Stability and absorbing set of parabolic chemotaxis model of Escherichia coli

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1 10 Nonlinear Analysis: Modelling and Control 013 Vol. 18 No Stability and absorbing set of parabolic chemotaxis model of Escherichia coli Salvatore Rionero a Maria Vitiello b1 a Department of Mathematics and Applications R. Caccioppoli University of Naples Federico II Via Cintia Monte S. Angelo 8016 Naples Italy rionero@unina.it b Department of Mechanics Mathematics and Management Politecnico di Bari Via Orabona 7015 Bari Italy vitiello@poliba.it Received: May 01 / Revised: 6 February 013 / Published online: 18 April 013 Abstract. This paper is devoted to model 1) for escherichia coli introduced in [1]. Based on the experimental observations of Budrene and Berg [3] Tyson and coworkers derived 1) n cell density c chemotrattactant concentration and s stimulant concentration. Our aim is to study the stability of constant meaning full solution and ultimately boundedness of the solutions. Precisely: i) linear and nonlinear stability is proved by using a peculiar Lyapunov function ii) the ultimately boundedness of the solutions in the L -norm is obtained iii) conditions guaranteeing the global stability are also obtained. Keywords: chemotaxis model linear and nonlinear stability absorbing set. 1 Introduction Mathematical models of the biological systems are an important tool. Much attention has been paid to pattern formation in the nature world. Especially patterns of bacteria colonies have long been investigated and mathematical models of pattern formation have been developed extensively. For an account of the state of art of chemotaxis and chemotaxis models we refer to [4 7] and references therein. The collection of diverse patterns observed by Budrene and Berg [ 3] is an interesting and well-documented example of complex pattern formation by bacteria. The most complex patterns are formed by Escherichia coli in semi-solid medium. These models are This work has been performed under the auspices of the GNFM of INDAM and was supported in part from the Leverhulme Trust Tipping points: mathematics metaphors and meanings. 1 Corresponding author. c Vilnius University 013

2 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 11 time-dependent systems of partial differential equations typically in two or three space dimensions which contain three distinct sets of terms modelling three distinct processes: reaction term diffusion term and chemotaxis term. Reaction diffusion equations including only the first two processes above have been widely studied both theoretically and numerically as models of many biological systems [6 8 9] cfr. also [10 11]). Here we describe dynamical behaviour of E. coli bacterial chemotaxis the best understood phenomena among pattern process in bioscience. A small initial inoculum of bacteria forms a swarm ring of high cell density which expands outward from the inoculum. The patterns are formed by fully-motile cells but a large portion of the cells becomes non-motile for some unknown reason and these maintain the pattern. The E. coli patterns are formed when the bacteria are exposed to intermediates of the tricarboxylic acid TCA) cycle principally succinate. In response the cell secrete aspartate which is a potent chemoattractant. A chemoattractant for E. coli is a chemical which the bacteria like in the sense that the bacterial cells tend to move up concentration gradients of the chemical a process known as chemotaxis. We choose a parabolic PDE model of reaction diffusion type which is modified from [1 1]. A model proposed by Tyson et al. [1] contains most of the relevant features observed in E. coli chemotaxis. Tyson and coworkers based their model on the experimental observations of Budrene and Berg [ 3] and they derived the following mathematical representation three variables the cell density n the chemoattractant concentration c and the stimulant concentration s [ ] k 1 n n t = d n n k + c) c n c t = d c c + k 5 s k 6 + n k 7nc s s t = d s s k 8 n k 9 + s. k4 s ) + k 3 n k 9 + s n The first term in 1) describes Fickian diffusion. In 1) 1 the second term denotes the chemotactic response and the third is for the growth of cells. Similarly the production of chemoattractant second term in 1) adds to its diffusion transport while the uptake of chemoattractant by the cells third term) reduces its extra cellular concentration. The rate of change of nutrient concentration 1) 3 is difference between the rates of diffusion and consumption. Tyson applied two simplifications: the succinate concentration s was assumed to be constant and used as a parameter and k 7 and k 8 were set to zero. This eliminate equation 1) 3 and reduces equations 1) 1 and 1). All these models contain more or less assumption yet none has been studied respect to all three pattern-forming processes when any realistic model of the system must reproduce them all. In this paper we present the mathematical model that captures all three observed pattern-forming processes and we analyze the linear and nonlinear L -stability of the solution of 1) under Neumann boundary data by following the methodology formulated in [13 17]. 1) Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

