Stochastic population growth in spatially heterogeneous environments
 Steven N. Evans,
 Peter L. Ralph,
 Sebastian J. Schreiber,
 Arnab Sen
 … show all 4 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average percapita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZ _{ t } = μ Z _{ t } dt + σ Z _{ t } dW _{ t }, t ≥ 0, where the conditional law of Z _{ t+Δt }− Z _{ t } given Z _{ t } = z has mean and variance approximately z μΔt and z ^{2} σ ^{2}Δt when the time increment Δt is small. The longterm stochastic growth rate \({\lim_{t \to \infty} t^{1}\log Z_t}\) for such a population equals \({\mu \frac{\sigma^2}{2}}\) . Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model \({{\bf X}_t = (X_t^1, \ldots, X_t^n)}\) , t ≥ 0, for the population abundances in n patches: the conditional law of X _{ t+Δt } given X _{ t } = x is such that the conditional mean of \({X_{t+\Delta t}^i  X_t^i}\) is approximately \({[x^i \mu_i + \sum_j (x^j D_{ji}  x^i D_{ij})] \Delta t}\) where μ _{ i } is the per capita growth rate in the ith patch and D _{ ij } is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of \({X_{t+\Delta t}^i  X_t^i}\) and \({X_{t + \Delta t}^j  X_t^j}\) is approximately x ^{ i } x ^{ j } σ _{ ij }Δt for some covariance matrix Σ = (σ _{ ij }). We show for such a spatially extended population that if \({S_t = X_t^1 + \cdots + X_t^n}\) denotes the total population abundance, then Y _{ t } = X _{ t }/S _{ t }, the vector of patch proportions, converges in law to a random vector Y _{∞} as \({t \to \infty}\) , and the stochastic growth rate \({\lim_{t \to \infty} t^{1}\log S_t}\) equals the spacetime average percapita growth rate \({\sum_i \mu_i \mathbb{E}[Y_\infty^i]}\) experienced by the population minus half of the spacetime average temporal variation \({\mathbb{E}[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j]}\) experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multiscale landscapes (e.g. insects on plants in meadows on islands). Our results provide fundamental insights into “ideal free” movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of densitydependent feedbacks, idealfree dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.
 blah 285_2012_514_MOESM1_ESM.zip (103KB)
 Adler, FR (1992) The effects of averaging on the basic reproduction ratio. Math Biosci 111: pp. 8998 CrossRef
 Bascompte, J, Possingham, H, Roughgarden, J (2002) Patchy populations in stochastic environments: critical number of patches for persistence. Am Nat 159: pp. 128137 CrossRef
 Benaïm, M, Schreiber, SJ (2009) Persistence of structured populations in random environments. Theor Popul Biol 76: pp. 1934 CrossRef
 Bhattacharya RN (1978) Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann Probab 6(4):541–553. ISSN 00911798. http://www.jstor.org/stable/2243121
 Bogachev VI, Röckner M, Stannat W (2002) Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Sbornik: Math 193(7):945. http://stacks.iop.org/10645616/193/i=7/a=A01
 Bogachev Vladimir I, Krylov Nikolai V, Röckner Michael (2009) Elliptic and parabolic equations for measures. Russ Math Surv 64(6):973. http://stacks.iop.org/00360279/64/i=6/a=R02
 Boyce, MS, Haridas, CV, Lee, CT, the NCEAS Stochastic Demography Working, Group (2006) Demography in an increasingly variable world. Trends Ecol Evol 21: pp. 141148 CrossRef
 Cantrell, RS, Cosner, C (1991) The effects of spatial heterogeneity in population dynamics. J Math Biol 29: pp. 315338 CrossRef
