Unique steady states for population models in a heterogeneous environment

TL;DR

This work addresses the question of whether positive steady states are unique in a two-patch reaction-diffusion population model with patch-dependent diffusion and growth. By recasting the PDE into planar Hamiltonian systems and employing a shooting method across the interface, the authors derive sufficient conditions—expressed as convexity/concavity constraints on the associated potentials and monotonicity properties of the resulting maps—that guarantee a unique positive steady state. These conditions are satisfied for generalized Richards growth (the logistic family with exponents $p^\pm\ge 1$), providing a robust theoretical guarantee for uniqueness in a broad ecological setting, while indicating potential non-uniqueness when $p^\pm\in(0,1)$. The analysis leverages Chicone's ideas on the monotonicity of a period map for planar Hamiltonian systems, embedded within an invariant-region and shooting-framework across the interface. The work clarifies the impact of habitat heterogeneity on steady-state uniqueness and offers a concrete set of verifiable criteria for ecological models of patchy environments.

Abstract

We revisit a model proposed by Freedman etal in \cite{freedman} which describes the dynamics of a population diffusing in a patchy environment. From their work it is known that positive steady states exist for this model, but not whether they are unique. Here, we provide sufficient conditions guaranteeing that steady states are unique. These conditions are satisfied when the reaction rates are generalized logistic growth rates. Our proofs critically exploit Chicone's ideas in \cite{chicone}, which were used to establish that the period map associated to a continuous family of periodic solutions of certain planar Hamiltonian systems is monotone.

Figures (5)

• Figure 1: Piecewise orbits of the Hamiltonian systems $(\ref{['ham-pm1']})-(\ref{['ham-pm2']})$. Reaction rates $f^{\pm}(u)=r^{\pm}u(1-u/K^{\pm})$, with $r^{\pm}=1$, $K^-=1$ and $K^+=2.2$. Diffusion constants $d^-=1.2$ and $d^+=2$. Domain lengths $L^-=1.0349$, $L^+=1.1671$. Top and bottom orbits (in red) represent the solution $(u^-(x),v^-(x))$, $x\in [-L^-,0]$, of the $H^-$-system and the solution $(u^+(x),v^+(x))$, $x\in [0,L^+]$, of the $H^+$-system, respectively. The jump in the states at $x=0$ of these solutions occurs along the vertical dashed line (in black) and expresses the continuity of the density $u^-(0)=u^+(0)$, and the flux $d^-v^-(0)=d^+v^+(0)$ at the interface. Other blue curves represent orbits of the Hamltonian systems, and the black dots their steady states.
• Figure 2: Orbits of the Hamiltonian system $(\ref{['ham1']})-(\ref{['ham2']})$. Reaction rate $f^+(u)=r^+u(1-u/K^+)$, with $r^+=1$, $K^+=2.2$ and diffusion constant $d^+=2$. Line segment $L_{u_0}$ for $u_0=1.1$.
• Figure 3: Orbits of the Hamiltonian system $(\ref{['ham1']})-(\ref{['ham2']})$. Reaction rate $f^+(u)=r^+u(1-u/K^+)$, with $r^+=1$, $K^+=2.2$ and diffusion constant $d^+=2$. Line segment $L_{v_0}$ for $v_0=0.4491$.
• Figure 4: Orbits of the Hamiltonian system $(\ref{['ham3']})-(\ref{['ham4']})$. Reaction rate $f^-(u)=r^-u(1-u/K^-)$, with $r^-=1$, $K^-=1$ and diffusion constant $d^-=1.2$. Line segment $L_{u_0}$ for $u_0=1.75$.
• Figure 5: Orbits of the Hamiltonian system $(\ref{['ham3']})-(\ref{['ham4']})$. Reaction rate $f^-(u)=r^-u(1-u/K^-)$, with $r^-=1$, $K^-=1$ and diffusion constant $d^-=1.2$. Line segment $L_{v_0}$ for $v_0=0.7348$.

Theorems & Definitions (20)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Corollary 1
• proof