Stability, Steady‐State Bifurcations, and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

Abstract

In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.

Pattern formation of a predator-prey model with the cost of anti-predator behaviors

“…By increasing the value of k 0 to k 0 = 20 while holding other parameters in Figure 4 unchanged, we obtain a figure similar to Figure 3 (omitted). Further check by substituting parameters into (35) and (36) gives a contradiction, which confirms the non-existence of pattern formation. Therefore, we conclude that large cost of anti-predator response of prey in its reproduction has a stabilizing effect by excluding the appearance of pattern formation and ensures the stability of the positive spatial homogeneous steady state.…”

“…In addition to pure random movement of prey and predators, directed movement of predators has attracted much attention in recent years and has inspired numerous research about the so called prey-taxis problems (see [1,8,18,37,41,43,35] for example). A common feature of the models in the aforementioned papers lies in that the movement of predators is affected by the density gradient of prey, in addition to random movement.…”

Pattern formation of a diffusive predator‐prey model with herd behavior and nonlocal prey competition

“…On the other hand, the appearance of Turing patterns generated by the presence of the prey taxis is confirmed in Song and Tang. 16 Furthermore, the occurrence of Hopf bifurcation generated by the presence of the time delay and the spatial diffusion is shown in Tang and Song. 18 The influence of the quadratic mortality and the prey taxis are explored in Liu et al, 13 where the existence of T-H bifurcation is shown.…”

“…Further, in Tang et al, the impact of the cross‐diffusion is studied where it has been proved that the presence of this last can lead to the presence of the T‐H bifurcation where the spatiotemporal behavior near this point is established. On the other hand, the appearance of Turing patterns generated by the presence of the prey taxis is confirmed in Song and Tang . Furthermore, the occurrence of Hopf bifurcation generated by the presence of the time delay and the spatial diffusion is shown in Tang and Song .…”

“…4,5 This functional response has been followed by numerous other interaction functionals by including many other factors such as searching for the prey pack by predators, saturation of predator consumption, herd shape, so on; see, for instance, other works. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] The impact of self-(resp. cross-)diffusion on the dynamics of predator-prey models has been always scrutinized since the pioneering work of Turing.…”