## Abstract

The increased frequency and intensity of extreme events are recognized among the most worrisome aspects of climate change. However, despite increased attention from scientists and conservationists, developing and testing general theories and hypotheses on the effects of extreme events on natural populations remains intrinsically challenging.

Using numerical simulations, I tested some of the hypotheses on risk of extinction and population and genetic dynamics in an environment in which both climate (e.g., temperature, rainfall) and point (e.g., fires, floods) extremes occur. A quantitative trait is selected for by a climate variable, but point extremes cause trait-independent massive mortalities.

I found additive effects between age at first reproduction and fecundity on risk of extinction. The extent of population bottlenecks (operationally, the number of years in which a population was at low numbers) was a good predictor of allelic richness for the quantitative trait selected for by the climate. Simple models including basic demographic and vital rates information of the species, along with climate/environmental measures, provided excellent predictions of contemporary risk of population extinction. Mean and minimum population size measured in a 10-year “observation window” were largely the most important predictors of risk of population extinction in the following 10-year “prediction window”.

## 1 Introduction

Extreme events are now recognized among the most worrisome aspects of climate change (IPCC 2007, 2012, Jentsch et al. 2007, One Earth editorial team 2020).

How to define extreme events is challenging, since they may be defined in terms of extreme values of a continuous climate variable (e.g., daily or mean summer temperature, rainfall) on the basis of the available climate record (Gutschick and BassiriRad 2010), or in the form of a “point” or discrete perturbation, such as a hurricane or a storm. The latter category also includes environmental extremes such as fires and floods, which may have catastrophic effects on the affected species and ecosystems despite being relatively frequent in certain habitats. Following Vincenzi (2014), throughout this work I will use the term climate extreme for extreme values of a continuous climate or environmental variable and point extremes for discrete extreme perturbations.

The genetic and demographic determinants of both adaptation to a more extreme environment and contemporary risk of extinction are receiving increasing interest from scientists and conservationists. However, despite the increased research and management focus on extremes, developing and testing general theories and hypotheses on the effects of extreme events remains more intrinsically challenging than for other aspects of climate and environmental change (van de Pol et al. 2017).

First, since the effects of extreme events are largely context-specific right, developing an overarching causal and predictive framework may be overly ambitious (Bailey and van de Pol 2016). For instance, the emergence of adaptations may depend on both the life histories of the species and the recurrence interval, intensity, and nature of extreme events. In addition, the demographic and genetic effects of climate and point extremes (e.g., population crashes, loss of genetic diversity, inbreeding and maladaptation, changes in population age and size structure, disruption in expression of quantitative traits) (Bryant and Meffert 1995, Kirkpatrick and Jarne 2000) are often the result of chance, and are thus not easily predictable or generalizable across species or habitats. Second, climate and point extremes that result in strong demographic and genetic responses are often rare events, whose occurrence may be also difficult to predict. Thus, when available, pre-disturbance empirical data is likely to have been collected by chance, and the studies on the effects of extreme events prone to be mostly opportunistic and anecdotal instead of planned and comprehensive.

Despite the intrinsic difficulties of developing and testing general theories on the effects of climate and point extremes on natural populations, there are a few theory-based predictions on the demographic and genetic consequences of extreme events that may quite general across species and habitats, although with varying degrees of modeling and empirical validation.

The first prediction is that by reducing population size, extreme events directly increase the risk of extinction of the affected populations (Willi et al. 2006, Frankham et al. 2014); common sense, mathematical modeling, and empirical observations suggest that smaller populations are more likely to go extinct than larger populations. After the occurrence extreme events, in some cases the extinction of the focal population is inevitable for numerical reasons (no individuals capable of reproducing survived), unless there is immigration from neighboring—either unaffected or less affected—populations (i.e., “demographic rescue”, Brown and Kodric-Brown 1977, Carlson et al. 2014).

Second, when the change in the environment is abrupt, but not causing mass mortalities, the evolution of fitness-determining traits might occur fast enough to stop population decline and allow population recovery before extinction (“evolutionary rescue”, Bell and Gonzalez 2009). However, population bottlenecks such as those caused by extreme events, especially when repeated over time, are predicted to decrease additive genetic variance and allelic diversity in the affected populations (Bouzat 2010). As adaptive potential tends to increase with genetic variability and genetic drift may overwhelm selection in small populations, the effects of extreme events on genetic variability are predicted to increase both short- and long-term risks of population extinction, with smaller chances of evolutionary rescue (Falconer and Mackay 1996, Willi and Hoffmann 2009).

