Bifurcation and Pattern formation in reaction-diffusion models with nonlocal advection and time delays

Reaction-diffusion equations with nonlocal advection and time delays arise naturally in ecological and biological systems where movement is influenced by spatially distributed information, such as perceived resource landscapes or memory of previously visited locations. These models incorporate cognitive or behavioral mechanisms by allowing individuals to respond to environmental signals averaged over spatial regions and past times. In this paper, we review several classes of reaction–diffusion models in which advective movement is governed by spatiotemporal information of population distributions. Different types of advection potentials lead to distinct forms of nonlocal advective forces. When the nonlocal interaction is sufficiently attractive or repulsive, symmetry-breaking bifurcations occur, producing aggregation or segregation patterns, as well as time-periodic solutions that persist as asymptotic states.