TRAVELING BANDS FOR THE KELLER-SEGEL MODEL WITH POPULATION GROWTH

This paper is concerned with the existence of the traveling bands to the Keller-Segel model with cell population growth in the form of a chemical uptake kinetics. We find that when the cell growth is considered, the profile of traveling bands, the minimum wave speed and the range of the chemical consumption rate for the existence of traveling wave solutions will change. Our results reveal that collective interaction of cell growth and chemical consumption rate plays an essential role in the generation of traveling bands. The research in the paper provides new insights into the mechanisms underlying the chemotactic pattern formation of wave bands.


1.
Introduction.The study of traveling waves of chemotaxis models began with the pioneering work of Keller and Segel [11] in which the following model was proposed and investigated: with the chemotactic sensitivity function ϕ(v) assumed to be logarithmic: and the chemical degradation (or death) rate function g(v) following a power law: where u(x, t) and v(x, t) represent the bacterial density and chemical concentration, respectively.χ is called the chemotactic sensitivity coefficient describing the strength of chemotaxis, d and ε denote the cell and chemical diffusion coefficients, respectively.The positive parameter m > 0 is called the chemical consumption rate.
When 0 ≤ m < 1, it was shown in [11] that model (1.1) with ε = 0 can generate traveling bands (traveling pulses, see an illustration in Fig. 1 (a)) which qualitatively were in satisfactory agreement with experimental observation of [1,2].Subsequently, a sequence of rigorous works on various aspects of traveling wave solutions of (1.1) with ε ≥ 0 had been carried out, cf.[22,24,[26][27][28]31] and references therein.When m > 1, the model (1.1) does not admit traveling wave solutions (e.g., see [31,34]), and the global solutions of (1.1) with other forms of chemotactic sensitivity function were studied in [5-7, 17, 32] in both bounded and unbounded domains.For the borderline case m = 1, the model (1.1) was used in [25] to describe the chemotactic boundary formation by bacterial population in response to the substrate consisting of nutrients if ε = 0, and recently in [16] to describe the directed migration of endothelial cells toward the signaling molecule vascular endothelial growth factor (VEGF) during the initiation of angiogenesis (see further references [5,6,15,16]), where u denotes the density of endothelial cells and v stands for the concentration of VEGF.The existence of traveling wavefronts of (1.1) with m = 1 was obtained in [35] for ε = 0 and in [33] for ε > 0. Though the existence of traveling wave solutions of the Keller-Segel model (1.1) has been extensively studied and well understood, the stability of traveling wave solutions is still a challenging problem due to the singular logarithmic sensitivity log v.The linear instability of traveling wave solutions to (1.1) in certain functional spaces was first obtained in [24] for a special case m = 0.The linear stability/instability of traveling wave solutions for m = 0 still remains open.The nonlinear stability of traveling wave solutions to (1.1) was not obtained until recently the last author with co-workers proved the nonlinear stability of traveling waves of (1.1) with ε = 0 in [9,19,20] and with ε > 0 in [18,21] for the borderline case m = 1.A kinetic description of chemotactic traveling bands can be found in [29,30].When g(v) is negative, results can be found in [4] and references therein.
