We modeled the population dynamics of M. acantholoba along the successional gradient through an integral projection model (IPM; Easterling, Ellner, & Dixon, 2000). IPMs allow analyzing the dynamics of populations structured by continuous stage variables (Ellner, Childs, & Rees, 2016). These are iterative models that describe the change of the population structure through a function k , called the kernel. They are represented through the equation

where n_t ₊₁(y ) is the size structure of the population, consisting of the number or individuals of size y (log-height, m) at successional age (years since plot abandonment) t+ 1, n_t (x ) is the size structure of individuals of size x at successional age t ,X is the range of all observed individual sizes, andk_t (y , x ) is the kernel function at time t , as explained below. Note that in this study we are considering a kernel that changes from one time unit to the next as the dynamics changes along succession.

where s (x , t ) is the probability of survival of an individual of size x at age t , g (x ,y , t ) is the probability that an individual of sizex at age t has of changing to a size y at aget +1, f ₁(x , t ) is the reproduction probability of an individual of size x at aget , f ₂(x , t ) is the average number of fruits an individual of size x has at successional aget , f ₃(x , t ) is the average number of seeds in a fruit of a size x individual at aget , f ₄(t ) is the average recruitment probability of a seed at age t , andf₅ (y ) is the size distribution of recruits originated from these seeds (Fig. 1).

To include resprouting in the analysis, we include the additional functions f ₆(t ), which represents the number of individuals recruited through resprouting at successional aget , and f ₇(y ), which represents the size y of these resprouts at the time of their incorporation into the population. Thus, the IPM used has the form

Each vital rate was modeled separately. To this end, we used generalized additive mixed models (GAMMs). For the binary variables, s andf ₁, a logit link function and a binomial distribution were used; for the continuous function g , we used an identity link function and a normal distribution; and for the count variables, f ₂ and f ₃, a log link function and a negative binomial distribution. In all these models, we considered as random effects the study plot, and for those vital rates for which more than one census year were available, the individual (nested in the plot) and the census year. The fitting of these models was performed in R (v. 4.1; R Core Team, 2023) using packages gamm4 (Wood and Scheipl, 2015) and brms (Bürkner, 2017; Bürkner, 2018). We considered the most complex models possible that could be fitted given the limited sample size, under the assumption that this type of models best reflects the reality of a complex system (Barr, Levy, Scheepers & Tily, 2013). Among them, the best-supported model was chosen through the sample-corrected Akaike’s information criterion (AICc) using the AICcmodavg package (Mazerolle & Mazerolle, 2017). In the case of the functions f ₂ andf ₃, in order to prevent the selection of models that extrapolated to biologically unrealistic values, we selected among those models that predicted a maximum value of less than twice the observed maximum of fruits and seeds, respectively.

Since no establishment data were recorded in the field, we estimated recruitment probability, f ₄, based on functionsf ₁, f ₂,f ₃ and the observed size structures,n _obs(x , t ), using inverse estimation as in González et al. (2016). For this purpose, we estimated, for each t , the total number of seeds produced per plot by individuals, distributed according to their size structure, as

Next, we obtained the total number of recruits from the following aget +1. Thus, f ₄(t ) was estimated as the proportion between the total number of recruits andb (t ), assuming a direct relation between the observed total number of recruits and the estimated number of seeds. Following this procedure, we obtained f ₄(t ) values for successional ages 2, 3, 4, 10, 24, 26, 27 and 64. With these values, we fitted a spline with time as the explanatory variable.

The identification of recruits (i.e., individuals originating from the germination of local seeds) incorporated into the population at a given year was inferred indirectly from the height and number of stems of individuals in the understory. We did this by performing a k-means clustering procedure, with k = 6 given by the within-clusters summed squares optimization method (Kriegel, Schubert & Zimek, 2017), using the size of individuals at their first record, and the individuals belonging to the group with the smallest sizes were classified as recruits. From the height of these individuals, the function describing recruit sizes, f ₅(y ), was calculated as the associated empirical probability density. To identify resprouts, the understory individuals were filtered by considering only those with 3 or more stems in their first observation; this criterion is similar to that used by Lebrija-Trejos (2004). From the number and heights of identified resprouts entering the population at each successional age, we obtained the function that describes the introduction of resprouts,f ₆(t ), and the associated empirical probability density of resprout height,f ₇(y ). Note that, since very few recruits and resprouts were recorded over the study years, we assume that bothf ₅(y ) and f ₇(y) do not change throughout succession.