Third, species may exhibit various adaptations to extreme events, some more predictable and general than others. For instance, extreme, but predictable variations in the flow regime of streams can select for fish life histories, such as spawning and emergence time, that are synchronized either to avoid or exploit the direct and indirect effects of extreme flows (Lytle and Poff 2004). On the contrary, unpredictable flow events such as flash floods may have low direct selective consequences for the affected populations, even though they might induce massive mortalities (Lytle 2000). In these cases, natural selection after the extreme event is predicted to favor individuals with a high capacity for increase in population size (*r* selection, Reznick et al. 2002). Intuitively, when after an extreme event the population is reduced to a few individuals, faster reproduction may be more critically needed than high fecundity, since the latter depends on being able to reproduce. However, the interaction effects between age at first reproduction and fecundity on risk of extinction in an extreme environment have been rarely investigated.

In this work, I test using a simulation approach some of the hypotheses on risk of extinction and population and genetic dynamics in an environment in which both climate and point extremes occur. In previous work, Vincenzi (2014) found that the interaction among climate trend (e.g., increase over years of average summer temperatures), climate variability and probability of point extremes (e.g., fires) had a minor effect on risk of extinction, time to extinction, and distribution of a quantitative trait selected for by climate after accounting for their independent effects. The survival chances of a population were found to decrease with increasing strength of selection, as well as with increasing climate trend (e.g., increasing or decreasing *n*-year moving average temperature, rainfall) and variability.

Here, I focus on prediction more than on inference: prediction of future observables has long been included as an aspect of biological and ecological studies, but as a methodological approach it has been much less prominent than description and statistical and causal inference. The work is also not purely theoretical in scope, but is motivated by the study of the population and genetic dynamics, and risk of extinction of marble trout *Salmo marmoratus* living in Slovenian streams (Vincenzi et al. 2016). Marble trout populations are affected by flash floods in autumn and droughts in summer, and the climate threat to the persistence of marble trout is likely to worsen with climate change; for instance, the increase in intra-annual variability in rainfall is predicted to increase the frequency of flash floods (Janža 2013). However, since there is a need for more general investigations of risk of population extinction and population and genetic dynamics in presence of climate and point extremes, I developed a more widely applicable modeling and conceptual framework.

I modeled an idealized situation of additive genes, a closed population of moderate size with random mating, and variable across simulations—but fixed within simulations—age at first reproduction and expected number of offspring per mating pair, climate trend and variability (i.e., the parameters of the distribution of climate variable that is selecting for a quantitative trait), selection strength, and frequency and severity of point extremes. I assume that point extremes cause massive mortalities, but no trait other than good fortune increases the survival chances of an individual when point extremes occur.

First, I tested whether in an extreme environment the extent of population bottlenecks, which I operationally defined as the total number of years with a “depressed” population size (e.g., ∼ 1/3 of the habitat carrying capacity), could help predict allelic richness at the end of simulation time for the quantitative trait selected for by the climate variable.

Second, I tested whether quantitative trait adaptation was correlated with the increased frequency of theoretically advantageous alleles (i.e., their allelic values are in the same direction of the change in the environment) and whether the genetic dynamics of the populations could be predicted using information on the climate and on the population life histories. Third, I tested for positive interaction effects between age at reproduction and fecundity on risk of population extinction. Finally, I tested whether it is possible to predict the extinction or survival of a population in a 10-year “prediction window” (i.e., contemporary extinction) when measuring or observing some of the environmental characteristics of the habitat (e.g., occurrence of extreme events) and of the population (e.g., population size, age at reproduction) during a 10-year “observation window” that immediately precedes the “prediction window”.

## 2 Material and methods

The model I use in this work is an extension of the model developed in Vincenzi (2014). The choice of parameter values for the present work was informed by the available literature and by the results of Vincenzi (2014).

