The aim of this paper is to investigate the existence, uniqueness and stability of mild solutions to a stochastic delayed reaction–diffusion equation with spatial non-locality. This equation can be used to model the spatial–temporal evolution for age-structured spices perturbed by some random effects or the stochastic neural networks. The Banach fixed point theorem and a truncation method are adopted to establish the existence and uniqueness of mild solutions under both global and local Lipschitz conditions. Then, we explore the mean square exponential stability and almost sure exponential stability by employing the inequality techniques, the stochastic analysis techniques together with the properties of the nonlocal delayed term. Furthermore, we obtain the critical value of time delay \(\tau \) that guarantees the stability of the mild solutions. At last, our theoretic results are illustrated by application to the stochastic non-local delayed Nicholson blowflies equation with numerical simulations.

本文的目的是研究具有空间非局域性的随机延迟反应扩散方程的温和解的存在性、唯一性和稳定性。该方程可用于模拟受一些随机效应或随机神经网络干扰的年龄结构香料的时空演变。在全局和局部 Lipschitz 条件下，采用 Banach 不动点定理和截断方法建立温和解的存在性和唯一性。然后，我们通过使用不等式技术、随机分析技术以及非局部延迟项的性质来探索均方指数稳定性和几乎肯定指数稳定性。此外，我们得到时间延迟的临界值\(\tau \)这保证了温和溶液的稳定性。最后，将我们的理论结果应用到随机非局部延迟 Nicholson 苍蝇方程并进行数值模拟。