Bifurcation analysis on a diffusive Holling–Tanner predator–prey model

Bifurcations and spatiotemporal patterns in a ratio‐dependent diffusive Holling‐Tanner system with time delay

“…The parameter r is the intrinsic growth ratio of the predator and prey, b stands for the saturation value of the prey, and a is the maximum value of prey consumed by per predator per unit time. This model has been widely investigated for pattern formation, such as the Hopf bifurcations, steady‐state bifurcations of simple and double eigenvalues, Turing patterns, and Turing‐Hopf patterns in previous studies . In particular, Chang et al unveiled six types of patterns exist in a corresponding delay system by the numerical method and the eigenvalue analysis of the Turing and Hopf bifurcation.…”

“…This model has been widely investigated for pattern formation, such as the Hopf bifurcations, steady-state bifurcations of simple and double eigenvalues, Turing patterns, and Turing-Hopf patterns in previous studies. [22][23][24][25] In particular, Chang et al 26 unveiled six types of patterns exist in a corresponding delay system by the numerical method and the eigenvalue analysis of the Turing and Hopf bifurcation. However, the theoretical results are scarce in both time-delay system and Turing-Hopf bifurcation.…”

Diffusion as a strategy for survival in an invasion predator–prey model

“…Although in several cases, diffusion has not given proper answers to questions about species propagation (Higgins & Richardson, 1996), it is still useful to understand the richness of the interaction between populations. For example, reaction-diffusion systems have made possible an understanding of the onset of spatial oscillations in predator-prey models (Ma & Li, 2013), the dynamics of propagating waves (Du, Duan, & Liao, 2014), and formation of spatial patterns in the medium (Hu, Li, & Wang, 2015;Murray, 2007;Sun, Jin, Liu, & Li, 2008).…”

Bifurcations in a diffusive predator–prey model with predator saturation and competition response

“…It has been observed that the spatially homogeneous and non‐homogeneous periodic solutions bifurcating from Hopf bifurcation and non‐constant steady state bifurcation in reaction–diffusion system are important spatiotemporal pattern mechanisms; hence, they have been investigated by many researchers . Recently, Ma and Li observed that the spatially homogeneous and non‐homogeneous Hopf bifurcations occur at the positive steady state of diffusive Holling–Tanner predator–prey model. Du and Wang studied the existence of multiple non‐homogeneous periodic solutions of 1 − D Lengyel–Epstein reaction diffusion model.…”

“…they have been investigated by many researchers [1,[4][5][6][7][8][9][10][11]. Recently, Ma and Li [7] observed that the spatially homogeneous and non-homogeneous Hopf bifurcations occur at the positive steady state of diffusive Holling-Tanner predator-prey model. Du and Wang [4] studied the existence of multiple non-homogeneous periodic solutions of 1 D Lengyel-Epstein reaction diffusion model.…”