### 2.1 Overview of the model

I consider a population of monoecious individuals living in a habitat whose population ceiling is *K* (Mangel and Tier 1993). The population is geographically isolated, with neither immigration nor emigration from or to neighboring populations. A single quantitative trait *a* corresponding to its breeding value for a phenotypic trait *z* characterizes the individuals. The population has discrete overlapping generations with *N*(*t*) total individuals, where *t* is time in years. The environment is described by an optimum phenotype *Θ*(*t*) that changes over time as a result from variations in a climate driver such as rainfall or average summer temperature, which selects for the trait *z*. The distance between the optimum phenotype *Θ*(*t*) and the trait *z*_{i}, (*Θ*(*t*) - *z*_{i}), of the individual *i* defines the maladaptation of the individual *i* with respect to the optimum phenotype (or, alternatively, it defines the “extremeness” of the climate event for the individual). Point extreme events such as floods or fires cause non-selective high mortality in the population, i.e. every phenotype has the same chances of surviving the event.

### 2.2 Optimum phenotype

The expected optimum phenotype *μ*_{Θ}(*t*) moves at a constant rate *β*_{μ},_{Θ} over time (i.e. trend), fluctuating randomly around its expected value *μ*_{Θ} (*t*). The optimum phenotype *Θ*(*t*) is randomly drawn at each time step *t* from a normal distribution ** Θ**(

*t*)

*∼ N*(

*μ*

_{Θ}(

*t*),

*σ*

_{Θ}(

*t*)). It is equivalent to consider

*Θ*(

*t*) as both the optimum phenotype and the value of a continuous climate variable (e.g., mean summer temperature or yearly rainfall), and I will use the two terms interchangeably throughout this work.

Mean and variance of the climate variable at time *t* are thus:
where *t*_{ch} is the time at which there is a change (*ch*) in the climate, `I` (•) is and indicator function equal to 1 when • is true and 0 otherwise. Eq. (1) indicates that the directional climate trend steadily increases through time after *t*_{ch} years and that the increase in variability starts after *t*_{ch} years, but stops after *t*_{ch}+*t*_{inc} years to avoid variability building up to unrealistic values over time. With the model formulation of Eq. (1), both the mean and variance of the distribution of the climate variable change over time so as to make the occurrence of events more likely after climate change (i.e., *t* > *t*_{ch}) than before, since after climate change, the realized (i.e., random draw from the statistical distribution of climate) climate is increasingly likely to be in the region of extremes (say, in the right 5% or 2.5% of the Gaussian distribution of climate) of the statistical distribution of climate before climate change.

Point extreme events *E* leading to trait-independent high mortalities occur with annual probability p(*E*_{b}) when *t* < *t*_{ch} (i.e. b - before climate change) and p(*E*_{a}) when *t* > *t*_{ch}.

### 2.3 Quantitative trait and survival

I model the phenotype *z* of an individual *i, z*_{i}, as the sum of its genotypic value *a*_{i} and a statistically-independent random environmental effect *e*_{i} drawn from :
The narrow sense heritability is the proportion of the phenotypic variance present in the population that is explained by the additive genetic variance (i.e. the variance of ** a** in the population).

For an individual *i*, the genetic value *a*_{i} is determined by *n*_{l} freely recombining diploid loci, with additive allelic effects within- and among-loci, that is where is the sum of the allelic values at locus *j*. For computational reasons, I chose *n*_{l} = 10. Allelic values are randomly drawn from a Gaussian distribution with mean of 0 and variance equal to . For simplicity, I did not model either dominance or epistatic variation or other complicating factors such as genotype-environment interaction and linkages. Likewise, I did not model mutation, since previous work has shown that mutation does not appear to have any effect short-term on extinction risk and the evolution of traits on contemporary temporal scales (Vincenzi 2014).

Stabilizing selection is modeled with a Gaussian function (Bürger and Lynch 1995, Zhang 2012), with fitness *W* (Endler 1986) for an individual with phenotypic trait *z*_{i} equal to:
and equivalent in this model to the annual survival probability of individual *i*. The curvature of the fitness function near its optimum increases with decreasing *ω*^{2}; it follows that that the smaller *ω*^{2}, the stronger is selection. Stabilizing selection is usually measured by the standardized quadratic selection gradient *γ*, which is defined as the regression of fitness *W* on the squared deviation of trait value from the mean (Lynch and Walsh 1998). An optimum phenotype in the tails of the distribution is likely to cause a large drop in population size and can be considered an extreme climate event (Fig. 1).

