Ecological Archives E081-027-A4

Frederick R. Adler, and Julio Mosquera. 2000. Is space necessary? Interference competition and limits to biodiversity. Ecology 81:3226-3232.

Appendix D: Conditions for existence of a two species coalition.

As in Appendix C, it can be shown that one member of any two species coalition must have mortality rate of 0. Suppose the coalition combines a fraction c0 of sites held by species 0 with c1 sites held by species z1, so that T=c0+c1. Necessary conditions for dynamic and evolutionary stability are

\begin{displaymath}\begin{array}{lrcll} \mbox{{\bf I.}} & 1-T -c_1 b(z_1) &=& ... ... 1 & \mbox{~~$f(m)$\space is decreasing at $m=0$ } \end{array}\end{displaymath}


The convexity assumption on b implies that b'(z1) < b'(0), which implies that c1 > c0. In any two species coalition, the species with the higher mortality rate is necessarily more common.

We write b(z)=sb0(z) where s is the slope of b at z=0 (which must be greater than 1 for such a coalition to exist). If we pick a value z1, we can try to solve for s, c0 and c1 for a given function b. Subtracting conditions I and II and simplifying gives

 
s T b0(z1) = z1 (1)

Then I gives

 \begin{displaymath}c_1 = \frac{T(1-T)}{z_1} . \end{displaymath} (2)

From III, we can solve for T with the equation

 \begin{displaymath}1-T = \frac{b_0(z_1)-z_1 b_0'(z_1)}{1- b_0'(z_1)} . \end{displaymath} (3)

Inequality IV requires that c1 > c0, or that c1/T > 1/2 or that

\begin{displaymath}\frac{1-T}{z_1} > \frac{1}{2} . \end{displaymath}


Plugging in equation 3 and simplifying gives the inequality

 
$\displaystyle b_0(z_1) - z_1 b_0'(z_1) > \frac{1}{2} z_1 (1-b_0'(z_1))$      
z1 - 2 b0(z1) + z1 b0'(z1) < 0 .     (4)

Now write b0(z)=zg(z) where g(0)=1 and g'(z)<0. Then b0'(z1) = g(z1) + z1 g'(z1), meaning that condition 4 can be rewritten

1 - g(z1) + z1 g'(z1) < 0 .


Finally, write g(z)=1-zh(z). The inequality simplifies to

h'(z1) > 0 .


For any value of z1 where h is increasing, there is a solution for s, c0 and c1 that satisfies all four conditions.

The condition h'(z1) > 0 is only a necessary condition for a stable two species coalition. To show that the coalition is indeed stable for small values of z1, and that those values correspond to values of the slope s slightly greater than 1, consider the first two terms in the series expansion of b0(z) around z=0 as

\begin{displaymath}b_0(z) = z - \beta \vert z\vert^{p-1} z \end{displaymath}


for some value $2 \leq p \leq 3$. The absolute value guarantees that this function is odd. With this function,

\begin{displaymath}h(z)=\beta \frac{\vert z\vert^{p-1}}{z} . \end{displaymath}


If p=2, then b0(z) has a discontinuous second derivative at z=0 and the function h(z) is discontinuous at z=0. If p>2, then b0(z) has a continuous second derivative at z=0 and h(z) is increasing for small values of z.

Solving the equations I - III and discarding higher order terms gives

 
T = $\displaystyle \frac{1}{s}$  
z1 = $\displaystyle \frac{p}{p-1} (1-T)$  
c1 = $\displaystyle \frac{p-1}{p} T$  
c0 = $\displaystyle \frac{1}{p} T .$ (5)

With p=3, the invasion function is

\begin{displaymath}f(m) = -\frac{m \beta (m-z_1)^2}{1-\beta z_1^2} \end{displaymath}


which is indeed 0 for m=0 and m=z1, and negative for all other positive arguments as long as z1 is sufficiently small. With 2 < p < 3, computer algebra (Maple Version 5) indicates that invasion by intermediate species is not possible. Therefore, for values of s just greater than 1, there is a stable two species coalition described by equation 5.

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