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

[EN] 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.This work has been partially supported by the European Union in the FP7-PEOPLE-2012-ITN program under Grant Agreement Number 304617 (FP7 Marie Curie Action, Project Multi-ITN STRIKE-Novel Methods in Computational Finance) and the Ministerio de Economía y Competitividad Spanish grant MTM 2013-41765-P.Piqueras-García, MÁ.; Company Rossi, R.; Jódar Sánchez, LA. (2017). A front-fixing numerical method for a free boundary nonlinear diffusion logistic population model. Journal of Computational and Applied Mathematics. 309:473-481. https://doi.org/10.1016/j.cam.2016.02.029S47348130

A sharp-front moving boundary model for malignant invasion

We analyse a novel mathematical model of malignant invasion which takes the form of a two-phase moving boundary problem describing the invasion of a population of malignant cells into a population of background tissue, such as skin. Cells in both populations undergo diffusive migration and logistic proliferation. The interface between the two populations moves according to a two-phase Stefan condition. Unlike many reaction-diffusion models of malignant invasion, the moving boundary model explicitly describes the motion of the sharp front between the cancer and surrounding tissues without needing to introduce degenerate nonlinear diffusion. Numerical simulations suggest the model gives rise to very interesting travelling wave solutions that move with speed $c$, and the model supports both malignant invasion and malignant retreat, where the travelling wave can move in either the positive or negative $x$-directions. Unlike the well-studied Fisher-Kolmogorov and Porous-Fisher models where travelling waves move with a minimum wave speed $c \ge c^* > 0$, the moving boundary model leads to travelling wave solutions with $|c| < c^{**}$. We interpret these travelling wave solutions in the phase plane and show that they are associated with several features of the classical Fisher-Kolmogorov phase plane that are often disregarded as being nonphysical. We show, numerically, that the phase plane analysis compares well with long time solutions from the full partial differential equation model as well as providing accurate perturbation approximations for the shape of the travelling waves.Comment: 48 pages, 21 figure

Exact sharp-fronted travelling wave solutions of the Fisher-KPP equation

A family of travelling wave solutions to the Fisher-KPP equation with speeds $c=\pm 5/\sqrt{6}$ can be expressed exactly using Weierstrass elliptic functions. The well-known solution for $c=5/\sqrt{6}$, which decays to zero in the far-field, is exceptional in the sense that it can be written simply in terms of an exponential function. This solution has the property that the phase-plane trajectory is a heteroclinic orbit beginning at a saddle point and ends at the origin. For $c=-5/\sqrt{6}$, there is also a trajectory that begins at the saddle point, but this solution is normally disregarded as being unphysical as it blows up for finite $z$. We reinterpret this special trajectory as an exact sharp-fronted travelling solution to a \textit{Fisher-Stefan} type moving boundary problem, where the population is receding from, instead of advancing into, an empty space. By simulating the full moving boundary problem numerically, we demonstrate how time-dependent solutions evolve to this exact travelling solution for large time. The relevance of such receding travelling waves to mathematical models for cell migration and cell proliferation is also discussed

Survival, extinction, and interface stability in a two--phase moving boundary model of biological invasion

We consider a moving boundary mathematical model of biological invasion. The model describes the spatiotemporal evolution of two populations: each population undergoes linear diffusion and logistic growth, and the boundary between the two populations evolves according to a two--phase Stefan condition. This mathematical model describes situations where one population invades into regions occupied by the other population, such as the spreading of a malignant tumour into surrounding tissues. Full time--dependent numerical solutions are obtained using a level--set numerical method. We use these numerical solutions to explore several properties of the model including: (i) survival and extinction of one population initially surrounded by the other; and (ii) linear stability of the moving front boundary in the context of a travelling wave solution subjected to transverse perturbations. Overall, we show that many features of the well--studied one--phase single population analogue of this model can be very different in the more realistic two--phase setting. These results are important because realistic examples of biological invasion involve interactions between multiple populations and so great care should be taken when extrapolating predictions from a one--phase single population model to cases for which multiple populations are present. Open source Julia--based software is available on GitHub to replicate all results in this study.Comment: 31 pages. 9 figure

Invading and receding sharp-fronted travelling waves

Biological invasion, whereby populations of motile and proliferative individuals lead to moving fronts that invade into vacant regions, are routinely studied using partial differential equation (PDE) models based upon the classical Fisher--KPP model. While the Fisher--KPP model and extensions have been successfully used to model a range of invasive phenomena, including ecological and cellular invasion, an often--overlooked limitation of the Fisher--KPP model is that it cannot be used to model biological recession where the spatial extent of the population decreases with time. In this work we study the \textit{Fisher--Stefan} model, which is a generalisation of the Fisher--KPP model obtained by reformulating the Fisher--KPP model as a moving boundary problem. The nondimensional Fisher--Stefan model involves just one single parameter, $\kappa$, which relates the shape of the density front at the moving boundary to the speed of the associated travelling wave, $c$. Using numerical simulation, phase plane and perturbation analysis, we construct approximate solutions of the Fisher--Stefan model for both slowly invading and slowly receding travelling waves, as well as for rapidly receding travelling waves. These approximations allow us to determine the relationship between $c$ and $\kappa$ so that commonly--reported experimental estimates of $c$ can be used to provide estimates of the unknown parameter $\kappa$. Interestingly, when we reinterpret the Fisher--KPP model as a moving boundary problem, many disregarded features of the classical Fisher--KPP phase plane take on a new interpretation since travelling waves solutions with $c < 2$ are not normally considered. This means that our analysis of the Fisher--Stefan model has both practical value and an inherent mathematical value.Comment: 47 pages, 13 figure

Integration factor combined with level set method for reaction-diffusion systems with free boundary in high spatial dimensions

For reaction-diffusion equations in irregular domain with moving boundaries, the numerical stability constraints from the reaction and diffusion terms often require very restricted time step size, while complex geometries may lead to difficulties in accuracy when discretizing the high-order derivatives on grid points near the boundary. It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties. Applying an implicit scheme may be able to remove the stability constraints on the time step, however, it usually requires solving a large global system of nonlinear equations for each time step, and the computational cost could be significant. Integration factor (IF) or exponential differencing time (ETD) methods are one of the popular methods for temporal partial differential equations (PDEs) among many other methods. In our paper, we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries. In particular, we rewrite all ETD schemes into a linear combination of specific {\phi}-functions and apply one start-of-the-art algorithm to compute the matrix-vector multiplications, which offers significant computational advantages with adaptive Krylov subspaces. In addition, we extend this method by incorporating the level set method to solve the free boundary problem. The accuracy, stability, and efficiency of the developed method are demonstrated by numerical examples.Comment: 20 pages, 6 figures, 2 table