3 1 S. Rionero M. Vitiello The plain of the paper is the following. Section is devoted to some preliminaries. In Section 3 we recall a Lyapunov functional introduced previously in [13] and analyze the linear stability of positive steady states. Section 4 is dedicated to the local) nonlinear stability. In Section 5 the existence of an absorbing set is shown. In Section 6 conditions for the global stability are derived. The paper ends an appendix in which the proof of Lemma 1 is sketched. Preliminaries By introducing the scalings [6] u = n n 0 d 1 = d n d c v = c k w = s k9 t = k 7 n 0 t = d c k 7 n 0 d 3 = d s α = k 1 ρ = k 3 d c d c k k 7 k9 δ = k 4 β = k 5 k = k 8 µ = k 6 n 0 k 7 k n 0 k 7 k9 n 0 the mathematical model dropping the stars in dimensionless form is [ ] u u t = d 1 u α 1 + v) v v t = v + βw u µ + u uv w t = d 3 w ku w 1 + w ) δw + ρu 1 + w u ) where u denotes bacterial cell density v the aspartate concentration and w the succinate concentration. We choose R 3 a bounded smooth domain and we refer here to the positive smooth solutions of ) under the smooth initial data ux 0) = u 0 x) vx 0) = v 0 x) wx 0) = w 0 x) x and we associate to ) the Neumann boundary conditions n being the outward normal to ) du dn = 0 dv dn = 0 dw dn = 0 on R+. 3) We denote by the scalar product in L ) the L )-norm

4 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 13 H 1 ) the Sobolev space such that { ψ H 1 ) ψ + ψ) L) dψ } = 0 on dn and recall that in H 1 ) holds the inequality [8] ψ ᾱ ψ 4) ᾱ = min H 1 ) ψ / ψ ) i.e. ᾱ is the lowest nonzero eigenvalue of Ψ = αψ in H 1 ). Remark 1. In the sequel we will denote by I 1 the set of solutions of ) initial data 0 v 0 w 0 ) and by I the set of solutions initial data u 0 v 0 w 0 ) u 0 const. 3 Equilibria and stability In this section we analyze the equilibrium points of the system ) and their stability. We observe that by inspection of ) one deduces that there are only two types of relevant equilibria given by E 0 = 0 0 0) E = 0 V W ) V and W positive constants. Let us consider the perturbation to the generic equilibrium E u = U v = V + V w = W + W the equations governing the perturbation U V W ) to the basic state E are U t = a 11 U + d 1 U + F 1 V t = a 1 U + a 3 W + V + F W t = a 31 U + d 3 W + F 3 5) where a 11 = ρδ a 1 = V a 3 = β a 31 = k [ ] [ U F 1 = α [1 + V + V )] V ρu [ U W ] F = βµ µ + U )µ W µ+u UV F 3 = δ 1 + W + W ) + U ] ku 1 + W + W ). 6) It can be sketched that the stability of the equilibria depends on the fulfillment of the initial conditions. We observe that if u 0 v 0 w 0 ) I 1 the stability analysis can be easily performed and simple stability is easily obtained. Of course this is not the situation we are interested in and hence our analysis will be focused on the case u 0 v 0 w 0 ) I Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

5 14 S. Rionero M. Vitiello and study the asymptotic) stability of the equilibrium E respect the perturbation U V W ) T U 0 0 i.e. in order to make the analysis consistent the experiments we consider initially nonzero cell density nonzero stimulant concentration and zero concentration of chemoattractant. In fact vx 0) is set to zero because initially bacteria have not secreted any chemoattractant [1]. In vectorial form system 5) can be expressed as U t = LU + F in R + a 11 + d U = U V W ) T F = F 1 F F 3 ) T L = a 1 d a 3 a 31 0 d 3 a ij = const R d i = const > 0 i j = 1 3. Setting b ii = a ii d i ᾱ i = 1 3 7) let us consider the system du To the matrix L we will apply the following Lemma. = LU 8) b L = a 1 b a 3. 9) a 31 0 b 33 Lemma 1. The Routh Hurwitz stability conditions of the matrix γ γ 1 γ γ 3 γ 31 γ 3 γ 33 γ ij real entries are γ 11 < 0 I = γ + γ 33 < 0 A = γ γ 33 γ 3 γ 3 > 0. 10) Proof. The proof is given in [18]. For the sake of completeness a sketch of the proof is given in the Appendix. Remark. Applying Lemma 1 to 9) it follows that the zero solution of 8) is asymptotically stable if and only if b 11 < 0 I = b + b 33 < 0 A = b b 33 > 0 and hence in view of a = a 33 = 0 if and only if b 11 < 0.

