Journal of Mathematical Biology

, Volume 74, Issue 1–2, pp 289–311 | Cite as

Species-abundance distributions under colored environmental noise

  • Tak FungEmail author
  • James P. O’Dwyer
  • Ryan A. Chisholm


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.


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

Mathematics Subject Classification

37N25 92D25 92D40 



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.

Supplementary material

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Supplementary material 1 (pdf 11089 KB)
285_2016_1022_MOESM2_ESM.nb (57 kb)
Supplementary material 2 (nb 57 KB)


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Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Tak Fung
    • 1
    Email author
  • James P. O’Dwyer
    • 2
  • Ryan A. Chisholm
    • 1
    • 3
  1. 1.Department of Biological SciencesNational University of SingaporeSingaporeSingapore
  2. 2.Department of Plant Biology, School of Integrative BiologyUniversity of IllinoisUrbanaUSA
  3. 3.Smithsonian Tropical Research InstituteBalboa, AncónRepublic of Panama

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