Mathematical Biosciences xxx (2012) xxx–xxx

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Mathematical Biosciences

journal homepage: www.elsevier .com/locate /mbs

Classification and stability of global inhomogeneous solutionsof a macroscopic model of cell motion

Richard Gejji a,c,⇑, Bogdan Kazmierczak b, Mark Alber c

a Mathematical Biosciences Institute, Ohio State University, Columbus, OH 43210, USAb Polish Academy of Sciences, Institute of Fundamental Technological Research, 02-106 Warszawa, Polandc Department of Mathematics, University of Notre Dame, Notre Dame 46656, USA

a r t i c l e i n f o

Article history:Received 13 February 2011Received in revised form 14 March 2012Accepted 27 March 2012Available online xxxx

Keywords:AggregationChemotaxisInhomogenous stabilityLyapunov functionalPlateau solutionsDictyostelium discoideum

0025-5564/$ - see front matter Published by Elsevierhttp://dx.doi.org/10.1016/j.mbs.2012.03.009

⇑ Corresponding author at: Mathematical BiosciUniversity, Columbus, OH 43210, USA.

E-mail addresses: richard.gejji@gmail.com (R.(B. Kazmierczak), malber@nd.edu (M. Alber).

Please cite this article in press as: R. Gejji et al., CBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2

a b s t r a c t

Many micro-organisms use chemotaxis for aggregation, resulting in stable patterns. In this paper, theamoeba Dictyostelium discoideum serves as a model organism for understanding the conditions for aggre-gation and classification of resulting patterns. To accomplish this, a 1D nonlinear diffusion equation withchemotaxis that models amoeba behavior is analyzed. A classification of the steady state solutions is pre-sented, and a Lyapunov functional is used to determine conditions for stability of inhomogenous solu-tions. Changing the chemical sensitivity, production rate of the chemical attractant, or domain lengthcan cause the system to transition from having an asymptotic steady state, to having asymptotically sta-ble single-step solution and multi-stepped stable plateau solutions.

Published by Elsevier Inc.

1. Introduction

As an initial step towards biofilm formation, a wide range ofmicroscopic organisms, including both innocuous amoeba andharmful pathogenic bacteria, are able to use a combination of cellto cell interactions and chemical signals to aggregate into mounds.These mounds are stable structures in the sense that if they are dis-turbed, the micro-organisms will sense this disturbance and re-form themselves into another mound like structure.

Dictyostelium discoideum is an amoeboid capable of demonstrat-ing fruiting body formation during starvation conditions. Theamoeba releases a chemical to signal to other amoebas its location.Using a combination of projecting pseudopodia and hydrostaticpressure, the amoeba is able to orient itself and follow local chem-ical gradients to find other amoebas [10,17]. With this procedure,the amoebas are able to aggregate together to form a slug.

Extensive past work on modeling cell aggregation of amoebawith chemotaxis provided significant insights into the mechanismsfor pattern formation due to the emergence of unstable perturba-tions [3,8,9,13]. However, much of this work implicitly allowedfor cells to overlap, and as a result, the models demonstratedblow-up when certain conditions were met. Works such as

Inc.

ences Institute, Ohio State

Gejji), bkazmier@ippt.gov.pl

lassification and stability of glo012.03.009

[14,11] use volume exclusion to implement non-overlapping. Inparticular, [11] derived a nonlinear diffusion PDE from a stochasticsystem, describing cells as extended objects with finite volumesand fluctuating membranes.

In this paper, we focus on modeling slug formation using anequation that was derived to model the density of the amoeba un-der excluded volume conditions [11]:

@tu ¼ r � hðuÞru½ � � r gðuÞrv½ � ð1Þ@tv ¼ Dcr2v þ au� cv : ð2Þ

where u and v represent the cell and chemical densities, respec-tively, and hðuÞ and gðuÞ represent cell diffusion and chemotacticrates, respectively. hðuÞ has a singularity which allows the equationto demonstrate so called fast diffusion, where the diffusion ap-proaches infinity for large enough cell density. It is known that (1,2) demonstrates non-trivial, non-homogeneous patterns, and thata similar 2D equation is capable of demonstrating aggregates, whichhave been reported to have similar structure to blood vessels [11]. Itis also known that linear stability analysis can be used to analyzethe stability of the homogeneous steady state under small perturba-tion, in particular, see Ref. [8].

