A dual-gradient chemotaxis system modeling the spontaneous aggregation of microglia in Alzheimer’s disease

In this paper, we consider the following dual-gradient chemotaxis model [Formula: see text] with [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text] and [Formula: see text]. The model was proposed to interpret the spontaneous aggregation of microglia in Alzheimer’s disease due to the interaction of attractive and repulsive chemicals released by the microglia. It has been shown in the literature that, when [Formula: see text], the solution of the model with homogeneous Neumann boundary conditions either blows up or asymptotically decays to a constant in multi-dimensions depending on the sign of [Formula: see text], which means there is no pattern formation. In this paper, we shall show as [Formula: see text], the uniformly-in-time bounded global classical solutions exist in multi-dimensions and hence pattern formation can develop. This is significantly different from the results for the case [Formula: see text]. We perform the numerical simulations to illustrate the various patterns generated by the model, verify our analytical results and predict some unsolved questions. Biological applications of our results are discussed and open problems are presented.