We develop a time and space-dependent predator—prey model. The predators' equation is a nonlocal hyperbolic balance law, while the diffusion of prey obeys a parabolic equation, so that predators “hunt” for prey, while prey diffuse. A control term allows to describe the use of predators as parasitoids to limit the growth of prey–parasites. The general well-posedness and stability results here obtained ensure the existence of optimal pest control strategies, as discussed through some numerical integrations. The specific example we have in mind is that of Trichopria drosophilæ used to fight against the spreading of Drosophila suzukii.

中文翻译：

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双曲线-抛物线寄生虫-寄生虫系统中的适定性和控制

我们开发了一个依赖时间和空间的捕食者——猎物模型。捕食者方程是一个非局部双曲平衡定律，而猎物的扩散服从抛物线方程，因此捕食者“猎杀”猎物，猎物扩散。控制术语允许将捕食者用作寄生虫来限制猎物寄生虫的生长。正如通过一些数值积分所讨论的，这里获得的一般适定性和稳定性结果确保了最佳害虫控制策略的存在。我们想到的具体例子是用于对抗Drosophila suzukii传播的Trichopria drosophilæ。