In this paper, we attempt to analyze the structure and stabilityof non-homogeneous patterns in a spatially one-dimensional casein order to understand whether or not structures resemblingmounds will occur and whether or not they are stable to perturba-tions. In particular, we establish that multi-stepped patterns will

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

2 R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx

start out unstable, but as the domain becomes larger, oscillatory,non-trivial patterns emerge and, under certain conditions, will be-come stable. For long enough domains, the stability of the patterncan be characterized by the maximum value of the density profile.Often times, stability can also be determined by whether or not thesolutions can be described as spikes or plateaus, as defined in [7,8].Qualitatively, there is no obvious distinction between plateaus andmounds, but it should be noted that the former is a mathematicalcharacterization, and the latter is an observed quality. In our case,we are able to show that for long enough domains, spikes corre-spond to unstable solutions.

The format of the paper is as follows. In the first section, we exam-ine an example system and perform phase plane analysis to catego-rize when the constant steady state solution(s) are saddles orcenters. Then, we describe the non-homogeneous stationary solu-tions as a single or multi-stepped patterns using a Hamiltonianframework in a similar manner as [15]. We then use the Hamiltonianto describe how single and multi-stepped solutions bifurcate from asteady state when the steady state is a center. In the following sec-tion, we examine the more general system and implement a Lyapu-nov functional, similar to the one described in [4], to classify severalconditions for instability. The existence of a bounded Lyapunovfunctional says that the solution will converge in time to an asymp-totically stable attractor contained in the union of all functions thatare local minimum values of the functional. These functions coincidewith the stationary solutions. By determining when stationary solu-tions are not local minima, it is possible to determine which patternsare unstable. Several conditions for instability and stability areestablished using various inequalities. In particular, we determinethe conditions when multi-stepped solutions can be constructed.We establish that each k-step solution, with given parameter valuesdiscussed below, becomes stable or unstable as the domain lengthapproaches infinity. After this, analysis is performed on the generalsystem to indicate when stationary solutions can be classified as pla-teaus or spikes. In the final section, numerics are performed to showthe existence of stable multi-step patterns.

2. Classification of stationary solutions

For the rest of the paper, we will derive results for a 1D versionof system (1, 2). However, many of the results are best expressedby analyzing and examining the specific example from [11],

ut ¼ D1þ u2

ð1� uÞ2ux

" #x

� vðuvxÞx ð3Þ

v t ¼ Dcvxx � cv þ au ð4Þ

on the interval ð0; LÞ; L > 0, satisfying the boundary conditionsuxð0; tÞ ¼ vxð0; tÞ ¼ uxðL; tÞ ¼ vxðL; tÞ ¼ 0: ð5Þ

In order to do so, we will first introduce a change of variables

q ¼ u; U ¼ ca

v ;

s ¼ffiffiffiffiffiffic

Dc

rx; s ¼ ct;

b ¼ DDc; x ¼ av

cDc; g ¼

ffiffiffiffiffiffic

Dc

rL

and consider the system

qs ¼ b1þ q2

ð1� qÞ2qs

" #s

�xðqUsÞs ð6Þ

Us ¼ Uss �Uþ q ð7Þ

on the interval ð0;gÞ;g > 0, satisfying the boundary conditions

qsð0; sÞ ¼ Usð0; sÞ ¼ qsðg; sÞ ¼ Usðg; sÞ ¼ 0: ð8Þ

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

Note that if we apply this change of variables to (3, 4), we can effec-tively set Dc;a; c ¼ 1 up to some scaling of h and g.

It is shown in [1,6] that this system admits unique, globally-bounded, non-negative solutions q;U 2 C2;1 ½0;g�; ½0;1Þð Þ, forsufficiently smooth initial conditions satisfying compatibilityconditions, and these solutions are capable of presenting non-homogeneous steady state solutions. Under these conditions, q isbounded between 0 and 1, and U is bounded between 0 andmaxðjjUðs;0ÞjjL1 ;1Þ. We now proceed to characterize the stationarybehaviour of the system through a series of manipulations, leadingup to a Hamiltonian formulation of the steady state. Thanks to suchmanipulations, we can express U as a function of q for stationarysolutions. First we set the left hand side of (6) to zero and integratefrom 0 to s to get

b1þ q2

ð1� qÞ2qs ¼ xqUs ð9Þ

where we set the integration constant to zero in order to satisfy theboundary conditions. Dividing both sides by xqðsÞ and then inte-grating from 0 to s, we get