6 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 15 Denoting now by µ i to be chosen suitably) positive rescaling constants and setting U = µ 1 U 1 V = µ U W = µ 3 U 3 b ij = µ j µ i a ij i j i j = 1 3 5) in view of 7) become du 1 = b 11 U 1 + d 1 U 1 + ᾱu 1 ) + 1 F1 µ 1 du = b 1 U 1 + b U + b 3 U 3 + U + ᾱu ) + 1 F µ du 3 = b 31 U 1 + b 33 U 3 + d 3 U 3 + ᾱu 3 ) + 1 F3 µ 3 F i = F i µ 1 U 1 µ U µ 3 U 3 ) i = 1 3. To 11) we associate the Lyapunov functional introduced by Rionero [13] 11) W t) = 1 U 1 + V V = 1 A U [ + U 3 ) + b U 3 b 3 U + b 3 U 3 b 33 U ]. The temporal derivative of W along the solutions of 11) is given by Ẇ = b 11 U 1 + IA U + U 3 ) + Φ 1 + Φ + Φ 3 1) Φ 1 = A 1 b 1 A 3 b 31 ) U 1 U + A b 31 A 3 b 1 ) U 1 U 3 Φ = U 1 d 1 U 1 + ᾱu 1 ) + A 1 U A 3 U 3 U + ᾱu + A U 3 A 3 U d 3 U 3 + ᾱu 3 ) Φ 3 = 1 U 1 F1 + 1 A 1 U A 3 U 3 F + 1 A U 3 A 3 U F3 µ 1 µ µ 3 and A 1 = A + b 33 ) A = A + b ) + b 3 ) A 3 = b 3 b 33. Lemma. Let ρδ < d 1 ᾱ 13) holds. Then Φ 1 [ 1 b11 U 1 + IA U + U 3 )]. 14) Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

7 16 S. Rionero M. Vitiello Proof. Choosing µ = µ 3 = 1 we observe that 6) and 13) imply 10) so that Lemma 3.3 in [13] can be applied. Then according to the procedure used in [13] Lemma 3. one obtains Lemma 3. Let holds. Then 1 + d 3 ) A 3 = A 1 A d 3 15) Φ 0. 16) Proof. In view of the boundary conditions it turns out that Φ = d 1 U1 + ᾱ U 1 ) + A 1 U + ᾱ U ) + A d 3 U3 + ᾱ U 3 ) d 3 )A 3 U U 3 ᾱ1 + d 3 )A 3 U U 3. By virtue of 15) it follows that Φ = d 1 U1 + ᾱ U 1 ) and hence 4) implies 16). [ A1 U ± d 3 A U 3 ) ᾱ A1 U ± d 3 A U 3 ] On linearizing 11) 1) reduces to Ẇ = b 11 U 1 + IA U + U 3 ) + Φ 1 + Φ and the following theorem holds true. Theorem 1. Let 13) and 15) hold. Then 0 V W ) is linearly asymptotically stable respect to L )-norm. Proof. In view of Lemma and Lemma 3 it follows that Ẇ 1 [ b11 U 1 + IA U + U 3 )]. 17) Setting it follows that and hence m = sup A 1 a 1 A 3 a 31 A a 31 A 3 a 1 ) Φ 1 mµ 1 U1 U + U 3 ) mµ 1 U1 U + U 3 )) m µ 1 IA U IA U + U 3 ) µ 1 = b 11IA m = 14).

