Species-abundance distributions under colored environmental noise

Abstract

Natural communities at all spatiotemporal scales are subjected to a wide variety of environmental pressures, resulting in random changes in the demographic rates of species populations. Previous analyses have examined the effects of such environmental variance on the long-term growth rate and time to extinction of single populations, but studies of its effects on the diversity of communities remain scarce. In this study, we construct a new master-equation model incorporating demographic and environmental variance and use it to examine how statistical patterns of diversity, as encapsulated by species-abundance distributions (SADs), are altered by environmental variance. Unlike previous diffusion models with environmental variance uncorrelated in time (white noise), our model allows environmental variance to be correlated at different timescales (colored noise), thus facilitating representation of phenomena such as yearly and decadal changes in climate. We derive an exact analytical expression for SADs predicted by our model together with a close approximation, and use them to show that the main effect of adding environmental variance is to increase the proportion of abundant species, thus flattening the SAD relative to the log-series form found in the neutral case. This flattening effect becomes more prominent when environmental variance is more correlated in time and has greater effects on species’ demographic rates, holding all other factors constant. Furthermore, we show how our model SADs are consistent with those from diffusion models near the white noise limit. The mathematical techniques we develop are catalysts for further theoretical work exploring the consequences of environmental variance for biodiversity.

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References

  1. Allen AP, Savage VM (2007) Setting the absolute tempo of biodiversity dynamics. Ecol. Lett. 10:637–646

    Article  Google Scholar 

  2. Ariño A, Pimm SL (1995) The nature of population extremes. Evol. Ecol. 9:429–443

    Article  Google Scholar 

  3. Azaele S, Pigolotti S, Banavar JR, Maritan A (2006) Dynamical evolution of ecosystems. Nature 444:926–928

    Article  Google Scholar 

  4. Bell G (2000) The distribution of abundance in neutral communities. Am. Nat. 155:606–617

    Article  Google Scholar 

  5. Bell G (2001) Neutral macroecology. Science 293:2413–2418

    Article  Google Scholar 

  6. Benton TG, Lapsley CT, Beckerman AP (2002) The population response to environmental noise: population size, variance and correlation in an experimental system. J. Anim. Ecol. 71:320–322

    Article  Google Scholar 

  7. Caswell H (1976) Community structure: a neutral model analysis. Ecol. Monogr. 46:327–354

    Article  Google Scholar 

  8. Caswell H, Cohen JE (1995) Red, white and blue: environmental variance spectra and coexistence in metapopulations. J. Theor. Biol. 176:301–316

    Article  Google Scholar 

  9. Chesson P (1991) Stochastic population models. In: Kolasa J, Pickett STA (eds) Ecological Heterogeneity. Springer-Verlag, New York, pp 123–143

    Chapter  Google Scholar 

  10. Chisholm RA, Lichstein JW (2009) Linking dispersal, immigration and scale in the neutral theory of biodiversity. Ecol. Lett. 12:1385–1393

    Article  Google Scholar 

  11. Chisholm RA, Pacala SW (2010) Niche and neutral models predict asymptotically equivalent species abundance distributions in high-diversity ecological communities. Proc. Natl. Acad. Sci. USA 107:15821–15825

    Article  Google Scholar 

  12. Chisholm RA, O’Dwyer JP (2014) Species ages in neutral biodiversity models. Theor. Popul. Biol. 93:85–94

    Article  MATH  Google Scholar 

  13. Chisholm RA et al (2014) Temporal variability of forest communities: empirical estimates of population change in 4000 tree species. Ecol. Lett. 17:855–865

    Article  Google Scholar 

  14. 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. Sim. 3:283–299

    MathSciNet  Article  MATH  Google Scholar 

  15. Engen S, Lande R (1996a) Population dynamic models generating species abundance distributions of the gamma type. J. Theor. Biol. 178:325–331

    Article  MATH  Google Scholar 

  16. Engen S, Lande R (1996b) Population dynamic models generating the lognormal species abundance distribution. Math. Biosci. 132:169–183

