Optimization Of The Survival Threshold For Anisotropic Logistic Equations With Mixed Boundary Conditions

TL;DR

This work analyzes a heterogeneous, anisotropic reaction-diffusion equation with mixed boundary conditions, establishing a survival threshold $d^*$ linked to a principal eigenvalue of the linearized operator. A parabolic problem with initial data is shown to converge to a unique positive stationary state when $0<d<d^*$ and to zero when $d\ge d^*$; the threshold is given by $d^* = 1/\lambda(m)$, where $\lambda(m)$ is the principal eigenvalue associated with a weight $m$. The paper also proves the existence of an optimal bang-bang weight $m_\omega$ minimizing $\lambda(m)$, attained on a set $\omega$ that is a superlevel of the eigenfunction, with a complete 1D analysis showing interval-type optimal habitats depending only on the boundary configuration. Together, these results link boundary conditions, diffusion anisotropy, and spatial resource placement to survival outcomes in heterogeneous environments.

Abstract

In this paper we study a reaction diffusion problem with anisotropic diffusion and mixed Dirichlet-Neumann boundary conditions on the boundary of the domain. First, we prove that the parabolic problem has a unique positive, bounded solution. Then, we show that this solution converges as t tends to infinity to the unique nonnegative solution of the elliptic associated problem. The existence of the unique positive solution to this problem depends on a principal eigenvalue of a suitable linearized problem with a sign-changing weights. Next, we study the minimization of such eigenvalue with respect to the sign-changing weight, showing that there exists an optimal bang-bang weight, namely a piece-wise constant weight that takes only two values. Finally, we completely solve the problem in dimension one.

Figures (3)

• Figure 1: Representation of the optimal weight $m_\omega$ in dimension 1, according to the mixed boundary conditions.
• Figure 2: Representation of the optimal weight $m_\omega$ in dimension 1, with homogeneous Neumann boundary conditions.
• Figure 3: Representation of the optimal weight $m_\omega$ in dimension 1, with homogeneous Dirichlet boundary conditions. Here $c_1=\frac{1-|\omega|}{a+b} a$ and $c_2=\frac{|\omega|b+a}{a+b}$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Definition 3.1
• Proposition 3.2
• proof
• Proposition 3.3