Abstract
Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of periodic matrix products. The perturbation analysis of periodic models must trace the effects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individuals are classified by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.
Highlights
Periodic matrix models are often used to study cyclical temporal variation or when multiple processes operate within a single projection interval
We present a general approach to the perturbation analysis of both linear and nonlinear periodic models
We present a new approach to perturbation analysis of periodic models, taking advantage of the ability of matrix calculus to compute derivatives of scalar, vector, or matrixvalued functions of scalar, vector, or matrix-valued arguments
Summary
Periodic matrix models are often used to study cyclical temporal variation (seasonal or interannual) or when multiple processes (e.g., demography and dispersal) operate within a single projection interval. The result is a periodic model that uses the vec-permutation matrix to generate a block-structured projection matrix over the entire interval This approach was introduced by Hunter and Caswell (2005b) to study classifications by stage and spatial location (see applications by Ozgul et al 2009, Goldberg et al 2010, Strasser et al 2010). Shyu et al (in prep.) have developed a nonlinear seasonal model of an invasive plant to account for the timing of both density effects and management actions within the year In such models, cyclic dynamics can be produced both by the environmental periodicity and the nonlinearities (e.g., Cushing 2006)
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