8 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 17 Further V is equivalent to the L -norm i.e. exist two positive constants K 1 K such that K 1 U + U 3 ) V K U + U 3 ) 18) K 1 = 1 A K = 1 A + b ) + b 3 ) + b 33 ). By virtue of 18) from 17) it turns out that i.e. Therefore it follows that Ẇ 1 b 11 U 1 IA K V Ẇ δw and the linear asymptotic stability is proved. 4 Nonlinear stability δ = inf b 11 IA ). K W W 0)e δt Setting bii = b ii + ᾱɛ di = d i ɛ ɛ = const > 0 i = ) it follows that 6) imply Ā = b b33 a 3 a 3 > 0 Ī = b + b 33 < 0. By virtue of 19) from 11) one obtains du 1 = b 11 U 1 + G F1 µ 1 du = b 1 U 1 + b U + b 3 U 3 + G + 1 F µ du 3 = b 31 U 1 + b 33 U 3 + G F3 µ 3 0) Let us define G i = d i U i + ᾱu i ) + ɛ U i i = 1 3. W t) = 1 U 1 + V Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

9 18 S. Rionero M. Vitiello V = 1 [Ā U + U 3 ) + b U 3 b 3 U + b 3 U 3 b 33 U ]. The time derivative of W is given by d W = b 11 U 1 + ĪĀ U + U 3 ) + Φ 1 + Φ + Φ 3 Φ 1 = Ā1b 1 Ā3b 31 ) U 1 U + Āb 31 Ā3b 1 ) U 1 U 3 Φ = U 1 G 1 + Ā1U Ā3U 3 G + ĀU 3 Ā3U G 3 Φ 3 = 1 U1 F Ā1 U µ 1 µ Ā3U 3 F 1 + Ā U 3 µ Ā3U F 3 3 Ā 1 = Ā + b 33 ) Ā = Ā + b ) + b 3 ) Ā 3 = b 3 b33. Let b 11 < 0 from Lemma Φ 1 is given by Φ 1 1 [ b11 U 1 + ĪĀ U + U 3 )]. 1) Moreover Φ = U 1 d 1 U 1 + ᾱu 1 ) + Ā 1 U Ā3U 3 d U + ᾱu ) + Ā U 3 Ā3U d 3 U 3 + ᾱu 3 ) + U 1 ɛ U 1 + Ā1U Ā3U 3 ɛ U + ĀU 3 Ā3U ɛ U 3 and following the procedure used in Lemma 3 one obtains d + d 3 ) Ā3 = Ā 1 Ā d d3 ) Φ ɛ U 1 ɛā1 U ɛā U 3 + ɛā3 U U 3. In view of ) it follows that Φ ɛ U 1 + d d 3 ) d + d Ā 1 U + d d 3 ) ) 3 d + d Ā U 3 3 ɛ δ U 1 + U + U 3 ) 3) δ = min {1 d d 3 ) d + d Ā 1 d d 3 ) } 3 d + d Ā. 3

10 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 19 In view of 1) and 3) one has d W 1 [ b11 U 1 + ĪĀ U + U 3 )] ɛ δ U 1 + U + U 3 ) + Φ 3. 4) It remains to estimate the term Φ 3 appearing in 4). Of course from the biological point of view we can confine ourselves to the positive solutions of 0) and hence to perturbations such that µ 3 U 3 > W. 5) Since U3 + W ) 0 U 3 W U 3 W by virtue of 5) it easily follows that Choosing µ = µ 3 = 1 from 6) and 6) it follows that 1 + U 3 + W ) 1. 6) U1 F1 C1 1 U 1 + C 1 U1 3 7) Ā1 U Ā3U 3 F C3 U 1 U + C 4 U U 3 + C 5 U1 U + C 6 U 1 U 3 + C 7 U C 8 U 3 U 8) Ā U 3 Ā3U F3 C9 U1 U 3 + C 10 U1 U 9) C 1 = ρδ C = ρµ 1 C 3 = β Ā1 W µ µ 1 + µ 1 Ā3 C 4 = β Ā1 C 5 = µ 1 Ā1 C 6 = β Ā3 µ 1W µ C 7 = β Ā3 C 8 = µ Ā3 C 9 = kµ 1 Ā C 10 = kµ 1 Ā3. By virtue of 7) 9) one obtains Φ 3 C 1 + C 9 + C ) 10 U 1 + C4 + + C 7 + C 9 C4 + C ) 10 U ) U 3 + C 1 U1 3 + C 3 U 1 U + C 5 U1 U + C6 U 1 U 3 + C 8 U 3 U. Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

