## 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.