Global existence and blowup of solutions to a free boundary problem for mutualistic model

Kim, KwangIk; Lin, Zhigui; Zhi, Liu

doi:10.1007/s11425-010-4007-6

Cited by 13 publications

(6 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The right spreading frontier is represented by the free boundary x = h(t). Assume that h(t) grows at a rate that is proportional to the population gradient at the frontier [7,11,17,18,28]. Then the conditions on the right frontier (free boundary) is…”

Section: Model Formulationmentioning

confidence: 99%

The spreading frontiers of avian-human influenza described by the free boundary

2013

Self Cite

View full text Add to dashboard Cite

In this paper, a reaction-diffusion system is proposed to investigate avian-human influenza. Two free boundaries are introduced to describe the spreading frontiers of the avian influenza. The basic reproduction numbers r F 0 (t) and R F 0 (t) are defined for the bird with the avian influenza and for the human with the mutant avian influenza of the free boundary problem, respectively. Properties of these two time-dependent basic reproduction numbers are obtained. Sufficient conditions both for spreading and for vanishing of the avian influenza are given. It is shown that if r F 0 (0) < 1 and the initial number of the infected birds is small, the avian influenza vanishes in the bird world. Furthermore, if r F 0 (0) < 1 and R F 0 (0) < 1, the avian influenza vanishes in the bird and human worlds. In the case that r F 0 (0) < 1 and R F 0 (0) > 1, spreading of the mutant avian influenza in the human world is possible. It is also shown that if r F 0 (t 0 ) 1 for any t 0 0, the avian influenza spreads in the bird world.

show abstract

Section: Model Formulationmentioning

confidence: 99%

The spreading frontiers of avian-human influenza described by the free boundary

2013

Self Cite

View full text Add to dashboard Cite

show abstract

“…Recently the free boundary has been introduced in many areas, especially the well-known Stefan condition has been used to describe the spreading process. For example, it was used in describing the melting of ice in contact with water [24], in the modeling of oxygen in the muscle [6], and in the dynamics of population [14,18,21,23]. There is a vast literature on the Stefan problem, and some recent and theoretically advanced results can be found in [4].…”

Section: Introductionmentioning

confidence: 99%

An SIR epidemic model with free boundary

Kim

Lin

Zhang

2013

Nonlinear Analysis: Real World Applications

Self Cite

View full text Add to dashboard Cite

An SIR epidemic model with free boundary is investigated.This model describes the transmission of diseases. The behavior of positive solutions to a reaction-diffusion system in a radially symmetric domain is investigated. The existence and uniqueness of the global solution are given by the contraction mapping theorem. Sufficient conditions for the disease vanishing or spreading are given. Our result shows that the disease will not spread to the whole area if the basic reproduction number R 0 < 1 or the initial infected radius h 0 is sufficiently small even that R 0 > 1. Moreover, we prove that the disease will spread to the whole area if R 0 > 1 and the initial infected radius h 0 is suitably large. IntroductionRecently epidemic model has been received a great attention in mathematical ecology. To describe the development of an infectious disease, compartmental models have been given to separate a population into various classes based on the stages of infection [2]. The classical SIR model is described by partitioning the population into susceptible, infectious and recovered individuals, denoted by S, I and R, respectively. Assume that the disease incubation period is negligible so that each susceptible individual becomes infectious and later recovers with a permanently or temporarily acquired immunity, then the SIR model is governed by the following system of differential equations:( 1.1) where the total population size has been normalized to one and the influx of the susceptible comes from a constant recruitment rate b. The death rate for the S, I and R class is, respectively, given by µ 1 , µ 2 and µ 3 . Biologically, it is natural to assume that µ 1 < min{µ 2 , µ 3 }. The standard incidence of disease is denoted by βSI, where β is the constant effective contact rate, which is the average number of contacts of the infectious per unit time. The recovery rate of the infectious is

show abstract

“…The last inequality of (9) follows from (26). In addition, (24) implies w 1 (0, x) ≥ u 0 (x) and w 2 (0, x) ≥ v 0 (x) for x ∈ [0, h 0 ].…”

Section: Sincementioning

confidence: 95%

“…Complete descriptions of spreading-vanishing dichotomy were provided for both weak and weak-strong competition case. Some predatorprey models and mutualistic models also use such free boundary conditions, such as [26,29,30,36,39,41,44,45,49]. More results can be found in the references cited therein.…”

mentioning

confidence: 99%

A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment

Duan¹,

Zhang²

2022

DCDS-B

View full text Add to dashboard Cite

In this paper, we investigate a reaction-diffusion-advection twospecies competition system with a free boundary in heterogeneous environment. The primary aim is to study the impact of small advection terms and heterogeneous environment, which is on two species' dynamics via a free boundary. The function m(x) represents heterogeneous environment, and it can satisfy positive everywhere condition or changeable sign condition. Firstly, on one hand, we provide long time behaviors of the solution in vanishing case when m(x) satisfies both conditions above; on the other hand, long time behaviors of the solution in spreading case are got when m(x) satisfies positive everywhere condition. Secondly, a spreading-vanishing dichotomy and several sufficient conditions through the initial data and the moving parameters are obtained to determine whether spreading or vanishing of two species happens when m(x) satisfies both conditions above. Furthermore, we derive estimates of spreading speed of the free boundary when m(x) satisfies positive everywhere condition and two species spreading occurs.

show abstract

Global existence and blowup of solutions to a free boundary problem for mutualistic model

Cited by 13 publications

References 17 publications

The spreading frontiers of avian-human influenza described by the free boundary

The spreading frontiers of avian-human influenza described by the free boundary

An SIR epidemic model with free boundary

A reaction-diffusion-advection two-species competition system with a free boundary in heterogeneous environment

Contact Info

Product

Resources

About