Nonconstant Positive Steady States and Pattern Formation of 1D Prey-Taxis Systems

Wang, Qi; Song, Yang; Shao, Lingjie

doi:10.1007/s00332-016-9326-5

Cited by 86 publications

(58 citation statements)

References 43 publications

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“…Their results coincide the same verdict of stability with our analysis, the spatial temporal behaviors of predators and preys are generated by more repulsive prey taxis components. In the existing works, either the group defense assumption (see Wang et al and Wang and Zou), repulsive prey taxis (see Wang et al()) or attractive predator taxis in our models was imposed for the taxis coefficient to ensure the instability of constant equilibrium.…”

Section: Application To Two‐predator and One‐prey Modelsmentioning

confidence: 99%

“…(I 10 ) ∶P 0,0,0 = ru * 1 u * 2 v * + (r + b)v * (u * 2 1 + u * 2 2 ) + u * 2 1 u * 2 + u * 2 2 u * 1 + r 2 v * (u * 1 + u * 2 ) + brv * 2 (u * 1 + u * 2 ) > 0. Hence, we have that (see Wang et al 26 and Wang and Zou 36 ), repulsive prey taxis (see Wang et al 10,11 ) or attractive predator taxis in our models was imposed for the taxis coefficient to ensure the instability of constant equilibrium.…”

Section: Examplementioning

confidence: 99%

“…() However, predators also move toward the gradient direction of prey and the pursuit between predator and prey has a strong impact on the movement pattern of predator and prey. () The first prey‐taxis model was derived in Kareiva and Odell, and it studied predator aggregation in high prey density areas, considerable efforts in recent years devote to the one‐predator one‐prey system with prey‐taxis():

({, \begin{matrix} array \frac{\partial u}{\partial t} = Δ u - χ \nabla \cdot (q (u) \nabla v) + c ϕ (u, v) - g (u), & array x \in Ω, t > 0, \\ array \frac{\partial v}{\partial t} = d Δ v + f (v) - ϕ (u, v), & array x \in Ω, t > 0, \\ array \frac{\partial u (x, t)}{\partial ν} = \frac{\partial v (x, t)}{\partial ν} = 0, & array x \in \partial Ω, t > 0, \\ array u (x, 0) = u_{0} (x) \geq 0, v (x, 0) = v_{0} (x) \geq 0, & array x \in Ω, \end{matrix})

where the global existence of weak and classical solutions, the spatiotemporal dynamics have been extensively studied.…”

Section: Introductionmentioning

confidence: 99%

“…[1][2][3] However, predators also move toward the gradient direction of prey and the pursuit between predator and prey has a strong impact on the movement pattern of predator and prey. [4][5][6] The first prey-taxis model was derived in Kareiva and Odell, 4 and it studied predator aggregation in high prey density areas, considerable efforts in recent years devote to the one-predator one-prey system with prey-taxis [7][8][9][10][11][12] :…”

Section: Introductionmentioning

confidence: 99%

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

Guo

Wang

2019

Math Methods in App Sciences

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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.

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Section: Application To Two‐predator and One‐prey Modelsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

({, \begin{matrix} array \frac{\partial u}{\partial t} = Δ u - χ \nabla \cdot (q (u) \nabla v) + c ϕ (u, v) - g (u), & array x \in Ω, t > 0, \\ array \frac{\partial v}{\partial t} = d Δ v + f (v) - ϕ (u, v), & array x \in Ω, t > 0, \\ array \frac{\partial u (x, t)}{\partial ν} = \frac{\partial v (x, t)}{\partial ν} = 0, & array x \in \partial Ω, t > 0, \\ array u (x, 0) = u_{0} (x) \geq 0, v (x, 0) = v_{0} (x) \geq 0, & array x \in Ω, \end{matrix})

where the global existence of weak and classical solutions, the spatiotemporal dynamics have been extensively studied.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Guo

Wang

2019

Math Methods in App Sciences

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show abstract

“…It is worthwhile to mention that besides the competition model, the advection, which is referred to as the prey-taxis, has been studied in predator-prey models by various authors. See [7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%