    Article  MATH  Google Scholar 

  17. Engen S, Lande R, Walla T, DeVries PJ (2002) Analyzing spatial structure of communities using the two-dimensional poisson lognormal species abundance model. Am. Nat. 160:60–73

    Article  Google Scholar 

  18. Engen S, Aagaard K, Bongard T (2011) Disentangling the effects of heterogeneity, stochastic dynamics and sampling in a community of aquatic insects. Ecol. Model. 222:1387–1393

    Article  Google Scholar 

  19. Etienne RS, Alonso D, McKane AJ (2007) The zero-sum assumption in neutral biodiversity theory. J. Theor. Biol. 248:522–536

    MathSciNet  Article  Google Scholar 

  20. Fisher RA, Corbet AS, Williams CB (1943) The relation between the number of species and the number of individuals in a random sample of an animal population. J. Anim. Ecol. 74:1131–1139

    Google Scholar 

  21. Foley P (1994) Predicting extinction times from environmental stochasticity and carrying capacity. Conserv. Biol. 8:124–137

    Article  Google Scholar 

  22. Gyllenberg M, Högnäs G, Koski T (1994) Population models with environmental stochasticity. J. Math. Biol. 32:93–108

    MathSciNet  Article  MATH  Google Scholar 

  23. Halley JM (1996) Ecology, evolution and 1/f-noise. Trends Ecol. Evol. 11:33–37

    Article  Google Scholar 

  24. Halley JM, Iwasa Y (1998) Extinction rate of a population under both demographic and environmental stochasticity. Theor. Popul. Biol. 53:1–15

    Article  MATH  Google Scholar 

  25. Halley JM, Iwasa Y (2011) Neutral theory as a predictor of avifaunal extinctions after habitat loss. Proc. Natl. Acad. Sci. USA 108:2316–2321

    Article  Google Scholar 

  26. Heino M, Ripa J, Kaitala V (2000) Extinction risk under coloured environmental noise. Ecography 23:177–184

    Article  Google Scholar 

  27. Heino M, Sabadell M (2003) Influence of coloured noise on the extinction risk in structured population models. Biol. Conserv. 110:315–325

    Article  Google Scholar 

  28. Hubbell SP (1997) A unified theory of biogeography and relative species abundance and its application to tropical rainforests and coral reefs. Coral Reefs 16(Suppl.):S9–S21

    Article  Google Scholar 

  29. Hubbell SP (2001) The Unified Neutral Theory of Biodiversity and Biogeography. Princeton University Press, Princeton

    Google Scholar 

  30. Inchausti P, Halley J (2002) The long-term temporal variability and spectral colour of animal populations. Evol. Ecol. Res. 4:1033–1048

    Google Scholar 

  31. Kaitala V, Ranta E (1996) Red/blue chaotic power spectra. Nature 381:198–199

    Article  Google Scholar 

  32. Kaitala V, Ylikarjula J, Ranta E, Lundberg P (1997) Population dynamics and the colour of environmental noise. Proc. R. Soc. Lond. B Biol. 264:943–948

    Article  Google Scholar 

  33. Kalyuzhny M, Schreiber Y, Chocron R, Flather CH, Kadmon R, Kessler DA, Shnerb NM (2014a) Temporal fluctuation scaling in populations and communities. Ecology 95:1701–1709

    Article  Google Scholar 

  34. Kalyuzhny M, Seri E, Chocron R, Flather CH, Kadmon R, Shnerb NM (2014b) Niche versus neutrality: a dynamical analysis. Am. Nat. 184:439–446

    Article  Google Scholar 

  35. Kalyuzhny M, Kadmon R, Shnerb NM (2015) A neutral theory with environmental stochasticity explains static and dynamic properties of ecological communities. Ecol. Lett. 18:572–580