 Cantrell, RS, Cosner, C, Lou, Y (2006) Movement toward better environments and the evolution of rapid diffusion. Math Biosci 204: pp. 199214 CrossRef
 Cantrell, RS, Cosner, C (1989) Diffusive logistic equations with indefinite weights: population models in disrupted environments. Proc Royal Soc Edinburgh Section A Math 112: pp. 293318 CrossRef
 Cantrell, RS, Cosner, C, Deangelis, DL, Padron, V (2007) The ideal free distribution as an evolutionarily stable strategy. J Biol Dyn 1: pp. 249271 CrossRef
 Chesson, PL (2000) General theory of competitive coexistence in spatiallyvarying environments. Theor Popul Biol 58: pp. 211237 CrossRef
 Da Prato, G, Zabczyk, J (1996) Ergodicity for InfiniteDimensional Systems, London Mathematical Society Lecture Note Series, vol 229. Cambridge University Press, Cambridge
 DeAngelis, DL, Wolkowicz, GSK, Lou, Y, Jiang, Y, Novak, M, Svanbäck, R, Araújo, MS, Jo, YS, Cleary, EA (2011) The effect of travel loss on evolutionarily stable distributions of populations in space. Am Nat 178: pp. 1529 CrossRef
 Delibes, M, Gaona, P, Ferreras, P (2001) Effects of an attractive sink leading into maladaptive habitat selection. Am Nat 158: pp. 277285 CrossRef
 Dennis, B, Munholland, PL, Scott, JM (1991) Estimation of growth and extinction parameters for endangered species. Ecol Monogr 61: pp. 115143 CrossRef
 Diaconis P (1988) Group representations in probability and statistics. Institute of Mathematical Statistics Lecture Notes—Monograph Series, vol 11. Institute of Mathematical Statistics, Hayward
 Diamond, JM (1975) The island dilemma: lessons of modern biogeographic studies for the design of natural reserves. Biol Conserv 7: pp. 129146 CrossRef
 Dias PC (1996) Sources and sinks in population biology. Trends Ecol Evol 11:326–330
 Dockery, J, Hutson, V, Mischaikow, K, Pernarowski, M (1998) The evolution of slow dispersal rates: a reaction diffusion model. J Math Biol 37: pp. 6183 CrossRef
 Durrett R, Remenik D (2012) Evolution of the dispersal distance. J Math Biol (in press)
 Fahrig, L (1997) Relative effects of habitat loss and fragmentation on population extinction. J Wildl Manag 61: pp. 603610 CrossRef
 Fahrig, L (2002) Effect of habitat fragmentation on the extinction threshold: a synthesis. Ecol Appl 12: pp. 346353
 Foley P (1994) Predicting extinction times from environmental stochasticity and carrying capacity. Conserv Biol 8:124–137
 Fretwell, SD, Lucas, HL (1970) On territorial behavior and other factors influencing habitat distribution in birds. Acta Biotheoretica 19: pp. 1636 CrossRef
 Gardiner, CW (2004) Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences, Series in synergetics, vol 13, 4th edn. Springer, Berlin
 Geiß, C, Manthey, R (1994) Comparison theorems for stochastic differential equations in finite and infinite dimensions. Stoch Processes Appl 53: pp. 2335 CrossRef
 Gilpin, ME (1988) A comment on Quinn and Hastings: extinction in subdivided habitats. Conserv Biol 2: pp. 290292 CrossRef
 Gonzalez, A, Holt, RD (2002) The inflationary effects of environmental fluctuations in sourcesink systems. Proc Natl Acad Sci 99: pp. 1487214877 CrossRef
 Gradshteyn IS, Ryzhik IM (2007) Table of integrals, series, and products. Elsevier/Academic Press, Amsterdam, 7th edn, 2007. Translated from the Russian, Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger
 Harper, DGC (1982) Competitive foraging in mallards: “ideal free” ducks. Animal Behav 30: pp. 575584 CrossRef
 Harrison, S, Quinn, JF (1989) Correlated environments and the persistence of metapopulations. Oikos 56: pp. 293298 CrossRef
