Unique steady states for population models in a heterogeneous environment

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.