It is evident that the cell growth (i.e. generation of biomass) was not considered in the Keller-Segel model (1.1).Since v is often a nutrient source (like energy or oxygen in the experiment of [1,2]), it is natural to consider the cell growth in the dynamics due to the consumption of nutrient as mentioned by Keller and Segel themselves in [11].Hence it would be of interest to investigate whether the cell growth plays a role in the generation of traveling bands.In other words, we are concerned with the traveling waves of the following Keller-Segel model with cell population growth The first work considering the cell population growth in chemotaxis models was presented in [13] where the chemotactic sensitivity function is assumed to be a receptor form: ϕ(v) = v v+k for some constant k > 0 and growth term f (u, v) = σu, and traveling bands (non-monotonic wavefronts, an illustration in Fig. 1 (b)) are numerically obtained at a specific growth rate σ > 0. When the chemotactic sensitivity is linear: ϕ(v) = v and cell growth is the Monod's model (namely f (u, v) = g(v)u with g(v) = kv K+v for some positive constants k and K), the existence of traveling bands was obtained in [12].Subsequently the same Monod's kinetics was numerically investigated in [14] for three main different types of chemotactic sensitivities (linear, logarithmic and receptor).When both the sensitivity function and chemical kinetics are linear, namely ϕ(v) = v and the term −uv m in (1.1) is replaced by u − v, the traveling wave solutions for the model (1.1) was studied in [8] for a bistable cell growth and in [23] for a logistic one.It can be clearly seen that all above-mentioned works considering traveling waves of the Keller-Segel model with cell growth either alter the chemotactic sensitivity function ϕ(v) or the chemical kinetics in (1.1).Hence a fundamental question rises as follows: • Is there an appropriate cell growth function which can be included into the Keller-Segel model (1.1) without changing any other terms such that the resulting model still admits the traveling bands?When the logistic growth is included into the first equation of (1.1) with m = 1, only monotonic traveling wavefronts can be obtained (see [3]).We stress here that it is important to keep the chemotactic sensitivity function ϕ(v) as the logarithm as in the original Keller-Segel model (1.1) since it has been confirmed recently by both experiments and model simulation in [10] that bacterial (like E. coli) cells do sense the spatial gradient of the logarithmic ligand concentration.Hence the logarithmic sensitivity has its fundamental biological relevance.Mathematically the logarithmic sensitivity function is much more challenging than other types of sensitivity (like linear or receptor) due to the singularity at v = 0.
Toward the basic question raised above, in this paper we shall include a nutrient uptake cell kinetics (meaning cells grow due to the nutrient uptake) into the Keller-Segel molde (1.1) directly and resulting model reads: where r (0 < r ≤ 1) is the conversion rate from the consumption of nutrient to the growth of cells.As we know, the chemical uptake kinetics in chemotaxis has not been studied before.However it is natural to consider such a kind of kinetics since the bacterial consume the energy and then increase its biomass.The main goal of this paper will be to find under what conditions for the parameter m > 0, the traveling bands of the model (1.2) exist and then discuss the differences of traveling bands generated by the model (1.1) with and without cell kinetics.Furthermore we shall discuss biological implications of our results.Since the model (1.2) is a system of two parabolic equations, it is generally nontrivial to obtain the traveling wave solutions.As the first step, we consider a simplified case ε = 0 (i.e., chemical diffusion is negligible) as treated in [11].Assume that (u, v)(x, t) = (U, V )(x − ct) is a traveling wave solution of (1.2), where c > 0 denotes the wave speed.With ϕ(v) = log v and ε = 0, the traveling wave solution (U, V ) satisfies the ODE system Here we are only interested in the case U ≥ 0, V ≥ 0 due to the biological relevance.Since V is an increasing wavefront, which can be seen from the second equation of (1.3), we assume that V (∞) = 1 without loss of generality and With these conditions, the integrated sum of equations of (1.3) gives Thus the travelling wave solutions (u, v) := (U, V ) satisfy the system and the conditions In the following we assume that d = 1 for simplicity.Clearly, when 0 ≤ m < 1, there is no solution to (1.4) -(1.5) since if there were such a solution (u, v) for some c > 0, we would have which contradicts u(ξ) → r as ξ → −∞.Therefore we just need to consider m ≥ 1.
Since the equilibrium points of (1.4) are different for m = 1 and for m > 1, we consider these two cases separately.When m = 1, (1.4) has one equilibrium When m > 1, (1.4) has two equilibria E 1 and E 3 for every c > 0. In Section 2, we consider the case m = 1 and show that for every c > max{2 √ r, 2 √ χr}, there is a unique (up to a translation) heteroclinic solution (u, v) of (1.4) connecting E * 1 to E 3 with u < 0 and v > 0. In Section 3, we consider m > 1 and show that there is a minimal value c 0 (m) with so that u < r along the whole orbit), and u changing sign exactly one time if 1 < m < 2 (so that u(ξ) > r as ξ → −∞); furthermore, c 0 (m) is a decreasing function of m ∈ (1, ∞).The precise statements of these results are given at the beginning of the corresponding sections.The proofs of these results are based on studying local dynamics near E 1 (or E * 1 ) and constructing positive invariant sets by making use of the existence result for m = 1 and the monotonic properties of the vector field of (1.4) in the region u > 0 and 0 < v < 1 with respect to m > 1 and c > 0. Since the system (1.4) for 1 < m < 2 is not smooth at E 1 , we cannot linearize (1.4) at E 1 , and hence cannot apply the unstable manifold theorem to prove that there exist solutions of (1.4) approaching E 1 as ξ → ∞.To resolve this problem, a shooting argument will be used.

