We study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t− (3/2) log t+x∞, the solution of the equation converges as t→ +∞ to a translate of the traveling wave corresponding to the minimal speed c∗ = 2...

The nonlocal Fisher equation is a diffusion-reaction equation where the reaction has a linear birth term and a nonlocal quadratic competition, which describes the reaction between distant individuals. This equation arises in evolutionary biological systems, where the arena for the dynamics is trait space, diffusion accounts for mutations and individuals with similar traits compete, resulting in...

We study the existence of heteroclinic solutions for semilinear secondorder difference equations related to the Fisher-Kolmogorov’s equation. The proof of the presented results are based on monotonicity and continuity arguments.

Approximate solutions of the Fisher equation obtained by different splitting methods are investigated. The error of this nonlinear problem is analyzed. The order of different splitting methods coupled with numerical methods of different order is calculated numerically and symbolically.

*Correspondence: [email protected] Department of Applied Mathematics, College of Science, China Agricultural University, Beijing, P.R. China Abstract In this paper, we consider positive solutions to the Fisher–KPP equation on complete Riemannian manifolds. We derive the gradient estimate. Using the estimate, we get the classic Harnack inequality which extends the recent result of Cao, Liu, Pendle...

We study the large time asymptotics of a solution of the Fisher-KPP reaction-diffusion equation, with an initial condition that is a compact perturbation of a step function. A well-known result of Bramson states that, in the reference frame moving as 2t− (3/2) log t+x∞, the solution of the equation converges as t→ +∞ to a translate of the traveling wave corresponding to the minimal speed c∗ = 2...

A generalized Fisher’s equation is solved by using the modified Adomian decomposition method MADM , variational iteration method VIM , homotopy analysis method HAM , and modified homotopy perturbation method MHPM . The approximation solution of this equation is calculated in the form of series whose components are computed easily. The existence, uniqueness, and convergence of the proposed metho...

In the study of pattern formation in bi–stable systems, the extended Fisher–Kolmogorov (EFK) equation plays an important role. In this paper, some a priori bounds are proved using Lyapunov functional. Further, existence, uniqueness and regularity results for the weak solutions are derived. Using C1-conforming finite element method, optimal error estimates are established for the semidiscrete ca...

We study the Fisher-KPP equation with a free boundary governed by a one-phase Stefan condition. Such a problem arises in the modeling of the propagation of a new or invasive species, with the free boundary representing the propagation front. In one space dimension and the radially symmetric case in higher space dimensions this problem was studied by some authors. In both cases a spreading-vanis...

Two analytical methods have been developed for constructing approximate solutions to a nonlocal generalization of the 1D Fisher– Kolmogorov–Petrovskii–Piskunov equation. This equation is of special interest in studying the pattern formation in microbiological populations. In the greater part of the paper, we consider in detail a semiclassical approximation method based on the WKB–Maslov theory ...