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### Open Access Journals

$\mathbb{R}^3$ |

$\left\{ \begin{array}{*{35}{l}} {{(\frac{\nabla }{i}-A(y))}^{2}}u+{{\lambda }_{1}}(|y|)u=|u{{|}^{2}}u+\beta |v{{|}^{2}}u,&x\in {{\mathbb{R}}^{3}}, \\ {{(\frac{\nabla }{i}-A(y))}^{2}}v+{{\lambda }_{2}}(|y|)v=|v{{|}^{2}}v+\beta |u{{|}^{2}}v,&x\in {{\mathbb{R}}^{3}}, \\\end{array} \right.$ |

$A(y)=A(|y|)∈ C^1(\mathbb{R}^3,\mathbb{R})$ |

$λ_1(|y|),λ_2(|y|)$ |

$β∈ \mathbb{R}$ |

$A(y),λ_1(y),λ_2(y)$ |

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.

$\left\{\begin{array}{ll}u_t=\nabla · (D(u) \nabla u-S(u) \nabla v)+u(1-u^γ),&x ∈ Ω,t>0, \\v_t=Δ v-v+u,&x ∈ Ω,t>0, \\\frac{\partial u}{\partial ν}=\frac{\partial v}{\partial ν}=0,&x∈\partial Ω,t>0\end{array}\right.$ |

$Ω \subset \mathbb R^N$ |

$N≥2$ |

$D(u)$ |

$S(u)$ |

$D(0)>0$ |

$D(u)≥ K_1u^{m_1}$ |

$S(u)≤ K_2u^{m_2}$ |

$\forall u≥0$ |

$K_i∈\mathbb R^+$ |

$m_i∈\mathbb R$ |

$i=1, 2$ |

$(m_1, m_2)$ |

$N≥3$ |

$γ≥1$ |

$m_1>γ-\frac{2}{N}$ |

$γ∈(0, 1)$ |

$m_1>γ-\frac{4}{N+2}$ |

$γ∈[1, ∞)$ |

$\frac{2}{N}$ |

## Year of publication

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