TRAVELING BANDS FOR THE KELLER-SEGEL MODEL WITH POPULATION GROWTH 5
2. Case of m = 1.Assume that m = 1.Then (1.4) reduces to the system Assuming c 2 > 4rχ, (2.1) has three equilibria ) and E 3 , where u * 1 and u * 2 are the solutions of χu 2 − c 2 u + c 2 r = 0 given by Note that u * 1 and u * 2 are strictly decreasing and increasing functions of c respectively, with r < u * 1 < 2r, 2r < u * 2 < ∞ and the asymptotic behavior In Lemma 2.2 below we will show that E * 1 is a saddle and E 3 is a stable node of (2.1).The features of these equilibria enable us to prove the following: Theorem 2.1.For every c > max{2 √ χr, 2 √ r }, there exists a unique (up to a translation) heteroclinic solution (u, v) of (2.1), with , and the following asymptotic formulas: for some positives constants C − and C + , where whose second component is negative, and where Remark 1. (i) Using a standard limiting procedure we can show the assertion of Theorem 2.1 for c = max{2 √ χr, 2 √ r }.This implies that the minimal speed c = 2 √ r is reached when χ ≤ 1 (since E 3 is a spiral of (1.4) if c < 2 √ r).(ii) It is easy to verify that E * 2 is a unstable node of (2.1).We can show that there are a continuum of infinitely many solutions (u, v) of (2.1) satisfying Due to the length of the paper, we will not give the proof here.
We need two lemmas to prove Theorem 2.1.The first one gives the local dynamics of (2.1) at E * 1 and 1 is a saddle point of (2.1), with the unstable manifold W u (E * 1 ) tangent to the vector V 1 defined in Theorem 2.1.
(ii) E 3 is a stable node of (2.1).
Proof.The Jacobian matrices of (2.1) at E * 1 and E 3 are, respectively, 1 and a positive eigenvalue Since the characteristic polynomial of J(E 3 ) is λ 2 + cλ + r = 0, we find that the eigenvalues of 4r , which are negative by virtue of c 2 > 4r.This yields the assertion (ii) from the stable manifold theorem.
, the intersection of the vertical line u = u * 1 with the line segment E 4 E 3 .Let R 1 be the region bounded by the arc E * 1 E 3 on the u-nullcline of (2.1) and the segments E * 1 E 5 and E 5 E 3 .Then R 1 is a positively invariant set of (2.1).(See Fig. 2.) Thus k 1 < 0. This shows (i).
To show (ii) let whose graph (a parabola) is the u-nullcline of (2.1).Since To show (iii), let (u, v) be an arbitrary point on int(E 5 E 3 ), which lies on the line v − 1 − ku = 0.If χ > 1, then k = k 1 , and using 0 < u < u * 1 , we have If χ ≤ 1, then k = k 2 , and using (1 − χ)uk 2 ≤ 0, we have This implies that the vector field of (2.1) points to the interior of R 1 .Since the vector field of (2.1) points strictly upper-ward on the arc int( E 3 E * 1 ), and u (2.1).This shows (iii), thereby completing the proof of Lemma 2.3.
We are now to prove Theorem 2.1.
) is bigger than the second component of V 1 and less than zero, it follows that the branch of W u (E * 1 ) with u < r lies in the interior of the region R 1 defined in Lemma 2.3 (iii).Let ϕ c = (u c , v c ) be a solution of (2.1) with ϕ(0) lying on this branch of W u (E * 1 ).The positive invariance of R 1 implies that ϕ c (ξ) is defined for all ξ ∈ (−∞, ∞) with ϕ c (ξ) ∈ int(R 1 ).The vector field of (2.1) in int(R 1 ) yields u c (ξ) < 0 and v c (ξ) > 0 or all ξ ∈ (−∞, ∞), and hence ϕ c (∞) = E 3 .
