SENSITIVITY ANALYSIS OF PERIODIC MATRIX MODELS

Periodic matrix models are used to describe the effects of cyclic environmental variation, both seasonal and interannual, on population dynamics. If the environmental cycle is of length m, with matrices B-(1), B-(2),..., B-(m) describing population growth during the m phases of the cycle, then population growth over the whole cycle is given by the product matrix A = B-(m)B-(m-1)...B-(1). The sensitivity analysis of such models is complicated because the entries in A are complicated combinations of the entries in the matrices B-(i), and thus do not correspond to easily interpreted life history parameters. In this paper we show how to calculate the sensitivity and elasticity of population growth rate to changes in the entries in the individual matrices B-(i) making up a periodic matrix product. These calculations reveal seasonal patterns in sensitivity that are impossible to detect with sensitivity analysis based on the matrix A. We also show that the vital rates interact in important ways: the sensitivity to changes in a rate at one point in the cycle may depend strongly on changes in other rates at other points in the cycle.