Stochastic Interacting Population Dynamics and Related Problems

连续态分支过程要么源于经典离散状态Galton-Watson分支过程的时间种群标度极限，要么源于谱正Lévy过程在达到0时停止的Lamperti变换。最近在引入种群模型方面取得了进展，该模型的繁殖机制取决于整个种群的特征。其中，Li(2019)引入了一类具有非线性分枝机制的CSBP。它们是由具有一般速率函数的谱正Lévy过程的广义Lamperti型随机时间变换得到的。直观地说，这种过程的分枝率取决于其当前的种群大小。结果，加性分支属性不再成立，许多用于CSBP的标准方法失败。非线性分支机制允许奇异的性质，如从无穷远处下来，这一点在傅卡特等人中进行了详细的研究。(2020)，对于某些速率函数类确定了从无穷大下降的速度。Li等人也对边界行为进行了研究。(2019a)，用于更一般的非线性CSBP。本着同样的精神，任等人。(2019年)提出并研究了两个种群的随机Lotka-Volterra种群动力学，该模型由两个随机微分方程组组成，由独立的Lévy噪声驱动，其中两个种群都根据Li等人的非线性CSBP进化。(2019a)，第二种群的分枝率取决于第一种群。我们计划继续研究非线性CSBPs和相关的相互作用的种群系统。我们对非线性CSBP的熄灭行为很感兴趣，并想知道当它发生时，这个过程接近0的速度有多慢。我们还想证明非线性CSBP的强Feller性质，这将有助于研究非线性CSBP的准平稳分布。我们将进一步研究Li等人的模型。(2019a)和Ren等人的模型。(2019年)。对于Li等人的非线性CSBP。(2019a)，一个更具挑战性的开放问题是建立边界分类的尖锐积分检验。对于Ren等人的随机种群动力学。(2019年)，我们计划引入和研究双向互动的人口模型。我们还建议探索将空间结构结合起来研究超过程和相关SPDEs与平均场相交的相似行为的可能性。这项研究有助于更好地理解种群内部和(或)种群之间的相互作用对种群动态极端行为的影响。此外，由于一般马尔可夫过程的边界行为和一般带跳的随机微分方程解的边界行为还没有被系统地研究，因此所提出的研究也有望对随机过程理论做出重大贡献。

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.