Using bifurcation theory, we prove the existence of spiky steady states and investigate the stability of bifurcating solutions of the one-dimensional continuous neighbour based chemotaxis model, in which the one-step jumping probability rate of cells is determined only by the chemoattractant concentration at the destination. These spiky steady states are crucial when we model cell aggregation, the most important phenomenon in chemotaxis.
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Figure 1. On the horizontal axis is the average mass of chemical substance, which is $ v* $. On the vertical axis is the space of $ (u, v) $. Every point on the cure is a solution of (1.7). On the "upper", branch, the solutions are decreasing in $ \Omega $ and vice versa. $ \bar{v}_1^* $ is a positive constant and will be given in Section 2
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On the horizontal axis is the average mass of chemical substance, which is