The median *γ* = -0.1 for stabilizing selection found by Kingsolver et al. (2001) corresponds to a value of , where is the variance of the environmental component of the phenotype defined in Eq. (2), when stabilizing selection is modeled using a Gaussian fitness function.

Eq. (3) can be written:
Where . With *γ* = -0.1, , and *h*^{2} = 0.2, the strength of selection *s* is about 0.08.

I assumed that both strength of selection *s* and environmental variance remain constant through time. When a point extreme occurs, the probability of yearly survival of individuals , where is mortality caused by the point extreme event.

### 2.4 Simulations

As this study focuses on the more immediate effects of climate change, the simulations last 250 years. Parents mate at time *t*-1, offspring are born at time *t* and become of age 1 at *t*+1. The sequence of operations is mortality of adults, mating and reproduction, mutation, mortality of offspring. At the start of each simulation, for each individual a value of *a* and *e* (Eq. 2) is randomly drawn from their initial distribution. A population is considered extinct if at any time during the simulation there are fewer than 2 individuals in the population. Parents form mating pairs starting at age *a*_{f} and produce a number of offspring randomly drawn from a Poisson distribution with intensity *λ*_{0}. I did not allow for genetic recombination. Offspring receive for the same locus one allele from each parent.

### 2.4.1 Parameter values

I reduced parameter space by fixing , and *t*_{inc} = 25 years. For the other parameters, I chose range of values that are both realistic for natural populations and instrumental for the main goal of the study, e.g., testing hypotheses on the effects of extreme events on risk of extinction and population and genetic dynamics.

I performed simulations with selection strength *s* equal to either 8 (average selection strength) or 1.1 10^{−2} (moderately strong selection). For the rate of increase in the mean of the climate variable, I used *β*_{μ,Θ} = 0 (base scenario) and 1.5 10^{−2}. I used rates of the increase in the standard deviation of the climate variable *β*_{σ,Θ} from 0 (base scenario) to 1.5 10^{−2}. According to Bürger and Lynch (1995), when the standard deviation of the distribution of the optimum *σ*_{θ} reaches the same order of magnitude as the width *ω* of the fitness function, the population is at risk of going suddenly extinct, with little role played by genetics. Therefore, I chose values of *β*_{σ,Θ} that strongly increase the probability of climate extremes, but did not inevitably make the population go extinct.

I used frequency of point extreme events p(*E*_{a}) of either 5 (no variation before and after climate change, corresponding to a recurrence interval of 20 years), 10 (i.e., recurrence interval is 10 years) or 15 10^{−2} (Table 1). I used moderate mortalities caused by point extremes (simulations with *m*_{E} equal to either 0.3, 0.5, 0.7) and moderate p(*E*_{a}), since with higher mortality induced by point extremes and higher probability of their occurrence the system will be largely driven by the point extremes, with no or little role of genetics and demography in determining population dynamics and risk of extinction.

For the Poisson distribution of the yearly number of offspring per mating pair, I used λ_{0} equal to either 1.0, 1.5, 2.0 or 2.5 and age at first reproduction from 1 to 4 years old with a step of 1. Parameter values are reported in Table 1.

### 2.4.2 Initialization

To reach mutation-selection-drift balance, I first let the population evolve for *t*_{ch} years in an environment in which mean and variance of the distribution of the optimal phenotype ** Θ** are constant. In preliminary simulations it was found that after

*t*

_{ch}∼ 100 years both phenotypic mean and variance remained basically constant. Then, the mean of

**increases for 150 years and variance of**

*Θ***for 25 years. The variance of**

*Θ***was then kept constant up to the end of the simulation.**

*Θ*I started every simulation replicate with 500 individuals. I modeled 10 alleles present in the population for each locus, which value was randomly drawn from a normal distribution . Since I set and , the narrow sense heritability *h*^{2} was around 0.2 at *t* = 1, close to what commonly observed for life-history traits (Lynch and Walsh 1998) and consistent with the Gaussian allelic approximation including only quasi-neutral and adaptive mutations, for which (Lande 1995).

