R code for Demographic consequences of changing environmental periodicity

These R scripts contain the code to replicate the analyses performed in<em> Demographic consequences of changing environmental periodicity</em>, Ecology. <br> <strong>Vital-rate estimation</strong> We used the demographic data of each species to model periodic differences in vital rates for each life-cycle stage using generalized linear models and mixed models (GLMs for the deterministic vital rates in dewy pines and GLMMs for all other vital rates). More specifically, we tested for the effect of season on marmot and meerkat vital rates and of post-fire habitat states (time since fire, TSF) on dewy pine vital rates. For all three species, we estimated stage-specific survival and probability of transition to another stage using a binomial distribution, and reproductive output using a Poisson distribution. In addition, we estimated meerkat helper emigration probability and dewy-pine flowering probability using a binomial distribution. In addition to fixed effects, we incorporated year random effects in all appropriate models to model year-specific differences among season-specific average vital-rates (for marmots and meerkats) or among vital-rate averages (for dewy pines). For the meerkats and dewy pines, we also incorporated the fixed effect of density on vital rates. For each modeled vital rate, we first selected the best random-effect structure where appropriate (i.e., testing whether a random effect on the average vital rate and the slope between seasons outperformed a random effect on the average vital rate only). We then selected the best fixed-effect variables using the Akaike Information Criterion corrected for small sample size (AICc). When the difference in AICc values was not significant (dAICc&lt;2), we picked the model with fewer parameters or the most biologically relevant. For models using a Poisson distribution, we tested for over- and under-dispersion in the best model and fitted over- and under-dispersed models with a quasi-Poisson distribution when necessary. <strong>Population projections</strong> We used the predictions of the vital-rate models described above to parameterize periodic matrix population models (MPMs) and project the population dynamics for 100 years according to two scenarios—control and perturbed—to assess the effect of changes in vital-rate periodicity on the stochastic growth rate (log <em>λ</em>_S), the variance of 100 annual growth rates (var(log <em>λ</em>)), and the probability of quasi-extinction (<em>p</em>_qext, i.e., the ratio of simulations leading to quasi-extinction out of 500 simulations, with a threshold set at 15% of the minimum observed abundance or number of reproductive individuals for the marmot and meerkat populations, and at 50% of the minimum observed aboveground or seed bank abundance for the dewy pines). <strong>Yellow-bellied marmot and meerkat (seasonal dynamics)</strong> For the meerkats and marmots, we projected population dynamics under changing patterns of vital rates for which the year random effect in the GLMM was applied both on the mean vital-rate estimate (i.e., the model intercept) and the difference between seasons. That is, for the marmots, we assessed the population responses to changes in the seasonal patterns of yearling and non-reproductive and reproductive adult survival. For the meerkats, we did so for subadult, helper, and dominant survival, helper emigration, transition from helper to dominant, and helper and dominant recruitment. We used the above models to predict season- and year-specific vital rates and parameterize periodic matrix population models (MPMs) for each season. We then simulated population dynamics by projecting MPMs representing low- (LS) and high-seasonality (HS) years for each of the vital rates. Half of all years in which a vital rate was estimated were considered as LS and the other half as HS. That is, for a given vital rate, we defined the threshold between LS and HS years as the 50th percentile of all year-specific absolute differences between seasons. In addition, we used a control scenario, where we projected the population dynamics using MPMs representing both high- and low-seasonality years indistinctly. We performed 500 simulations, each starting with the same population vector, and projected the population dynamics for 100 years. For each simulation, we randomly selected 100 years corresponding to each scenario (LS, HS, or control). In each step of the simulation, all vital rates were predicted based on the same randomly selected year. This allowed us to maintain within-year vital-rate correlation. The predicted vital rates were then used to build the corresponding period-specific MPM.<br> <strong>Dewy pine (Multi-year habitat succession and periodicity in fire regimes)</strong> For dewy pines, we simulated two distinct types of fire regimes, each with two frequencies: (1) periodic burning occurring systematically every 15 or 30 years, and (2) stochastic fires occurring on average every 15 or 30 years. In all simulations, dewy pines transition deterministically through the first four post-fire states, TSF₀ to TSF₃. Under periodic fires, once in the fifth state, TSF_&gt;3, the population remains in that state until the next fire (15 or 30 years after the previous fire) and then transitions to TSF₀. Under stochastic fires, the population can transition from TSF_&gt;3 to TSF₀conditional on fire frequency (<em>p</em>). This is done using a Markov-chain approach. In addition to the different periodic patterns in fire regimes, we perturbed periodic patterns in habitat succession in natural heathlands by introducing an additional human disturbance (i.e., using vital rates from populations under a browsing-induced disturbance) first only in the years of the last post-fire state (TSF_&gt;3), and then increasingly in the previous states (i.e., in TSF_{3 and &gt;3}, in TSF_{2, 3, and &gt;3}, etc.) until all post-fire states of a natural population were perturbed. We used TSF-specific vital-rate predictions from our models and rates on seed-bank dynamics to build MPMs for each TSF. We used these MPMs to project population dynamics for 100 years using 500 simulations, in each iteration randomly sampling among MPMs describing one of the two natural populations. In addition, for each iteration in the stochastic post-fire state TSF_&gt;3, we randomly sampled a year-specific MPM. We increased browsing pressure by replacing MPMs associated with natural habitat conditions by MPMs parameterized with vital rates estimated from human-disturbed populations, for any given TSF. We then compared scenarios of increasing anthropogenic pressures to the control scenario (i.e., natural populations with no browsing perturbation) in each fire regime. <strong>Population responses to vital-rate periodic patterns under density dependence</strong> For the meerkats and dewy pines, projections of population dynamics incorporated density dependence. That is, at each iteration of the simulations, population density was estimated and used to predict vital rates and parameterize an MPM from these predictions at the next iteration. We then compared density-dependent projections to ones where the density input during parameter estimation was fixed at constant average values obtained from the observed data. <strong>Analysis of the simulations results</strong> For all three systems and for each simulation, we recorded the stochastic growth rate log λ_S. We also investigated the effect of changes in vital-rate patterns on the variance in 100 annual log <em>λ</em>, var(log <em>λ</em>), and the quasi-extinction probability <em>p</em>_qext. We checked the overlap of the distributions of each metric (i.e., the mean and the 2.5th and 97.5th percentiles) across the 500 simulations between two scenarios. We considered a metric to differ between scenarios when 95% of the distributions (i.e., between the 2.5th and 97.5th percentiles) did not overlap. <strong>Comparing population sensitivity to changes in periodic environmental patterns vs. stochastic environmental variation</strong> We compared our results from periodic population models to “classic” assessments of population fitness sensitivity to stochastic environmental variation. To do so, for marmots and meerkats, we computed the stochastic elasticities of the population growth rate to changes in the mean and standard deviation of vital rates. We then calculated the relative importance of the stochastic elasticity of the growth rate due to changes in the variability of vital rates compared to changes in their mean. For dewy pines, as the effects of periodic patterns consisted of changing the sequences of post-fire habitat states, we explored the link between the effects of human-induced disturbances in various post-fire habitat states and the role of these states in shaping population dynamics. We thus used the megamatrix approach to calculate the elasticity of the population growth rate to each post-fire habitat state.

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