Stationary Solutions of a Volume-Filling Chemotaxis Model with Logistic Growth and Their Stability

Abstract

Abstract. In this paper, we derive the conditions for the existence of stationary solutions (i.e., nonconstant steady states) of a volume-filling chemotaxis model with logistic growth over a bounded domain subject to homogeneous Neumann boundary conditions. At the same time, we show that the same system without the chemotaxis term does not admit pattern formations. Moreover, based on an explicit formula for the stationary solutions, which is derived by asymptotic bifurcation analysis, we establish the stability criteria and find a selection mechanism of the principal wave modes for the stable stationary solution by estimating the leading term of the principal eigenvalue. We show that all bifurcations except the one at the first location of the bifurcation parameter are unstable, and if the pattern is stable, then its principal wave mode must be a positive integer which minimizes the bifurcation parameter. For a special case where the carrying capacity is one half, we find a necessary and sufficient condition for the stability of pattern solutions. Numerical simulations are presented, on the one hand, to illustrate and fit our analytical results and, on the other hand, to demonstrate a variety of interesting spatio-temporal patterns, such as chaotic dynamics and the merging process, which motivate an interesting direction to pursue in the future. 1. Introduction. The process of generation of spontaneous patterns (i.e., selforganization) involves many instances where symmetry is broken or a more symmetric state develops into a less symmetric one. It is striking that many of these breaks in symmetry have no external trigger, but are instigated internally. Hence a natural question arises: how can this process be done? The significant progress toward this question was made in 1952 by Alan Turing, whose paper [29] was considered one of the most influential works in theoretical biology. Turing thought about a simple model with two morphogens (chemical species): a short-range activator and a long-range inhibitor. Both morphogens diffuse in space, but at different rates, and the inhibitor diffuses faster than the activator. The combination of local strong activation and longrange inhibition is able to instigate a spatial patterning process. Turing proposed a mathematical model based on a couple of partial differential equations encapsulating these elements, known as a reaction-diffusion model (see, e.g., [18]). Turing's revolutionary contribution was that passive diffusion could interact with chemical reaction

Pattern Formation in Keller–Segel Chemotaxis Models with Logistic Growth

“…This provides a selection mechanism of stable wavemode to which the homogeneous solution (ū,v) loses its stability. The same results have been obtained in [Ma, et al , 2012] for chemotaxis system with volume-filling effect over 1D (though it is easy to see that their results carry over for more general systems). On one hand, our existence and stability results extend theirs to multi-dimension.…”

“…We also want to mention that numerical studied in 1D model with volume filling effect has been performed in [Ma, et al, 2012]. We are interested in patterns with concentrating properties in Figure 2: Emergence of single boundary spike at corner (1, 1).…”

“…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. They applied degree method to obtain nonconstant positive steady states and established a selection mechanism of the wave modes for Ω being an interval in R 1 .…”

Basic Mechanisms Driving Complex Spike Dynamics in a Chemotaxis Model with Logistic Growth

“…Expand in the power series in ε as (20) λ = λ 0 + ελ 1 + · · · , Φ = Φ 0 + εΦ 1 + · · · , P = P 0 + εP 1 + · · · so that to leading order we get…”

Existence and stability of nonconstant positive steady states of morphogenesis models

“…First of all, we show that the existence of nonconstant positive steady states of system is induced by the presence of chemotactic effect. To this end, we study the dynamics of system with χ = 0, that is, the following system without the chemotactic term We have the following global stability result of the trivial steady state and the proof of this result is similar as that of Proposition 2.2 in . Proposition The positive equilibrium of is asymptotically stable and system does not have any nonconstant steady state. Proof First of all, the corresponding linearized system of at has a Jacobian matrix of the following form which has a positive discriminant .…”