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Large time behavior of solutions to a cooperative model with population flux by attractive transition

Ryuichi Kato, Kousuke Kuto

TL;DR

This work analyzes a two-species cooperative diffusion system with population flux by attractive transition on a bounded convex domain under Neumann boundaries. By leveraging equal diffusion rates and a weakened cooperative condition, it first proves global existence of classical solutions for spatial dimension $N\le 3$ and establishes an $L^{\infty}$ absorbing set, using an $L^{\infty}$ bound for $w=u+\gamma v$ and higher-regularity estimates. It then shows that, when a positive constant steady state exists, all positive solutions converge to it as $t\to\infty$ provided the diffusion rate is sufficiently large, via a Lyapunov functional and energy-dissipation analysis. The paper also connects the time-nonstationary behavior to stationary bifurcation results, illustrating how small diffusion admits nonconstant steady states while large diffusion yields global attraction to the homogeneous steady state, thereby clarifying the large-time dynamics of nonlinear cross-diffusion ecological models.

Abstract

This paper is concerned with a diffusive Lotka-Volterra cooperative modelwith population flux by attractive transition. We study the time-global well-posedness and the large time behavior of solutions in a case where the habitat is a bounded convex domain and random diffusion rates equal to each other. A main result shows that when the spatial dimension is less than or equal to 3, under a weaker cooperative condition, a classical solution exists globally in time if the initial data belongs to a suitable functional space. Furthermore, it is shown that if there exists a positive steady state and the equal random diffusion rate is sufficiently large, then all positive solutions asymptotically approach the positive steady state as time tends to infinity.

Large time behavior of solutions to a cooperative model with population flux by attractive transition

TL;DR

This work analyzes a two-species cooperative diffusion system with population flux by attractive transition on a bounded convex domain under Neumann boundaries. By leveraging equal diffusion rates and a weakened cooperative condition, it first proves global existence of classical solutions for spatial dimension and establishes an absorbing set, using an bound for and higher-regularity estimates. It then shows that, when a positive constant steady state exists, all positive solutions converge to it as provided the diffusion rate is sufficiently large, via a Lyapunov functional and energy-dissipation analysis. The paper also connects the time-nonstationary behavior to stationary bifurcation results, illustrating how small diffusion admits nonconstant steady states while large diffusion yields global attraction to the homogeneous steady state, thereby clarifying the large-time dynamics of nonlinear cross-diffusion ecological models.

Abstract

This paper is concerned with a diffusive Lotka-Volterra cooperative modelwith population flux by attractive transition. We study the time-global well-posedness and the large time behavior of solutions in a case where the habitat is a bounded convex domain and random diffusion rates equal to each other. A main result shows that when the spatial dimension is less than or equal to 3, under a weaker cooperative condition, a classical solution exists globally in time if the initial data belongs to a suitable functional space. Furthermore, it is shown that if there exists a positive steady state and the equal random diffusion rate is sufficiently large, then all positive solutions asymptotically approach the positive steady state as time tends to infinity.

Paper Structure

This paper contains 8 sections, 11 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Existence of the $L^{\infty}(\Omega )$ absorbing set
  4. A priori estimates for $w=u+\gamma v$
  5. Hölder estimate
  6. $W^{1,4}(\Omega )$ estimate
  7. Global asymptotic stability of the positive constant steady state
  8. Relationship between stationary and nonstationary solutions

Key Result

Theorem 1.1

Assume that $u_{0}$ and $v_{0}$ are nonnegative functions belonging to $W^{1,\infty}(\Omega)$. Then there exist $T_{{\rm max}}\in (0,\infty]$ and nonnegative which solves para classically in $\overline{\Omega}\times (0,T_{{\rm max}})$. Furthermore, if $T_{{\rm max}}<\infty$, then

Figures (2)

  • Figure 1: $\mathcal{R}_{j}$ and $\gamma =A\tau^{*}$ on the $\alpha\beta$ plane
  • Figure 2: Bifurcation branches of solutions to \ref{['SP']} with bifurcation parameter $d$ in case $(\alpha, \beta, a_{1}, a_{2}, b_{1}, b_{2}, c_{1},c_{2})= (2,1,1,-1,4,5,2,3)$

Theorems & Definitions (18)

  • Theorem 1.1: Am3
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4: SSW
  • ...and 8 more