    Article  Google Scholar 

  36. Kamenev A, Meerson B, Shklovskii B (2008) How colored environmental noise affects population extinction. Phys. Rev. Lett. 101:268103

    Article  Google Scholar 

  37. Kemeny JG, Snell JL (1960) Finite Markov Chains. Van Norstrand, Princeton

    MATH  Google Scholar 

  38. Kessler DA, Shnerb NM (2014) Neutral-like abundance distributions in the presence of selection in a continuous fitness landscape. J. Theor. Biol. 345:1–11

    MathSciNet  Article  MATH  Google Scholar 

  39. Lande R (1993) Risks of population extinction from demographic and environmental stochasticity and random catastrophes. Am. Nat. 142:911–927

    MathSciNet  Article  Google Scholar 

  40. Lande R, Engen S, Sæther BE (2003) Stochastic Population Dynamics in Ecology and Conservation. Oxford University Press, Oxford

    Book  MATH  Google Scholar 

  41. Leigh EG (1981) The average lifetime of a population in a varying environment. J. Theor. Biol. 90:213–239

    MathSciNet  Article  Google Scholar 

  42. Lotka AJ (1920) Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 24:254–262

    Google Scholar 

  43. Lotka AJ (1925) Elements of Physical Biology. Williams and Wilkins, Baltimore

    MATH  Google Scholar 

  44. MacArthur R, Levins R (1967) The limiting similarity, convergence, and divergence of coexisting species. Am. Nat. 101:377–385

    Article  Google Scholar 

  45. Manrubia SC, Zanette DH (2002) At the boundary between biological and cultural evolution: the origin of surname distributions. J. Theor. Biol. 216:461–477

    MathSciNet  Article  Google Scholar 

  46. Mantua, N.J.: Pacific-Decadal Oscillation (PDO). In: MacCracken, M.C., Perry, J.S. (eds.) The Earth System, Physical and Chemical Dimensions of Global Environmental Change. Wiley, Chichester, pp. 592–594 (2002)

  47. Maruvka YE, Shnerb NM, Kessler DA (2010) Universal features of surname distribution in a subsample of a growing population. J. Theor. Biol. 262:245–256

    MathSciNet  Article  Google Scholar 

  48. Maruvka YE, Shnerb NM, Kessler DA (2011) The birth-death-mutation process: a new paradigm for fat tailed distributions. PLoS One 6:e26480

    Article  Google Scholar 

  49. Maruvka YE, Shnerb NM, Kessler DA, Ricklefs RE (2013) Model for macroevolutionary dynamics. Proc. Natl. Acad. Sci. USA 27:E2460–E2469

    Article  Google Scholar 

  50. Melbourne BA, Hastings A (2008) Extinction risk depends strongly on factors contributing to stochasticity. Nature 454:100–103

    Article  Google Scholar 

  51. Méndez V, Llopis I, Campos D, Horsthemke W (2010) Extinction conditions for isolated populations affected by environmental stochasticity. Theor. Popul. Biol. 77:250–256

    Article  Google Scholar 

  52. Mode CJ, Jacobson ME (1987a) A study of the impact of environmental stochasticity on extinction probabilities by Monte Carlo integration. Math. Biosci. 83:105–125

    MathSciNet  Article  MATH  Google Scholar 

  53. Mode CJ, Jacobson ME (1987b) On estimating critical population size for an endangered species in the presence of environmental stochasticity. Math. Biosci. 85:185–209

    MathSciNet  Article  MATH  Google Scholar 

  54. Morales JM (1999) Viability in a pink environment: why “white noise” models can be dangerous. Ecol. Lett. 2:228–232

    Article  Google Scholar 

  55. Nee S (2005) The neutral theory of biodiversity: do the numbers add up? Funct. Ecol. 19:173–176

    Article  Google Scholar 

  56. O’Dwyer JP, Green JL (2010) Field theory for biogeography: a spatially explicit model for predicting patterns of biodiversity. Ecol. Lett. 13:87–95