 Hastings, A (1983) Can spatial variation alone lead to selection for dispersal?. Theor Popul Biol 24: pp. 244251 CrossRef
 Holt, RD (1985) Patch dynamics in twopatch environments: some anomalous consequences of an optimal habitat distribtuion. Theor Popul Biol 28: pp. 181208 CrossRef
 Holt, RD (1997) On the evolutionary stability of sink populations. Evol Ecol 11: pp. 723731 CrossRef
 Holt, RD, McPeek, MA (1996) Chaotic population dynamics favors the evolution of dispersal. Am Nat 148: pp. 709718 CrossRef
 Hutson, V, Mischaikow, K, Poláčik, P (2001) The evolution of dispersal rates in a heterogeneous timeperiodic environment. J Math Biol 43: pp. 501533 CrossRef
 Ikeda, N, Watanabe, S (1989) Stochastic differential equations and diffusion processes, NorthHolland Mathematical Library, vol 24, 2nd edn. NorthHolland Publishing Co, Amsterdam
 Jansen, VAA, Yoshimura, J (1998) Populations can persist in an environment consisting of sink habitats only. Proc Natl Acad Sci USA 95: pp. 36963698 CrossRef
 Keagy, J, Schreiber, SJ, Cristol, DA (2005) Replacing sources with sinks: when do populations go down the drain?. Restor Ecol 13: pp. 529535 CrossRef
 Khas’minskii RZ (1960) Ergodic properties of recurrent diffusion processes and stabilization of the solution to the cauchy problem for parabolic equations. Theory Probab Appl 5(2):179–196, 1960. ISSN 0040585X. doi:10.1137/1105016.
 Kirkland, S, Li, CK, Schreiber, SJ (2006) On the evolution of dispersal in patchy landscapes. SIAM J Appl Math 66: pp. 13661382 CrossRef
 Kreuzer, MP, Huntly, NJ (2003) Habitatspecific demography: evidence for sourcesink population structure in a mammal, the pika. Oecologia 134: pp. 343349
 Lande R, Engen S, Sæther BE (2003) Stochastic population dynamics in ecology and conservation: an introduction. Oxford University Press, Oxford
 Levin, SA, Cohen, D, Hastings, A (1984) Dispersal strategies in patchy environments. Theor Popul Biol 26: pp. 165191 CrossRef
 Levins, R (1969) Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull ESA 15: pp. 237240
 Lonsdale, WM (1993) Rates of spread of an invading speciesmimosa pigra in northern Australia. J Ecol 81: pp. 513521 CrossRef
 Lundberg, P, Ranta, E, Ripa, J, Kaitala, V (2000) Population variability in space and time. Trends Ecol Evol 15: pp. 460464 CrossRef
 Matthews, DP, Gonzalez, A (2007) The inflationary effects of environmental fluctuations ensure the persistence of sink metapopulations. Ecology 88: pp. 28482856 CrossRef
 May, RM (1975) Stability and complexity in model ecosystems. Princeton University Press, Princeton
 McPeek, MA, Holt, RD (1992) The evolution of dispersal in spatially and temporally varying environments. Am Nat 6: pp. 10101027 CrossRef
 Metz, JAJ, Jong, TJ, Klinkhamer, PGL (1983) What are the advantages of dispersing; a paper by Kuno extended. Oecologia 57: pp. 166169 CrossRef
 Murphy, MT (2001) Sourcesink dynamics of a declining eastern kingbird population and the value of sink habitats. Conserv Biol 15: pp. 737748 CrossRef
 Oksanen, T, Power, ME, Oksanen, L (1995) Ideal free habitat selection and consumerresource dynamics. Am Nat 146: pp. 565585 CrossRef
 Petchey, OL, Gonzalez, A, Wilson, HB (1997) Effects on population persistence: the interaction between environmental noise colour, intraspecific competition and space. Proc Biol Sci 264: pp. 18411847 CrossRef
 Philippi, T, Seger, J (1989) Hedging one’s evolutionary bets, revisited. Trends Ecol Evol 4: pp. 4144 CrossRef
