Conjecture on ideal-free reducible strategies in source-sink landscapes

Assume $1\le k<n$, assumptions $\mathrm{(A2)}$–$\mathrm{(A4)}$ hold, $(d_{ij})$ is ideal-free on patches $1\le i\le k$ and satisfies $d_{ij}=0$ whenever $i>k$ or $j>k$, and $(D_{ij})$ satisfies $\mathrm{(A1)}$. Let $u_i^*>0$ be the unique solution to $g_i(u_i)=1$ for $1\le i\le k$, and let $u_i^*=0$ for $i>k$. The source-sink stability conjecture. Then $(u^*,0)$ is globally asymptotically stable among positive initial data for the two-species model. This conjecture would extend the preceding theorem from sedentary populations to reducible dispersal strategies that are ideal-free on the source patches and never disperse into or from sink patches.