Dynamics and pattern formations in diffusive predator‐prey models with two prey‐taxis

Abstract

A reaction‐diffusion two‐predator‐one‐prey system with prey‐taxis describes the spatial interaction and random movement of predator and prey species, as well as the spatial movement of predators pursuing preys. The global existence and boundedness of solutions of the system in bounded domains of arbitrary spatial dimension and any small prey‐taxis sensitivity coefficient are investigated by the semigroup theory. The spatial pattern formation induced by the prey‐taxis is characterized by the Turing type linear instability of homogeneous state; it is shown that prey‐taxis can both compress and prompt the spatial patterns produced through diffusion‐induced instability in two‐predator‐one‐prey systems.

Existence of local and global solution for a spatio-temporal predator-prey model

“…Iterative sequence: Let (u 0 , v 0 ) ∈ 0 L 2 0 0, T 2 ; H 2 (Ω) × L 2 0, T ; H 2 (Ω) be a weak solution of (7) and (8), with u t ,v t ∈ L 0, T ; L 2 (Ω)…”

“…The technique of upper and lower solutions is used in [7], to show the existence and uniqueness of a classical global time-dependent solution and its asymptotic relation with the steady-state solutions. Another line of research is presented in [8], where the global existence and boundedness of classical solutions is studied for a predator-prey model via semigroup theory. The existence and uniqueness of weak solutions of the system (1) in Ω ⊂ n with homogeneous Dirichlet boundary conditions have been studied in [9] using the semi-implicit Rothe method.…”

Steady states of a diffusive predator-prey model with prey-taxis and fear effect

“…When |h| is small, (u i (h, x), v i (h, x)) is the approximate solution near (u, v, χ i ) of model ( 13), see Fig. The main part of (ui(h, x), vi(h, x)) of model (13) in the interval of (0, 1), the first line is the image of prey corresponding to i = 1, 2, 3 with h = ±0.02, ±0.01 and h = 0, and the second line is predator.…”

“…Moreover, the prey can also adjust the corresponding position to reduce the risk of being caught, which is called predator-taxis [12]. And for the systems with three species [13,14], the authors investigated the global existence of the system solutions and the local stability conditions of the equilibria.…”

Dynamics and pattern formation in a cross-diffusion model with stage structure for predators

“…It can find that prey-taxis has extremely important significance on the dynamic behavior: attractive prey-taxis has stabilization effect, while repulsive prey-taxis can generate pattern formation, see, for example [24,21]. For three-species predator-prey systems with prey-taxis, the dynamics becomes more complicated, which strongly depends on the interaction between predators and prey [8,23,22].…”