### 2.4.3 Characterization of simulations

At the level of single replicate, to characterize the behavior of the simulated populations I tracked or recorded (among other results): (*a*) whether the population was extinct or still persisting at the end of the simulation time (0 for persistence and 1 for extinction, in the latter case I also recorded the year of extinction). At each time *t*, I then recorded: (*b*) the distribution of the trait *z* in the population and individual maladaptation; (*c*) population size *N* after mortality of adults; (*d*) total number of alleles and allelic frequencies (the latter every 25 years).

For an ensemble of realizations (10 replicates for a fixed set of parameters) I also computed the frequency of population extinction as the number of replicates in which the population went below two individuals during simulation time.

### 2.5 Statistical analysis

I used simulation results as pseudo-empirical data and analyzed them with standard statistical and machine learning modeling. The main focus of the statistical analyses and modeling was more on prediction than on inference or traditional *p*-value hypothesis-testing.

I estimated parameters of Generalized Additive Models (GAMs, Wood 2006), Generalized Linear Models (or Ordinary Least-Square regression models, McCullagh and Nelder 1989), and Random Forests (Breiman 2001a) using as response variable either (*i*) the total number of alleles (overall allelic richness) at the end of simulation time for the populations that persisted, (*ii*) the difference in mean allelic frequency of the top 10% and bottom 10% (according to their allelic value, top 10% were the alleles with the bigger allelic value and *vice versa* for the bottom 10%) of alleles. For (*i*), in particular I tested whether the number of consecutive years (*n_low*, the only variable in the GAM models for which I hypothesized a non-linear effect on the total number of alleles) under a low population size threshold (*tr_low*) contributed to predict the number of alleles at the end of simulation time, in addition to climate variables and vital rates such as expected intensity and frequency of extreme events, variance of the optimum, expected yearly number of offspring per mating pair and age at first reproduction. I calculated *tr_low* using either 150 or 100 individuals as threshold (results were very similar when using either 150 or 100 individuals). For (*ii*), when the climate is changing (i.e., *β*_{μ},_{Θ} > 0), alleles with positive allelic values are expected to be more adaptive than those with negative (or positive, but smaller) allelic values. I tested whether a model including climate variables and vital rates trained on data from 80% of the replicates that did not go extinct could predict the difference in mean allelic frequency at year 250 (end of simulation time) of the top and bottom 10% of alleles according to their allelic values in the test data set (20% replicates that did not go extinct). I chose the top and bottom 10% of the alleles (10 alleles in the top and bottom sets), since the fate of single alleles is more likely to be affected by chance than that of a group of advantageous alleles.

Then, I investigated whether a combination of demographic and environmental factors measured or estimated in a short time window (“observation window”) can predict the risk the risk of extinction of the population in the following years (“prediction window”). First, I set aside a balanced test data set of simulation replicates (50% that went extinct and 50% that survived)—these simulations were not used in any phase of model development and training. Second, since the number of simulations that went extinct was approximately one third of the number of those that survived, I augmented the data set by replicating 3 times the simulations that went extinct and were not included in the test data set.

Third, I fitted GLMs and GAMs with binomial error distribution (i.e., logistic regression), and classification Random Forests (RFs) with population extinction (1) or persistence (0) between (*t*_{ext} - *u*) as response variable, where *t*_{ext} is either (a) the time at extinction for the replicate that went extinct, or (b) a random deviate from a uniform distribution bounded between 20 and 240 for the replicates that survived. *u* is a random deviate from a uniform distribution bounded between 1 and 10 years. This way, I am trying to model extinction or persistence not at a specific time, but in a “prediction window” of 10 years. I used as candidate predictors, as measured in the 10 years (“observation window”) before the “prediction window”, minimum and mean population size *N*, the maximum value of the optimum phenotype, the expected yearly number of offspring per mating pair, age at first reproduction, and maximum and mean distance over the observation window between the mean phenotype and the optimum (i.e., maximum and mean population-level maladaptation, or maximum and mean “extremeness” of the climate). In other words, I wanted to test whether a model including climate and population traits measured over a limited time frame could predict the extinction or persistence of the population in the years immediately following the end of the “observation window”.

