# 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.

## Keywords

Demographic stochasticity Environmental change Stationary Non-stationary Homogeneous Nonhomogeneous Environment-dependent demography Stochastic simulation algorithm Poisson process## Notes

### Acknowledgements

GL conceived of the study and ran the simulations. Authors GL and BAM did the analysis of the simulations and wrote the paper. The Janus supercomputer is a joint effort of the University of Colorado Boulder, the University of Colorado Denver and the National Center for Atmospheric Research. We thank the ESA Theoretical Ecology section for a poster award to GL, which provided early support for the project.

### Funding information

In this study, preliminary simulation work used the Janus supercomputer, which is supported by the National Science Foundation (DEB, award number 1457660) and the University of Colorado Boulder.

## References

- Adolph SC, Porter WP (1993) Temperature, activity, and lizard life histories. Am Nat 142(2):273–295PubMedCrossRefGoogle Scholar
- Allen E (2016) Environmental variability and mean-reverting processes. Discrete & Continuous Dynamical Systems - B 21:2073–2089CrossRefGoogle Scholar
- Allen EJ, Allen LJ, Schurz H (2005) A comparison of persistence-time estimation for discrete and continuous stochastic population models that include demographic and environmental variability. Math Biosci 196(1):14–38PubMedCrossRefGoogle Scholar
- Angert AL, Huxman TE, Barron-Gafford GA, Gerst KL, Venable DL (2007) Linking growth strategies to long-term population dynamics in a guild of desert annuals. J Ecol 95(2):321–331CrossRefGoogle Scholar
- Angilletta M (2009) Thermal adaptation: a theoretical and empirical synthesis. Oxford University Press, LondonCrossRefGoogle Scholar
- Bartlett MS (1955) An introduction to stochastic processes: with special reference to methods and applications. Cambridge University Press, CambridgeGoogle Scholar
- Bartlett MS (1957) Measles periodicity and community size. J R Stat Soc Ser A (General) 120(1):48–70CrossRefGoogle Scholar
- Black AJ, McKane AJ (2012) Stochastic formulation of ecological models and their applications. Trends Ecol Evol 27(6):337–345PubMedCrossRefGoogle Scholar
- Boguñá M, Lafuerza LF, Toral R, Serrano MA (2014) Simulating non-Markovian stochastic processes. Phys Rev E 90:042,108CrossRefGoogle Scholar
- Buffon GLL (1774) Histoire naturelle, générale et particulière servant de suite à la théorie de la terre, et d’introduction à l’histoire des miné raux. De l’Imprimerie royale á ParisGoogle Scholar
- Collins M, Knutti R, Arblaster J, Dufresne JL, Fichefet T, Friedlingstein P, Gao X, Gutowski W, Johns T, Krinner G, Shongwe M, Tebaldi C, Weaver A, Wehner M (2013) Long-term climate change: projections, commitments and irreversibility. In: Stocker T, Qin D, Plattner G K, Tignor M, Allen S, Boschung J, Nauels A, Xia Y, Bex V, Midgley P (eds) Climate change 2013: the physical science basis. Contribution of working Group I to the fifth assessment report of the intergovernmental panel on climate change. Cambridge University Press, pp 1029–1136, Chap 12Google Scholar
- Cox DR (1955) Some statistical methods connected with series of events. J R Stat Soc Ser B Methodol 17 (2):129–164Google Scholar
- Cox DR, Isham V (1980) Point processes. Chapman and Hall, LondonGoogle Scholar
- Crow LH (1974) Reliability analysis for complex repairable systems. Reliability and Biometry 13:379–410Google Scholar
- Davidson J, Andrewartha HG (1948) The influence of rainfall, evaporation and atmospheric temperature on fluctuations in the size of a natural population of Thrips imaginis (Thysanoptera). J Anim Ecol 17(2):200–222CrossRefGoogle Scholar
