A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model

Abstract

The spatial-temporal spreading of a new invasive species in a habitat has interest in ecology and is modeled by a moving boundary diffusion logistic partial differential problem, where the moving boundary represents the unknown expanding front of the species. In this paper a front-fixing approach is applied in order to transform the original moving boundary problem into a fixed boundary one.A finite difference method preserving qualitative properties of the theoretical solution is proposed. Results are illustrated with numerical experiments.

Propagation, diffusion and free boundaries

“…To overcome this shortcoming of (1.3), Du and Lin [29] introduced a free boundary version of the Fisher equation (1.1), where the equation for u(t, x) is satisfied over a changing interval (g(t), h(t)), representing the population range at time t, together with the boundary condition uðt; xÞ ¼ 0 for x 2 fgðtÞ; hðtÞg, and free boundary condition They showed that this modified model always has a unique solution and as time goes to infinity, the population u(t, x) exhibits a ''spreading-vanishing dichotomy'', namely it either vanishes or converges to 1; moreover, in the latter case, a finite spreading speed can be determined. This work has motivated considerable research, and the ''spreading-vanishing dichotomy'' discovered in [29] has been shown to occur in a variety of similar models; see, for example, extensions to equations with a more general nonlinear term f(u) [31,59,60] etc., extensions to equations with advection [52,58,83] etc., extensions to systems of population or epidemic models [1,30,40,53,68,80,82] etc., and development of numerical methods for treating some of these free boundary problems [69,70,72] etc.…”

Development and Design of the Application of Logistic Growth Curve for Mobile Terminal

“…To explore the influence of the change of a parameter on the number of population, we design an experiment to explore the three factors, that is, the initial population size, the population growth rate and the environmental maximum capacity [4]. We will take paramecium as the experimental biology sample.…”

Numerical study of vanishing and spreading dynamics of chemotaxis systems with logistic source and a free boundary

“…In general, it is always difficult to handle the attraction term in chemotaxis system which may lead to convection dominant in the system [4], [20], [21], [28]. However, in the system (1), we have extra numerical challenges in efficiently and accurately handling the moving boundaries [26]. These two challenges require us to construct a new numerical algorithm in the numerical study.…”