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Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

A. E. Kulagin, A. V. Shapovalov

Abstract

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.

Quasiparticles for the one-dimensional nonlocal Fisher-Kolmogorov-Petrovskii-Piskunov equation

Abstract

We construct quasiparticles-like solutions to the one-dimensional Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) with a nonlocal nonlinearity using the method of semiclassically concentrated states in the weak diffusion approximation. Such solutions are of use for predicting the dynamics of population patterns. The interaction of quasiparticles stems from nonlocal competitive losses in the FKPP model. We developed the formalism of our approach relying on ideas of the Maslov method. The construction of the asymptotic expansion of a solution to the original nonlinear evolution equation is based on solutions to an auxiliary dynamical system of ODEs. The asymptotic solutions for various specific cases corresponding to various spatial profiles of the reproduction rate and nonlocal competitive losses are studied within the framework of the approach proposed.
Paper Structure (16 sections, 1 theorem, 87 equations, 2 figures)

This paper contains 16 sections, 1 theorem, 87 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The nonlocal 1D FKPP equation
  3. Classes of functions $\mathcal{P}_{s,t}^D$
  4. Estimates
  5. Moments
  6. Expansions of coefficients
  7. Decomposition of the FKPP equation
  8. Equations for $\sigma_{u_s}(t,D)$
  9. Equations for $x_{u_s}(t,D)$
  10. Operator $\hat{T}_s$
  11. Equations for $\alpha^{(n)}_{u_s}(t,D)$, $n\geq 2$.
  12. Semiclassical reduction of the FKPPDS to an auxiliary associated linear system
  13. The Cauchy problem
  14. The Green function
  15. Asymptotic solutions
  16. ...and 1 more sections

Key Result

Proposition 4.1

Let $u_s(x,t,D)\in {\mathcal{P}}_{s,t}^D = {\mathcal{P}}_t^D\left(X_s(t,D),S_s(t,D)\right)$, $s=1,2$, be solutions to the equation dec-2 and $v_s^{(k)}(x,t,D,\mathbf{C})$, $s=1,2$, be solutions to the equations als-7. Let $\mathbf{C}$ be the integration constants determined by alg-1b and the initia Then, satisfy the equations dec-2 with the right-hand side accuracy of $\bar{O}(D^{(M+1)/2})$ and

Figures (2)

  • Figure 1: Dependence of $u(x,t)$ on $x$ for various $t$ when the reproduction rate is constant. The blue line is for $u^{(0)}$ (leading term of asymptotics), yellow line is for $u^{(1)}$, green line is for $u^{(2)}$, and black dashed line is for $u_{num}$.
  • Figure 2: Dependence of $u(x,t)$ on $x$ for various $t$ when the reproduction rate depends on $x$. The blue line is for $u^{(0)}$ (leading term of asymptotics), yellow line is for $u^{(1)}$, green line is for $u^{(2)}$, and black dashed line is for $u_{num}$.

Theorems & Definitions (1)

  • Proposition 4.1