Hyperstate matrix models: extending demographic state spaces to higher dimensions

Summary Demographic models describe population dynamics in terms of the movement of individuals among states (e.g. size, age, developmental stage, parity, frailty, physiological condition). Matrix population models originally classified individuals by a single characteristic. This was enlarged to two characteristics in vec‐permutation models and the closely related megamatrix models. However, it has been recognised that the interplay of more than two characteristics can affect the population dynamics. Here, we present a framework, called hyperstate matrix model , in which individuals may be classified by any number of characteristics, using the generalisation of the vec‐permutation approach to hypermatrices. These models are constructed from a simple block‐diagonal matrix formulation of the movement of individuals among each dimension of the i ‐state. This framework provides a step‐by‐step construction and makes available the usual demographic analysis developed for classical matrix models. In particular, we derive a general formula for the sensitivity of any output of the hyperstate matrix model, to any vector of parameters. In spite of the technicalities underlying these models, implementation is straightforward and we provide the MATLAB code to carry it out. We apply this approach to a three‐dimensional example in which individuals are classified by developmental stage, age and heterogeneity classes. The analysis of this model provides insights into how the heritability of the heterogeneity classes affects the long‐term growth rate of the population. As the questions in conservation biology become more sophisticated and data on threatened species become more detailed, multiple dimensions in demographic models will become increasingly important. Hyperstate matrix methods will make such analyses possible and directly applicable to conservation and population management.