Global existence for chemotaxis with finite sampling radius

Abstract

Abstract. Migrating cells measure the external environment through receptorbinding of specific chemicals at their outer cell membrane. In this paper this non-local sampling is incorporated into a chemotactic model. The existence of global bounded solutions of the non-local model is proven for bounded and unbounded domains in any space dimension. According to a recent classification of spikes and plateaus, it is shown that steady state solutions cannot be of spike-type. Finally, numerical simulations support the theoretical results, illustrating the ability of the model to give rise to pattern formation. Some biologically relevant extensions of the model are also considered.1. Introduction. Chemotaxis, the active orientation of cells and organisms along chemical gradients, plays a crucial role in many biological processes, including embryonic development, immunology and cancer growth. Accordingly, a vast amount of research, both experimental and theoretical, has been devoted to understanding the mechanistic basis of chemotaxis.In 1953, Patlak [28] introduced the first mathematical model for chemotaxis. A similar model was derived by Keller and Segel in 1970 [16], albeit under different assumptions. These pioneering works have initiated an intensive mathematical investigation of the Patlak-Keller-Segel (PKS) model over the last 30 years. Of particular interest is the following special case, which we refer to as the classical chemotaxis model:

A user’s guide to PDE models for chemotaxis

“…3. Notice that in [37] we claimed thatv is a plateau for model (M4). Unfortunately, our proof of Theorem 3 in [37] is wrong, and we use an Appendix of this paper to correct this.…”

“…In either case, the signal detected by the cell is intrinsically non-local and it may therefore be appropriate to consider movement based on a non-local gradient by the integration of the signal by the cell over some region. A model incorporating this was postulated in [77] and a detailed derivation, analytical and numerical study has been carried out by [37].…”

“…A detailed derivation of the above model has been carried out by [37] where for ρ > 0 it was shown that the above equations have globally existing solutions in all space dimensions. In addition, detailed stability and numerical analyses were performed.…”

“…The first method can be summarised in the following Lemma 1 from [37]. [7] (M2b) β = 0 Global existence below threshold 6.1.1. in Horstmann [40] (M3a) Global existence in nD Hillen and Painter [36] and Wrzosek [109] (M3b) Global existence in nD Velazquez [102] and Lemma 5 (M4) Global existence in nD Hillen et al [37] (M5) Global existence in nD Kowalczyk [50] (M6) Global existence in nD Horstmann [39] (M7) Global existence in nD Lemma 3…”

“…The method of uniform estimates in higher dimensions has been applied, for example, to the volume-filling model (M3a) in [36], and the non-local model (M4) in [37]. It also applies to model (M7), since the chemotaxis term contains = F c , with F c given in Eq.…”