Sensitivity analysis of equilibrium in density-dependent matrix population models

We consider the effects of parameter perturbations on a density-dependent population at equilibrium. Such perturbations change the dominant eigenvalue lambda of the projection matrix evaluated at the equilibrium as well as the equilibrium itself. We show that, regardless of the functional form of density dependence, the sensitivity of lambda is equal to the sensitivity of an effective equilibrium density (N) over tilde, which is a weighted combination of the equilibrium stage densities. The weights measure the contributions of each stage to density dependence and their effects on demography. Thus, (N) over tilde is in general more relevant than total density, which simply adds all stages regardless of their ecological properties. As log lambda is the invasion exponent, our results show that successful invasion will increase (N) over tilde, and that an evolutionary stable strategy will maximize (N) over tilde. Our results imply that eigenvalue sensitivity analysis of a population projection matrix that is evaluated near equilibrium can give useful information about the sensitivity of the equilibrium population, even if no data on density dependence are available.