FINITE TRAVELING WAVE SOLUTIONS IN A DEGENERATE CROSS-DIFFUSION MODEL FOR BACTERIAL COLONY WITH VOLUME FILLING

. This work deals with the properties of the traveling wave solutions of a double degenerate cross-diﬀusion model where p ≥ 0 ,q > 1 ,l > 1. This system accounts for degenerate diﬀusion at the population density n = b = 0 and b = 1 modeling the growth of certain bacteria colony with volume ﬁlling. The existence of the ﬁnite traveling wave solutions is proven which provides partial answers to the spatial patterns of the colony. In order to overcome the diﬃculty of traditional phase plane analysis on higher dimension, we use Schauder ﬁxed point theorem and shooting arguments in our paper.


(Communicated by Yuan Lou)
Abstract. This work deals with the properties of the traveling wave solutions of a double degenerate cross-diffusion model ∂b ∂t where p ≥ 0, q > 1, l > 1. This system accounts for degenerate diffusion at the population density n = b = 0 and b = 1 modeling the growth of certain bacteria colony with volume filling. The existence of the finite traveling wave solutions is proven which provides partial answers to the spatial patterns of the colony. In order to overcome the difficulty of traditional phase plane analysis on higher dimension, we use Schauder fixed point theorem and shooting arguments in our paper.

1.
Introduction. Bacteria grown on the surface of thin agar plates develops colonies of various spatial patterns, such as fractal morphogenesis, dense-branching pattern, depending on both species and environmental conditions [7], [8]. Kawasaki et al. [12] proposed a degenerate parabolic system with cross diffusion that captures the qualitative features of the growth patterns. The model is as following: with initial data b(x, y, 0) = b 0 (x), n(x, y, 0) = n 0 , (3) here b represents the bacterial density and n represents the nutrient density. b 0 (x) is usually a compactly supported function and n 0 is a constant. D b and D n are positive constants that represent the diffusivity of bacterial and nutrient respectively.
When considering Model (1), (2) in one dimensional case with special case D n = 0, Maini et al. [15] used phase plane analysis method to study the existence and uniqueness of the traveling wave solutions (b(ξ), n(ξ)), where ξ = x − ct is the usual wave coordinate. In their paper, they found that existence of the traveling wave solution depends on the traveling speed c and the traveling wave with the minimum speed has sharp profile. However when the speeds are greater than the minimum speed, the traveling waves are smooth.
Recently, P. Feng and Z. Zhou [9] considered the existence of the traveling wave solutions for the case: where p ≥ 0, q > 1, l > 1. In their paper, by using Schauder fixed point theorem and shooting arguments, they established the existence of the finite traveling wave solution to Problem (6), (7). However, the movement of the bacterial may be affected by its space or volume in the neighborhood. In a recent paper by L. Bao and Z. Zhou [4], by using the technique of scaling of the population density and biased random walk in probability, they introduced the following single population model in one dimension: where g(b) is a logistic like birth term and The same idea also appears in [1], [2] and [3]. As far as we know, high population density may cause the effect of slow diffusion which is not considered in the population systems so far. In our paper, we introduce the following population model which accounts for the slow diffusion both at low and high population density: where p ≥ 0, q > 1, l > 1. Here the cross diffusion coefficient D(n, b) = n p b(1 − b) means slow diffusion when the nutrient is poor, the bacterial density is small and large.
The paper is organized as follows. In section 2, we derive some properties of the traveling wave solutions such as the positivity of the speed c and the monotonicity of the traveling wave solutions. In section 3, we prove the existence of the traveling wave solution b for a given n when c ≥ c * . In section 4, we study the existence of the traveling wave n for the determined traveling wave solution b. In the last section, by using Schauder fixed point theorem, we derive the following existence of finite traveling wave solution result: There exists a constant velocity c * such that the system (10), (11) admits a unique traveling wave solution (b(ξ), n(ξ)) where b(ξ) is a finite monotone traveling wave solution and n(ξ) is a monotone classical traveling wave solution.
Here ξ = x − ct is the usual wave coordinate and by finite traveling wave we mean 2. Preliminary results. We recall that the traveling wave solutions for the system (10), (11) are solutions of the form (b(x, t), n(x, t)) = (b(ξ), n(ξ)), where ξ = x − ct for some constant traveling speed c. The system we have to deal with is changed to where stands for derivation with respect to the wave variable ξ = x − ct. Let D n = 1 and D b = D, which is the non-dimensional diffusion coefficient of bacteria. We denote the spatially uniform steady states by (b, n) = (b s , 0), (0, n s ), where b s and n s are some arbitrary constants. The laboratory set-up consists of a nutrientenriched agar plate on which an initial inoculum of bacteria is placed. Therefore the appropriate steady state to be considered initially is the one given by The biological interpretation of this condition is that the nutrient is at a uniform concentration level of 1 (non-dimensionalised) and there are no bacteria present [15]. We shall also assume that lim Condition (15) is motivated by the possible occurrence of sharp type or finite type profiles, that is, solutions reaching the equilibria at a finite value ξ 1 or ξ 2 . However, when the existence interval is the whole real line, then Condition (15) is automatically satisfied and the sharp type is reduced to the classical one which is the front type traveling wave solution.
