Spatial pattern formation in a chemotaxis–diffusion–growth model

Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth

“…These patterns are heuristic in understanding the dynamics of system (1.1) with logistic growth, demonstrating that it can develop many other complicated and interesting structures besides those established in previous numerical experiments. The numerical results and choice of system parameters in this figure are motivated by those obtained in [Kuto, et al, 2012]. For all plots in Figure 6, the initial data are taken to be u 0 =ū + 0.05e In (i), we choose that D 1 = D 2 = 0.25, µ = 5,ū = 3, χ = 10; stable steady solution emerge with multi-spikes, boundary and interior.…”

“…System (1.2) has been studied by various authors. For D 1 = D 2 = α = 1, φ(u, v) = f (u) = u and, Tello and Winkler [2007] obtained infinitely many branches of local bifurcation solutions to (1.2) with µ > 0 if N ≤ 4 and with µ > Kuto et al [2012] constructed local bifurcation branches of strip and hexagonal steady states when the domain Ω is a rectangle in R 2 . Ma et al [2012] studied the model with a volume-filling effect with φ(u, v) = u(1 − u) and f (u) = βu, β being a positive constant.…”

Convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source

“…A direct calculation shows that the parameters b, χ satisfy (1.3) or (1.4) only for χ a b − a + 1 ≥ 1, i.e., b ≤ χ. Kuto et al [9] got with additional assumption a = b > 0 that the system admits special stationary patterns such as stripe or hexagonal with the parameters b, χ and the domain Ω appropriately chosen. For example, for Ω = (0, π l ) × (0, π √ 3l ), there are stationary stripe [9, Theorem 5.1] and stationary hexagonal [9, Theorem 6.1] bifurcating from the constant steady state (χ(m, n), 1, 1) with various χ(m, n)…”

Transition of patterns in the cell-chemotaxis system with proliferation source

“…It should be mentioned that the bifurcation analysis of the diffusion-chemotaxis models with proliferation source has been conducted in several works, see [8] (and references therein) where the bifurcation problem of the stationary system has been studied; also see [16,11] and references therein. It is also known that hexagonal patterns can emerge from three unstable modes with no restriction on the geometry of the spatial domain; and, in fact, the existence of hexagonal patterns with three unstable modes has been recently studied in [13].…”