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 λ 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 λ is equal to the sensitivity of an effective equilibrium density inline image, 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, inline image is in general more relevant than total density, which simply adds all stages regardless of their ecological properties. As  log λ is the invasion exponent, our results show that successful invasion will increase inline image, and that an evolutionary stable strategy will maximize inline image. 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.