Stochastic Interacting Population Dynamics and Related Problems

随机相互作用的种群动态及相关问题

• 批准号: RGPIN-2021-04100
• 负责人: Zhou, Xiaowen
• 金额: $ 1.75万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742870
• 关键词: Stochastic; Interacting; Population; Dynamics; Related

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.

连续状态分支过程（简称CSBP）要么源于经典离散状态Galton-Watson分支过程的时间总体标度极限，要么源于光谱正Lévy过程的Lamperti变换，每当达到0时就会停止。最近取得了进展Li (2019) 引入了种群模型，其繁殖机制取决于整个种群的特征，其中引入了一类具有非线性分支机制的 CSBP。具有一般速率函数的谱正 Lévy 过程的 Lamperti 型随机时间变换直观地讲，此类过程的分支率取决于其当前的总体规模，因此，加性分支属性不再成立，并且许多 CSBP 的标准方法都失败了。非线性分支机制允许奇异的特性，例如从无穷大下降，Foucart 等人（2020）对此进行了详细研究，并为某些类别的速率函数确定了从无穷大下降的速度。 Li 等人 (2019a) 也研究了更一般的非线性 CSBP，Ren 等人 (2019) 提出并研究了由系统建模的两个种群的随机 Lotka-Volterra 种群动态。由独立 Lévy 噪声驱动的两个随机微分方程，其中两个种群均根据 Li 等人 (2019a) 的非线性 CSBP 演化，并且第二个种群的分支率取决于第一个种群。我们计划继续研究非线性 CSBP 和相关的相互作用的总体系统。我们对非线性 CSBP 的消失行为很感兴趣，并想知道当它发生时，该过程接近 0 的速度有多慢。希望证明非线性 CSBP 的强 Feller 性质，我们相信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 (2019a) 中的模型。对于 Ren 等人 (2019a) 的非线性 CSBP，一个更具挑战性的开放问题是针对 Ren 等人的随机总体动态建立尖锐的积分检验。 （2019），我们计划引入和研究具有双向相互作用的群体模型，我们还建议探索结合空间结构来研究具有平均场交叉的超级过程和相关 SPDE 的相似行为的可能性。有助于更好地理解群体内部和（或）群体之间的相互作用对群体动态极端行为的影响。此外，由于一般马尔可夫过程和具有跳跃的一般 SDE 的解决方案的边界行为尚未得到系统研究。拟议的研究预计还将对随机过程理论做出重大贡献。