- Davidson AC, Hinkley DV (1997) Bootstrap methods and their application. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Dell AI, Pawar S, Savage VM (2011) Systematic variation in the temperature dependence of physiological and ecological traits. Proc Natl Acad Sci U S A 108(26):10,591–10,596CrossRefGoogle Scholar
- Deutsch CA, Tewksbury JJ, Huey RB, Sheldon KS, Ghalambor CK, Haak DC, Martin PR (2008) Impacts of climate warming on terrestrial ectotherms across latitude. Proc Natl Acad Sci U S A 105(18):6668–6672PubMedPubMedCentralCrossRefGoogle Scholar
- Devroye L (1986) Non-uniform random variate generation. Springer, BerlinCrossRefGoogle Scholar
- Doering CR, Sargsyan KV, Sander LM (2005) Extinction times for birth-death processes: exact results, continuum asymptotics, and the failure of the Fokker–Planck approximation. Multiscale Model Simul 3(2):283–299CrossRefGoogle Scholar
- Donat MG, Alexander LV (2012) The shifting probability distribution of global daytime and night-time temperatures. Geophys Res Lett 39(14):L14707CrossRefGoogle Scholar
- Duan Q, Liu J (2015) A first step to implement Gillespie’s algorithm with rejection sampling. Statistical Methods & Applications 24(1):85–95CrossRefGoogle Scholar
- Estay SA, Clavijo-Baquet S, Lima M, Bozinovic F (2011) Beyond average: an experimental test of temperature variability on the population dynamics of Tribolium confusum. Popul Ecol 53(1):53–58CrossRefGoogle Scholar
- Estay SA, Lima M, Bozinovic F (2014) The role of temperature variability on insect performance and population dynamics in a warming world. Oikos 123(2):131–140CrossRefGoogle Scholar
- Fay PA, Carlisle JD, Knapp AK, Blair JM, Collins SL (2003) Productivity responses to altered rainfall patterns in a c4-dominated grassland. Oecologia 137(2):245–251PubMedCrossRefGoogle Scholar
- Finkelstein JM (1976) Confidence bounds on the parameters of the weibull process. Technometrics 18(1):115–117CrossRefGoogle Scholar
- Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361CrossRefGoogle Scholar
- Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115(4):1716– 1733CrossRefGoogle Scholar
- Gokhale CS, Papkou A, Traulsen A, Schulenburg H (2013) Lotka–Volterra dynamics kills the Red Queen: population size fluctuations and associated stochasticity dramatically change host-parasite coevolution. BMC Evol Biol 13(1):254PubMedPubMedCentralCrossRefGoogle Scholar
- Grandell J (1976) Doubly stochastic Poisson processes. Springer, BerlinCrossRefGoogle Scholar
- Hart SP, Schreiber SJ, Levine JM (2016) How variation between individuals affects species coexistence. Ecol Lett 19(8):825– 838PubMedCrossRefGoogle Scholar
- Hartmann D, Klein Tank A, Rusticucci M, Alexander L, Bronnimann S, Charabi Y, Dentener F, Dlugokencky E, Easterling D, Kaplan A, Soden B, Thorne P, Wild M, Zhai P (2013) Observations: atmosphere and surface. In: Stocker T, Qin D, Plattner GK, Tignor M, Allen S, Boschung J, Nauels A, Xia Y, Bex V, Midgley P (eds) Climate change 2013: the physical science basis. Contribution of working Group I to the fifth assessment report of the intergovernmental panel on climate change, vol 2. Cambridge University Press, Cambridge. Book Section, pp 159– 254Google Scholar
- Heisler-White JL, Knapp AK, Kelly EF (2008) Increasing precipitation event size increases aboveground net primary productivity in a semi-arid grassland. Oecologia 158(1):129–140PubMedCrossRefGoogle Scholar
- Henson SM, Costantino RF, Cushing JM, Desharnais RA, Dennis B, King AA (2001) Lattice effects observed in chaotic dynamics of experimental populations. Science 294(5542):602–605PubMedCrossRefGoogle Scholar
- Huang W, Hauert C, Traulsen A (2015) Stochastic game dynamics under demographic fluctuations. Proc Natl Acad Sci U S A 112(29):9064–9069PubMedPubMedCentralCrossRefGoogle Scholar
- Huey RB, Kingsolver JG (1989) Evolution of thermal sensitivity of ectotherm performance. Trends Ecol Evol 4(5):131–135PubMedCrossRefGoogle Scholar