For the GAMs and GLMs, I estimated the optimal cutoff given equal weight to sensitivity (probability that the model predicts extinction when the replicate went extinct) and specificity (probability that the model predicts persistence when the replicate persisted). Then, I tested the model by predicting population extinction and persistence on the test dataset using the computed optimal cutoff. For the classification RF models, I directly used the binary prediction (population going extinct or surviving) from the models. I used different modeling approaches because I did not explicitly modeled any mechanism or process that can be hypothesized to lead to extinction (i.e., models are correlative and not mechanistic), and different modeling approaches can give different insights on how contemporary extinctions are predicted (e.g., tree-based models like RFs provide a measure of variable importance for predicting the target variable, Breiman 2001a, and GAMs can model non-linear relationships between predictors and target variable using semi-parametric estimation). For the GAMs and GLMs, I centered and scaled the predictors in order to compare their importance (Schielzeth 2010). As I use realistic variable ranges representing the variability that may be observed in nature, some of the estimated parameters can be compared in terms of effect on a standardized scale. I also fitted the same models using non-standardized predictors to test for possible data leakage between training and testing data sets (results were directionally the same). As interactions among predictors did not improve model performance, I did not include them in the models in order to improve the interpretability of results. I visually checked residuals for variation to model assumptions.

## 3 Results

Results are fully reproducible. Data and R code are at https://github.com/simonevincenzi/Contemporary_Extinction.

After year 100, the directional trend, the increase in variability of climate, and the increased occurrence and severity of point extremes led to noticeable fluctuations in population size over time (Fig. 1). Twenty-one per cent of the 34 560 simulation replicates went extinct. As expected, risk of extinction increased with higher frequency and severity of point extreme events, older age at first reproduction, and fewer offspring produced per mating pair (Fig. 2). Given an expected yearly number of offspring per mating pair, the proportion of replicates that went extinct increased approximately linearly with age at first reproduction (Fig. 2a). When considering all replicates or only the most extreme scenario, for a fixed expected number of offspring produced per mating pair, increasing age at first reproduction by one year would increase the probability of going extinct by approximately 10% (Fig. 2a). The combination of relative high frequency and high severity of extreme events led to a noticeably higher proportion of replicates that went extinct (Fig. 2b). Among the replicates that went extinct, 16% (34%) of them were not affected by a point extreme in the 10 years before extinction and 34% in the 5 years preceding extinction.

In populations that persisted until the end of simulation time, the GLM and GAM models that included *n_low* provided a good prediction of the total number of alleles at the end of simulation time (Table 2). The total number of years with population size smaller than 150 individuals had a strong, negative non-linear effect on the total number of alleles at the end of simulation time (Fig. 3). Models that did not include *n_low* had much lower predictive performance (Table 2).

Models for differences in allelic frequency of more or less theoretically advantageous alleles were able to explain less than 5% of the variance of the target variable (Fig. 4). The average value of population-mean phenotype in the last ten years of simulation time was negatively correlated with the difference in frequency of top and bottom 10% of alleles according to their allelic value at the end of simulation time (*r* = 0.71, *p* < 0.01).

The GLM, GAM, and RF models fitted on the training data sets had similar high predictive accuracy and low false positive and negative rates when predicting extinction or survival in the 10-year “prediction window” (Table 3). For more than 98% of replicates included in the test data set, the GLM, GAM, and RF models provided the same prediction of either extinction or persistence (Fig. 5). In the GAM model, only minimum population size in the observation window had a strong non-linear (negative) effect on the log-odds of extinction. Likewise, minimum population size was the most important predictor in the RF model (Fig. 6a). The models without minimum and mean population size as predictors had fairly low accuracy (Table 3). The RF model without minimum and mean population size as predictors found age at reproduction and expected yearly number of offspring per mating pair as the most important predictors of contemporary extinction (Fig. 6b).

## 4 Discussion

Understanding and predicting the effects of extreme events on risk of extinction and population and genetic dynamics of natural populations is critical for both population forecasting and managing human intervention in an increasingly more extreme world.

I found additive effects—with largely no interaction effects—between age at first reproduction and fecundity on risk of extinction for the range of values I simulated. In the replicates that survived up to the end of simulation time, the total number of years in which the population was at a small size was a good predictor of allelic richness for the quantitative trait selected for. The population frequency of theoretically advantageous alleles was strongly correlated with the mean value of the phenotype under selection, but was otherwise largely unpredictable. Last, simple models including basic demographic and vital rates information, along with climate/environmental measures, provided excellent predictions of contemporary risk of population extinction.