 Pulliam, HR (1988) Sources, sinks, and population regulation. Am Nat 132: pp. 652661 CrossRef
 Quinn, JF, Hastings, A (1987) Extinction in subdivided habitats. Conserv Biol 1: pp. 198209 CrossRef
 Remeš, V (2000) How can maladaptive habitat choice generate sourcesink population dynamics?. Oikos 91: pp. 579582 CrossRef
 Roy, M, Holt, RD, Barfield, M (2005) Temporal autocorrelation can enhance the persistence and abundance of metapopulations comprised of coupled sinks. Am Nat 166: pp. 246261 CrossRef
 Ruelle, D (1979) Analycity properties of the characteristic exponents of random matrix products. Adv Math 32: pp. 6880 CrossRef
 Schmidt, KA (2004) Site fidelity in temporally correlated environments enhances population persistence. Ecol Lett 7: pp. 176184 CrossRef
 Schreiber, SJ (2010) Interactive effects of temporal correlations, spatial heterogeneity, and dispersal on population persistence. Proc Royal Soc Biol Sci 277: pp. 19071914 CrossRef
 Schreiber, SJ, LloydSmith, JO (2009) Invasion dynamics in spatially heterogenous environments. Am Nat 174: pp. 490505 CrossRef
 Schreiber, SJ, Saltzman, E (2009) Evolution of predator and prey movement into sink habitats. Am Nat 174: pp. 6881 CrossRef
 Schreiber, SJ, Vejdani, M (2006) Handling time promotes the coevolution of aggregation in predator–prey systems. Proc Royal Soc Biol Sci 273: pp. 185191 CrossRef
 Schreiber SJ, Fox LR, Getz WM (2000) Coevolution of contrary choices in hostparasitoid systems. American Naturalist 637–648
 Serre JP (1977) Linear representations of finite groups. Springer, New York. Translated from the second French edition by Leonard L. Scott, Graduate Texts in Mathematics, vol 42
 Slatkin, M (1974) Hedging one’s evolutionary bets. Nature 250: pp. 704705
 Talay Denis (1991) Approximation of upper Lyapunov exponents of bilinear stochastic differential systems. SIAM J Numer Anal 28(4):1141–1164. ISSN 00361429. http://www.jstor.org/stable/2157791
 Tuljapurkar, S (1990) Population dynamics in variable environments. Springer, New York
 Turelli, M (1978) Random environments and stochastic calculus. Theor Popul Biol 12: pp. 140178 CrossRef
 Baalen, M, Sabelis, MW (1999) Nonequilibrium population dynamics of “ideal and free” prey and predators. Am Nat 154: pp. 6988 CrossRef
 Wilbur, HM, Rudolf, VHW (2006) Lifehistory evolution in uncertain environments: bet hedging in time. Am Nat 168: pp. 398411 CrossRef
 Wilcox, BA, Murphy, DD (1985) Conservation strategy: the effects of fragmentation on extinction. Am Nat 125: pp. 879887 CrossRef
 Title
 Stochastic population growth in spatially heterogeneous environments
 Journal

Journal of Mathematical Biology
Volume 66, Issue 3 , pp 423476
 Cover Date
 20130201
 DOI
 10.1007/s0028501205140
 Print ISSN
 03036812
 Online ISSN
 14321416
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Stochastic population growth
 Spatial and temporal heterogeneity
 Dominant Lyapunov exponent
 Ideal free movement
 Evolution of dispersal
 Single large or several small debate
 Habitat fragmentation
 92D25
 37H15
 60H10
 Authors

 Steven N. Evans ^{(1)}
 Peter L. Ralph ^{(2)}
 Sebastian J. Schreiber ^{(2)}
 Arnab Sen ^{(3)}
 Author Affiliations

 1. Department of Statistics #3860, University of California, 367 Evans Hall, Berkeley, CA, 947203860, USA
 2. Department of Evolution and Ecology, University of California, Davis, CA, 956116, USA
 3. Statistical Laboratory, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 0WB, UK