Kyungpook Mathematical Journal 2023; 63(2): 141-154
Published online June 30, 2023
Copyright © Kyungpook Mathematical Journal.
Numerical Nonlinear Stability of TravelingWaves for a Chemotaxis Model
Department of Mathematics, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : email@example.com
Received: March 26, 2023; Revised: April 8, 2023; Accepted: April 20, 2023
We study the stability of traveling waves of a certain chemotaxis model. The traveling wave solution is a central object of study in a chemotaxis model. Kim et al. [
Keywords: traveling waves, stability, chemotaxis
In this paper, we study the stability of traveling waves of a certain chemotaxis model. In a chemotaxis model, formation of traveling fronts is a central theme of the study: let a long tube be filled with a nutrient that is to be consumed by a certain micro-organism, such as bacteria. Let a certain amount of bacteria population be placed at the leftmost entrance of the tube. Conversion of nutrient to a bacteria population at some rate would take place and eventually results in the exhastion of nutrient at the place. If the bacteria is capable of moving from one place to another, then those that move right will find a new opportunity of further growth with a high nutrient density at the new place. As a proof that this does take place, traveling wave solutions where the nutrient front retreats, and that of the bacteria population propagates, comes with an important physical relevance.
Kim et al.  introduced a following chemotaxis model:
This chemotaxis model has been suggested from the consideration of non-fickian flux law for diffusion, which is an independent and substantial subject of the random walk in a heterogeneous medium. See the series of studies [7, 5, 6]. The key difference from the classical Keller-Segel model  is to have a flux function as in the first equation of (1.1)
As a consequence, migration of population is from the place where
The measurements of this
Consider one space dimensional Cauchy problem
The existence of traveling waves are proved by phase space analysis of (1.6).
Because those traveling waves are constructed by solving the system (1.6), not the system (1.3), once existence is established, the stability against a small perturbation as a solution of (1.3) is subjected to a further study. Any of traveling waves that is not robust against a small perturbation would have little physical significance.
In general, nonlinear stability is a difficult problem, in particular around this non-steady state solution. The objective of this paper is thus to study numerical nonlinear stability of traveling waves: we numerically construct the traveling wave solutions of (1.6) and the computed data are prepared as an initial data of the system (1.3). In this preparation, a small perturbation is added. The system (1.3) with the perturbed initial data is then numerically solved until a finite time
The remainder of this paper is organised as follows. In Section 2 we summarize the existence results of traveling waves presented in  for completeness of our discussion. In Section 3 we present numerical integration results done by python SciPy Library for a few relevant instances of traveling waves. In Section 4 we present the numerical stability results, and finally we give conclusions in Section 5.
2. Traveling Waves and Phase Space Analysis for (1.6)
for some integration constant
Here, we have used the assumption (1.2) that
2.1. Equilibrium points and linearization
It is straightforward to verify that (2.1) has only two equilibrium points that are
This implies that
Linearization around each equilibrium point is summarized below.
Phase space analysis  based on the discussion so far gives the following result:
Theorem 2.1. Fix
3. Numerical Construction of Traveling Waves
As stated in Theorem 2.1, the traveling wave exists as a heterocilinic orbit of (2.1) that is a saddle-to-stable node connection. If
Based on that, in this section we numerically compute the saddle-to-stable node connection. We specify the procedure below.
1. We first let
We present three instances of traveling waves. As discussed earlier,
Figure 1. Traveling waves of cases A, B, and C.
Considering the above, those three instances are specified below, including various wave speeds and parameters.
Numerically captured heteroclinic orbits are presented in Figure 1. In every case,
4. Numerical stability of traveling waves
Any traveling wave solution that is not robust under a perturbation has little physical significance. In this section, we conduct the numerical nonlinear stability tests on three captured data in the preceding section. The specification of stability test procedure is presented below.
4.1. Procedure of numerical stability test
1. Points on the heteroclinic orbit are collected, for
Note that the heteroclinic orbits have been computed on sufficiently large interval [0,
2. Preparation of
3. Preparation of initial data
4. Preparation of a perturbed initial data
5. Finally, we numerically solve (1.3) in
For the parabolic solver, we simply used the explicit scheme:
Figure 2. Test of wave A against the sinusoidal absolute additive noise. For the computation,
h=0.1, k=0.005, have been used.
Figure 3. Test of wave A against the sinusoidal absolute additive noise. For the computation,
h=0.1, k=0.005, have been used.
We first consider the severer absolute noises. Figure 2 presents snapshots of
This illustrates that the shape of traveling wave is maintained for a short time in a stable manner, but due to the reaction of the system, traveling wave can be oscillatory in later time.
Note that the two end states at
In Figure 4, 5, and 6 are tests against relative additive noises of 10% levels. We conducted computations until
Figure 4. Snapshots of
Figure 5. Snapshots of
Figure 6. Snapshots of
We see in Figure 4 (a) the farily clean traveling wave fronts of
In Figure 6 (a) for wave C, we presented the profile of
In this paper, numerical stability of traveling waves of (1.3) against perturbations has been tested on three instances. The traveling waves themselves were computed by Python SciPy Library, and are assigned as initial data for the system (1.3) after adding a perturbation. We tested both the absolute and relative perturbations. Considering that the end states of traveling waves contain the zero state in one of
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