UðsÞ ¼Z s

0

bx

1þ qðfÞ2

qðfÞ½1� qðfÞ�2qfðfÞdfþUð0Þ: ð10Þ

Evaluating the integral gives the following system for the stationarysolutions

U ¼ QðqÞ � K ð11ÞUss ¼ U� q ð12Þ

where

QðqÞ :¼ bx

lnðqÞ þ 21� q

� �and

K ¼ Qðqð0ÞÞ �Uð0Þ: ð13Þ

Let us note that Q 0ðqÞ > 0 for q 2 ð0;1Þ with RðQÞ ¼ ð�1;1Þ, so bythe inverse function theorem (11) is invertible. For bounded sta-tionary solutions and fixed K, we can define the inversefK : ð�1;1Þ ! ð0;1Þ. Below for brevity of notation, we will omitthe index K and write simply f instead of fK . For given f, we can de-fine qðUÞ :¼ f ðUÞ. f ðUÞ will be used extensively when we define theHamiltonian for stationary solutions. Also, let us note that if

M :¼ 1g

Z g

0qðsÞds; ð14Þ

where M is the total cell density and is a conserved quantity withrespect to time, then

K ¼ bxg

Z g

0lnðqðsÞÞ þ 2

1� qðsÞ

� �ds�M: ð15Þ

As q depends on U, then we will write K ¼ K½U�. In the case when qand U are constant, we haveK ¼ QðMÞ �M: ð16Þ

2.1. Phase plane analysis

To analyze the properties of spatially inhomogeneous steadystate solutions, we construct a new system of equations. To doso, we treat K½v � as a given constant, substitute q ¼ f ðUÞ into (11,12) and set w :¼ Us. The stationary solution can then be describedwith the 2D system of autonomous ODEs:

Us ¼ w ð17Þws ¼ U� f ðUÞ ð18ÞUsð0Þ ¼ UsðgÞ ¼ 0: ð19Þ

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R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx 3

This is a system of Hamiltonian equations with the Hamiltonian

H ¼ �w2

2þ 1

2U2 � FðUÞ ð20Þ

where dFðUÞdU ¼ f ðUÞ. Looking at the Jacobian of the system at a fixed

point reveals that the trace is zero, so all fixed points are either sad-dles or centers.

Alternatively, we can formulate a Hamiltonian in terms of q. Bysubstituting U ¼ QðqÞ � K into (18) and multiplying by QðqÞs to get

0 ¼ QðqÞssQðqÞs � QðqÞsQðqÞ þ KQðqÞs þ QðqÞsq

and taking the integral with respect to q gives

eH ¼ QðqÞ2s2� QðqÞ2

2þ KQðqÞ � BðqÞ þ QðqÞq ð21Þ

where BðqÞ is the integral of QðqÞ:

BðqÞ :¼ bx

q lnðqÞ � q� 2 lnð1� qÞ½ � ð22Þ

The system (17–19) has fixed points ðU;wÞ ¼ ðf ðUÞ;0Þ and itsdynamics depend not only on the parameters, but also on the valueof K½U�. Since QðqÞ � K : ½0;1� ! ½�1;1� is monotonically increas-ing, the inverse, f ðUÞ, is defined and sigmoid in shape with asymp-totes f ð�1Þ ¼ 0 and f ð1Þ ¼ 1.

Typically, the equation U ¼ f ðUÞ has either one or three roots,Ui, which correspond to fixed points Si of system (17, 18). Thiscan be observed by numerically solving for solutions to the PDEsystem (see numerics section below) and examining resulting stea-dy states. If we solve the PDE and look the resulting steady stateswith M ¼ 0:25; g ¼ 40; b ¼ 0:1, and x ¼ 0:7, and compare it tothe results with x ¼ 1, using initial conditions described in thenumerics section, it is possible to see that the former convergesto a constant steady state solution, while the latter results in aninhomogenous solution. In particular, the former approaches asteady state solution with ðqðgÞ;UðgÞÞ ¼ ð0:25;0:25Þ and the latterwith ðqðgÞ;UðgÞÞ ¼ ð0:0499;0:0499Þ. Using (13) and the definitionof f we can examine how many roots exist for f ðUÞ �U for thesetwo choices of solutions (see Fig. 1). The Ui generally have to benumerically solved for in order to determine their values. Thiscan be accomplished by inverting f back and solvingQðUÞ � K ¼ U (see numerics section below for an example of this).