It remains to show the asymptotic formulas for ϕ c as stated in Theorem 2.1.Since E * 1 is saddle, the asymptotic formula for ϕ c (ξ) as ξ → −∞ follows directly from the stable manifold theorem.Recall that the Jacobian matrix of (2.1) at E 3 has two eigenvalues λ ± with associated eigenvectors [cλ ± , 1] .Therefore, E 3 is a stable node of (2.1) with 1-dimensional strongly stable manifold W ss (E 3 ) tangent to the eigenvector [cλ − , 1] at E 3 .To derive the asymptotic formula for ϕ c (ξ) as ξ → ∞ it suffices to show that the orbit of ϕ c does not lie on W ss (E 3 ).If this is false, since ϕ c lies entirely inside the region R 1 connecting E * 1 and E 3 , R 1 is positively invariant for the orbits of (2.1), and W ss (E 3 ) lies above the eigenvector [cλ + , 1] , it follows (using different orbits cannot intersect) that all orbits of (2.1 starting from interior of the segment E * 1 E 5 (see Fig. 2) go to E 3 as ξ → ∞ tangentially to W ss (E 3 ).This is impossible since there is at most one orbit of (2.1) lying in R 1 that is allowed to have such a tangential behavior at E 3 .This shows that ϕ c must be tangent to the eigenvector [cλ + , 1] at E 3 , yielding its asymptotic formula as ξ → ∞ as stated in Theorem 2.1.This completes the proof of Theorem 2.1.
(ii) When 1 < m < 2, we find that the component u c of the wave solution (u c , v c ) obtained in Theorem 3.1 is a profile of non-monotonic wavefront and hence generates a traveling band; however the upper bound for its maximum value u c (ξ 0 ) shows that u c (ξ 0 ) → r as c → ∞.
We need a series of lemmas to prove Theorem 3.1.The lemma 3.2 below will be used in the following subsections, and Lemma 3.3 shows that c 0 (m) > √ χr when 1 < m ≤ 2.
3.1.Existence of solutions of (1.4) approaching E 1 as ξ → −∞.We assume that m > 1.The goal in this section is to prove the existence of solutions (u, v) of (1.4) satisfying (u(ξ), v(ξ)) → E 1 as ξ → −∞.As mentioned in the introduction, since v m−1 is not differentiable at v = 0 for 1 < m < 2, we cannot linearize (1.4) at E 1 and then apply the unstable manifold theorem.We shall directly prove the existence of the desired solutions.For this, we need to study the vector filed of (1.4).Solving the u-nuclline equation c 2 (r − u − rv) + χu 2 v m−1 = 0 for u in a neighborhood of E 1 with v ≥ 0 gives a unique solution where and by virtue of (3.5) to get, for sufficiently small v > 0, the following asymptotic formulas hold as ξ → −∞: 3.2.Proof of Lemma 3.4 (i) for 1 < m < 2 and m = 2 with c < √ χr.Proof of Lemma 3.4 (i) for 1 < m < 2 and for m = 2 with c < √ χr.We divide the proof into several steps.
Step 1. Choose v0 ∈ (0, v 0 ) sufficiently small such that u = −crv + χr 2 v m−1 > 0 from (1.4) on the segment E 1 A 1 − {E 1 } where A 1 = (r, v0 ), and u − (v) > 0 for v ∈ (0, v0 ] from (3.7).Let A 2 = (u − (v 0 ), v0 ).See Fig. 3 (a), where the arc follows from the continuous dependence of ϕ A on A and the connectedness of int(A 1 A 2 ) that there exist A 3 and A 4 (with A 3 lying to the left of A 4 if A 3 = A 4 ) such that the backward flow of ϕ A leaves the region R 0 bounded by , and remains in int(R 0 ) for each A ∈ A 3 A 4 over the left maximal interval (ξ A , 0] of its existence.Since the vector field of (1.4) in int(R 0 ) satisfies u > 0 and v > 0, it follows that lim ξ→ξA ϕ A (ξ A ) = E 1 for every A ∈ A 3 A 4 .Since (1.4) is not smooth at E 1 , we cannot conclude from the general global existence theorem that ξ A = −∞; instead, we prove this in the next step.
Step 2. Let A ∈ A Since v(ξ) → 0 as ξ → ξ A , we conclude from the above equation that ξ A = −∞ and the asymptotic formula for v as stated in (3.8).
