A fractional attraction-repulsion chemotaxis system with generalized logistic source and nonlinear productions

Abstract

This paper studies a fractional attraction-repulsion system with generalized logistic source and nonlinear productions: \begin{equation*} \left\{ \begin{aligned} &u_t = -(-\Delta)^\alpha u - \chi_1 \nabla \cdot (u \nabla v) + \chi_2 \nabla \cdot (u \nabla w) + au - bu^\gamma, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = \Delta v - \lambda_1 v + \mu_1 u^k, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = \Delta w - \lambda_2 w + \mu_2 u^k, &x \in \mathbb{R}^N, \, t > 0. \end{aligned} \right. \end{equation*} We first establish the global boundedness of classical solutions with nonnegative bounded and uniformly continuous initial data in two different cases: $\gamma \geq k + 1$ and $\gamma < k + 1$, respectively. Next, we show the asymptotic behavior of the global solutions for both cases $\gamma = k + 1$ and $\gamma \neq k + 1$. Finally, we obtain the spreading speed of solutions. In particular, when $\gamma = k + 1$, the upper bound of the spreading speed increases monotonically with $k$. If the condition of balanced attraction-repulsion intensities is further specified, the spreading speed will be equal to $\frac{a}{N + 2\alpha}$.