Anna L. W. Sears and Peter Chesson. 2007. New methods for quantifying the spatial storage effect: an illustration with desert annuals. Ecology 88:2240–2247.

Appendix A. The theory behind the formulas.

The theory on which this work is based is given in detail in Chesson (2000) and more explicitly for this setting in Chesson (2007). Some of the calculations that we need here depend on statistical formulae, several of which are discussed in this context in Chesson et al (2005). Here we give the details for our present needs. First, we need the formula for the average yield when the yield at a particular location is given by Eq. 3 of the text, i.e., . This formula is a product of two separate quantities. The standard statistical result discussed for ecological applications in Chesson et al. (2005) is that the average of a product is equal to the product of the averages of each quantity separately, plus the covariance between them. Thus, we obtain the formula,

which is Eq. 4 of the text. This equation is exact for any situation. In particular, there is no normality assumption in Eq. 4. We now wish to find approximations to the three different parts of the formula for . To do this, we do use normal approximations. First, note that if X is a normal random variable, then

(A.1)

 
for example see Johnson et al. (1994). This formula is exact for normal X, and approximately true in other circumstances, and is usually much better than the more common approximation derived from Taylor series, viz , applicable for small variance. Applying formula (A.1) separately to X = E_jx and X = -C_jx, we see that

	(A.2)
and so
	(A.3)

 
Applying formula (A.1) to X-Y, for the case where (X, Y) is bivariate normal, we see that

(A.4)

       
Subtracting (A.3) from (A.4), and rearranging, gives the following exact covariance formula for bivariate normal (E_jx, C_jx):

(A.5)

 
Linearly approximating the exponential in (A.5) above, this formula in turn gives the small variance approximation

(A.6)

which is applicable regardless of normality. Expressions (A.3) and (A.5) combine to give Eq. 5 of the text for the average yield. Equations 9 and 10 follow respectively for the cases of no competition, and no covariance between environment and competition. These equations for the mean yield, however, assume that the errors are true sampling errors, rather than environmental effects operating on patches that cause deviations from the strict linear relationship between E_x and C_x. In the latter case, Eq. 9 is multiplied by and Eqs. 5 and 10 are multiplied by . These modifications, however, do not affect the relative heights of the critical comparison of yield with and without covariance between environment and competition.

LITERATURE CITED

Chesson, P. 2000. General theory of competitive coexistence in spatially-varying environments. Theoretical Population Biology 58:211–237.

Chesson, P., M. Donahue, B. Melbourne, and A. Sears. 2005. Scale transition theory for understanding mechanisms in metacommunities. Pages 279–306 in M. Holyoak, M. A. Leibold, and R. D. Holt, editors. Metacommunities: spatial dynamics and ecological communities.

Chesson, P. 2007. Quantifying and testing species coexistence mechanisms. In F. Valladares, editor. Unity in Diversity: Ecological reflections as a tribute to Margalef. In press.