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

**Appendix E:** Results with non-spatial model.

We assume that the function is
decreasing and that there exists a value *s*_{max} that
solves the equation
(the species with the largest value
of *s* that could replace itself in the absence of competition). We also
assume that the second derivative of is negative for *x* >
0.

First, we show that the non-spatial model supports a continuum of species if
the competition function is discontinuous with the form

All members of a coalition must satisfy

where the integral is taken over all members of the coalition. We show that
the coalition consists of all species with
. The condition that *f*(*s*)=1
can be rewritten as
where
. Differentiating, we find that
. This solution is the uninvadible coalition.

Second, suppose that and are both analytic. The integral
in the condition

is a convolution (as in Appendix B), hence both
sides are analytic. Therefore, if the coalition includes an interval, both sides
must be equal over their entire domains. However,
if
*s* > *s*_{max}, while the right hand side must
always exceed 1. Therefore, the coalition can include no intervals.

As in Appendix C, any uninvadible single species
must be at the minimum value of *s* (this requires that
for *x* > 0). Similarly, it can be shown that one
member of any two species coalition must have *s*=0. Suppose the coalition
consists of *N*_{0} seeds of the species with *s*=0 and *N*_{1}
seeds of a species with *s*=*z*_{1}. For convenience, and
without loss of generality, we scale the parameter *a* to be equal to 1.
Necessary conditions for dynamic and evolutionary stability are

Suppose that

(6) |

where *A*(*x*) is odd and *A*'(0)=1Then
. The first
three conditions can be solved for , *N*_{0} and *N*_{1}, and the result
substituted into inequality IV, which becomes

If
, this condition simplifies to

This condition on the function *A* exactly matches the condition on the
function *b*_{0} in equation 4 from Appendix
D. Therefore, the conditions for a existence of a two species coalition
are identical to those in the spatially-structured model, which implies also
that *N*_{1} > *N*_{0}.

If we Taylor expand the function (assuming
it is differentiable) around *x*=0, the first two terms are given by equation
6. Similarly, if we Taylor expand the function
around *s*=0, the first two terms are given by the linear
form. Therefore, near *s*=0 where the bifurcation takes place, condition
4 determines whether a two species coalition will exist.

The necessary occurrence of the bifurcation from one species to two might seem
in conflict with some results in Rees and Westoby (1997). With their first model,
which takes the form
when written in terms of seed size, coexistence is impossible.
However, this function is not a function of *u*-*s*, has a slope that
does not increase continuously as increases, and has the wrong concavity when .

More importantly, this competitiveness function can be written as a ratio

where
in this case. The expression for the per seed reproduction *f*(*s*)
can be rewritten

All species experience the same value of the integral which can thought of as a surrogate for resource depletion (Adler, 1990). The species that can persist with the maximum level of resource depletion (which in this case maximizes ) is uninvadible. The fact that can be written as a ratio, not that it is unbounded (as argued by Rees and Westoby (1997)), leads to the lack of coexistence.

*Literature Cited*

Adler, F. R. 1990. Coexistence of two types on a single resource in discrete
time. Journal of Mathematical Biology 28: 695-713.

Rees, M., and M. Westoby. 1997. Game-theoretical evolution of seed mass in multi-species
ecological models. Oikos 78: 116-126.