## A Comment on How Biased Dispersal can Preclude Competitive Exclusion

Let $n(t)$ be a vector for population abundances at time $t$ accross a set of interconnected patches, with $n_i(t)$ being the number in patch $i$. Let's adopt the minimal patch-dynamics model \begin{gather} \dot{n}_i = r n_i ( 1 - n_i/K_i ) + ( D n )_i \end{gather} where $r$ is the net proliferation rate, $K_i$ is the carrying capacity in patch $i$, and the dispersal-rate matrix $D$ is a column-stochastic rate matrix ($-D$ is a Z-matrix, under the definition of Horn and Johnson, with columns that sum to 0). It should have been established using convexity and monotonicity that as long as $D$ communinicates ($e^D$ is strictly positive), there is one steady state $n=0$ and a positive steady-state $n = n^* > 0$.

Now, assume, we introduce a new species with abundances $c$. This species can invade at low densities if and only if \begin{gather} \dot{c}_i = r_c c_i (1- n_i^*/K_i) + (D_c c)_i \end{gather} grows from a small initial condition. It's well known that this is the case if $r_c > r$, but what if $c$ is an inferior competitor and $r_c Because the columns of$D$sum to$0$, then at steady-state, $\sum_i \dot{n}^*_i = r \sum_i n^*_i (1-n^*_i/K_i) = 0.$ Since $( \forall i, n^*_i > K_i ) \rightarrow \sum_i \dot{n}^*_i < 0$ and $( \forall i, 0 < n^*_i < K_i ) \rightarrow \sum_i \dot{n}^*_i > 0$, either$ \forall i, n^*_i = K_i$or$\exists (j, \ell) : n^*_j > K_j$and$n^*_{\ell} < K_{\ell}$. Under what conditions is$n^*_i = K_i \forall i$? Well, \begin{gather} 0 = r K \circ ( 1 - K/K ) + ( D K ) = D K, \end{gather} so$K$must be in the right nullspace of$D$. In particular, if the environment is homogeneous {\it e.g.} carrying capacity is constant in all patches, and$D$is also a row-stochastic rate matrix (including symmetric matrices where$D = D^T$, coorsponding to balanced dispersal). But under balanced dispersal in a heterogeneous environment, unbalanced dispersal in a homogeneous environment, and almost all cases of unbalanced dispersal in a heterogeneous environment, there is some patch$\ell$where$n^*_{\ell} < K_{\ell}$. It follows, then, that \begin{gather} \dot{c}_{\ell} = r_c c_{\ell} (1- n_{\ell}^*/K_{\ell}) + (D_c c)_{\ell} > 0 \end{gather} such that the inferior competitor can coexist with the dominate competitor, provided dispersal is sufficiently slow ($D_c \approx [[0]]\$).

See Hastings (1983) and McPeek (1992) allong with the related literature for more information.