# Demography and Dispersal: Calculation and Sensitivity Analysis of Invasion Speed for Structured Populations

Neubert and Caswell provide a brief introduction to integrodifference equations before introducing stage structure to the typical equation. The stage structure is easily incorporated using a density-dependent projection matrix for population growth and a matrix of the same shape containing dispersal kernels which represent the dispersal as individuals transition from one stage to the next. In order to determine the speed of spread in the stage structured model they use the linear conjecture (much like the non stage-structured integrodifference equations) to simplify the rate of spread of a population to depend on the linearization of population growth near n=0. Using this simplification, and a few variable substitutions, the authors generate the Matrix **H**(s) which incorporates population growth and the m.g.f.s of dispersal for each class transition where s determines the shape of the leading edge of the wave. The wave speed depends on the dominant eigenvalue of **H**(s) and therefore on the value of s itself. The asymptotic wave speed can be determined by finding the minimum speed (c*) given any value of s. They also provide analytical solutions for the sensitivity and elasticity of c to changes in demographic parameters and dispersal kernels.

The authors then provide two examples of these methods applied to data from two different species of plant. In the first species (Dipsacus sylvestris) dispersal occurs as seeds are produced, while in the second (Calathea ovandensis) dispersal occurs prior to seeds transitioning into plants. In both plant species wave speed tends to increase with the overall population growth rate. The sensitivity and elasticity of c are also highly correlated with the same measures for total population growth rate in both species. In the case of the second species, which is generally ant dispersed, a rare fall into a river may produce much longer distance dispersal. They demonstrate that the rate of spread is largely dependent only on the longest distance dispersal and that c is not very sensitive to changes in the frequency of long-distance dispersal events (even in orders of magnitude). Finally they clearly demonstrate that non-stage structured models, parameterized for both species, clearly over-estimate the speed of spread as compared to the stage structured models. If these stage structured models are in fact more accurately representing the wave speed then this difference seems sufficient motivation for the use of stage structured models.

Neubert, Michael G., and Hal Caswell. “Demography and dispersal: calculation and sensitivity analysis of invasion speed for structured populations.” *Ecology* 81.6 (2000): 1613-1628.