As in [15], the system (10), (11) are to be solved subject to the following condi- so that the wave is propagating into the fresh nutrient region of the plate while behind the wave b → b s , n → n s as ξ → −∞, that is, the nutrient concentration and bacterial density have relaxed to spatially uniform values after the wave has passed.
2.1. Some properties of the traveling wave solutions. In this section, we derive some important properties of the traveling wave solutions which may be used in the proof of our main theorem. P1. There are no traveling wave solutions with b(ξ) ≡ 0 or n(ξ) ≡ 1.
Proof. If not, there exists a traveling wave solution such that n(ξ) ≡ 1, then n = n = 0. From Equation (13), we can get n q b l = 0 which leads to the trivial solution b ≡ 0 and it is a contradiction.
P2. We require that b s = 1 and n s = 0.
Proof. In Equation (13), letting ξ → −∞, we find n q s b l s = 0 which leads to n s = 0 or b s = 0. If we integrate Equation (12) from −∞ to ξ, we have Substituting Equation (19) to (18) and passing ξ to +∞, we have If n, b are the traveling wave solutions, then n is monotone increasing and b is monotone decreasing if 0 < b(ξ) < 1.
Suppose there exist b (ξ 0 ) = 0 for some ξ 0 , then from Equation (12) which is a contradiction to the definition of minimum.
Now we prove the monotonicity of n. From Equation (13), we obtain (n e cξ ) ≥ 0 and n e cξ > 0 by integrating the inequality from −∞ to ξ. Hence n > 0 for all ξ.
3. The existence of b for a given n. In this section, we will prove the existence of the traveling wave solution b for any given n. The strategies is using the monotonicity property of b for any given n, then transforming the solvability of the problem to an equivalent singular boundary value problem. For the possibility of the finite or sharp type traveling wave solution for b, we denote here(ξ 1 , ξ 2 ) is the minimum interval for 0 < b < 1. Because for the given n, b(ξ) is a decreasing function on (ξ 1 , ξ 2 ), so the inverse function ξ = ξ(b) is well defined on (0, 1) and takes value on (ξ 1 , ξ 2 ). Therefore we may define and Differentiating both sides of Equation (24) with respect to ξ, we have from (15) we can z(0 + ) = z(1 − ). Next we will consider the corresponding singular boundary value problem (25). Let n(b) ∈ V where V is the closed convex set of the Banach space C 0 ([0, 1]) defined by where L is a sufficiently large constant and will be chosen later. We shall now consider the solvability of the singular problem (P * ) here n(b) ∈ V and V is defined as above. We first introduce the following lemma.
In view of the arguments above, we have proved that z is well defined in (0, b 0 ). Now we may define the maximal existence interval (0, b α ) for solution z. The aim is to show that b α = 1 for some α ∈ [ν(b 0 ), 0). Let z 1 and z 2 be two distinct solutions corresponding to initial value α 1 and α 2 respectively. Suppose for definiteness that . We now claim that if |α| is sufficiently small, then b α < 1 and z(b − α ) = 0. Since that lim there exists a sufficiently small constant M > 0 and λ < 2M such that for all −λ < z < 0 and b 0 ≤ b < b 0 + λ. Let α > −λ and define which solves the following initial problem: By the choice of α and λ,we have Applying a similar comparison argument as before, we conclude that z , then b α * = 1, therefore the corresponding solution z is defined and negative on (0, 1) and z > ν in (0, 1) and z(0 + ) = 0. This complete the proof.
In the following, we prove the solvability result for Problem P * .
Theorem 3.2. There exists c * > 0 such that for all c ≥ c * , Problem P * has a unique negative solution.
Proof. We first show Problem P * is solvable for c sufficiently large. To this aim, we let v = sup s∈(0,1) for all b ∈ (0, 1). Hence ν(b) satisfies condition of Lemma 3.1, and therefore Problem P * is solvable for every c > 2 √ v. We now show that Problem P * is not solvable for c = 0. Otherwise, let z solves that is defined on some interval (α, 1) with 0 < α < 1 and z(b) < 0 for all b ∈ (α, 1). Integrating the equation above in [b, b] with a < b < b < 1, we obtain

LIANZHANG BAO AND WENJIE GAO
Therefore, if z(1 − ) = 0, we have which implies z(0 + ) < 0, a contradiction. We now let c * = inf{c : P * is solvable} which is well defined and c * > 0 based on the observation above.
