#### Posts

2017-08-22: A rebuttal on the beauty in applying math

2017-04-22: Free googles book library

2016-11-02: In search of Theodore von Karman

2016-09-25: Amath Timeline

2016-02-24: Math errors and risk reporting

2016-02-20: Apple VS FBI

2016-02-19: More Zika may be better than less

2016-01-14: Life at the multifurcation

2015-09-28: AI ain't that smart

2015-06-24: MathEpi citation tree

2015-03-31: Too much STEM is bad

2015-03-24: Dawn of the CRISPR age

2015-02-09: Hamilton's selfish-herd paradox

2015-02-08: Risks and values of microparasite research

2014-11-10: Vaccine mandates and bioethics

2014-10-18: Ebola, travel, president

2014-10-12: Ebola numbers

2014-09-23: More stochastic than?

2014-08-17: Feynman's missing method for third-orders?

2014-07-31: CIA spies even on congress

2014-07-16: Rehm on vaccines

2014-06-20: Random dispersal speeds invasions

2014-04-14: More on fairer markets

2014-02-17: Is life a simulation or a dream?

2014-01-30: PSU should be infosocialist

2014-01-12: The dark house of math

2014-01-11: Inconsistencies hinder pylab adoption

2013-12-24: Cuvier and the birth of extinction

2013-12-17: Risk Resonance

2013-12-15: The cult of the Levy flight

2013-12-09: 2013 Flu Shots at PSU

2013-12-02: Amazon sucker-punches 60 minutes

2013-11-26: Zombies are REAL, Dr. Tyson!

2013-11-22: Crying wolf over synthetic biology?

2013-11-21: Tilting Drake's Equation

2013-11-18: Why $1^\infty != 1$

2013-11-14: 60 Minutes misreport on Benghazi

2013-11-11: Making fairer trading markets

2013-11-09: Using infinitessimals in vector calculus

2013-11-08: Functional Calculus

## 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.