Ecological Archives E084-076-A1

Eric W. Seabloom and Shane A. Richards. 2003. Multiple stable equilibria in grasslands mediated by herbivore population dynamics and foraging behavior. Ecology 84:2891–2904.

Appendix A. A description of the conditions promoting herbivore persistence.

Here we elucidate conditions promoting herbivore persistence. Suppose herbivores always attack patches of type x (x = b, a, p) and females are not male-limited, then the mean number of offspring a female will produce in her lifetime is rx(1-mx)/mx. If this expectation is greater than or equal to two then the population can persist on the patch type, otherwise it will eventually be driven to extinction. 

Now, suppose herbivores always forage in patches that are in state i (i.e., there is no feedback between territory state, succession and foraging). In this case the fraction of territories occupied at equilibrium would be

,
(A.1)

where

(A.2)

is the probability a herbivore survives a time step and,

(A.3)

is the probability a herbivore chosen at random is female, survives a time step, and has an offspring.  Here we have assumed the probability offspring successfully settle in a territory is proportional to the fraction of unoccupied patches. 

Deriving analytic conditions for herbivore persistence for the full dynamic model is difficult; however, numerical analyses suggest that typically there is a unique stable equilibrium distribution and herbivore persistence is enhanced with increasing territory size (N), increasing reproduction (rx), decreasing mortality (mx), and increasing colonization rates of the preferred vegetation type, which in this paper is annuals (Ca). 



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