To evaluate the effect of persistently fast growers on population growth, an integral projection model (IPM) was constructed that included past growth. The basis for this model was an age- and size-dependent IPM (Childs et al. 2003; Ellner & Rees 2006), which was adapted such that population dynamics depended on size (stem length) and past growth rate (in stem length). In an age-size IPM, population dynamics are described as (Ellner & Rees 2006):
- (eqn 1a)
- (eqn 1b)
in which x is size at time t, y is size at time t + 1 and Ω the set of all possible sizes. The probability density function n a (y,t) describes the state of the population of individuals of age a. F a (x,y) and P a (x,y) are the fecundity and survival-growth function, respectively, and m is the maximum age. Applying the midpoint rule (Easterling, Ellner & Dixon 2000), this model can be transformed into a set of large transition matrixes (one transition matrix per age), where Ω is now divided into very narrow size classes. In an age-size IPM, the functions F a (x,y) and P a (x,y) are based on continuous functions that relate vital rates (growth, survival and reproduction) to both size and age. Lifetime past growth rate (p) can be expressed as a function of size (x) and age (a) as . Therefore, in a linear example case, vital rate (v) can be related to size and past growth as:
- (eqn 2)
where α, β and γ are regression coefficients. Note that an age-term (δ*a) can be added to (eqn 2) in case age per se explains (additional) variation in vital rate v. As (eqn 2) is a function of size and age, incorporating such a function in F a (x,y) and P a (x,y) allows applying the analyses outlined by Ellner & Rees (2006). A detailed explanation and R code are included in Appendix S1 in the Supporting Information.
We used regression equations for vital rates to construct F a (x,y) and P a (x,y). We assumed no pollen limitation, and therefore, we based the model on female palms only. As we lacked data on the influence of past growth rate on the performance of seedlings and individuals <10 cm stem length, these size classes were not included. New stemmed individuals entered the model with a size distribution based on the growth rate distribution of individuals smaller than 10 cm, which was determined from the internode data, see Appendix S1. To construct F a (x,y), we averaged values of the three annual reproduction probability functions and multiplied this by the seed production function and by the average number of seedlings per seed. We applied a normal distribution in P a (x,y) to describe the variation in growth rate. As growth variation was independent of stem length or past growth rate, we used mean variation. As we did not find a significant contribution of size to survival (see 'Results'), we used the average adult survival in P a (x,y). Maximum size in the model was 1.1 times the maximum observed stem length, minimum size 0.9 times the minimum observed stem length. The maximum age was taken to be 30 years, as very few individuals exceed this age. All individuals smaller than 11 cm were considered to be non-reproductive, as we did not observe any smaller individual with flowers or fruits. Two hundred points were used when applying the midpoint rule to construct the transition matrix. To verify whether including persistent growth differences in IPMs changes population growth rate, we also constructed IPMs based on regression equations with only size as an explanatory variable.