11 0 S. Rionero M. Vitiello The Hölder inequality implies Φ 3 C 1 + C 9 + C ) 10 U 1 + C4 + + C 7 + C 9 C4 + C ) 10 U ) U 3 + U 1 + U + U 3 ) 1/ [ C + C 3 + C 6 ) U C 5 + ) U 4 + C 8 + ) U 3 4 and in view of the embedding inequality it turns out that f 4 K f + f ) K = K) = const > 0 Φ 3 H 1 U 1 + H U + U 3 ) + KM U 1 + U + U 3 + U 1 + U + U 3 ) U 1 + U + U 3 ) 1/ H 1 = C 1 + C 9 + C 10 H = max { C4 + C 10 C 4 + C 7 + C 9 } M = max{c + C 3 + C 6 C 5 + C 8 + }. Since Ā > 0 an inequality like 18) holds also for W K 1 K replaced by two positive constant K 1 K. Hence on taking into account 30) 4) implies d W [ ) 1 b 11 H 1 U 1 + ĪĀ ) U H + U 3 )] [ ) 1/ ] ɛ δ KM U K U1 V + U + U 3 ) ) 3/ + KM U K V ] 30) i.e. d W δ 1 δ W 1/ ) W δ3 δ 4 W 1/ ) U 1 + U + U 3 ) { 1 δ 1 = inf δ = KM 1 K ĪĀ H ) } b 11 H 1 ) 3/ K δ 3 = ɛ δ δ 4 = KM K ) 1/.

12 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 1 By recursive argument implies ) δ1 W 0 < inf[ δ δ3 δ 4 ) ] W W 1/ δ1 δ W 0 e 0 )t. We can summarize the results of this section in the following theorem. Theorem. Let b 11 < 0 ) and 1 b 11 H 1 > 0 ĪĀ H > 0 hold. Then the zero solution of 0) is asymptotically stable respect to the L )-norm. 5 Absorbing set Let us consider the energy E = 1 u + v + w ) as a measure in the phase space of the distance from the origin and denote by the measure of. Along the solutions of ) belonging to I one has de = d 1 u u d + v v d + d 3 w w d [ ] ) u δw α 1 + v) v u d + ρu 1 + w u d u + β wv µ + u d uv d k uw w 1 + w d. By virtue of the boundary conditions 3) and 4) it follows that [ ] de d 1ᾱ u ᾱ v d 3 ᾱ w u α 1 + v) v u d + ρδ βµ [ u 1 1 ] 1 + w d ρ wv µ + u d u 3 d + β wv d uv d k uw w 1 + w d. Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

13 S. Rionero M. Vitiello Moreover in view of 3) one obtains [ ] u α 1 + v) v u d = α [ u 1 + v) 1] u d = α u u 1 + v) 1 d = α u u)1 + v) 1 d = α u) 1 + v) 1 d + α u u1 + v) 1 d. 31) Considering the biologically meaningful concentration u > 0 v > 0 w > 0) one has that 1 + v > 1 and in view of 31) one obtains [ ] u α 1 + v) v u d α u + α u u d 0. Since ρδ u ρ 1 u 3 ρδ ) ρδɛ1 1 1 u + ρ u 3 ɛ 1 ɛ 1 positive constant choosing ɛ 1 = δ it turns out that de d 1ᾱ u ᾱ v d 3 ᾱ w + ρδ 1 u + β 4 On the other hand therefore for one obtains β v w de d 1ᾱ u β ɛ v + βɛ w ɛ 3 = d 3 ᾱ vw d. ρδ 4 1 u ρ δ 4 3ɛ 3 + ɛ 3 u ᾱ β ) v d 3 ᾱ βɛ ) w + ρ δ 4 ɛ 3d 1 ᾱ i.e. de C 1 u + v + w ) + C