Lemma 2.1. If (17, 18) has a single fixed point, it is a saddle. If it hasthree fixed points with U1 < U2 < U3, then U2 is a center and theother two fixed points are saddles.

Proof. Taking the trace and determinant of the Jacobian of thesystem, we see the eigenvalues, m�, of the fixed points can bedetermined from the equations

0 0.5 10

0.2

0.4

0.6

0.8

1

Φ0 0.5 1

0

0.2

0.4

0.6

0.8

1

Φ

f(Φ)Φ

Fig. 1. With K and ðqðgÞ;UðgÞÞ calculated from the convergent steady state solutionof the PDE, the equation U ¼ f ðUÞ demonstrates (a) one root forðqðgÞ;UðgÞÞ ¼ ð0:25;0:25Þ and x ¼ 0:7 or (b) three roots forðqðgÞ;UðgÞÞ ¼ ð0:0499;0:0499Þ and x ¼ 1.

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

mþ þ m� ¼ 0 ð23Þmþm� ¼ f 0ðUiÞ � 1: ð24Þ

Consider a fixed point Ui. If f 0UiÞ < 1, then the fixed point is a sad-dle. If f 0ðUiÞ > 1, then the fixed point is a center. Note that if wehave f 0ðUiÞ > 1, then, at Ui, we can use the fact that f ðUÞ �U isincreasing and the asymptotic behaviour of f ðUÞ to deduce thatthere are saddles, U1 and U3, such that U1 < Ui < U3. In otherwords, in the case of one root, we must always have a saddle. Ifwe have three roots, we have a center along with two saddles, sincethe slope of the middle root is larger than the slope of the line (seeFig. 1(b)). h

As will be seen later, inhomogeneous patterns can only occur ifwe have a center, so we will consider the case where we have threeroots, U1 < U2 < U3.

2.2. Hamiltonian characterization

Additional information about the structure of the steady statesolutions of (17)–(19) can be gained by examining the Hamiltonian(20). In particular, since the Hamiltonian is independent of s-vari-able, for fixed Hamiltonian constant H, we can examine the Ham-iltonian curves and solve for w as a function of U to get

w ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 �FðUÞ þU2

2� H

!vuut : ð25Þ

The plus and minus signs correspond to the positive and negativesolutions for w. It is also seen that U is symmetric across thew ¼ 0 axis, and we are free to choose which sign we examine. SeeFig. 2 for sample Hamiltonian curves with x ¼ 10; b ¼ 1, andK ¼ �0:1392. In this case ðU1;U2;U3Þ is approximately equal toð0:05;0:318;0:625Þ.

Unless otherwise specified, from this point forward, we onlyconsider the positive part of the curve where w P 0. In this case,with w ¼ Us, we can solve for s as a function of U along the positivecurve by inverting both sides of equation (25) and integrating withrespect to U. Doing this we get

s ¼Z UðsÞ

Uð0Þ

dUffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�FðUÞ þ U2

2 � HÞq : ð26Þ

(26) describes the s-location where the specified U occurs withgiven Hamiltonian energy H.

The authors would like to note at this time that equations (25,26) are only currently known to be well defined and real for a givennon-constant stationary solution, ðUðsÞ;wðsÞÞ. If H or Uð0Þ are

0 0.2 0.4 0.6−0.2

−0.1

0

0.1

0.2

Φ

ψ

Fig. 2. Graph of sample Hamiltonian curves using x ¼ 1; b ¼ 0:1; K ¼ �0:1392,and various Uð0Þ (corresponding to various energy constants H). Trajectories thatpossibly can and definitely cannot generate stationary solutions to the PDE problem(6)–(8) for finite g are colored blue and red respectively. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version ofthis article.)

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

4 R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx

chosen arbitrarily, then the value inside the square-root (25) maybecome negative, and (26) will no longer be real, if defined at all.

