x) is strictly monotone increasing in $[0,\infty )$. Hence Eq. (12) has only one positive root, and this completes the proof of part (i).

Finally, if $s_2<0$ and $s_3<0$ besides $s_4<0$, then the total number of sign changes is equal to one, which implies that there is exactly one positive real root in Eq. (12). This proves (ii). Therefore the proof of the lemma is completed. $\square$