Accounting for environmental change in continuous-time stochastic population models

Abstract

The demographic rates (e.g., birth, death, migration) of many organisms have been shown to respond strongly to short- and long-term environmental change, including variation in temperature and precipitation. While ecologists have long accounted for such nonhomogeneous demography in deterministic population models, nonhomogeneous stochastic population models are largely absent from the literature. This is especially the case for models that use exact stochastic methods, such as Gillespie’s stochastic simulation algorithm (SSA), which commonly assumes that demographic rates do not respond to external environmental change (i.e., assumes homogeneous demography). In other words, ecologists are currently accounting for the effects of demographic stochasticity or environmental variability, but not both. In this paper, we describe an extension of Gillespie’s SSA (SSA + ) that allows for nonhomogeneous demography and examine how its predictions differ from a method that is partly naive to environmental change (SSAn) for two fundamental ecological models (exponential and logistic growth). We find important differences in the predicted population sizes of SSA + versus SSAn simulations, particularly when demography responds to fluctuating and irregularly changing environments. Further, we outline a computationally inexpensive approach for estimating when and under what circumstances it can be important to fully account for nonhomogeneous demography for any class of model.

Appendices

Appendix A: Diagnostic tests of SSA + method

To ensure we were generating appropriate inter-event times, $\tau$, using our implementation of the SSA + method, we performed two diagnostic tests:

• 1. We simulated a non-stationary exponential growth model using our SSA + implementation with a rate function of $F(t)=\alpha t^{-\beta }$ (a power law function) and compared our results to taking samples from a Weibull distribution with parameters $\alpha$ and $\beta$, as discussed in the main text. These results should align nearly exactly if our implementation is correct, which appears to be the case (Fig. 5).

• 2. We set the environment function to a constant value (i.e., demography was stationary) and compared the results of 50,000 simulations of the SSA + method to an independent SSA implementation (since demography is stationary in this case, we refer to it simply as the SSA rather than the SSAn, though both use the same methods). With a constant environment, the SSA + method should match the SSA method, which appears to be the case (Fig. 6).

Fig. 5

The probability density of 1,000,000 arrival times, obtained by sampling from a Weibull distribution with scale parameter α and shape parameter β (black), or using our SSA + method for a non-stationary stochastic process with rate function F(t) = αt^−β (blue). In both cases, α = 1 and β = 1.5

Fig. 6

The frequency distribution of population sizes at t = 100 for 50,000 simulations of exponential growth (a) and logistic growth (b), simulated using the SSA (red) or SSA + (blue) methods with a constant environment. Colors are transparent, so the color purple indicates overlap between the SSA and SSA + methods. Also displayed are the (overlapping) expected values, $\bar {N}$ (rounded) for each simulation method

Finally, we also examined the extent to which 10,000 simulations was sufficient to capture the ensemble mean and variance of the stochastic simulations. To do so, we examined how the means and variances of the exponential growth models were affected by the number of simulations used. We found only very small differences in the means and variances from 1000 to 10,000 simulations used (see Fig. 7 for this comparison).

Fig. 7

The expected population sizes at t = 100 of the SSA + method (exponential growth) for different numbers of simulations (x-axis). Displayed are the population sizes for the increasing (a), fluctuating (b), and irregularly changing (c) environment functions. Points represent the mean population size (across simulations), while bars are the standard deviations of the simulations

Appendix B: R code for SSA + method

This section contains sample code for simulating 1 run of the Gillespie algorithm (SSA or SSAn in the main text) and 1 run of the extended, nonhomogeneous (i.e., non-stationary) version of the algorithm (SSA + ) for an exponential growth model. The code is specifically designed to allow for environment- or time-dependent demography, but in this example the environment is constant. Additional code and more detailed examples are available at https://github.com/legault/SSAplus.

Appendix C: Comparison of SSAn and SSA + for slower environment functions

In Fig. 8, we compare SSAn and SSA + predictions for the slower environment functions, specifically “increasing 1”, “fluctuating 1”, and “irregular 1”. The qualitative results are similar to those in Fig. 3 (in main text), albeit with more overlap between the distributions.

Fig. 8

The frequency distribution of population sizes at t = 100 for the exponential growth (a–c) and logistic growth (d–f) models, simulated using either the SSAn (red) or SSA + (blue) method. Displayed are the population size distributions for environment functions: increasing 1 (a, d), fluctuating 1 (b, e), and irregular 1 (c, f). Colors are transparent, so purple indicates overlap between the SSAn and SSA + methods. Also displayed are the expected values, $\bar {N}$ (rounded to nearest integer), for each simulation method (same coloration as above). Note the different scales on both axes for each panel

Appendix D: CDFs beginning at different time points

In Fig. 9, we compare the CDFs for the exponential growth model assuming the simulations began at $t = 3$ (Fig. 9a–c) or $t = 5$ (Fig. 9d–f), with respect to the environment functions. For the fluctuating and irregular environments, these CDFs differ from those displayed in Fig. 2 of the main text. For example, at $t = 3$ environment function “fluctuating 1” is decreasing rapidly thereby lowering the birth rate and consequently the CDF in Fig. 9b. In contrast, the function “fluctuating 2” is increasing rapidly at $t = 3$ and the resulting CDF is elevated compared to the SSAn CDF.

Fig. 9

Cumulative probabilities over time (starting at t = 3 for (a–c); at t = 5 for (d–f)) that the next demographic event will occur for the exponential growth model under increasing (a, d), fluctuating (b, e), and irregular (c, f) environments. The solid lines represent the probabilities for the SSAn method, which assumes that demographic processes occur at constant rates over time (in this case, the rate at t = 3 or t = 5). Dashed and dotted lines represent the probabilities obtained from the SSA + method, which accurately accounts for nonhomogeneous demographic rates (compare to Fig. 1). As in Fig. 2, the CDFs assume an “initial” (i.e., beginning at t = 3 or t = 5) population size of 100