    Article  Google Scholar 

  57. O’Dwyer JP, Chisholm R (2014) A mean field model for competition: from neutral ecology to the Red Queen. Ecol. Lett. 17:961–969

    Article  Google Scholar 

  58. Ovaskainen O, Meerson B (2010) Stochastic models of population extinction. Trends Ecol. Evol. 25:643–652

    Article  Google Scholar 

  59. Petchey OL (2000) Environmental colour affects aspects of single-species population dynamics. Proc. R. Soc. Lond. B Biol. 267:747–754

    Article  Google Scholar 

  60. Pimm SL, Redfearn A (1988) The variability of population densities. Nature 334:613–614

    Article  Google Scholar 

  61. Preston FW (1948) The commonness and rarity of species. Ecology 29:254–283

    Article  Google Scholar 

  62. Preston FW (1962a) The canonical distribution of commonness and rarity: part I. Ecology 43:185–215

    Article  Google Scholar 

  63. Preston FW (1962b) The canonical distribution of commonness and rarity: part II. Ecology 43:410–432

    Article  Google Scholar 

  64. Reuman DC, Desharnais RA, Costantino RF, Ahmad OS, Cohen JE (2006) Power spectra reveal the influence of stochasticity on nonlinear population dynamics. Proc. Natl. Acad. Sci. USA 103:18860–18865

    Article  Google Scholar 

  65. Ripa J, Lundberg P (1996) Noise colour and the risk of population extinctions. Proc. R. Soc. Lond. B Biol. 263:1751–1753

    Article  Google Scholar 

  66. Ruokolainen L, Linden A, Kaitala V, Fowler MS (2009) Ecological and evolutionary dynamics under coloured environmental variation. Trends Ecol. Evol. 24:555–563

    Article  Google Scholar 

  67. Schreiber SJ (2010) Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence. Proc. R. Soc. Lond. B Biol. 277:1907–1914

    Article  Google Scholar 

  68. Schwager M, Johst K, Jeltsch R (2006) Does red noise increase or decrease extinction risk? Single extreme events versus series of unfavorable conditions. Am. Nat. 167:879–888

    Article  Google Scholar 

  69. Steele JH (1985) A comparison of terrestrial and marine ecological systems. Nature 313:355–358

    Article  Google Scholar 

  70. Vallade M, Houchmandzadeh B (2003) Analytical solution of a neutral model of biodiversity. Phys. Rev. E 68:061902

    Article  Google Scholar 

  71. Volkov I, Banavar JR, Hubbell SP, Maritan A (2003) Neutral theory and relative species abundance in ecology. Nature 424:1035–1037

    Article  Google Scholar 

  72. Volkov I, Banavar JR, Hubbell SP, Maritan A (2007) Patterns of relative species abundance in rainforests and coral reefs. Nature 450:45–49

    Article  Google Scholar 

  73. Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memorie Della R. Academia Nazionale Dei Lincei 2, 31–113 (1926)

  74. Xu C-L, Li Z-Z (2003) Population dynamics and the color of environmental noise: a study on a three-species food chain system. Ecol. Res. 18:145–154

    Article  Google Scholar 

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Acknowledgments

TF and RAC are supported by the National University of Singapore grants WBS R-154-000-603-112 and R-154-000-560-651. JPOD acknowledges support from the Templeton World Charity Foundation grant TWCF0079/AB47. We also thank Jin Yi Lau, Felix Lim and Francesca McGrath for constructive discussions on the work presented.

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Correspondence to Tak Fung.

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Fung, T., O’Dwyer, J.P. & Chisholm, R.A. Species-abundance distributions under colored environmental noise. J. Math. Biol. 74, 289–311 (2017). https://doi.org/10.1007/s00285-016-1022-4

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Keywords

  • Colored noise
  • Demographic variance
  • Environmental variance
  • Fokker–Planck equation
  • Master equation
  • Species-abundance distribution

Mathematics Subject Classification

  • 37N25
  • 92D25
  • 92D40