Two kinds of free boundary problems for the diffusive prey–predator model

Wang, Mingxin; Zhang, Yang

doi:10.1016/j.nonrwa.2015.01.004

Cited by 60 publications

(42 citation statements)

References 24 publications

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“…The main objective of this paper is to study the spreading and vanishing phenomena and long time behavior of solution to (1.2). This work can be regarded as a continuation of our previous papers [23,25,26,28,29] in order to have a better understanding of the dynamics for the above two realistic phenomena (i) and (ii). To compare the present work with our previous papers, here we briefly state the main contents of [23,25,26,28,29].…”

Section: Introductionmentioning

confidence: 83%

“…This work can be regarded as a continuation of our previous papers [23,25,26,28,29] in order to have a better understanding of the dynamics for the above two realistic phenomena (i) and (ii). To compare the present work with our previous papers, here we briefly state the main contents of [23,25,26,28,29]. In [28], the author and Zhao investigated a free boundary problem of (1.1) with double free boundaries in which the prey lives in the whole space but the predator lives in a region enclosed by free boundaries, and in [29] we generalized the results of [28] to higher dimension and heterogeneous environment case.…”

Section: Introductionmentioning

confidence: 83%

“…In [28], the author and Zhao investigated a free boundary problem of (1.1) with double free boundaries in which the prey lives in the whole space but the predator lives in a region enclosed by free boundaries, and in [29] we generalized the results of [28] to higher dimension and heterogeneous environment case. For the case that b 3 ¼ 0, and either a 1 ¼ a 2 ¼ 0 or b 1 ¼ b 2 ¼ 0, the problem (1.2) has been studied by the author in [25], and in [23] the free boundary condition h 0 ðtÞ ¼ Àlu x at x ¼ hðtÞ was replaced by h 0 ðtÞ ¼ Àlðu x þ qv x Þ with q > 0. In [26], the author discussed a similar problem with variable intrinsic growth rate for predator and with boundary conditions…”

Section: Introductionmentioning

confidence: 99%

“…In general, the Robin boundary conditions utilized here are more complicated than the Dirichlet and Neumann boundary conditions. The problem (1.2) will exhibit different phenomena from those of [23,25,26,28,29] in the following three aspects:…”

Section: Introductionmentioning

confidence: 99%

“…The one-phase Stefan free boundary condition has also been used in the prey-predator and competition models to describe spreading of two species [8,11,17,23,[25][26][27][28]. A two-phase Stefan condition was also used for the reaction diffusion systems over a bounded domain, where the free boundary separates two species from each other in the domain [12,13,18].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Spreading and vanishing in the diffusive prey–predator model with a free boundary

Wang

2015

Communications in Nonlinear Science and Numerical Simulation

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Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 83%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Spreading and vanishing in the diffusive prey–predator model with a free boundary

Wang

2015

Communications in Nonlinear Science and Numerical Simulation

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The diffusive Beddington‐DeAngelis predator‐prey model with nonlinear prey‐taxis and free boundary

Wang

2018

Math Methods in App Sciences

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The diffusive Beddington-DeAngelis predator-prey model with nonlinear prey-taxis and free boundary is considered. We investigate the existence, uniqueness, regularity, uniform estimates, and long-time behavior of the global solution. Some sufficient conditions for both spreading and vanishing are also established. To our best knowledge, this is the first work concerning the dynamics of the cross-diffusion systems with free boundaries. KEYWORDSdiffusive predator-prey model, free boundary, long-time behavior, nonlinear prey-taxis, spreading and vanishing Math Meth Appl Sci. 2018;41:6741-6762.wileyonlinelibrary.com/journal/mma

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A semilinear parabolic system with a free boundary

Wang

Zhao

2015

Z. Angew. Math. Phys.

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This paper deals with a semilinear parabolic system with free boundary in one space dimension. We suppose that unknown functions $u$ and $v$ undergo nonlinear reactions $u^q$ and $v^p$, and exist initially in a interval $\{0\leq x\leq s(0)\}$, but expand to the right with spreading front $\{x=s(t)\}$, with $s(t)$ evolving according to the free boundary condition $s'(t)=-\mu (u_x+\rho v_x)$, where $p,\, q,\, \mu, \,\rho$ are given positive constants. The main purpose of this paper is to understand the existence, uniqueness, regularity and long time behavior of positive solution or maximal positive solution. Firstly, we prove that this problem has a unique positive solution $(u,v,s)$ defined in the maximal existence interval $[0,T_{\max})$ when $p,\,q\geq 1$, while it has a unique maximal positive solution $(u,v,s)$ defined in the maximal existence interval $[0,T_{\max})$ when $p<1$ or $q<1$. Moreover, $(u,v,s)$ and $T_{\max}$ have property that either (i) $T_{\max}=+\infty$, or (ii) $T_{\max}<+\infty$ and $$ \limsup_{T\nearrow T_{\max}}\|u,\,v\|_{L^{\infty}([0,T]\times[0,s(t)])}=+\infty.$$ Then we study the regularity of $(u,v)$ and $s$. At last, we discuss the global existence ($T_{\max}=+\infty$), finite time blow-up ($T_{\max}<+\infty$), and long time behavior of bounded global solution.Comment: 26 page

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Two kinds of free boundary problems for the diffusive prey–predator model

Cited by 60 publications

References 24 publications

Spreading and vanishing in the diffusive prey–predator model with a free boundary

Spreading and vanishing in the diffusive prey–predator model with a free boundary

The diffusive Beddington‐DeAngelis predator‐prey model with nonlinear prey‐taxis and free boundary

A semilinear parabolic system with a free boundary

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