Global existence of solutions and uniform persistence of a diffusive predator–prey model with prey-taxis

Wu, Sainan; Shi, Junping; Wu, Boying

doi:10.1016/j.jde.2015.12.024

Cited by 215 publications

(115 citation statements)

References 51 publications

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“…The third equation of can be rewritten as

\frac{\partial v}{\partial t} = d_{3} normalΔ v - v + φ false(u_{1}, u_{2}, v false),

where φ ( u 1 , u 2 , v ) = h ( v ) + v − f 1 ( u 1 , u 2 , v ) − f 2 ( u 1 , u 2 , v ). Then, from the variation of constants formula for , we have

v (\cdot, t) = e^{- t (A_{d_{3}} + 1)} v_{0} + \int_{0}^{t} e^{- (t - s) (A_{d_{3}} + 1)} φ (u_{1} (\cdot, t), u_{2} (\cdot, t), v (\cdot, t)) d s .

From lemma 2.3 in Wu et al, we have

\begin{matrix} right | | v (\cdot, t) | {false|}_{1, \infty} & left \leq C | | {(A_{d_{3}} + 1)}^{θ} v (\cdot, t) | {false|}_{q} & right \\ right & left \leq C \int_{0}^{t} {(t - s)}^{- θ} e^{- γ (t - s)} {‖φ (u_{1} (\cdot, t), u_{2} (\cdot, t), v (\cdot, t))‖}_{q} d s + C t^{- θ} e^{- γ t} \end{matrix}

…”

Section: Existence and Boundednessmentioning

confidence: 92%

“…One is to construct Lyapunov functional and the other is to use Gagliardo‐Nirenberg type estimate through Moser iteration. Here, we follow the Moser Alikakos L p iteration technique, which is also developed in Wu et al and Wang et al to prove the global existence and boundedness of classical solutions. In , the prey growth rate h ( v ), the predator mortality rate g i ( u i )( i = 1,2), and the sensitivity function q ( u i )( i = 1,2) are similar to the ones in Wu et al The interaction between two predators can be cooperation, competition and consumer‐resource type, and examples are shown as in Saleem et al and Pang and Wang :

f_{1} false(u_{1}, u_{2}, v false) = u_{1} \frac{u_{2} v}{u_{1} + u_{2}}, f_{2} false(u_{1}, u_{2}, v false) = u_{2} \frac{β u_{1} v}{u_{1} + u_{2}}, β > 0 .

…”

Section: Existence and Boundednessmentioning

confidence: 99%

“…From , and , we obtain that

\begin{matrix} right & left \frac{{(d_{3} + d_{1})}^{2}}{k - 1} \int_{normalΩ} u_{1}^{k} \frac{{φ^{'}}^{2} (v)}{φ (v)} | \nabla v {false|}^{2} + χ_{1}^{2} (k - 1) \int_{normalΩ} u_{1}^{k} φ (v) | \nabla v {false|}^{2} + χ_{1} \int_{normalΩ} u_{1}^{k} φ^{'} (v) | \nabla v {false|}^{2} & right \\ right & left \leq \frac{d_{3}}{k} \int_{normalΩ} u_{1}^{k} φ^{''} (v) | \nabla v {false|}^{2} . \end{matrix}

Substituting into , we get

\frac{1}{k} \frac{d}{d t} \int_{Ω} u_{1}^{k} φ false(v false) + \frac{false(k - 1 false) false(2 d_{1} - 1 false)}{2} \int_{Ω} u_{1}^{k - 2} φ false(v false) false| \nabla u_{1} |^{2} \leq ((), R_{1} + \frac{2 ξ^{2} N_{0}^{2} D}{k}) \int_{Ω} u_{1}^{k} φ false(v false) .

From Poincaré's inequality of lemma 2.4 in Wu et al and Gagliardo‐Nirenberg interpolation inequality of lemma 2.5 in Wu et al, we find

\begin{matrix} right & left \int_{normalΩ} u_{1}^{k} φ (v<...> \end{matrix}

…”

Section: Existence and Boundednessmentioning

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%

See 3 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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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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“…The third equation of can be rewritten as

\frac{\partial v}{\partial t} = d_{3} normalΔ v - v + φ false(u_{1}, u_{2}, v false),

where φ ( u 1 , u 2 , v ) = h ( v ) + v − f 1 ( u 1 , u 2 , v ) − f 2 ( u 1 , u 2 , v ). Then, from the variation of constants formula for , we have

v (\cdot, t) = e^{- t (A_{d_{3}} + 1)} v_{0} + \int_{0}^{t} e^{- (t - s) (A_{d_{3}} + 1)} φ (u_{1} (\cdot, t), u_{2} (\cdot, t), v (\cdot, t)) d s .

From lemma 2.3 in Wu et al, we have

\begin{matrix} right | | v (\cdot, t) | {false|}_{1, \infty} & left \leq C | | {(A_{d_{3}} + 1)}^{θ} v (\cdot, t) | {false|}_{q} & right \\ right & left \leq C \int_{0}^{t} {(t - s)}^{- θ} e^{- γ (t - s)} {‖φ (u_{1} (\cdot, t), u_{2} (\cdot, t), v (\cdot, t))‖}_{q} d s + C t^{- θ} e^{- γ t} \end{matrix}

…”

Section: Existence and Boundednessmentioning

confidence: 92%

f_{1} false(u_{1}, u_{2}, v false) = u_{1} \frac{u_{2} v}{u_{1} + u_{2}}, f_{2} false(u_{1}, u_{2}, v false) = u_{2} \frac{β u_{1} v}{u_{1} + u_{2}}, β > 0 .

…”

Section: Existence and Boundednessmentioning

confidence: 99%

“…From , and , we obtain that

\begin{matrix} right & left \frac{{(d_{3} + d_{1})}^{2}}{k - 1} \int_{normalΩ} u_{1}^{k} \frac{{φ^{'}}^{2} (v)}{φ (v)} | \nabla v {false|}^{2} + χ_{1}^{2} (k - 1) \int_{normalΩ} u_{1}^{k} φ (v) | \nabla v {false|}^{2} + χ_{1} \int_{normalΩ} u_{1}^{k} φ^{'} (v) | \nabla v {false|}^{2} & right \\ right & left \leq \frac{d_{3}}{k} \int_{normalΩ} u_{1}^{k} φ^{''} (v) | \nabla v {false|}^{2} . \end{matrix}

Substituting into , we get

\frac{1}{k} \frac{d}{d t} \int_{Ω} u_{1}^{k} φ false(v false) + \frac{false(k - 1 false) false(2 d_{1} - 1 false)}{2} \int_{Ω} u_{1}^{k - 2} φ false(v false) false| \nabla u_{1} |^{2} \leq ((), R_{1} + \frac{2 ξ^{2} N_{0}^{2} D}{k}) \int_{Ω} u_{1}^{k} φ false(v false) .

From Poincaré's inequality of lemma 2.4 in Wu et al and Gagliardo‐Nirenberg interpolation inequality of lemma 2.5 in Wu et al, we find

\begin{matrix} right & left \int_{normalΩ} u_{1}^{k} φ (v<...> \end{matrix}

…”

Section: Existence and Boundednessmentioning

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%

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

View full text Add to dashboard Cite

show abstract

“…That is modeled by a prey-taxis term χ i ∇(S(v i )∇u), i = 1, 2, respectively, where the χ i are the prey-taxis coefficients, and the movement is decided also by the predator's density, which is indicated by the function S(v i ). As pointed in [12], the sensitivity function S(u) satisfies the general hypotheses:…”

Section: Introductionmentioning

confidence: 99%