Journal of Statistical Physics

, Volume 173, Issue 1, pp 182–221 | Cite as

Exact Steady-State Distributions of Multispecies Birth–Death–Immigration Processes: Effects of Mutations and Carrying Capacity on Diversity

  • Renaud Dessalles
  • Maria D’Orsogna
  • Tom ChouEmail author


Stochastic models that incorporate birth, death and immigration (also called birth–death and innovation models) are ubiquitous and applicable to many problems such as quantifying species sizes in ecological populations, describing gene family sizes, modeling lymphocyte evolution in the body. Many of these applications involve the immigration of new species into the system. We consider the full high-dimensional stochastic process associated with multispecies birth–death–immigration and present a number of exact and asymptotic results at steady state. We further include random mutations or interactions through a carrying capacity and find the statistics of the total number of individuals, the total number of species, the species size distribution, and various diversity indices. Our results include a rigorous analysis of the behavior of these systems in the fast immigration limit which shows that of the different diversity indices, the species richness is best able to distinguish different types of birth–death–immigration models. We also find that detailed balance is preserved in the simple noninteracting birth–death–immigration model and the birth–death–immigration model with carrying capacity implemented through death. Surprisingly, when carrying capacity is implemented through the birth rate, detailed balance is violated.


Birth–death–immigration processes Multispecies Steady-state probability distributions Diversity Mutations 



This work was supported in part by an INRA Contrat Jeune Scientifique Award (RD) and by the National Science Foundation through grants DMS-1814364 (TC) and DMS-1814090 (MD). The authors also thank Song Xu for clarifying discussions.

Supplementary material


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Authors and Affiliations

  1. 1.Department of BiomathematicsUniversity of CaliforniaLos AngelesUSA
  2. 2.MaIAGE, INRA, Université Paris-SaclayJouy-en-JosasFrance
  3. 3.Department of MathematicsCalifornia State UniversityNorthridgeUSA
  4. 4.Department of MathematicsUniversity of CaliforniaLos AngelesUSA

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