**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 *c*_{0} of sites held
by species 0 with *c*_{1} sites held by species *z*_{1},
so that *T*=*c*_{0}+*c*_{1}. Necessary conditions
for dynamic and evolutionary stability are

The convexity assumption on *b* implies that
*b*'(*z*_{1}) < *b*'(0), which implies that *c*_{1}
> *c*_{0}. In any two species coalition, the species with the
higher mortality rate is necessarily more common.

We write
*b*(*z*)=*sb*_{0}(*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 *z*_{1}, we can try to solve for
*s*, *c*_{0} and *c*_{1} for a given function
*b*. Subtracting conditions **I** and **II** and simplifying gives

s T b_{0}(z_{1})
= z_{1} |
(1) |

Then **I** gives

(2) |

From **III**, we can solve for *T* with the equation

(3) |

Inequality **IV** requires that *c*_{1} > *c*_{0},
or that
*c*_{1}/*T* > 1/2 or that

Plugging in equation 3 and simplifying gives the inequality

z_{1} - 2 b_{0}(z_{1})
+ z_{1} b_{0}'(z_{1}) <
0 . |
(4) |

Now write
*b*_{0}(*z*)=*zg*(*z*) where *g*(0)=1 and *g*'(*z*)<0.
Then
*b*_{0}'(*z*_{1}) = *g*(*z*_{1})
+ *z*_{1} *g*'(*z*_{1}), meaning that condition
4 can be rewritten

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

For any value of *z*_{1} where *h* is increasing, there is
a solution for *s*, *c*_{0} and *c*_{1} that
satisfies all four conditions.

The condition
*h*'(*z*_{1}) > 0 is only a necessary condition for a stable
two species coalition. To show that the coalition is indeed stable for small
values of *z*_{1}, 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 *b*_{0}(*z*) around *z*=0 as

for some value
. The absolute value guarantees that this function is odd.
With this function,

If *p*=2, then *b*_{0}(*z*) has a discontinuous second
derivative at *z*=0 and the function *h*(*z*) is discontinuous
at *z*=0. If *p*>2, then *b*_{0}(*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 |
= | ||

z_{1} |
= | ||

c_{1} |
= | ||

c_{0} |
= | (5) |

With *p*=3, the invasion function is

which is indeed 0 for *m*=0 and *m*=*z*_{1}, and negative
for all other positive arguments as long as *z*_{1} 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.