Size-Specific Sensitivity: Applying a New Structured Population Model

Abstract

Matrix population models require the population to be divided into discrete stage classes. In many cases, especially when classes are defined by a continuous variable, such as length or mass, there are no natural breakpoints, and the division is artificial. We introduce the “integral projection model,” which eliminates the need for division into discrete classes, without requiring any additional biological assumptions. Like a traditional matrix model, the integral projection model provides estimates of the asymptotic growth rate, stable size distribution, reproductive values, and sensitivities of the growth rate to changes in vital rates. However, where the matrix model represents the size distributions, reproductive value, and sensitivities as step functions (constant within a stage class), the integral projection model yields smooth curves for each of these as a function of individual size. We describe a method for fitting the model to data, and we apply this method to data on an endangered plant species, northern monkshood (Aconitum noveboracense), with individuals classified by stem diameter. The matrix and integral models yield similar estimates of the asymptotic growth rate, but the reproductive values and sensitivities in the matrix model are sensitive to the choice of stage classes. The integral projection model avoids this problem and yields size‐specific sensitivities that are not affected by stage duration. These general properties of the integral projection model will make it advantageous for other populations where there is no natural division of individuals into stage classes.

Evolution of flowering strategies inOenothera glazioviana: an integral projection model approach

“…Given the kernel we can calculate the various demographic quantities, such as the nite rate of increase, l. Using the methods described in Easterling et al (2000) we nd l = 1.041, in excellent agreement with the value obtained by Kachi & Hirose (1985) using an individualbased simulation (l = 1.04). The right eigenvector corresponding to the dominant eigenvalue gives the stable size distribution, w( y).…”

“…The integral projection model can be used to describe how a continuously size-structured population changes in discrete time (Easterling et al 2000). The state of the population is described by a probability density function, n(x,t), which can intuitively be thought of as the proportion of individuals of size x at time t. The integral projection model for the proportion of individuals of size y at time t 1 1, one year later, is then given by…”

“…These methods eliminate the need to divide data into discrete classes, without requiring any extra biological assumptions (Easterling et al 2000). Equally importantly, the data required to parameterize integral projection models are no more complicated than those needed to construct an individual-based or matrix model of the same system.…”

“…Equally importantly, the data required to parameterize integral projection models are no more complicated than those needed to construct an individual-based or matrix model of the same system. Underlying the integral projection model is a stochastic individual-based model, making the integral projection model a good alternative to simulation (Easterling et al 2000). Integral projection models have many properties in common with matrix models, for example they allow the calculation of the stable size distribution, population growth rate l, and sensitivities and elasticities of l.…”

Simple Adaptive Control for Positive Linear Systems with Applications to Pest Management

“…where, for simplicity say, Ω is the closure of some bounded set in R n , n ∈ N. IPMs were introduced by [47] (see also [48,49,50]) as a tool for population modelling where the discrete age-or stage-class variable of a matrix model is replaced by a continuous variable, such as stem width of a plant species. …”