First, we prove that for every c > c * , P * is solvable. Given c > c * , take c such that P * is solvable with c < c and the unique solution z for c. Since hence z satisfies condition of Lemma 3.1. Therefore, we conclude the solvability of P * for c. Secondly, we prove Problem P * is solvable for c = c * . Let (c n ) be a sequence of speeds decreasing to c * and let z n be the unique solution of Problem P * for c = c n . Since for all b ∈ (0, 1). Then z n+1 is a strict upper solution for Problem P * with c = c n . Hence, by using Lemma 3.1 again, we deduce that z n (b) ≥ z n+1 (b) for all b ∈ [0, 1] and n ∈ N. Therefore (z n ) is a decreasing sequence. Moreover, since z n (0 + ) = 0 and z n (b) ≥ c n > c 1 , it holds Hence we can define z * (b) = inf n∈N z n (b) for all b ∈ [0, 1]. The monotone convergence theorem ensure that z * (b) is a solution of Problem P * for c = c * . Also from equation (34), we can have z * (0 + ) = 0. Finally, we apply the shooting argument and comparison techniques to the sequence (w * m ) of solution of the Cauchy problems for m ∈ N, we can get that z * (1 − ) = 0 and then z * is the required solution of Problem P * for c = c * Finally, we prove that P * admits at most one solution. Suppose for contradiction that z 1 and z 2 are two distinct solution of P 1 . For definiteness, we assume . Therefore, it is impossible that This complete the proof.
In the following, we will show some properties of the traveling wave solution.
Lemma 3.3. Let z ∈ C 1 (0, 1) be the solution of Problem P * for every c > c * . The following limit Proof. Let z(b) be the solution of Problem P * satisfying z(0 + ) = z(1 − ) = 0. Assume by contradiction that Let χ ∈ (l, L) and let {b n } be an decreasing sequence converging to 0 such that Passing to the limit as n → +∞, since χ < 0, we have Similarly, we can choose an decreasing sequence {ν n } converging to 0, such that we can deduce By the arbitrariness of χ ∈ (l, L), we conclude that Given c > c * , let z(b) and z * (b) be the solutions of Problem P * on [0, 1] with z(0) = z(1) = 0, respectively, for c and c * . Assuming the existence of b ∈ (0, 1) satisfying z * (b) ≥ z(b), from P * we then havė This implies the contradictory conclusion 0 for all c ≥ c * by the same arguments.
Let b(t) be the finite traveling solution of (36) defined in its maximal existence interval (t 1 , t 2 ), with −∞ ≤ t 1 < t 2 < +∞. So we can see b(t) is a solution of (12) in (t 1 , t 2 ). Observe that b (t) < 0 for every t ∈ (t 1 , t 2 ), so there exists the limit (ξ) = 0, and the solution could be continued in the whole half-line (t 2 , +∞), in contradiction with the maximality of the interval (t 1 , t 2 ), so z (0 + ) should be −c * .
One the other hand, if z (0 + ) = −c * , we have which implies that b is a finite traveling wave solution.
Remark 2. From the above discussions, we can see the finite traveling wave solution occurs only at the minimum traveling speed c * .
Furthermore, we have the following lemma.
Lemma 3.5. There exist a sequence (h n ) n of positive real numbers decreasing to 0, and a sequence (ψ n ) n of continuous functions respectively defined on [0, u n ] with u n ≤ u n+1 ≤ 1 for all n ∈ N, satisfying the following properties: The strategies of the proof is similar to [lemma 13, [14]] with the difference in Dn p+q (b)b 1+l (1 − b) and we omit the proof here.
Applying the monotone convergence theorem to (ψ n ), we can prove that ψ is a solution of the equation u]. Moreover, by the monotonicity of (ψ n ), we get From lemma 3.4, this implies (z * ) (0) = −c * . Using Lemma 3.4 again, we can further have the following results: Lemma 3.6. For the solution of Problem P * , there exists C 1 < 0, C 2 < 0 such that for b sufficiently close to 1.
Proof. We only need to show that it is impossible to find negative constants Otherwise, we either have for b sufficiently close to 1. This is a contradiction to Remark 1.
Or we have as b → 1 − . This contradicts to Remark 1 again.
We also have the following result.
4. The existence of the finite traveling wave solution. We have shown that for any n ∈ V , there exists c * that depends on the choice of n such that b is a finite traveling wave, we may assume that b(ξ) = 0 for ξ ≥ 0, which lead to n satisfies n + cn = 0 for ξ ≥ 0.
Proof. The phase plane analysis shows that every trajectory of the following ODE system: can intersect the n − axis at most once. Hence p changes sign at most once and consequently n(+∞) exists. Let n(+∞) = v and we can get We showed before that b(ξ) decrease monotonically from 1 to 0 as ξ varies from −∞ to 0. Therefore we can define ξ = ξ(b) as the inverse function of b(ξ).
where b varies from 0 to 1 and ξ takes value in (−∞, 0). We define and n(b) = n(ξ(b)). Since we can have the following two equivalent problems: