The branching process with logistic growth

By Amaury Lambert. 2005

168 citations recorded by the source corpus.

In order to model random density-dependence in population dynamics, we construct the random analogue of the well-known logistic process in the branching processes’ framework. This density-dependence corresponds to intraspecific competition pressure, which is ubiquitous in ecology, and translates mathematically into a quadratic death rate. The logistic branching process, or LB-process, can thus be seen as (the mass of) a fragmentation process (corresponding to the branching mechanism) combined with constant coagulation rate (the death rate is proportional to the number of possible coalescing pairs). In the continuous state-space setting, the LB-process is a time-changed (in Lamperti’s fashion) Ornstein-Uhlenbeck type process. We obtain similar results for both constructions: when natural deaths do not occur, the LB-process converges to a specified distribution; otherwise, it goes extinct a.s. In the latter case, we provide the expectation and the Laplace transform of the absorption time, as a function of the solution of a Riccati dierential equation. We also show that the quadratic regulatory term allows the LB-process to start at infinity, despite the fact that births occur infinitely often as the initial state goes to 1. This result can be viewed as an extension of the pure death process starting from infinity associated to Kingman’s coalescent, when some fragmentation is added.

Subfields: Mathematical Physics, Public Health, Environmental and Occupational Health, Water Science and Technology

View Paper: https://www.semanticscholar.org/paper/34dc0e2367e5c7eee27ce004b889a6019af3a353

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Public Health, Environmental and Occupational Health

In Water Science and Technology

Citations in the corpus

Ergodic behavior of locally regulated branching populations (2005, background): “An example is the so-called logistic branching process [15] in which, in addition to the “natural” births and deaths in a super-critical branching mechanism, there are deaths resulting from a competition between any two individuals in the population.”
Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients (2009, unknown): “ay choose β = 2 and α = 1. The constant C in equation (32) can be chosen as C = max σ, 2 λ , 2η+σ2 λ . 11 Feller diﬀusion with logistic growth The branching process with logistic growth (see e.g. [16]) is a stochastic Verhulst equation with Feller noise. It solves dXt = λXt (K − Xt)dt+σ p Xt dWt, ”
Continuous-State Branching Processes with Immigration (2019, methodology): “Kyprianou (2014) and Li (2011). In the recent work of Pardoux(2016),morecomplicatedprobabilisticpopulationmodelsinvolvingcompetition were studied, which extend the stochastic logistic growth model of Lambert (2005). A continuous CBI-process with subcritical branching mechanism was used by Cox et al.”
Quasi-stationary distributions and diffusion models in population dynamics (2007, methodology): “When h is concave quadratic, we get the logistic Feller diffusion [6, 18].”
Lyapunov criteria for uniform convergence of conditional distributions of absorbed Markov processes (2017, methodology): “easures on E, dened by kk TV = sup f2L1(E);kfk11 j(f)j. We apply this result to two standard models in ecology and evolution, called Lotka-Volterra (or logistic) birth-death or diusion processes [27, 6, 7], which have attracted a lot of attention in the past and for which the question of uniform con”
Quasi-stationary distributions and population processes (2011, methodology): “In the general case where c 6= 0, it defines the so-called logistic Feller diffusion process following the terminology introduced by Etheridge [20] and Lambert [42].”
Exchangeable Fragmentation-Coalescence Processes and their Equilibrium Measures (2004, background): “sion time such that #π(c) |[k+n](t (k) 1 ) ≤ k. Hence Gc k ⊆ {t (k) i s.t. #π (c) |[k+n] (t(k) i ) ≤ k}. As this is true for each n almost surely, the conclusion follows. As recently noted by Lambert [18], there is an interpretation of some EFC processes in terms of population dynamics. More precise”
Stochastic Models for Structured Populations: Scaling Limits and Long Time Behavior (2015, background): “When c = 0, we get the Feller di↵usion equation dXt = p 2 XtdBt +Xt( µ)dt. (3.12) If c 6= 0, Equation (3.11) is called by extension the logistic Feller di↵usion equation (see Etheridge [32] and Lambert [55]).”
Population Dynamics and Random Genealogies (2008, methodology): “Since the properties of the LB-process reviewed hereafter have rather technical proofs, we merely state them and refer the reader to Ref.[79] for details.”