TRAVELING WAVE SOLUTIONS FOR A BACTERIA SYSTEM WITH DENSITY-SUPPRESSED MOTILITY

. In 2011, Liu et. al. proposed a three-component reaction-diﬀusion system to model the spread of bacteria and its signaling molecules (AHL) in an expanding cell population. At high AHL levels the bacteria are immotile, but diﬀuse with a positive diﬀusion constant at low distributions of AHL. In 2012, Fu et. al. studied a reduced system without considering nutrition and made heuristic arguments about the existence of traveling wave solutions. In this paper we provide rigorous proofs of the existence of traveling wave solutions for the reduced system under some simple conditions of the model parameters.


1.
Introduction. It is well known that spatial patterns are ubiquitous in living organisms. For many years, scientists are intrigued by these patterns and have developed many mathematical models trying to explain them [5]. In the paper [4], the authors described a genetic circuit to suppress the motility of Escherichia coli cells at high cell level. They were able to observe periodic stripe patterns in their experiments. In the same paper, the authors also developed a three-component reaction-diffusion system involving the cell density and densities of the signaling molecules AHL and the nutrients to explain the observed phenomena. In the paper [3], the authors studied a reduced two-component system and gave heuristic arguments that traveling wave solutions exist. It is the purpose of this paper to give a rigorous proof of the existence of traveling wave solutions. The plots of the cell and AHL densities in [3] look remarkably similar to our traveling wave solutions. The reduced-system studied in [3] is the following, where µ(h) is the bacteria diffusion which decreases from D ρ to a smaller value D ρ,0 as h increases pass the threshold value h 0 . In the above model, ρ is the density of the bacteria Escherichia Coli, which follow logistic growth. The bacteria excrete a small and rapidly degraded signaling molecules acyl-homoserine lactone (AHL) represented by h in the above model. At AHL level below h 0 , the bacteria perform random walk via their run-and-tumble motion and are motile. At AHL level above h 0 , the bacteria tumble incessantly and may be considered as immotile [1,2]. In this paper, we assume that D ρ,0 = 0 and let µ(h) = D ρ for h < h 0 and µ(h) = 0 for h > h 0 . We first scale the system by dividing the first equation by h 0 , the second equation by ρ s . Then we scale time and space by letting Suppose the location where h = 1 moves at a constant speed c > 0. We look for traveling wave solutions of the formĥ(z) =ĥ(x − ct),ρ(z) =ρ(x − ct) that are nonconstant on the intervals (−∞, 0) and (0, ∞), and have constant limits as z → ±∞. The constant c is called the wave speed and has to be determined together witĥ h(z) andρ(z). Substituting into (1.1), we havê In what follows, we drop the 'hat' sign in equations (1.2) and (1.3). Let We assume that h is a continuous function and ρ is discontinuous only at z = 0. We let and assume that Note that from (1.5), h(0) = 1. The positive constants ρ −0 , α, γ and D are considered as model parameters in this paper. The paper is organized as follows: In Section 2, we consider solutions of equations (1.2) and (1.3). In Section 3, we derive an equation the wave speed must satisfy and also give sufficient conditions that this equation has a root. The main results are given in Section 4. In Theorem 4.2, we give a sufficient condition for h(z) ≤ 1 when z ≥ 0, which together with the existence of wave speed imply the existence of traveling waves. In Theorem 4.3, we give sufficient condition that given c greater than the minimum wave speed, there exists a unique ρ −0 (see (1.4)) such that traveling wave solutions exist with wave speed c. Section 5 is discussion.

The function h. Consider the equation
Since h(z) is continuous, integrating (2.4) from −ε to ε and letting ε ↓ 0, we have h ′ (0−) = h ′ (0+) so that h is differentiable. Clearly, h is not C 2 . Consider the homogeneous differential equation We denote the characteristic roots by Applying the variation of constant formula to (2.4), we have where Here we have used the facts that Since A(0) = 0, B(0) = 0, the condition h(0) = 1 implies that C 1 + C 2 = 1 so that In order for h to be bounded for z > 0, since λ + > 0, we need C 1 = −A(∞). Next we confirm that (C 1 + A(z))e λ+z in (2.5) converges as z → ∞. Since λ + > 0, converges to zero if ρ(z) is integrable near infinity. Now since λ − < 0, C 2 e λ−z goes to zero as z → ∞. Also Since ρ(∞) = 0, the dominated convergence theorem implies that the above term goes to zero as z → ∞. Hence we obtain lim z→∞ h(z) = 0.
From the above, we see that h(z) can also be written as 3. The wave speed c. In this section, we first derive an equation for the wave speed, which is the equation (3.1) below. We then give sufficient conditions for equation (3.1) to have a root.
Proposition 3.1. If c > 0 is such that h(z) is differentiable at the origin, then c satisfies the equation Proof. Multiplying equation (2.4) by e −λ+z and integrating the result from 0 to ∞, we have, Similarly, multiplying equation (2.4) by e −λ−z and integrating the result from −∞ to 0, we have Setting the two derivatives of h at the origin equal and using the fact λ + − λ − = √ c 2 + 1, we obtain equation (3.1). The proof of the proposition is complete.
Proof. Let u(z) = h ′ (z). Then u satisfies u ′′ + cu ′ − u > 0 for z < 0. Thus u cannot have a positive local maximum. Every horizontal line above the z-axis can only intersect the graph of u once. Since u(0) < 0 by our assumption (4.2) and h(z) is bounded, u(z) < 0 on (−∞, 0). Now we show that h(−∞) = α. The first term of the right hand side of (3.4) can be rewritten as For the second term, we have The proof of the lemma is complete.

3)
Then there exists a traveling wave solution with speed c which satisfies h(z) ≤ 1 for any z ≥ 0.
The proof of the theorem is complete. Figure 1 shows the profile of a traveling wave solution with parameter values and wave speed that satisfy equation (3.1) and inequality (4.3). The parameter values and wave speed are given in the caption of Figure 1.