The Generalized Ronce-Kirkpatrick Model with Predation

Ronce and Kirkpatrick (2001) derived the model on which our first model (eqq. [1], [2]) is based, and our equation for evolution is identical to theirs. Our ecological model differs because we include nonlinear density dependence and mortality due to predation. It is useful to express the ecological model in terms of more familiar ecological and evolutionary parameters to point out a difference in the normalization that we employ, because we include nonlinear density dependence. Two habitats (denoted by , 2) have equal area, with prey at i p 1 density ni in habitat i (table A1 summarizes variables and parameters). The trait under selection has a normal distribution with mean in habitat i (phenotypic variance is assumed constant and equal in both habitats). An 2 z̄ j i p individual of phenotype zi in habitat i has fitness (per capita growth rate) of v 2 r [1 (n /K) ] g(z z ) /2 0 i i opt, i , where is the optimum phenotype in habitat i, K is carrying capacity, r0 is intrinsic growth rate at low d z i opt, i density of an individual with the optimum phenotype, g is the strength of stabilizing selection, v is a density dependence parameter (all assumed equal across habitats), and di is per capita predator-induced mortality in habitat i. (This expression with and matches one in Ronce and Kirkpatrick 2001.) The form of v p 1 d p 0 i density dependence is theta logistic (Sæther and Engen 2002; Filin et al. 2008). The larger v is, the weaker density dependence is at low density. Without predation, a population with its mean phenotype at the optimum for habitat i has a maximal growth rate of (the second term reflects mean fitness depression due ′ 2 r p r gj /2 0 p to phenotypic variation around the optimum). We scale time by maximal growth rate ( ) and normalize ′ T p r t prey densities with . We express mean phenotypes as normalized deviations from the habitat ′ 1/v N p (n /K)(r /r ) i i 0 optima ( , where is genetic variance, which is used to normalize phenotypes as it determines 2 ̄ Y p Fz z F/j j i i opt, i g g their rate of change). The per capita rate of dispersal between the two habitats is m (i.e., the probability an individual moves in a short time interval Dt is mDt), again scaled by . The dimensionless parameters of the ′ r model are (1) , strength of selection; (2) , difference in phenotypic optima; and 2 ′ G p j g/r H p (z z )/j g opt, 2 opt, 1 g (3) , movement rate. Predator-induced mortality is scaled by , so (this can be a fixed ′ ′ ′ M p m/r r D p d /r i i parameter or dynamic; we consider both in the main text). With these rescalings, the equation for prey population dynamics in habitat i is