The minimal and maximal values of U;Umin and Umax, occur atthe turning points of the Hamiltonian, when w ¼ 0, i.e., can befound from the equality:

0 ¼ �FðUÞ þU2

2� H: ð27Þ

If we instead use the alternate Hamiltonian, eH, we can use the chainrule and similarly solve for

s ¼Z qðsÞ

qð0Þ

Q 0ðqÞdqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðeH þ ðQðqÞ2=2� KQðqÞÞ þ ðBðqÞ � qQðqÞÞÞ

q ; ð28Þ

where we use the fact that Q 0ðqÞ > 0 on the domain, and the mini-mal and maximal values of q occur when the denominator evalu-ates to 0.

If we use Umin and Umax (obtained by solving (27)) as parame-ters, then the duration of the half curve’s orbit, TðUminÞ, satisfies

T ¼Z Umax

Umin

dUffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�FðUÞ þ FðUminÞ þ

U2�U2min

2 Þq ð29Þ

Duration will become important as we link solutions to the Hamil-tonian equation to solutions satisfying the boundary value problemof (17)–(19).

These results can be put in a more concrete form with the fol-lowing Theorem.

Theorem 2.2. For any given qð0Þ > 0, andUð0Þ > 0; f ðUÞ 2 C1ð½�1;1�Þ is well defined andFðUÞ 2 C2ð½�1;1�Þ is defined up to a constant. Assuming that theequation f ðUÞ �U ¼ 0 has three distinct roots, U1;U2, and U3, andthat the constant H can be chosen in such a way that Eq. (27) haspositive roots Umin;Umax satisfying U1 < Umin < U2 < Umax < U3.Then:

1. It is possible to define ewðUÞ 2 C2ððUmin;UmaxÞÞ by (25) andUðsÞ 2 C2ðð0;gÞÞ implicitly by (26).

2. For wðsÞ ¼ ewðUðsÞÞ; ðUðsÞ;wðsÞÞ are non-homogenous single-stepsolutions to the ODE described in (17, 18) and there is a uniqueT such that ðUðsÞ;wðsÞÞ also satisfy the boundary conditionsdescribed in (19) using the chosen domain length g ¼T.

3. We can define qðsÞ ¼ f ðUðsÞÞ to get step-like solutionsðqðsÞ;UðsÞÞ that are unique steady state solutions to (6), (7) sat-isfying boundary conditions (8) with the specified T.

4. The reflection of ðq;UÞ over the domain is also a steady statesolution to (6), (7) that satisfies the boundary conditions (8)with the specified T.

Proof. Given ðqð0Þ;Uð0ÞÞ, we define K using (13). Having K, it ispossible to define f ðUÞ as the inverse of (11) by the inverse func-tion Theorem. Since f ðUÞ is C1 class, the indefinite integral FðUÞis well defined up to a constant of integration and is C2 class. Wecan assume without loss of generality that Umin :¼ Uð0Þ. Usingthe assumption that f ðUÞ �U has three roots, and the fact thatf ð�1Þ ¼ 0; f ð1Þ ¼ 1, we have,

f ðUÞ �U < 0 for U 2 ðU1;U2Þf ðUÞ �U > 0 for U 2 ðU2;U3Þ

Integrating the function f ðUÞ �U with respect to U and defining Husing (27), we see that it is possible to define ew : ðUmin;UmaxÞ ! Rþby choosing the positive sign in (25). Also, from the above state-ments, it follows the ew is C2 class in ðUmin;UmaxÞ.

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

Since ew P 0 for U 2 ðUmin;UmaxÞ, we can define, using (26), afunction sðUÞ, with domain DðsÞ ¼ ðUmin;UmaxÞ and rangeRðsÞ ¼ ð0; sðUmaxÞÞ. sðUÞ is well defined since for Umin < U <

Umax <1 we have some constant eC > 0 such that:

s ¼Z U

Umin

dUffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ð�FðUÞ þ U2

2 � HÞq ð30Þ

6

Z U

Umin

dUffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieCðU�UminÞðUmax �UÞq <1: ð31Þ

Note that the second line follows from the fact that the roots of�FðUÞ þ U2

2 � H at Umin and Umax are simple. The derivative of thisfunction is �f ðUÞ þU, which by the above statements and assump-tions is not zero at the roots because Umin;Umax – U1;U2;U3. Other-wise, our trajectory would be a fixed point or tend to a fixed point.Since sðUÞ is monotonically increasing, it has a monotonicallyincreasing inverse UðsÞ. This establishes part 1.

