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

   Renaud Dessalles, Maria D’Orsogna (October 2018, Journal of statistical physics)

   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. 

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

   Renaud Dessalles, Maria D'Orsogna (October 2018, Journal of statistical physics)

   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. 

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3. Dynamic contagion in a banking system with births and defaults

   https://doi.org/10.1007/s10436-019-00351-2  

   Ichiba, Tomoyuki; Ludkovski, Michael; Sarantsev, Andrey. (January 2019, Annals of finance)

   We consider a dynamic model of interconnected banks. New banks can emerge, and existing banks can default, creating a birth-and-death setup. Microscopically, banks evolve as independent geometric Brownian motions. Systemic effects are captured through default contagion: as one bank defaults, reserves of other banks are reduced by a random proportion. After examining the long-term stability of this system, we investigate mean-field limits as the number of banks tends to infinity. Our main results concern the measure-valued scaling limit which is governed by a McKean–Vlasov jump-diffusion. The default impact creates a mean-field drift, while the births and defaults introduce jump terms tied to the current distribution of the process. Individual dynamics in the limit is described by the propagation of chaos phenomenon. In certain cases, we explicitly characterize the limiting average reserves. 

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4. Exponential ergodicity and steady-state approximations for a class of markov processes under fast regime switching

   https://doi.org/10.1017/apr.2020.47  

   Arapostathis, Ari; Pang, Guodong; Zheng, Yi (March 2021, Advances in Applied Probability)

   null (Ed.)

   Abstract We study ergodic properties of a class of Markov-modulated general birth–death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken to be large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the ‘averaged’ fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov-modulated birth–death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied. 

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5. A class of identifiable phylogenetic birth–death models

   https://doi.org/10.1073/pnas.2119513119  

   Legried, Brandon; Terhorst, Jonathan (August 2022, Proceedings of the National Academy of Sciences)

   In a striking result, Louca and Pennell [S. Louca, M. W. Pennell, Nature 580, 502–505 (2020)] recently proved that a large class of phylogenetic birth–death models is statistically unidentifiable from lineage-through-time (LTT) data: Any pair of sufficiently smooth birth and death rate functions is “congruent” to an infinite collection of other rate functions, all of which have the same likelihood for any LTT vector of any dimension. As Louca and Pennell argue, this fact has distressing implications for the thousands of studies that have utilized birth–death models to study evolution. In this paper, we qualify their finding by proving that an alternative and widely used class of birth–death models is indeed identifiable. Specifically, we show that piecewise constant birth–death models can, in principle, be consistently estimated and distinguished from one another, given a sufficiently large extant timetree and some knowledge of the present-day population. Subject to mild regularity conditions, we further show that any unidentifiable birth–death model class can be arbitrarily closely approximated by a class of identifiable models. The sampling requirements needed for our results to hold are explicit and are expected to be satisfied in many contexts such as the phylodynamic analysis of a global pandemic. 

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