We prove Freidlin-Wentzell type large deviation principles for various rescaled models in populations dynamics that have immigration and possibly harvesting: birth-death processes, Galton-Watson trees, epidemic SI models, and prey-predator models. The proofs are carried out using a general analytic approach based on the well-posedness of a class of associated Hamilton-Jacobi equations. The notable feature for these Hamilton-Jacobi equations is that the Hamiltonian can be discontinuous at the boundary. We prove a well-posedness result for a large class of Hamilton-Jacobi equations corresponding to one-dimensional models, and give partial results for the multi-dimensional setting.

中文翻译：

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Hamilton-Jacobi 方程在种群动力学中的适定性及其对大偏差的应用

我们证明了 Freidlin-Wentzell 型大偏差原则适用于具有移民和可能收获的种群动态中的各种重新缩放模型：出生-死亡过程、高尔顿-沃森树、流行病 SI 模型和猎物-捕食者模型。证明是使用基于一类关联 Hamilton-Jacobi 方程的适定性的一般分析方法进行的。这些 Hamilton-Jacobi 方程的显着特征是哈密顿量可以在边界处不连续。我们证明了对应于一维模型的一大类 Hamilton-Jacobi 方程的适定性结果，并给出了多维设置的部分结果。