Using part 1, we can now define the pull back wðsÞ :¼ ewðUðsÞÞ.By computing the derivative of sU by means of (26) and inverting it,it is straightforward to see that

dUðsÞds

¼ wðsÞ:

Furthermore, taking the derivative of both sides of (25) (with ‘plus’sign) we get that w satisfies (18). Since Umax <1, there is a uniqueT such that UðTÞ ¼ Umax. In this caseewðUminÞ ¼ ewðUmaxÞ ¼ 0

and

Umin ¼ Uð0Þ with Umax ¼ UðgÞ

satisfying the boundary conditions.Part 3 follows from the fact that we can define qðsÞ ¼ f ðUðsÞÞ,

and taking the derivative of qðsÞ gives

qs ¼ f 0ðUÞUs ¼xb

qð1� qÞ2

1þ q2

!Us:

This equation is equivalent to (9) which we can take the derivativeof to get (6) with qs ¼ 0 and (7) follows from part 2. The boundaryconditions for q are satisfied using (9) and are satisfied for U by part2. Uniqueness follows from [1], establishing part 3.

Using a change of variables, s! �sþ g, it is possible to see thatthe reflection of the solution across the domain also satisfies thestationary equation to (6), (7) with the caveat that w changes sign.This yields part 4. h

Remark 2.3. If Umin or Umax equals either U1;U2, or U3, then it isinstead possible to define UðsÞ as the appropriate constant func-tion, q ¼ U, and w ¼ 0 which satisfies the conditions of being a sta-tionary solution.

Corollary 2.4. All positive non-homogenous steady state solutions to(6) satisfying boundary conditions (8) are a sequence of single-stepsalternating between specific values of Umin and Umax.

Proof. Given a positive non-homogenous bounded solutionðqðsÞ;UðsÞÞ, let s1 > 0 be the first point after s ¼ 0 where either qs

or Us is zero. By (7, 9), if Us ¼ 0, then qs ¼ 0. If qs ¼ 0, thenUs ¼ 0 or q ¼ 0 where the latter result is ruled out by positivity.We can therefore restrict the domain to ½0; s1� with ðqðsÞ;UðsÞÞ asthe unique step solution to the restricted boundary value problem.At this point, we see from the Hamiltonian dynamics given in thephase plane analysis that having a non-constant steady state

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx 5

solution means that f ðUÞ �U is sigmoid with three roots. In orderto satisfy the boundary conditions, wðUÞ must have two roots. Bythe above Theorem, since we have K ¼ Qðqð0ÞÞ �Uð0Þ, and byuniqueness, we must have Uðs1Þ either be Umax or Umin. In eithercase, if s1 ¼ g, we are done. Otherwise, we can repeat the argumentfor the next critical point, s2. Notice that K stays the same for therestricted solution with domain ½s1; s2�, since K is an integrationconstant with K ¼ QðqðsÞÞ �UðsÞ, and s1 is in both the first and sec-ond domains. Furthermore, since K stays the same, we have thesame Umin and Umax, and their corresponding q values are alsothe same. By part 4 of the Theorem and uniqueness, the secondstep must be a reflection of the previous step because the stepsshare the same minimum, maximum, and K value. Also note thatbecause it is a reflection, the second step has the same step lengthas the first step with s2 ¼ s1 þT. Repeating this argument untilsn ¼ g, for some n > 0, gives T divides g, and the steps must alter-nate between reflections of the first step. h

2.3. Generation of multi-stepped solutions

To get spatially non-homogenous solutions along the centermanifold that corresponds to stationary solution of (6), we needthe trajectory to begin and end at w ¼ 0 after duration g by theboundary conditions wð0Þ ¼ wðgÞ ¼ 0. Note that this condition can-not be satisfied if we only had a single saddle as a fixed point. Usingthe symmetry of U across the w-axis, it is possible to see that fol-lowing the ðU;wÞ trajectory curves gives the condition g ¼ kT forsome integer k. To parametrize possible curves, let us vary thepoint of intersection on the w-axis, ðUmax;0Þ, whereU2 6 Umax 6 U3 and let T be the duration of the half curve’s orbitthat ends at ðUmax; 0Þ. As Umax increases, the curves move awayfrom the fixed point. Eventually we will approach either a hetero-clinic orbit connecting S1 and S3 or a homoclinic orbit connectingone of the roots to itself. Using (27), it may or may not be possibleto solve for U – U1 such that

0 ¼ FðUÞ �U2

2� FðU1Þ þ

U21

2:

If no such U exists, then the Hamiltonian curve containing S1 doesnot connect back to the w-axis, which occurs when S3 has a homo-clinic orbit connecting to itself. If such a U exists, then we eitherhave a heteroclinic orbit, with U ¼ U3, or a homoclinic orbit con-necting S1 to itself with U < U3. In all of these cases, for large en-ough Umax, the duration for the half curve’s orbit approaches 1.

