A Nonlocal and Delayed Predator—Prey Model

Abstract

The celebrated Lotka—Volterra model proposed by Lotka [155] in the context of chemical reactions and by Volterra [251] for prey—predator dynamics has been generalized in several directions: to include many species with complicated interactions, to include spatial effects in either a discrete way or a continuous way, and to include delays or internal population structure. Sometimes these generalizations combine diffusion and delays (see, e.g., [267]). While most of the delayed diffusion equations in the literature are local, nonlocal effects very naturally appear in diffusive prey—predator models with delays if one carefully models the delay as condensation of the underlying retarding process and takes into account that individuals move during this process (see [97]). In the predator equation, the delay is often caused by the conversion of consumed prey biomass into predator biomass, whether in the form of body size growth or of reproduction.