### 4.1 Life histories

Life-history theory predicts a prevalence of fast life histories in environments in which extreme events occur (Winemiller 2005). Fast life histories, which are often defined as including a combination of faster body growth early in life, younger age at maturity, and higher reproductive effort early in the reproductive life of the individual, should allow for faster population growth rate after a drastic reduction in population size, an adaptive life-history strategy when the population is at risk of extinction.

For many animal species that commonly experience the often-catastrophic effects of extreme climate or point events—and have limited movement range for physiological and behavioral reasons or for the environment they inhabit (e.g., freshwater fish in mountain streams, insects)—offspring production is typically very high compared to the habitat carrying capacity. Thus, offspring production is not usually what is limiting population recovery, i.e. the re-establishment of the pre-event population size. However, when after a climate or point extreme event the population is reduced to such low numbers that the population is at immediate risk of extinction, younger age at first reproduction for the surviving individuals could be what makes the difference between population persistence or extinction. Vincenzi et al. (2017a) found that fish born after flash floods had younger mean age at reproduction than fish born before flash floods, and hypothesized that younger age at reproduction after flash floods was due to a combination of faster growth due to lower population density and fewer older fish competing for mates. However, younger age at sexual maturity and higher energetic investment in offspring production often comes at the cost of shorter life expectancy (Fay et al. 2016), and life histories that are adaptive after population crashes can show lower fitness in steady-state conditions (Vincenzi et al. 2012, 2014).

### 4.2 Prediction of genetic dynamics and of population extinction

It is expected that persisting populations experiencing recurrent bottlenecks should have their genetic pool eroded over time. The erosion of the genetic pool should should be noticeable in particular in the loss of allelic richness, even after a single bottleneck event (Allendorf 1986). I found that the extent of population bottlenecks, which I operationalized as the number of years in which population size was relatively small with respect to the habitat carrying capacity and species numerical potential, was a strong predictor of allelic richness for the quantitative trait under selection. These results appear to be at least partially consistent with recent empirical results. Vincenzi et al. (2017a) found an increase in the proportion of fixed alleles in year-classes of two trout populations born after flash floods that caused massive mortalities. Poff et al. (2018) tested predictions about population genomic change in aquatic insects living in Colorado, US, mountain streams after a one in 500-year rainfall event. They found that allelic richness at presumably neutral loci declined after the event only in two out of six species analyzed. Moderate reduction of allelic diversity after strong bottlenecks might be attributable to the particular demographic history of the populations that are investigated; according to Bouzat (2010), one can expect that populations experiencing recurrent bottlenecks might have had their genetic pool already eroded over time, which would decrease the effectiveness of both purifying selection and random allele loss. In addition, the loss of alleles could be greater in species—like salmonids——with high variance in reproductive success among adults (i.e., greater than Poisson variance in reproductive success), although variance in reproductive success has been found to bias (i.e., make false positives more likely) empirical investigations of genetic bottlenecks (Hoban et al. 2013). Moreover, in our simulations, we started from 10 unique alleles for each of the 10 loci, and the high initial allelic diversity may explain the strong relationship that was found between the temporal extent of bottlenecks and allelic richness.

On the other hand, a model including age at first reproduction and fecundity of the species and some traits of the environment was not able to predict the dynamics of the population frequencies of more advantageous alleles for the quantitative trait under selection. However, a strong correlation was found between the frequencies of more or less advantageous alleles at the end of simulation time and the average value of the phenotype under selection. This result seems to at least suggest that, although the fate of alleles is difficult to predict from just the coarse-grain description of the environment and of the species, even in an extreme environment that also causes trait-independent mass mortalities, a shift in the phenotype is likely to be caused by the increased prevalence of the most advantageous alleles.

