Asymptotic profile of a mutualistic model on a periodically evolving domain

Abstract

In order to investigate the impact of periodically evolving domain on the mutualism interaction of two species, we study a mutualistic model on a periodically evolving domain. To overcome the difficulty caused by the advection and dilution terms, we transform the model to a reaction–diffusion problem in a fixed domain. By means of eigenvalue problems, the threshold parameters are introduced. The asymptotic profiles of the solutions on an evolving domain are studied by using the threshold parameters and the upper and lower solutions method. The impact of the domain evolution rate on the persistence or extinction of species is analyzed. Numerical simulations are performed to illustrate our analytical results.

The Diffusive Model for West Nile Virus on a Periodically Evolving Domain

“…In [31], Jiang and Wang studied the impact of periodic evolution on the singlespecies diffusion logistic model. Asymptotic profile of a mutualistic model on a periodically evolving domain has been investigated by Adam et al in [33]. e diffusive model for Aedes aegypti mosquito on a periodically evolving domain has been considered by Zhang and Lin in [32].…”

“…eir results indicated that the asymptotic spreading speed of the WNv model with free boundary is strictly less than that of the corresponding model in Lewis et al [4]. For the domain changing with known boundary, there are also some papers, for instance, [27][28][29][30] for a growing domain and [31][32][33][34] for a periodically evolving domain. In [31], the authors introduced the periodically evolving domain into a single-species diffusion logistic model and studied the influence of periodic evolution on the survival and extinction of species.…”

“…eir results indicated that the periodic domain evolution has a significant impact on the dispersal of Aedes aegypti mosquito. To investigate the impact of periodically evolving domain on the mutualism interaction of two species, Adam et al [33] studied a mutualistic model on the periodic evolving domain. ey suggested that the periodic evolution of domain places significant influence on the interaction of two species.…”

“…In this paper, we will consider the impact of the periodic evolving domain on the dynamics of a diffusive WNv model corresponding to system (1). We followed the methods of Adam et al [33], Zhang and Lin [32], and Zhu et al [34]. e rest of this paper is organized as follows.…”

Spatial diffusion and periodic evolving of domain in an SIS epidemic model

“…Owing to the turnover of the four seasons, the evolving of domain is usually periodic. In recent years, some scholars have studied mathematical models on a periodically evolving domain, for example, an SIS epidemic model [28], a mutualistic model [1], an Aedes aegypti mosquito model [40], a dengue fever model [41], and so on. In this paper, we consider the following problem        S t − d S ∆S = a(x)S − b(x)S 2 − β(x)f (S, I)I + γ(x)I, x ∈ Ω, t > 0, I t − d I ∆I = β(x)f (S, I)I − γ(x)I, x ∈ Ω, t > 0, ∂S ∂ν = ∂I ∂ν = 0, x ∈ ∂Ω, t > 0, S(x, 0) = S 0 (x), I(x, 0) = I 0 (x),…”