Appendix A. Exact and approximate stream-scale models of periphyton dynamics.
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We begin with the exact model of Eq. 3 in the main text:
From the definition of covariance:
(see, for example, Rice 1995 or Chesson et al. 2005). By definition, the scale transition in periphyton growth rate Tg is the difference between the full spatial model and the mean-field model:
Similarly, the scale transition in the functional response Tf is
By substitution into Eq. A.1:
which is Eq. 4 in the main text, where Cov(f(R),C) = TfC.
To find an approximate stream-scale model, we use the delta method to approximate and , by approximating g(R) and f(R) as second order Taylor polynomials (see, for example, Rice 1995, appendix in Chesson 1998, or Chesson et al. 2005):
To approximate the covariance term, recall that
Substituting the approximations
into Eq. A.8 we obtain
where f'(R) is the first derivative of f(R). The approximation in Eq. A.11 has the same order of accuracy as the approximations in Eq. A.6 and A.7 (Chesson et al. 2005). By substitution of Eq. A.6, A.7 and A.11 into Eq. A.2 and A.1, we obtain:
which is Eq. 5 in the main text.
Chesson, P. 1998. Spatial scales in the study of reef fishes: a theoretical perspective. Australian Journal of Ecology 23:209215.
Chesson, P., M. J. Donahue, B. A. Melbourne, and A. L. Sears. 2005. Scale transition theory for understanding mechanisms in metacommunities. Pages 279306 in M. Holyoak, M. A. Leibold, and R. D. Holt, editors. Metacommunities: Spatial Dynamics and Ecological Communities. University of Chicago Press, Chicago, Illinois, USA.
Rice, J. A. 1995. Mathematical Statistics and Data Analysis, Second Edition. Duxbury Press, Belmont, California, USA.