- Huey RB, Stevenson R (1979) Integrating thermal physiology and ecology of ectotherms: a discussion of approaches. Am Zool 19(1):357–366CrossRefGoogle Scholar
- Huntingford C, Jones PD, Livina VN, Lenton TM, Cox PM (2013) No increase in global temperature variability despite changing regional patterns. Nature 500(7462):327–330PubMedCrossRefGoogle Scholar
- Huxman TE, Snyder KA, Tissue D, Leffler AJ, Ogle K, Pockman WT, Sandquist DR, Potts DL, Schwinning S (2004) Precipitation pulses and carbon fluxes in semiarid and arid ecosystems. Oecologia 141 (2):254–268PubMedCrossRefGoogle Scholar
- Kaplan N (1973) A continuous time Markov branching model with random environments. Adv Appl Probab 5(1):37–54CrossRefGoogle Scholar
- Keeling M, Ross J (2008) On methods for studying stochastic disease dynamics. J R Soc Interface 5 (19):171–181PubMedCrossRefGoogle Scholar
- Kendall DG (1948) On the generalized “birth-and-death” process. Ann Math Stat 19(1):1–15CrossRefGoogle Scholar
- Kessler DA, Shnerb NM (2007) Extinction rates for fluctuation-induced metastabilities: a real-space WKB approach. J Stat Phys 127(5):861–886CrossRefGoogle Scholar
- Kingsolver JG, Diamond SE, Buckley LB (2013) Heat stress and the fitness consequences of climate change for terrestrial ectotherms. Funct Ecol 27(6):1415–1423CrossRefGoogle Scholar
- Kingsolver JG, Higgins JK, Augustine KE (2015) Fluctuating temperatures and ectotherm growth: distinguishing non-linear and time-dependent effects. J Exp Biol 218(14):2218– 2225PubMedCrossRefGoogle Scholar
- Knapp AK, Smith MD (2001) Variation among biomes in temporal dynamics of aboveground primary production. Science 291(5503):481–484PubMedCrossRefGoogle Scholar
- Kolmogoroff A (1931) ÜBer die analytischen methoden in der wahrscheinlichkeitsrechnung. Math Ann 104:415–458CrossRefGoogle Scholar
- Kolpas A, Nisbet RM (2010) Effects of demographic stochasticity on population persistence in advective media. Bull Math Biol 72(5):1254–1270PubMedCrossRefGoogle Scholar
- Kramer AM, Drake JM (2010) Experimental demonstration of population extinction due to a predator-driven Allee effect. J Anim Ecol 79(3):633–639PubMedCrossRefGoogle Scholar
- Kramer AM, Drake JM (2014) Time to competitive exclusion. Ecosphere 5(5):1–16CrossRefGoogle Scholar
- Lande R (1993) Risks of population extinction from demographic and environmental stochasticity and random catastrophes. Am Nat 142(6):911–927PubMedCrossRefGoogle Scholar
- Loik ME, Breshears DD, Lauenroth WK, Belnap J (2004) A multi-scale perspective of water pulses in dryland ecosystems: climatology and ecohydrology of the Western USA. Oecologia 141(2):269–281PubMedCrossRefGoogle Scholar
- Mangel M, Tier C (1993) A simple direct method for finding persistence times of populations and application to conservation problems. Proc Natl Acad Sci U S A 90(3):1083–1086PubMedPubMedCentralCrossRefGoogle Scholar
- Marion G, Renshaw E, Gibson G (2000) Stochastic modelling of environmental variation for biological populations. Theor Popul Biol 57(3):197–217PubMedCrossRefGoogle Scholar
- Meisner MH, Harmon JP, Ives AR (2014) Temperature effects on long-term population dynamics in a parasitoid–host system. Ecol Monogr 84(3):457–476CrossRefGoogle Scholar
- Miquel J, Lundgren PR, Bensch KG, Atlan H (1976) Effects of temperature on the life span, vitality and fine structure of drosophila melanogaster. Mech Ageing Dev 5:347–370PubMedCrossRefGoogle Scholar
- Nisbet RM, Martin BT, de Roos AM (2016) Integrating ecological insight derived from individual-based simulations and physiologically structured population models. Ecol Model 326:101– 112CrossRefGoogle Scholar
- Novoplansky A, Goldberg DE (2001) Effects of water pulsing on individual performance and competitive hierarchies in plants. J Veg Sci 12(2):199–208CrossRefGoogle Scholar
- Okuyama T (2015) Demographic stochasticity alters the outcome of exploitation competition. J Theor Biol 365:347–351PubMedCrossRefGoogle Scholar