14 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 3 Now assuming { d1 ᾱ C 1 = inf ᾱ β ) d 3 ᾱ βɛ )} C = ρ δ 4 ɛ 3d 1 ᾱ. and choosing ɛ in such a way that ᾱ d 3 > β 4 β ᾱ < ɛ < d 3ᾱ β it follows that d Et) + C 1Et) C 33) C 1 C positive constants independent of the initial data u 0 x) v 0 x) w 0 x)). Hence the following theorem holds Theorem 3. Let 3) holds. Then it turns out that for any δ > 0 the ball S δ) centered at the origin 0 0) defined by { S δ) = u v w): E 1 + δ) C } C 1 is an absorbing set. Proof. First of all we observe that S δ) is invariant in fact from by virtue of 33) one obtains ) de t= t Moreover from 33) it turns out that E t) = 1 + δ) C C 1 < C C δ) C C 1 < δc < 0. Et) E0)e C1t + C C 1. Thus denoting by Γ a bounded set of the phase-space there exists a positive constant M such that sup Et) M. Γ From Me C1t + C = 1 + C δ) C 1 C 1 we obtain that for any t > t t = 1 C 1 log MC 1 δc any trajectory starting initially in Γ belongs to S δ). 3) Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

15 4 S. Rionero M. Vitiello 6 Global nonlinear asymptotic stability If one is able to control the functional Φ 3 in 1) then conditions for the global stability may be obtained. This is the main goal of this section. By virtue of the ultimately boundedness L -norm of the solutions of ) it follows that we can assume that exists a positive constant H such that U 1 U U 3 < H a.e. in R +. In this way by virtue of 7) 9) we get the estimate Φ 3 M 1 U 1 + M U + U 3 ) and M 1 = C 1 + C + C 3 + C 6 )H + C 9 + C 10 { C4 M = max + C 5 H + C 10 C 4 + C 7 + C 8 H + C } 9 C 1 = ρδ C = ρµ 1 C3 = β A 1 W µ µ 1 + µ 1 A 3 C 4 = β A 1 C5 = µ 1 A 1 C6 = β A 3 µ 1W µ C 7 = β A 3 C8 = µ A 3 C9 = kµ 1 A C10 = kµ 1 A 3. Hence the following theorem holds Theorem 4. Let 13) 15) 3) and 1 b 11 M 1 > 0 IA M > 0 34) hold. Then the zero solution of 11) is globally asymptotically stable respect to the L -norm. Proof. On taking into account 14) 16) and 18) 1) implies Ẇ 1 b 11 U 1 1 IA U + U 3 ) i.e. and hence + M 1 U 1 + M U + U 3 ) Ẇ χ 1 W { } 1 χ 1 = inf b 1 ) 11 M 1 IA M K W t) W 0)e χ1t.

16 Stability and absorbing set of parabolic chemotaxis model of Escherichia coli 5 Appendix. Proof of Lemma 1 Since the invariants I 1 I I 3 of γ γ 1 γ γ 3 35) γ 31 γ 3 γ 33 and I 1 I I 3 are given by I 1 = γ 11 + I I = γ 11 I + A I 3 = γ 11 A ) I 1 I I 3 = γ 11 I I + γ 11 + A γ 11 36) it immediately follows that 10) imply the Routh Hurwitz stability condition for 35) I 1 < 0 I 3 < 0 I 1 I I 3 < 0. 37) Vice versa let 37) hold. Since one easily verifies that γ 11 is a real root of 35) by virtue of 37) it follows that γ 11 < 0. Then 36) 3 and 37) imply A > 0. It remains to obtain I < 0. But 37) implies and the roots of ) γ 11 )I I + γ 11 + A = γ 11 )I γ11 + A ) I > 0 γ 11 γ 11 )I γ 11 + A ) I = 0 are 0 and γ 11 + A)/ γ 11 > 0 hence I / [0 γ 11 + A)/γ 11 ]. Since γ 11 + A)/γ 11 > γ 11 37) 1 does not allow in view of 36) 1 I > γ 11 hence I < 0. References 1. R. Tyson S.R. Lubkin J.D. Murray A minimal mechanism for bacterial pattern formation Proc. R. Soc. Lond. Ser. B ): E.O. Budrene H.C. Berg Complex patterns formed by motile cells of Escherichia coli Nature 349: E.O. Budrene H.C. Berg Dynamics of formation of symmetrical patterns by chemotactic bacteria Nature 376: K. Kawasaki A. Mochizuki M. Matsushita T. Umeda N. Shigesada Modeling spatiotemporal patterns generated by Bacillus Subtilis J. Theor. Biol. 188): E.F. Keller L.A. Segel Model for chemotaxis J. Theor. Biol. 30): Nonlinear Anal. Model. Control 013 Vol. 18 No. 10 6

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