[Paper Review] Asymptotic solutions of the 1D nonlocal Fisher-KPP equation

А. В. Шаповалов, A. Yu. Trifonov|arXiv (Cornell University)|Aug 27, 2014

Mathematical and Theoretical Epidemiology and Ecology Models3 citations

TL;DR

This paper develops two analytical methods—semiclassical asymptotics via WKB-Maslov theory and large-time perturbative asymptotics—for approximating solutions to the 1D nonlocal Fisher-KPP equation, a model for pattern formation in microbial populations. The semiclassical approach yields a countable family of leading-order asymptotic solutions valid over finite time intervals, while the large-time method captures multimodal pattern formation via small perturbations on a quasi-stationary solution.

ABSTRACT

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 under the supposition of weak diffusion. The semiclassical asymptotics are sought in a class of trajectory concentrated functions. Such functions are localized in a neighborhood of a point moving in space. In terms of the semiclassical formalism developed, the original nonlinear equation is reduced to an associated linear partial differential equation and some algebraic equations for the coefficients of the linear equation with a given accuracy of the asymptotic parameter. The solutions of the nonlinear equation are constructed from the solutions of both the linear equation and the algebraic equations. A countable family of the leading terms of the semiclassical asymptotics is constructed in explicit form. The semiclassical asymptotics are valid by construction in a finite time interval which can be small in the sense that a pattern has no time to form in this interval. In the final part of the paper, we have constructed asymptotics which are different from the semiclassical ones and can describe the evolution of the solutions of the Fisher-Kolmogorov-Petrovskii-Piskunov equation at large times. These asymptotics represent small perturbations on the background of an exact quasi-stationary solution. In the example considered, an initial unimodal distribution becomes multimodal, which can be treated as pattern formation.

Motivation & Objective

• To develop analytical approximation methods for the nonlocal Fisher-KPP equation, which models pattern formation in microbial populations.
• To extend semiclassical asymptotic techniques, based on WKB-Maslov theory, to a nonlocal, nonlinear PDE with weak diffusion.
• To construct solutions valid over finite time intervals where patterns have not yet fully formed.
• To analyze long-time behavior by deriving asymptotics that describe the emergence of multimodal patterns from unimodal initial conditions.
• To provide explicit, analytical expressions for leading-order asymptotic solutions in both short- and long-time regimes.

Proposed method

• Employ the WKB-Maslov semiclassical formalism to reduce the nonlinear nonlocal Fisher-KPP equation to an associated linear PDE and algebraic equations for coefficients.
• Seek solutions in the class of trajectory-concentrated functions, localized near a moving point in space, to model localized population dynamics.
• Construct a countable family of leading-order asymptotic solutions using the semiclassical framework with a given asymptotic accuracy in the small parameter.
• For large times, derive perturbative asymptotics as small deviations from an exact quasi-stationary solution to capture long-term pattern evolution.
• Use the resulting asymptotic expansions to analyze the transition from unimodal to multimodal population distributions.
• Ensure the validity of the semiclassical asymptotics over a finite time interval, consistent with the absence of pattern formation during that period.

Experimental results

Research questions

• RQ1 How can semiclassical asymptotic methods be adapted to solve the nonlocal Fisher-KPP equation with weak diffusion?
• RQ2 What is the structure of the leading-order asymptotic solutions in the semiclassical regime, and how are they constructed from linear and algebraic equations?
• RQ3 Can the semiclassical approximation describe the evolution of solutions before pattern formation occurs?
• RQ4 How do the solutions evolve at large times, and what mechanisms lead to the emergence of multimodal distributions?
• RQ5 What role does the quasi-stationary solution play in the long-time asymptotic behavior of the nonlocal Fisher-KPP equation?

Key findings

• A countable family of leading-order semiclassical asymptotic solutions is constructed explicitly in terms of trajectory-concentrated functions.
• The semiclassical asymptotics are valid over a finite time interval, consistent with the absence of pattern formation during that period.
• The method reduces the original nonlinear nonlocal PDE to a linear PDE and algebraic equations for coefficients, enabling systematic approximation.
• At large times, the asymptotics describe the emergence of multimodal patterns through small perturbations on a quasi-stationary solution.
• The transition from an initial unimodal distribution to a multimodal one is analytically captured, indicating pattern formation in the long-time regime.
• The asymptotic solutions provide a framework for understanding both transient dynamics and long-term spatial organization in microbial populations.