Next, we show the asymptotic formula of u.Since u > r, v > 0, v > 0, u = r + o(1), and v = o(1) as ξ → −∞, we regard u as a function of v with v > 0 small to get, for 1 < m < 2 and c > 0, and for m = 2 and c < √ χr, Note that, for m > 1 and c > 0,

TRAVELING BANDS FOR THE KELLER-SEGEL MODEL WITH POPULATION GROWTH 13
Applying the variation of constants formula to (3.9) we have, for 1 < m < 2 and c > 0, frow which we conclude Similarly, by applying the variation of constants formula to (3.10) we get for m = 2 and c < √ χr, This shows the asymptotic formula for u as stated in (3.8).
Step 3. We prove the uniqueness of the solution as stated in the lemma.It suffices to show from Steps 1 and 2 that A 3 = A 4 .Let (u 3 , v 3 ) := ϕ A3 and (u 4 , v 4 ) := ϕ A4 .Since v 3 > 0 and v 4 > 0, we can regard u 3 and u 4 as a function of v ∈ (0, v0 ), both satisfying the scalar equation

Assume by contradiction that
for v > 0 sufficiently small, we have, for sufficiently small ṽ0 < v0 and 0 < v ≤ ṽ0 , Upon an integration over [v, ṽ0 ] gives The proof of Lemma 3.4 (i) for 1 < m < 2 is complete.The proof is similar to that for 1 < m < 2 in the previous subsection.
Step 1. Choose v0 ∈ (0, v 0 ) sufficiently small such that u = −crv + χr 2 v m−1 < 0 on the segment E 1 A 1 − {E 1 } where A 1 = (r, v0 ), and u − (v) > 0 for v ∈ (0, v0 ] from (3.7).Let A 2 = (u − (v 0 ), v0 ).See Fig. 3 (b), where the arc E 1 A 2 is the graph of u = u − (v) for v ∈ [0, v0 ].It follows from the same reasoning as in the previous proof that there exist A 3 and A 4 (with A 3 lying to the left of A 4 if A 3 = A 4 ) such that the backward flow of ϕ A for A ∈ A 3 A 4 stays in int(R 0 ) for A ∈ A 3 A 4 where the region R 0 bounded by Since the vector field of (1.4) in int(R 0 ) is C 1 smooth (including on its boundary) and satisfies u < 0 and v > 0, it follows that ϕ A is defined on (−∞, 0] with ϕ A (−∞) = E 1 .The same proof in Step 3 in the previous subsection for 1 < m < 2 can be used to show that A 3 = A 4 for the present case, which gives the uniqueness of the solution claimed in Lemma 3.4 (i) for m ≥ 2. From the u nullcline equation of (1.4), it follows that E 1 A 2 lies above the segment E 1 E 3 , and hence the solution we found satisfies u(ξ Step 2. It remains to show the asymptotic formula (3.8) for m > 2. The formula for v is obtained by the same proof for 1 < m < 2; so is the asymptotic formula for u when m = 2 and c > √ χr.The asymptotic formula for u with m > 2 needs a slight modification of the proof there.In this case, since u < 0, u < r, v > 0 and (u, v) → E 1 as ξ → −∞, (3.9) becomes Applying the variation of constants formula and using (3.11) we have, as in (3.12), and thus as desired.This completes the proof of Lemma 3.4 (i) under the current assumptions on m and c.
Upon substraction, we have

TRAVELING BANDS FOR THE KELLER-SEGEL MODEL WITH POPULATION GROWTH 15
where Applying the variation of constants formula gives Note that p(v) ≥ χ/v and so exp − Upon substraction, we have where Applying the same argument we get (3.14) in which p and q are replaced by p 1 and q 1 respectively.Since q 1 (v) < 0 because of c 1 > c 2 and r − u 1 − rv < 0, it follows that u 1 (v 0 ) − u 2 (v 0 ) < 0 for every small v 0 > 0, that is, u 1 (v) − u 2 (v) < 0 for every small v > 0. This proves (ii) (b).
