A Lotka-Volterra competition model with nonlocal diffusion and free boundaries

Cao, Jia-Feng; Li, Wan–Tong; Wang, Jie

doi:10.48550/arxiv.1905.09584

Cited by 2 publications

(4 citation statements)

References 41 publications

(44 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this section, we always assume that (J) and (H) hold. The following proposition directly holds from Theorem 3.5 in [6].…”

Section: Spreading-vanishingmentioning

confidence: 88%

“…It is easy to find that the above articles are devoted to the study of SIS models on a fixed domain. In real life, the movement of species leads to changes in biological habitats, and in mathematics, the free boundary can be used to describe this phenomenon, such as the healing of wounds [10] and the expansion of new species or invasive species [6,14,27,38]. Free boundary problems can also be used to describe the transmission of disease, such as the SIRS model [7], SIS model [22], SIR model [23,47] and references therein.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, nonlocal diffusion equations have attracted extensive attention and have been used to characterize long-range dispersal in population ecology [6,26]. In addition, scholars have extensively investigated infectious disease models with nonlocal diffusion, such as the West Nile virus model [15], SIS epidemic model [17,44], and SIR reaction-diffusion model [18,45].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Threshold dynamics of a nonlocal dispersal SIS epidemic model with free boundaries

Tong¹,

Ahn²,

Zhang³

2023

Preprint

View full text Add to dashboard Cite

To study the influence of the moving front of the infected interval and the spatial movement of individuals on the spreading or vanishing of infectious disease, we consider a nonlocal SIS (susceptible-infected-susceptible) reaction-diffusion model with media coverage, hospital bed numbers and free boundaries. The principal eigenvalue of the integral operator is defined, and the impacts of the diffusion rate of infected individuals and interval length on the principal eigenvalue are analyzed. Furthermore, sufficient conditions for spreading and vanishing of the disease are derived. Our results show that large media coverage and hospital bed numbers are beneficial to the prevention and control of disease. The difference between the model with nonlocal diffusion and that with local diffusion is also discussed and nonlocal diffusion leads to more possibilities.

show abstract

“…In this section, we always assume that (J) and (H) hold. The following proposition directly holds from Theorem 3.5 in [6].…”

Section: Spreading-vanishingmentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Threshold dynamics of a nonlocal dispersal SIS epidemic model with free boundaries

Tong¹,

Ahn²,

Zhang³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…They also obtained well-posedness of solutions and spreading-vanishing results. Moreover, Cao et al [4] recently considered a nonlocal diffusion Lotka-Volterra type competition model with free boundaries in the homogeneous environment, which consists of a native species distributing in the whole space R and an invasive species.…”

Section: Introductionmentioning

confidence: 99%

A reaction–diffusion–advection competition model with two free boundaries in heterogeneous time-periodic environment

Chen

Wang³

2017

IMA J Appl Math

View full text Add to dashboard Cite

This paper is concerned with a Lotka-Volterra type competition model with free boundaries in time-periodic environment. One species is assumed to adopt nonlocal dispersal and the other one adopt mixed dispersal, which is a combination of both random dispersal and nonlocal dispersal. We show that this free boundary problem with more general growth functions admits a unique solution defined for all time. A spreading-vanishing dichotomy is obtained and criteria for spreading and vanishing are provided.which has been studied in [3]. Problem (1.3) is a nature extension of the local diffusion model with free boundary in [10], and similar results including the existence and uniqueness of global solutions for more general growth function f (t, x, u) and the spreading-vanishing results in the homogeneous environment were obtained in [3], from which one can see that the nonlocal diffusion brings many essential difficulties in analysis.Since the work of Du and Lin [10], the local diffusion models with free boundary(ies) have been studied extensively. For example, the model in [10] has been extended to other situations of single species model such as in higher dimensional space, heterogeneous environment, timeperiodic environment, or with other boundary conditions, general nonlinear term, advection term, we refer the readers to [2, 6, 9, 11, 13-15, 19, 23, 25, 27, 28, 33, 39, 41] and references therein. Moreover, two-species Lotka-Volterra type competition problems and predator-prey problems with free boundary(ies) have also been considered in the homogeneous environment or heterogeneous time-periodic environment, e.g., [7,12,16,20, 21,30,32,[35][36][37][38]40]. The epidemic models with free boundary(ies) have also been considered in [5,18,26] The aim of this paper is to study the well-posedness and long-time behaviors of solutions to problem (1.1). We first investigate the existence and uniqueness of solutions to (1.1) with more general growth functions. To achieve it, we shall establish the maximum principle for linear parabolic equations with mixed dispersal, and prove that the nonlinear parabolic equations with mixed dispersal (see (2.5)) admit a unique positive solution under the assumption that g ′ (t), h ′ (t) and u(t, x) are only continuous functions by approximation method, which plays an important role in the process of using the fixed point theorem. Then we establish a spreading-vanishing dichotomy and criteria for spreading and vanishing. To discuss the spreading and vanishing, we need to consider the existence and properties of principle eigenvalue of time-periodic parabolic-type eigenvalue problems with random/mixed dispersal. Since the intrinsic growth rates a(t) and c(t)

show abstract

A Lotka-Volterra competition model with nonlocal diffusion and free boundaries

Cited by 2 publications

References 41 publications

Threshold dynamics of a nonlocal dispersal SIS epidemic model with free boundaries

Threshold dynamics of a nonlocal dispersal SIS epidemic model with free boundaries

A reaction–diffusion–advection competition model with two free boundaries in heterogeneous time-periodic environment

Contact Info

Product

Resources

About