We now calculate how the duration behaves as Umax approachesU2 from above, (Umax & U2). The periodic solution is non-homoge-neous, but its variation tends to zero, that is to saymaxsUðsÞ �minsUðsÞ ! 0 and jUðsÞ �U2j ! 0 for all s 2 ð�1;1Þ.By using (11) we conclude that jqðsÞ � q2j ! 0 for alls 2 ð�1;1Þ with q2 ¼ U2. Hence by (12), we infer thatUssðsÞ ! 0, and consequently that UsðsÞ ! 0 for all s 2 ð�1;1Þ.Knowing this, we can approximate the solution by the solution ofthe linearization of system (17, 18) around ðU;wÞ ¼ S2. Solvingthe linearized system yields that the U is periodic with half-periodT� if T�jmþj ¼T�jm�j ¼ p, where mþ and m� ¼ �mþ withmþ ¼ i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0ðU2Þ � 1

pare the strictly imaginary eigenvalues of the

Jacobian of (17, 18) at point S2. This gives us the limiting duration

limUmax&U2

TðUmaxÞ ¼T� ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0ðU2Þ � 1

p : ð32Þ

We will later show that given a domain of length g >T�, there is atleast one non-homogenous solution. For g <T�, we cannot rule outthe possibility of non-homogenous solutions existing, as we cannotestablish the monotinicity of TðUminÞ.

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

In order to describe the generation of multi-step solutions, wefirst recall that g ¼ kT, where k is the number of half cycles, orsteps, the solution contains. Multi-stepped stationary solutions of(17)–(19) for k P 1 can only exist with duration, T ¼ g=k. So thegeneration of new stationary solutions corresponds to when thefirst inhomogenous solution of (17)–(19) with domain length kTappears. This is equivalent to satisfying the linearized condition

T� ¼ g=k ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0ðU2Þ � 1

p :

This process of bifurcating new periodic solutions due to changes indomain length relates to the generation of new eigenvalues of anelliptic operator. For further information on branching solutions,see Ref. [5].

In general, f 0ðU2Þ is hard to determine since we have to solve forU2. However, for the constant solution, U ¼ eM , we recall that f ðUÞdefines the inverse with respect to q. According to (11), we have

f 0ðU2Þ ¼xb

eMð1� eMÞ21þ eM2

" #ð33Þ

where eM satisfies 0 ¼ Qð eMÞ � eM � K from (16).We now use the chemotactic strength, x, as a parameter and

the fact that in the limit of zero amplitude oscillations we can sub-stitute (33) into (32) to get the following result.

Lemma 2.5. Assuming the same conditions as Theorem 2.2, a k-stepstationary solution to (6) exists for x > xk when

xk ¼bð1þ eM2Þððkpg Þ

2 þ 1ÞeMð1� eMÞ2 : ð34Þ

Proof. Recall that by the assumptions, the system (17–19) isHamiltonian with a center and two saddles. The phase plane con-tains Hamiltonian curves that connect the w ¼ 0 axis to itself. Bythe above calculation in (32), there are orbits whose duration isarbitrarily close to the duration T� for given x and since there iseither an orbit homoclinic to the point S1 or S3 or a heteroclinicorbit connecting S1 with S3. Both kinds of orbits have infinite dura-tion. The function (29) is continuous with respect toUmin 2 ðU1;U2Þ due to the continuity of the integrand. So, for anyduration T 2 ðT�;1Þ, there exists at least one correspondingHamiltonian curve whose orbit has that duration. Since each Ham-iltonian curve satisfies the boundary conditions for some domainlength g, it remains to show there is a duration Tk ¼ g=k >T�

for there to exist a k-step solution on domain length L. Letx ¼ xk þ R;R > 0, and using (32), (33), and f 0ðU2Þ > 0 we have,

T� ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 0ðv2Þ � 1

p ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkpg Þ

2 þ Rf 0ðU2Þq <

pffiffiffiffiffiffiffiffiffiffiffiðkpg Þ

2q ¼ g=k ¼Tk

We note that there is a unique correspondence between our choiceof K and the value of eM since eM ¼ U2. This correspondence allowsus to switch the roles of eM and K, making eM a parameter for deter-mining the dynamics of the system.

