Appendix A. I used the "Partial patch recovery" model of Wright et al. (2004) to link the population dynamics of beaver to the dynamics of beaver-modified patches. This model describes a landscape consisting of 4 patch types that can potentially be affected by an ecosystem engineer. Patches designated A (active) are currently in use by an ecosystem engineer. Patches designated D (degraded) have been abandoned by the ecosystem engineer and cannot be colonized. Patches designated P (partially recovered) can be recolonized but have not reached their pre-disturbance state. Patches in the F state are fully recovered. In the context of a beaver-modified landscape, A represents active ponds, D represents young meadows, P represents older meadows with recovered food resources around the perimeter, and F represents forested riparian zone.
If we consider the variables A, D, P, and F to represent the fraction of the modifiable landscape in each state, we can represent the dynamics of the system by the following set of differential equations:
The parameter n represents the per patch production rate of new colonists. The parameter ζ is the degree of discrimination for (ζ > 1) or against (ζ < 1) partially recovered sites relative to fully recovered sites. The parameter δ is the rate at which active patches (A) decay into degraded patches (D). The parameter ρ is the rate at which degraded patches (D) recover into partially recovered patches (P). The parameter r is the recovery rate of patches from the partially recovered state (P) to the fully recovered state (F).
Wright, J. P., W. S. C. Gurney, and C. G. Jones. 2004. Patch dynamics in a landscape modified by ecosystem engineers. Oikos 105:336–348.