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A stationary population model with an interior interface-type boundary

Pablo Alvarez-Caudevilla, Cristina Brändle

Abstract

We propose a stationary system that might be regarded as a migration model of some population abandoning their original place of abode and becoming part of another population, once they reach the interface boundary. To do so, we show a model where each population follows a logistic equation in their own environment while assuming spatial heterogeneities. Moreover, both populations are coupled through the common boundary, which acts as a permeable membrane on which their flow moves in and out. The main goal we face in this work will be to describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.

A stationary population model with an interior interface-type boundary

Abstract

We propose a stationary system that might be regarded as a migration model of some population abandoning their original place of abode and becoming part of another population, once they reach the interface boundary. To do so, we show a model where each population follows a logistic equation in their own environment while assuming spatial heterogeneities. Moreover, both populations are coupled through the common boundary, which acts as a permeable membrane on which their flow moves in and out. The main goal we face in this work will be to describe the precise interplay between the stationary solutions with respect to the parameters involved in the problem, in particular the growth rate of the populations and the coupling parameter involved on the boundary where the interchange of flux is taking place.
Paper Structure (11 sections, 18 theorems, 159 equations, 1 figure)

This paper contains 11 sections, 18 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The model
  4. Basic notation and functional setting.
  5. Auxiliary problems
  6. A linear problem
  7. A weighted linear eigenvalue problem
  8. Asymptotic behaviour of a spatially heterogeneous linear problem
  9. Existence of solutions for the Non-degenerate case
  10. Existence of solutions for the degenerate case
  11. Conclusions and further work

Key Result

Theorem 3.1

If $F\in L^\infty(\Omega_1)\times L^{\infty}(\Omega_2)$ and $G\in\mathcal{L}$, then mainlinear--mainlinearbdy has a unique weak solution $U$. Moreover, there exits a positive constant $C$, such that

Figures (1)

  • Figure 1: A possible configuration of the domain.

Theorems & Definitions (44)

  • Definition 2.1
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • Definition 3.4
  • Theorem 3.5
  • Definition 3.6
  • Lemma 3.7
  • ...and 34 more