(ii) We now consider m ≥ 2 or, m = 2 and c > √ χr.We have u (ξ) < 0 for sufficiently negative ξ from Lemma 3.4 (i).Thus, assume on the contrary that there is ξ 0 such that u < 0 on (−∞, ξ 0 ) and u (ξ) = 0. Using a similar argument to the above we derive that u (ξ) > 0 for all ξ > ξ 0 , which contradicts the assumption that u(∞) = 0. Therefore we must have u (ξ) < 0 for all ξ ∈ (−∞, ∞).This shows (ii) of the lemma.Proof.Fix m and c satisfying the conditions as stated in the lemma.Let R be the region bounded by where the arc E * 1 E 3 is the connecting orbit of (2.1) from E * 1 to E 3 obtained in Theorem 2.1 (i) (see Fig. 4).We claim that R Then ϕ m,c0 is a solution of (1.4)-(1.5)with the properties descried in Theorem 3.1.
Proof.We assume that 1 < m < 2. The case for m ≥ 2 can be similarly proved.It follows from Lemma 3.6 that c 0 := c 0 (m) is well defined.The dentition of c 0 implies that there exists a sequence {c n } ∞ n=1 such that c n > c 0 , c n → c 0 as n → ∞, and ϕ cn = (u n , v n ) := ϕ m,cn is a solution of (1.4)-(1.5)associated with c = c n .Since u n (−∞) = r and u n (∞) = 0 and u n (ξ) > 0 for sufficiently negative ξ, it follows from Lemma 3.5 that each u n changes sign exactly once and u n = r exactly once.By translation invariance we may assume that u n (0) = r.Since u n (0) < 0, it follows from (1.4) that v n (0) > χr c 2 n .Furthermore, since 0 < u n ≤ 2r and 0 < v n < 1, it follows that ϕ cn is uniformly bounded, so are ϕ cn from (1.4) and ϕ cn via differentiating (1.4).Applying Arzela-Ascoli's theorem yields that there exists a subsequence of {ϕ cn }, which is still denoted by {ϕ cn }, and C 1 functions u 0 and v 0 defined on (−∞, ∞) such that {ϕ cn } converges to (u 0 , v 0 ) and {ϕ cn } to (u 0 , v 0 ) uniformly on every compact interval of (−∞, ∞), and (u 0 , v 0 ) is a solution of (1.4) with c = c 0 on (−∞, ∞), satisfying u 0 . We claim that u 0 (ξ) > 0 for ξ ∈ (−∞, ∞).For otherwise at the first point u 0 = 0, we would have v 0 < 1, and u 0 = cr(1 − v 0 ) > 0, which is contradiction to u 0 ≥ 0. This claim also gives The uniqueness of the solution of (1.4) approaching E 1 as ξ → −∞ from Lemma 3.4 (i) implies that (u 0 , v 0 ) is a translation of ϕ m,c0 .Then applying Lemma 3.5 yields the assertions of the lemma.R. Taken an arbitrary point (u, v) on int( E 1 E 3 ), the outer normal at (u, v) is given by − gives what we claimed.Now, it follows from Lemma 3.4 (i) and (ii) (b) that ϕ c lies inside R near E 1 , hence the full orbit of ϕ c stays in R by the positive invariance of R, yielding that ϕ c is defined on (−∞, ∞) with u m,c > 0 and v m,c > 0, so that ϕ c (∞) = E 3 .It then follows from Lemma 3.5 and Lemma 3.4, ϕ c is a solution of (1.4)-(1.5)with described properties in Theorem 3.1 (i), except for the asymptotic formulas given in (3.2) as ξ → ∞.We show these formulas as follows.
