Stochastic Interacting Population Dynamics and Related Problems

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Continuous-state branching process (CSBP for short) arises either from time-population scaling limit of the classical discrete-state Galton-Watson branching processes or from the Lamperti transform of a spectrally positive Lévy process stopped whenever reaching 0. Recent progresses have been made in introducing population models whose reproduction mechanisms depend on features of the entire populations. Among them a class of CSBPs with nonlinear branching mechanisms is introduced in Li (2019). They are obtained by generalized Lamperti type random time transformations from spectrally positive Lévy processes with general rate functions. Intuitively, the branching rate for such a process depends on its current population size. As a result, the additive branching property does not hold anymore and many standard methods for CSBPs fail. The nonlinear branching mechanism allows exotic properties such as coming down from infinity, which is investigated in detail in Foucart et al. (2020) with speeds of coming down from infinity identified for certain classes of rate functions. The boundary behaviors are also investigated in Li et al. (2019a) for more general nonlinear CSBPs. In the same spirit, Ren et al. (2019) propose and study a stochastic Lotka-Volterra population dynamics of two populations modeled by a system of two stochastic differential equations driven by independent Lévy noises, where both populations evolve according to nonlinear CSBPs of Li et al. (2019a) and the branching rates of the second population depend on the first population. We plan to continue with the study on the nonlinear CSBPs and the related interacting population systems. We are interested in the extinguishing behaviors of the nonlinear CSBP and want to know, when it occurs, how slowly the process approaches to 0. We also want to prove the strong Feller property for nonlinear CSBPs, which we believe will help to investigate the quasi-stationary distributions of the nonlinear CSBPs. We are going to further study the models in Li et al. (2019a) and the models in Ren et al. (2019). For the nonlinear CSBPs of Li et al. (2019a), a more challenging open problem is to establish sharp integral tests on boundary classification. For the stochastic population dynamics of Ren et al. (2019), we plan to introduce and study population models with two-way interactions. We also propose to explore the possibility of incorporating the spatial structures to study the similar behaviors for superprocesses and related SPDEs with mean field intersections. The proposed research helps to better understand the effects of interactions within and (or) between populations on the extreme behaviors of the population dynamics. In addition, since the boundary behaviors for general Markov processes and for solutions to general SDEs with jumps have not been systematically investigated, the proposed research is also expected to make significant contributions to the theory of stochastic processes.