# Perturbation Analysis of Nonlinear Matrix Population Models

By Caswell, Hal | Demographic Research, January-June 2008 | Go to article overview

# Perturbation Analysis of Nonlinear Matrix Population Models

Caswell, Hal, Demographic Research

Abstract

Perturbation analysis examines the response of a model to changes in its parameters. It is commonly applied to population growth rates calculated from linear models, but there has been no general approach to the analysis of nonlinear models. Nonlinearities in demographic models may arise due to density-dependence, frequency-dependence (in 2-sex models), feedback through the environment or the economy, and recruitment subsidy due to immigration, or from the scaling inherent in calculations of proportional population structure. This paper uses matrix calculus to derive the sensitivity and elasticity of equilibria, cycles, ratios (e.g., dependency ratios), age averages and variances, temporal averages and variances, life expectancies, and population growth rates, for both age-classified and stage-classified models. Examples are presented, applying the results to both human and non-human populations.

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1. Introduction

The goal of this paper is to present a new approach to the perturbation analysis of nonlinear population models, providing the sensitivity and elasticity of a wide range of demographic quantities.

1.1 Perturbation analysis

The output of any model depends on the values of its parameters. Perturbation analysis asks how changes in one or more parameters will affect the output. Widely used by demographers of all types, perturbation analysis is important in evolutionary biology (where the perturbations are produced by mutation or recombination), conservation, pest control, and and population policy (where the concern is with management manipulations), and sampling theory (the parameters to which a quantity is most sensitive are those that must be estimated most precisely). The results of perturbation analysis are often expressed as sensitivities (the sensitivity of y to x is the derivative dy/dx) and elasticities (the elasticity of y to x is (x=y)dy/dx).

The perturbation analysis of linear demographic models has focused on the sensitivity of λ or r (e.g., Keyfitz 1971, Hamilton 1966, Caswell 1978, Baudisch 2005), of the stable age or stage distribution (Coale 1957, 1972, Caswell 1982), and of life expectancy (Key-fitz 1977, Pollard 1982, Vaupel 1986, Vaupel and Romo 2003, Caswell 2006). The perturbation analysis of short-term transient dynamics has recently been presented (Caswell 2007a).

This paper presents new methods for perturbation analysis of nonlinear models, using matrix calculus. It uses those methods to analyze the sensitivity of a selection of important nonlinear models: density-dependent, environment-dependent, subsidized, two-sex, and proportional structure models.

Plant and animal demographers have recognized the need for sensitivity analysis of nonlinear models (e.g., Grant and Benton 2000, 2003), but until now there has been no general perturbation analysis for such models. Instead, most studies have relied on numerical calculations using difference quotients. This is a notoriously unstable method for computing derivatives, requires lots of computation, and provides no analytical insight into the structure of the sensitivities.

Yearsley et al. (2003) used an analytical approach, analyzing a model with a known characteristic equation, in which the vital rates depend only on total density. They derived the sensitivity of the equilibrium density by implicit differentiation of the characteristic equation. There exists a related but distinct set of results in evolutionary biodemography that analyze the sensitivity of the invasion exponent in density-dependent models. The sensitivity of this exponent to a parameter is the selection gradient on that parameter; together with a measure of genetic variation it determines the rate of phenotypic change under selection. The sensitivity of the invasion exponent to parameter changes has been shown to be equal to the sensitivity of a kind of weighted average population density to those parameter changes (Takada and Nakajima 1992, 1998, Caswell et al.

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