Selection pressures on vital rates in density-dependent populations

For density-independent populations, the sensitivity of population growth rate to changes in individual vital rates indicates the strength of selection on different parts of the life history. Here I show how this approach may be extended to any density-dependent and/or stochastic population model, including those that show cyclic, quasi-periodic and chaotic dynamics. One calculates the influence of individual vital rates on the outcome of competition between two almost identical life histories. The outcome of this competition is determined by the invasion exponent theta introduced by Rand. This is the Lyapunov exponent of the linearized system describing the invasion of a population With one life history by a variant type with another. Demographic sensitivities are given by the partial derivatives of theta with respect to the individual vital rates of the invading type. The density-independent analysis is a special case of this general framework. Sensitivities can often be obtained analytically when the population has a stable equilibrium point, and can be calculated by numerical differentiation in other cases. One can also use the methodology to examine selection pressures on the parameters describing density dependence and, if there are trade-offs between vital rates, it can be used to determine optimal life histories. A two age-class example shows that the occurrence of nonlinear dynamics can markedly alter selection pressures on a life history from those which operate when the population has a stable equilibrium point.