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Logistic growth in seasonally changing environments

Daniel Daners, Zeaiter Zeaiter

TL;DR

This work analyzes a time-periodic, degenerate logistic reaction-diffusion problem with seasonal refuges modeled by a nonnegative weight $b(x,t)$ in a $T$-periodic framework. Using an abstract evolution-system approach, principal periodic-parabolic eigenvalues $\mu_1(b)$ and their monotone limits $\mu^*(b)$ govern the bifurcation and existence of positive $T$-periodic solutions, which exist if $\mu_1(0)<\mu<\mu^*(b)$ and are unique and linearly stable. As $\mu$ approaches the critical value $\mu^*(b)$, the solution blows up on regions where the limit eigenfunction $\varphi_\infty>0$, while remaining finite elsewhere; a subsequent local analysis shows the blow-up can be localized to subsets of the domain, with a local weak limit $u_\infty$ on the non-blow-up set. Under a diffusion-dominated, growth-assuming regime, the authors prove local boundedness of blow-up solutions and describe their partial blow-up geometry, highlighting essential differences from the elliptic counterpart. Overall, the results extend prior work to degenerate and non-smooth settings, offering a rigorous framework for understanding seasonally driven refuges and their impact on long-term population dynamics.

Abstract

We consider a parameter dependent periodic-logistic problem with a logistic term involving a degeneracy that replicates time dependent refuges in the habitat of a population. Working under no or very minimal assumptions on the boundary regularity of the domain we show the existence of a time-periodic solution which bifurcates with respect to the parameter and show their stability. We show that under suitable assumptions that the periodic solution blows up on part of the domain and remains finite on other parts when the parameter approaches a critical value.

Logistic growth in seasonally changing environments

TL;DR

This work analyzes a time-periodic, degenerate logistic reaction-diffusion problem with seasonal refuges modeled by a nonnegative weight in a -periodic framework. Using an abstract evolution-system approach, principal periodic-parabolic eigenvalues and their monotone limits govern the bifurcation and existence of positive -periodic solutions, which exist if and are unique and linearly stable. As approaches the critical value , the solution blows up on regions where the limit eigenfunction , while remaining finite elsewhere; a subsequent local analysis shows the blow-up can be localized to subsets of the domain, with a local weak limit on the non-blow-up set. Under a diffusion-dominated, growth-assuming regime, the authors prove local boundedness of blow-up solutions and describe their partial blow-up geometry, highlighting essential differences from the elliptic counterpart. Overall, the results extend prior work to degenerate and non-smooth settings, offering a rigorous framework for understanding seasonally driven refuges and their impact on long-term population dynamics.

Abstract

We consider a parameter dependent periodic-logistic problem with a logistic term involving a degeneracy that replicates time dependent refuges in the habitat of a population. Working under no or very minimal assumptions on the boundary regularity of the domain we show the existence of a time-periodic solution which bifurcates with respect to the parameter and show their stability. We show that under suitable assumptions that the periodic solution blows up on part of the domain and remains finite on other parts when the parameter approaches a critical value.
Paper Structure (8 sections, 31 theorems, 163 equations, 2 figures)

This paper contains 8 sections, 31 theorems, 163 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Precise assumptions
  3. Evolution systems and their properties
  4. Periodic-parabolic eigenvalue problems
  5. Comparison of Principal Eigenfunctions
  6. The periodic-parabolic logistic equation
  7. Smoothness with with respect to the parameter
  8. Local boundedness of blow up solutions

Key Result

Theorem 1.1

The equation eq:pp-logistic has a non-trivial positive weak solution if and only if $\mu_1(0)<\mu<\mu^*(b)$. In that case this solution is unique and linearly stable.

Figures (2)

  • Figure 4.1: Example of $b$ with $\mu^*(b)<\infty$.
  • Figure 8.1: $Q_b$ (shaded) with overlapping sub-cylinders covering part of $Q_0$

Theorems & Definitions (62)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Proposition 2.2: A priori estimates
  • proof
  • Remark 2.3
  • Theorem 2.4
  • Theorem 2.5: perturbation result
  • proof
  • Proposition 3.1: Properties of evolution system
  • ...and 52 more