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.

Population dynamics of the cactus Mammillaria gaumeri: an integral projection model approach

“…Population dynamics: the integral projection model An integral projection model describes how a population structured by a continuous individual-level state variable changes in discrete time intervals (Easterling et al 2000).…”

“…The fecundity entries in a matrix model are typically along the top row of the matrix, representing the contributions of each age class to the stage class containing newborns (e.g., the smallest size class). The analog for the integral model would be a function f(x, y) that is positive for large x (parents) and small y (offspring) and is zero in other cases (Easterling et al 2000). The integral projection model has a dominant eigenvalue k that represents the population's asymptotic growth rate under biological assumptions that are no more restrictive than those required in the matrix model.…”

“…The integration must run over all possible sizes, not just those observed in the data. We set the limits of integration based on the variance of growth (described extensively in Easterling et al 2000). The lower limit of integration was set at the minimum observed size x min minus three standard deviations of the growth increment at x min , and the upper limit was set at the maximum observed size x max plus three standard deviations of the growth increment at x max .…”

“…Integral projection models are extensions of matrix models that yield similar outputs (i.e., population growth, sensitivity, elasticity), but use continuous relations of vital rates (growth, survival, reproduction) versus size (or age) as inputs instead of category-specific values to calculate vital rates (Easterling et al 2000;Ellner and Rees 2006). We studied the rare globular cactus Mammillaria gaumeri (Britton & Rose) Orcutt, using this approach.…”

Ruderality in extreme‐desert cacti? Population effects of chronic anthropogenic disturbance on Echinocereus lindsayi

“…No evidence for density-dependent population regulation was found (ESM), so it was not included in the model. The four functions were then incorporated into an integral projection model (IPM), which projects a population in discrete time through the function (Easterling et al 2000;Ellner and Rees 2007) where x is the size of an individual plant at time t and y is its size at time t ? 1; e is the probability that a seed gets established (i.e., that it germinates and survives till March); k is a variable that has a value of 1 if the year is appropriate for establishment with a probability p y and 0 otherwise; X is the whole interval of sizes that a plant may have, K i (x,y,e,k) is the kernel, a function describing the population dynamics for the ith-type year (i = 1, 2, 3 or 4, chosen randomly; see below for details on stochastic simulations), and n t (x) is a function describing the size-structure of the population at time t, and that has the property that R r q n t ðxÞd x equals the number of individuals at time t that have a size between q and r. Thus, the population size is N t ¼ R X n t x ð Þd x.…”