- Orrock JL, Fletcher RJ (2005) Changes in community size affect the outcome of competition. Am Nat 166 (1):107–111PubMedCrossRefGoogle Scholar
- Orrock JL, Watling JI (2010) Local community size mediates ecological drift and competition in metacommunities. Proc R Soc Lond B Biol Sci 277(1691):2185–2191CrossRefGoogle Scholar
- Ovaskainen O, Meerson B (2010) Stochastic models of population extinction. Trends Ecol Evol 25(11):643–652PubMedCrossRefGoogle Scholar
- Paaijmans KP, Heinig RL, Seliga RA, Blanford JI, Blanford S, Murdock CC, Thomas MB (2013) Temperature variation makes ectotherms more sensitive to climate change. Glob Chang Biol 19(8):2373–2380PubMedPubMedCentralCrossRefGoogle Scholar
- Palamara GM, Carrara F, Smith MJ, Petchey OL (2016) The effects of demographic stochasticity and parameter uncertainty on predicting the establishment of introduced species. Ecol Evol 6(23):8440–8451PubMedPubMedCentralCrossRefGoogle Scholar
- Parmesan C (2006) Ecological and evolutionary responses to recent climate change. Annu Rev Ecol Evol Syst 37(1):637–669CrossRefGoogle Scholar
- Pearl R, Reed LJ (1920) On the rate of growth of the population of the United States since 1790 and its mathematical representation. Proc Natl Acad Sci U S A 6(6):275–288PubMedPubMedCentralCrossRefGoogle Scholar
- Pedruski MT, Fussmann GF, Gonzalez A (2015) Predicting the outcome of competition when fitness inequality is variable. Royal Society Open Science 2(8):150,274CrossRefGoogle Scholar
- R Core Team (2017) R: a language and environment for statistical computing r foundation for statistical computing, Vienna, AustriaGoogle Scholar
- Ross S (2014) Introduction to probability models, 11th edn. Academic, New YorkGoogle Scholar
- Shaffer ML (1981) Minimum population sizes for species conservation. Bioscience 31(2):131–134CrossRefGoogle Scholar
- Simonis JL (2012) Demographic stochasticity reduces the synchronizing effect of dispersal in predator–prey metapopulations. Ecol 93(7):1517–1524CrossRefGoogle Scholar
- Soetaert K, Petzoldt T, Setzer RW (2010) Solving differential equations in R: package desolve. J Stat Softw 33(9):1–25CrossRefGoogle Scholar
- Stroustrup N, Anthony WE, Nash ZM, Gowda V, Gomez A, López-Moyado IF, Apfeld J, Fontana W (2016) The temporal scaling of
*Caenorhabditis elegans*ageing. Nature 530:103– 107PubMedPubMedCentralCrossRefGoogle Scholar - Uhlenbeck GE, Ornstein LS (1930) On the theory of the Brownian motion. Phys Rev 36:823–841CrossRefGoogle Scholar
- van den Broek J, Heesterbeek H (2007) Nonhomogeneous birth and death models for epidemic outbreak data. Biostatistics 8(2):453–467CrossRefGoogle Scholar
- van Kampen NG (1992) Stochastic processes in physics and chemistry. Elsevier, AmsterdamGoogle Scholar
- Varughese M, Fatti L (2008) Incorporating environmental stochasticity within a biological population model. Theor Popul Biol 74(1):115–129PubMedCrossRefGoogle Scholar
- Verhulst P (1845) Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux mémoires de l’Académie Royale des Sciences et Belles-Lettres de Bruxelles 18:14–54Google Scholar
- Vestergaard CL, Génois M (2015) Temporal gillespie algorithm: fast simulation of contagion processes on time-varying networks. PLoS Comput Biol 11(10):1–28CrossRefGoogle Scholar
- Von Neumann J (1951) Various techniques used in connection with random digits. Appl Math Ser 12:36–38Google Scholar
- Wilcox C, Possingham H (2002) Do life history traits affect the accuracy of diffusion approximations for mean time to extinction? Ecol Appl 12(4):1163–1179CrossRefGoogle Scholar
- Yaari G, Ben-Zion Y, Shnerb NM, Vasseur DA (2012) Consistent scaling of persistence time in metapopulations. Ecol 93(5):1214–1227CrossRefGoogle Scholar
- Yule GU (1925) A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S. Philos Trans R Soc Lond B: Biol. Sci. 213(402–410):21–87CrossRefGoogle Scholar