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 = -(-Δ)^αu - χ_1 \nabla \cdot (u \nabla v) + χ_2 \nabla \cdot (u \nabla w) + au - bu^γ, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = Δv - λ_1 v + μ_1 u^k, &x \in \mathbb{R}^N, \, t > 0, \\ &0 = Δw - λ_2 w + μ_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: ${\hat{I}}^3\ge k+1$ and ${\hat{I}}^3<k+1$, respectively. Next, we show the asymptotic behavior of the global solutions for both cases ${\hat{I}}^3=k+1$ and ${\hat{I}}^3\ne k+1$. Finally, we obtain the spreading speed of solutions. In particular, when ${\hat{I}}^3=k+1$, 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 $\frac{a}{N+2\hat{I}\pm }$.