The diffusive competition model with
a free boundary: Invasion of a superior or inferior competitor

Du, Yihong; Lin, Zhigui

doi:10.3934/dcdsb.2014.19.3105

Cited by 131 publications

(138 citation statements)

References 13 publications

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“…It is clear that (u, v; g, h) ∈ D is a fixed point of F if and only if it solves (8). By (11) and (12), we see that…”

mentioning

confidence: 92%

Dynamics of a nonlocal SIS epidemic model with free boundary

Cao¹,

Li²,

Yang³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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This paper is concerned with the spreading or vanishing of a epidemic disease which is characterized by a diffusion SIS model with nonlocal incidence rate and double free boundaries. We get the full information about the sufficient conditions that ensure the disease spreading or vanishing, which exhibits a detailed description of the communicable mechanism of the disease. Our results imply that the nonlocal interaction may enhance the spread of the disease.2010 Mathematics Subject Classification. 35K57, 35R20, 92D25. Key words and phrases. SIS model, reaction-diffusion, free boundary, spreading-vanishing dichotomy, nonlocal incidence rate. 247 248 JIA-FENG CAO, WAN-TONG LI AND FEI-YING YANG t −∞ +∞ −∞ K(x−y, t−s)I(y, t)dyds, and obtained the threshold dynamics for the spread of the disease (see [42] for the detailed description about the kernel function K). For more relevant work on the existence of traveling waves of reactiondiffusion equations with nonlocal interaction and time delay, we refer readers to Ducrot et al. [15], Faria et al. [16] and references cited therein. We also refer to Lou [31] for some challenging mathematical problems in evolution of dispersal and population dynamics.Recently, free boundary problems have been studied intensively in many fields. In particular, the well-known Stefan condition has been used to describe the spreading front in many applied problems. For example, it was used to describe the melting of ice in contact with water [35], the wound healing [6], the tumor growth [7] and so on. In order to get a more precise prediction of the location of the spreading front of an invading species, Du et al. [10] firstly studied the spreading-vanishing dichotomy of some invasion species which is described by a diffusive logistic model in the homogenous environment of one dimensional space. Since then, more results for more general free boundary problems have been obtained, for example, see

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“…It is clear that (u, v; g, h) ∈ D is a fixed point of F if and only if it solves (8). By (11) and (12), we see that…”

mentioning

confidence: 92%

Dynamics of a nonlocal SIS epidemic model with free boundary

Cao¹,

Li²,

Yang³

2017

Discrete &Amp; Continuous Dynamical Systems - B

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“…For the existence and uniqueness of local solution, in previous work, for example in the references [17,18,24], the embedding theorem…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…Very recently, Wang has overcome this loophole and given a strict proof in [31]. So the existence and uniqueness of local solution can be obtained from the proof of Theorem 1.1 in [31], and the local solution can be extended to by Theorem 2.4 in [17], we omit the details. Therefore, we will only show the inequalities (6) and (7).…”

Section: Existence and Uniquenessmentioning

confidence: 99%

“…For instance, it was used to describe the melting of ice in contact with water [12], the modeling of oxygen in the muscle [13], the wound healing [14], the tumor growth [15], and so on. As far as population models are concerned, Wang and Zhao [16] used such a condition for a predator-prey system with double free boundaries in one dimension, in which the prey lives in the whole space but the predator lives in a bounded area at the initial state; in [17,18], a Stefan condition was used for a competition system and a predatorprey system in radially symmetric setting, respectively, in which one species lives in the whole space but the other lives in a bounded area at the initial state. They established the spreading-vanishing dichotomy, long time behavior of the solution and sharp criteria for spreading and vanishing.…”

Section: Introductionmentioning

confidence: 99%

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A diffusive stage-structured model with a free boundary

2018

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In this paper we mainly consider a free boundary problem for a single-species model with stage structure in a radially symmetric setting. In our model, the individuals of a new or invasive species are classified as belonging either to the immature or to the mature cases. We firstly study the asymptotic behavior of the solution to the corresponding initial problem, then obtain a spreading-vanishing dichotomy and give sharp criteria governing spreading and vanishing for the free boundary problem.

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“…To the best of our knowledge, this is the first work concerning the free boundary problem of chemotaxis systems. Other works for various species competition systems with free boundaries can be found in [8,10,12,13,40,47,49]. The authors of [11,18,19,30] considered the transmission of viruses and diseases by using a free boundary.…”

Section: Introductionmentioning

confidence: 99%

A free boundary problem for an attraction–repulsion chemotaxis system

2018

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In this paper we study an attraction-repulsion chemotaxis system with a free boundary in one space dimension. First, under some conditions, we investigate existence, uniqueness and uniform estimates of the global solution. Next, we prove a spreading-vanishing dichotomy for this model. In the vanishing case, the species fail to establish and die out in the long run. In the spreading case, we provide some sufficient conditions to prove that the species successfully spread to infinity as t → ∞ and stabilize at a constant equilibrium state. The criteria for spreading and vanishing are also obtained.

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The diffusive competition model with a free boundary: Invasion of a superior or inferior competitor

Cited by 131 publications

References 13 publications

Dynamics of a nonlocal SIS epidemic model with free boundary

Dynamics of a nonlocal SIS epidemic model with free boundary

A diffusive stage-structured model with a free boundary

A free boundary problem for an attraction–repulsion chemotaxis system

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