We linearize (1.4) at E 3 and find that the linearized system has the coefficient matrix J(E 3 ) given in (2.2) with two negative eigenvalues λ ± = 1 2 (−c ± √ c 2 − 4r) and associated eigenvectors (cλ ± , 1) respectively.The stable manifold theorem implies that the 1-dimensional strongly stable manifold W ss c (E 3 ) of (1.4) is tangent to the eigenvector (cλ − , 1) .To show the asymptotic formulas as ξ → ∞ in (3.2) for ϕ c = (u c , v c ), it suffices to show that ϕ c cannot lie on W ss c (E 3 ).This follows from the following two facts.Fact 1.For 2 √ r < c 2 < c 1 , the portion of W ss c2 (E 3 ) near E 3 in the first quadrant lies below the corresponding portion of W ss c1 (E 3 ).This is because that the slope of the tangent vector (cλ − , 1) given by is negative and decreasing as c decreases.Fact 2. For c 0 (m) < c 2 < c 1 , the orbit of ϕ c2 lies above the orbit of ϕ c1 .This fact can be proved by using Lemma 3.4 (ii) and the argument as used in the first paragraph of this proof.Based on these two facts, if for some c 0 (m) < c 1 the orbit of ϕ c1 lies on W ss c1 (E 3 ), then for any c 2 with c 0 (m) < c 2 < c 1 , since the eigenvector [c 2 λ + , 1] lies below W ss c2 (E 3 ), the orbit of ϕ c2 cannot approach to E 3 as ξ → ∞.This contradicts the definition of ϕ c2 .Therefore, the orbit of ϕ c for any c > c 0 (m) must be tangent to the eigenvector [cλ + , 1] as ξ → ∞, which yields its asymptotic formula as stated in 3.1 (i).We thus complete the proof Theorem 3.1 (i).

TRAVELING BANDS FOR THE KELLER-SEGEL MODEL WITH POPULATION GROWTH 19
Step 2. It is clear that the assertion (a) in Theorem 3.1 (ii) follows Lemmas 3.6 and 3.7, and the assertion (b) follows Lemma 3.3.It remains to show the assertion (c), i.e., the monotonicity of c 0 (m).Let m 1 > m 2 > 1.To show that c 0 (m 1 ) ≤ c 0 (m 2 ), it suffice to show that ϕ c,m1 is a solution of (1.4)- (1.5) where c := c 0 (m 2 ).To this end, we again use the positive invariant set R as defined above except that its boundary curve int( E 1 E 3 ) is the orbit of ϕ c,m2 (see Fig. 5).The positive invariance of R for flows of (1.It follows from Lemma 3.4 (i) and (ii) (a) that ϕ c,m1 lies inside R near E 1 , hence the full orbit of ϕ c,m1 stays in R. It follows that ϕ c,m1 is defined for all ξ ∈ (−∞, ∞) with u c,m1 > 0 and v c,m1 > 0, so that ϕ c,m1 (∞) = E 3 .Applying Lemma 3.5 gives that ϕ c,m1 is a solution of (1.4)- (1.5).This completes the proof of Theorem 3.1 (ii).4. Discussion.The propagation of traveling bands of bacterial chemotaxis was the typical picture in the chemotactic pattern formation of bacterial (see [1,2]).When r = 0 (without cell growth), it has been shown (see the references mentioned in the introduction) that the Keller-Segel model (1.2) will produce the traveling bands (i.e.traveling pulses) only if 0 ≤ m < 1, where the wave speed has a minimum value independent of the consumption rate parameter m (e.g., see [33]).Our results in this paper show that if a chemical uptake cell kinetics is included (i.e.r > 0), the resulting model (1.2)can produce the traveling bands (i.e., nonmonotonic wavefronts) only if 1 < m < 2, where the minimum wave speed exists but depends on the parameter m.As we know, this is the first result that includes the cell growth into the Keller-Segel model (1.1) directly such that the resulting model still can generate the traveling bands to recover the original motivation of Keller-Segel model.We find that the profile of traveling bands, the range of the parameter m and minimum wave speed for the existence of traveling bands are significantly different between the cases r = 0 and r > 0. These differences imply that the collective interaction between the cell kinetics and chemical consumption rate is vital to generate traveling bands.In particular there are two biological implications of our results.First if the uptake type cell growth occurs, the traveling bands can be generated by increasing chemical consumption rate.On the other hand if the traveling bands are formed by the non-monotonic wavefronts, the cell growth must be considered and then the chemical consumption rate will be important to determine the nature of wave propagation such as the wave speed and wave profile.Our research provides a new perspective to understand the role of cell growth in wave band formation in (bacterial) chemotaxis.

Figure 1 .
Figure 1.An illustration of two distinct profiles of a traveling band: (a) traveling pulse; (b) non-monotonic wavefront.

3. 6 . 1 E 3 E u v 1 E 3 EFigure 5 .
Figure 5.A sketch of positive invariant sets R used in the Proof of Theorem 3.1.