We study an integral-differential equation that models a pure birth-jump process, where birth and dispersal cannot be decoupled. A case has been made that these processes are more suitable for phenomena such as plant dynamics, fire propagation, and cancer cell dynamics. We contrast the dynamics of this equation with those of the classical reaction-diffusion equation, where the reaction term models either logistic growth or a strong Allee effect. Recent evidence of an Allee effect has been found in plant dynamics during the germination process (due to seed predation) but not in the generation of seeds. This motivates where the Allee effect is included in our model. We prove the global existence and uniqueness of solutions with bounded initial data and analyze some properties of the solutions. Additionally, we prove results related to the persistence or extinction of a species, which are analogous to those of the classical reaction-diffusion equation. A key finding is that in some cases a population which is initially below the Allee threshold in some area, even if small, will actually survive. This is in contrast to solutions of the classical reaction-diffusion with the same initial data. Another difference of note is the lack of regularization and an infinite number of discontinuous equilibrium solutions to the birth-jump model.

我们研究了积分微分方程，该方程对纯出生跃迁过程进行建模，在该过程中出生与扩散无法分离。已经证明这些过程更适合诸如植物动态，火灾传播和癌细胞动态等现象。我们将该方程的动力学与经典反应扩散方程的动力学进行对比，在经典反应扩散方程中，反应项模拟逻辑增长或强阿利效应。在发芽过程中（由于种子捕食）在植物动态中发现了Allee效应的最新证据，但在种子的产生中却没有发现。这会激发在模型中包括Allee效应的位置。我们用有限的初始数据证明了解的全局性和唯一性，并分析了解的某些性质。另外，我们证明了与物种的持久性或灭绝有关的结果，类似于经典的反应扩散方程式。一个关键发现是，在某些情况下，即使在很小的地方，最初在某些地区甚至低于Allee阈值的人口实际上仍可以生存。这与具有相同初始数据的经典反应扩散的解决方案相反。另一个值得注意的区别是出生跳模型缺乏正则化和无穷多个不连续平衡解。这与具有相同初始数据的经典反应扩散的解决方案相反。另一个值得注意的区别是出生跳模型缺乏正则化和无穷多个不连续平衡解。这与具有相同初始数据的经典反应扩散的解决方案相反。另一个值得注意的区别是出生跳模型缺乏正则化和无穷多个不连续平衡解。