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This paper investigates the formation of time-periodic and stable patterns of a two-competingspecies Keller-Segel chemotaxis model with a focus on the effect of cellular growth. We carry out rigorous Hopf bifurcation analysis to obtain the bifurcation values, spatial profiles and time period associated with these oscillating patterns. Moreover, the stability of the periodic solutions is investigated and it provides a selection mechanism of stable time-periodic mode which suggests that only large domains support the formation of these periodic patterns. Another main result of this paper reveals that cellular growth is responsible for the emergence and stabilization of the oscillating patterns observed in the 3 × 3 system, while the system admits a Lyapunov functional in the absence of cellular growth. Global existence and boundedness of the system in 2D are proved thanks to this Lyapunov functional. Finally, we provide some numerical simulations to illustrate and support our theoretical findings.

In this paper, we study two PDEs that generalize the urban crime model proposed by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249-1267. Our modifications are made under assumption of the spatial heterogeneity of both the near-repeat victimization effect and the dispersal strategy of criminal agents. We investigate pattern formations in the reaction-advection-diffusion systems with nonlinear diffusion over multi-dimensional bounded domains subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as the intrinsic near-repeat victimization rate decreases and spatially nonconstant solutions emerge through bifurcation. Moreover, we find the wavemode selection mechanism through rigorous stability analysis of these nontrivial patterns, which shows that the only stable pattern must have wavenumber that maximizes the bifurcation value. Based on this wavemode selection mechanism, we will be able to precisely predict the formation of stable aggregates of the house attractiveness and criminal population density, at least when the diffusion rate is around the principal bifurcation value. Our theoretical results also suggest that large domains support more stable aggregates than small domains. Finally, we perform extensive numerical simulations over 1D intervals and 2D squares to illustrate and verify our theoretical findings. Our numerics also include some interesting phenomena such as the merging of two interior spikes and the emerging of new spikes, etc. These nontrivial solutions can model the well observed aggregation phenomenon in urban criminal activities.