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Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth

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    * Corresponding author 

QW is partially supported by NSF-China (Grant 11501460)

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  • This paper investigates the formation of time-periodic and stable patterns of a two-competing-species 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.

    Mathematics Subject Classification: Primary:92C17, 35B10, 35B32, 35B35, 35B36, 35K20, 37K45, 37K50, 92D25.

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  • Figure 1.  Bifurcation diagrams of $\rho_k(s)$ around $(\bar u,\bar v,\bar w)$. The stable bifurcation curve is plotted in solid lines and the unstable bifurcation curve is plotted in imaginary line. The branch $\rho_{k}(s)$ around $(\bar u,\bar v,\bar w,\chi_{k})$ is always unstable if $k\neq k_0$, while the turning direction of $\rho_{k_0}(s)$ determines its stability

    Figure 2.  Initiation and development of time-periodic spatial patterns to (1.1) over $(0,6)$ with initial data being small perturbations of $(\bar u,\bar v,\bar w)$. System parameters are chosen to be $d_1=5$, $d_2=0.1$, $\mu_1=\mu_2=1$, $\lambda=5$, $\xi=0.1$ and $\chi=80$. Our theoretical results indicate that the homogeneous equilibrium loses its stability at $\chi_0=\chi^H_{2}\approx 63.2$ through Hopf bifurcation to a stable time-periodic pattern which has spatial profile $\cos \frac{\pi x}{3}$ and period $T\approx 8$. Space and time grid sizes are $\Delta x=0.02$ and $\Delta t=0.05$. The numerical simulations are in good agreement with our theoretical findings

    Figure 3.  In each subfigure, we plot in the 3D $u$-$v$-$w$ phase space the trajectories for specific locations $x=1,2,...6$ which converge to enclosed orbits. $\Delta x=0.02$ and $\Delta t=0.05$

    Figure 4.  Effect of cellular growth on the pattern formation of $u$-species, where we choose $\mu_1=\mu_2$. System parameters are chosen to be $d_1=8$, $d_2=0.5$, $\chi=130$ and $\xi=0.4$. Initial data are taken to be small perturbations of $(\bar u,\bar v,\bar w)$. Space and time grid sizes are $\Delta x=L/500=0.012$ and $\Delta t=0.05$. We observe that the cellular growth rate $\mu$ supports the formation of periodic patterns. However, the periodic pattern disappears at $\mu\approx 2.1$, for which we surmise that the oscillating solutions become unstable and develop into a stable stationary pattern

    Figure 5.  Effect of domain size on the pattern formation of $u$-species. We choose the system parameters to be the same as those in Figure 3 except that $\chi$ is slightly larger than $\chi_{k_0}$. $\Delta x=L/500$ and $\Delta t=0.05$ in each graph. Our simulations support our theoretical findings that large domains support periodic patterns with higher modes, however when the domain size is small, therefore does not exist time-periodic solutions that bifurcate from the homogeneous solution

    Figure 6.  Pattern formation of $u$-species in (2.1) when chemotaxis rate $\chi$ is far away from $\chi_{k_0}$=63.2

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