One of the foundational tenets of conservation biology is that small, fragmented populations should be considered locally vulnerable to extinction—even more critically so when affected by highly variable climatic conditions and other environmental disturbances. The conditions that led to the extinction of a population or species can almost always be understood retrospectively, but forecasting extinction, especially over contemporary time horizons, is much more challenging. In this work, I found that the most important predictors of contemporary extinctions were mean and minimum population size measured in the few years before the “prediction window”. However, it was not uncommon for populations to swiftly rebound after collapses in numbers, and age at reproduction and yearly fecundity were the most important predictors of extinction when measures of population size were not included in the model. This result highlights the importance of age at first reproduction and yearly fecundity for population persistence in highly stochastic environments, as also suggested by theoretical (Bürger and Lynch 1995) and experimental (Griffen and Drake 2008) studies. However, other genetic challenges not accounted for in my simulation model are likely to be encountered by populations that decline to very small numbers, such as a reduction of viability and/or fecundity due to either inbreeding or the expression of deleterious alleles (Willi et al. 2006).

Although it may appear from the simulation results that vital rates and environmental conditions play a small role in models that predict contemporary risk of extinction, those rates and conditions heavily contribute to determining population size in the “observation window”; for example, populations with higher fecundity and younger age at first reproduction are less likely to remain at low population sizes than populations with lower fecundity and delayed sexual maturity. Likewise, a more extreme environment (e.g., greater selection strength and more frequent and/or severe climate and point extremes) tends to decrease average population size, either due to acute events that kill individuals or constant recovery from population crashes.

Luck plays also plays a major role in determining whether a population will recover after a population crash. Vincenzi et al. (2017a) found that the almost-complete recovery of a trout populations that was reduced to a handful of individuals after a flash flood was due the large production of offspring by a single mating pair. Considering the high variance in adult reproductive success in trout population—which is at least partially due to differences in individual “quality” (Auld et al. 2019)—had the mating pair been killed or displaced during the flash flood, population recovery would have suddenly become much less likely.

### 4.3 Modeling considerations

Modeling and simulation approaches can help understand the effects of multiple extreme stressors on the contemporary risk of extinction of species, and can be used to guide or support both the set-up of ecologically relevant experimental designs and the interpretation of biological responses to multiple stressors.

However, models of population and genetic dynamics are limited in their scope of prediction by both the understanding of the biology and ecology of the species and the availability of data to parametrize, train, and test models. Although often intuitive, it is nevertheless important to remind ourselves that the simulation results of modeling exercises depend on, first, our biological and ecological understanding, and, second, the simplified modeling of the species and the environment they inhabit. For instance, general trade-offs between allocation of resources to competing physiological functions are not only often intrinsically challenging to model, but they may also vary over time and space for the same species or population. That is, even when there is qualitative understanding of some biological or ecological process, the parameterization and choice of parameter values for the model may be too uncertain too provide actionable predictions.

Trade-offs between model accuracy and interpretability also need to be taken into account when developing models of population and genetic dynamics. Accuracy describes the ability of a model to explain observed data and make correct predictions, while interpretability concerns to what degree the model allows for understanding processes. Often, a trade-off exists between accuracy and interpretability: more complex models are usually opaque, while more interpretable models often do not provide the same accuracy or predictive power of more complex models (Breiman 2001b). Then, although intuitively more complex models are expected to provide more accurate predictions of risk of extinction or population and genetic dynamics, this is not always the case. For instance, Ward et al. (2014) tested the predictive performance of short-term forecasting models of population abundance of varying complexity. They found that more complex models often performed worse than simpler models, which simply treated the most recent observation as the forecast. In their case, the estimation of even a small number of parameters imposed a high cost while providing little benefit for short-term forecasting of species. However, when there was a clear signal of cyclic dynamics, more complex models were able to more accurately predict future population sizes.

As always, the purpose of a scientific investigation should drive model formulation, the type and amount of data collected, and the acceptable prediction uncertainty.

## Ethics

Not applicable

## Data accessibility

Data and relevant code for this research work are stored in GitHub: https://github.com/simonevincenzi/Contemporary_Extinction

## Authors’ Contributions

Simone Vincenzi conceived the ideas, designed the methods, run the analyses, and wrote the manuscript.

## Competing interests

The author declares no competing interests

## Funding

Not applicable

## Acknowledgements

Simone Vincenzi developed the idea behind this work while walking the Malecón of La Habana, Cuba, and finished writing the manuscript after the 2020 California fires that imperiled the town where he resides (Santa Cruz, CA, US) and the COVID-19 pandemic highlighted the importance of proper simulation modeling for basic science and policy, and the increasing threat of extreme events for people and nature.