%0 Journal Article %1 Lambert:2005bq %A Lambert, A %D 2005 %J Annals of Applied Probability %K branching_processes density_dependence fragmentation_coalescence logistic_growth population_dynamics %P 1506-1535 %T The branching process with logistic growth %U http://arxiv.org/pdf/math.PR/0505249 %V 15 %X In order to model random density-dependence in population dynamics, we construct the random analogue of the well-known logistic process in the branching process' 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 functional of the solution of a Riccati differential 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 ∞. This result can be viewed as an extension of the pure-death process starting from infinity associated to Kingman's coalescent, when some independent fragmentation is added.

@article{Lambert:2005bq, abstract = {In order to model random density-dependence in population dynamics, we construct the random analogue of the well-known logistic process in the branching process' 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 functional of the solution of a Riccati differential 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 \∞. This result can be viewed as an extension of the pure-death process starting from infinity associated to Kingman's coalescent, when some independent fragmentation is added.}, added-at = {2009-01-15T22:24:48.000+0100}, author = {Lambert, A}, biburl = {https://www.bibsonomy.org/bibtex/24bd0e181d93790472096d75fac171165/peter.ralph}, description = {alexeilist1}, interhash = {29fbeb09caf1394f60e9bc55038b1a1b}, intrahash = {4bd0e181d93790472096d75fac171165}, journal = {Annals of Applied Probability}, keywords = {branching_processes density_dependence fragmentation_coalescence logistic_growth population_dynamics}, mrclass = {60J80 (60J60 60J70 92D15 92D25 92D40)}, mrnumber = {MR2134113 (2006h:60137)}, mrreviewer = {D. R. Grey}, pages = {1506-1535}, timestamp = {2009-01-15T22:24:48.000+0100}, title = {The branching process with logistic growth}, url = {http://arxiv.org/pdf/math.PR/0505249}, volume = 15, year = 2005 }