## 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 \(r > r_c\) ? In the
absence of dispersal, we expect competitive exclusion of the less-efficient
species. Can dispersal change this?
What if the competitor can exploit a different dispersal pattern than the
dominate species?

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 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.