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Range expansion by growth and congestion

Henri Berestycki, Antoine Mellet

Abstract

We introduce a nonlinear and nonlocal model that describes the range expansion of a population resulting from growth and competition for space. This type of phenomenon underlies the expansion of colonies of immotile cells which motivated this work; Similar mechanisms are at play in urban sprawling which we briefly discuss as well. We rigorously derive a singular limit of this model corresponding to a regime where dispersal occurs only from saturated areas. The limiting model, which has the structure of an obstacle free boundary problem in time, provides an effective approach to the description of the range expansion of a population as a result of growth, saturation and dispersion. We then establish the main mathematical properties of this singular problem. In particular, we characterize the evolution of a free boundary that delimits the saturated area. We identify traveling wave solutions and characterize the asymptotic speed of spreading of compactly supported solutions.

Range expansion by growth and congestion

Abstract

We introduce a nonlinear and nonlocal model that describes the range expansion of a population resulting from growth and competition for space. This type of phenomenon underlies the expansion of colonies of immotile cells which motivated this work; Similar mechanisms are at play in urban sprawling which we briefly discuss as well. We rigorously derive a singular limit of this model corresponding to a regime where dispersal occurs only from saturated areas. The limiting model, which has the structure of an obstacle free boundary problem in time, provides an effective approach to the description of the range expansion of a population as a result of growth, saturation and dispersion. We then establish the main mathematical properties of this singular problem. In particular, we characterize the evolution of a free boundary that delimits the saturated area. We identify traveling wave solutions and characterize the asymptotic speed of spreading of compactly supported solutions.
Paper Structure (16 sections, 20 theorems, 146 equations, 1 figure)

This paper contains 16 sections, 20 theorems, 146 equations, 1 figure.

Table of Contents

  1. Derivation of the model
  2. A nonlinear nonlocal equation for range expansion
  3. Properties of \ref{['eq:main1']}
  4. Singular limit
  5. Assumptions and main results
  6. Assumptions and definition
  7. Main results
  8. Further considerations.
  9. Generalizations
  10. Urban Sprawl
  11. Singular limit: Proof of Theorem \ref{['thm:limit']}
  12. Comparison principle and uniqueness
  13. Traveling waves: Proof of Theorem \ref{['thm:TW']}
  14. Spreading of compactly supported solutions
  15. Spreading of the support and the saturated set
  16. ...and 1 more sections

Key Result

Theorem 2.2

Assume that $K$ and $g$ satisfy respectively eq:K1 and eq:f1. For all $\gamma\geq 1$, Equation eq:gamma has a unique solution $u_\gamma(t,x)$ satisfying $u_\gamma(0,x)= u^{in}(x)$. When $\gamma\to+\infty$, $u_\gamma$ converges, up to a subsequence, locally uniformly in $\mathbb{R}_+\times \mathbb{R}

Figures (1)

  • Figure 1: Illustration of the limit in \ref{['eq:limR']}. The spherical cap $C=H\cap B_\ell(s)$ (blue and red regions) corresponds to the planar traveling front. The blue region $E=D_R\cap B_\ell(s)$ corresponds to the spherical symmetric solution. The region in red is $C\setminus E$. When $K$ is given by \ref{['eq:KC']}, \ref{['eq:limR']} states that the ratio $\frac{|C\setminus E|}{|C|}$ goes to zero as $R\to\infty$, uniformly with respect to $\ell$.

Theorems & Definitions (35)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3: Comparison principle
  • Corollary 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Theorem 2.7
  • Corollary 2.8
  • Proposition 2.9
  • Theorem 2.10
  • ...and 25 more