Notice that (34) can also be used to count the number of sta-tionary solutions that the given parameters can produce. Givenxk < x < xkþ1, there are k known curves that satisfy the condi-tions for the given eM . However, if we were to try and leave eM arbi-trary, then we cannot, at this time, discount the possibility formultiple k-step solutions with the same value of eM .

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

6 R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx

3. Stability analysis of solutions

3.1. Lyapunov stability condition

In order to analyze the stability of multiple stepped equations,we will define a Lyapunov functional and describe the conditionswhen the stationary solutions are local minimums. Here we con-sider the system

qs ¼ ðhðqÞqsÞs � ðgðqÞUsÞs ð35ÞUs ¼ Uss �Uþ q ð36Þqsð0Þ ¼ Usð0Þ ¼ qsðgÞ ¼ UsðgÞ ¼ 0: ð37Þ

We define a generalized entropic functional

S ¼ �Z g

0CðqðxÞÞdx ð38Þ

where CðqÞ is the convex function determined by the equality:

C 00ðqÞ ¼ hðqÞgðqÞ : ð39Þ

Let us note that the function Cð�Þ is defined only up to a termc1qþ c2. However, integration of this term over space gives a con-stant independent of s, due to the mass conservationðgÞ�1 R g

0 qðs; sÞds ¼ M. For definiteness, we will take c1 ¼ c2 ¼ 0here. Let us also define the generalized free energy

F ¼Z g

0

12ðUsÞ2dsþ

Z g

0

12

U2ds�Z g

0qUdsþ

Z g

0CðqÞds: ð40Þ

Theorem 3.1. If q 2 ðqmin;qmaxÞ, where 0 < qmin < qmax; gðqÞ> 0;hðqÞP 0, and U 6 0. Then F is bounded below and is a Lyaponuvfunctional for the system (35, 36).

Proof. Since qmin < q < qmax, it follows that 12 U2 � qU reaches a

critical point with respect to U at � 12 q2 when U ¼ q. This critical

point is a minimum since the second derivative yields 1 > 0.Since C00ðqÞP 0, it follows that CðqÞP 0. As a consequence, the

functional is bounded below by

FðqÞP �Z g

0

12q2ds ð41Þ

P �g2q2

max ð42Þ

Note that for (6, 7), the functional is bounded below by � g2 since

q < 1.Taking the time derivative, it is possible to show that for non-

stationary q and U, the free energy

F ¼Z g

0

12ðUsÞ2dsþ

Z g

0

12

U2ds�Z g

0qUdsþ

Z g

0CðqÞds

is decreasing with respect to time.dFds¼Z g

0UsUssdsþ

Z g

0UUsds�

Z g

0qsU�qUsdsþ

Z g

0C 0ðqÞqsds ð43Þ

¼Z g

0ð�UssþUþqÞUsdsþ

Z g

0ð�UþC 0ðqÞÞqsds ð44Þ

¼ �Z g

0U2

sdsþZ g

0ð�Uþ C 0ðqÞÞðhðqÞqs � gðqÞUsÞsds ð45Þ

¼ �Z g

0U2

sds�Z g

0ð�Us þ C 00ðqÞqsÞðhðqÞqs þ gðqÞUsÞds ð46Þ

¼ �Z g

0U2

sds�Z g

0gðqÞð�Us þ C 00ðqÞqsÞ

2ds ð47Þ

6 0 ð48Þ

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

where we used integration by parts and the no-flux boundary con-ditions to get the second and fourth lines, and substitution with(35) to get the third line. The last line follows from the positivityof gðqÞ. Notice that _F ¼ 0 if and only if Us ¼ 0 and�gðqÞUs þ hðqÞqs ¼ 0. The latter condition implies qs ¼ 0. Ifqs ¼ 0 we have that �gðqÞUs þ hðqÞqs ¼ const. And by boundaryconditions, we must have the constant be zero. _F < 0 if and onlyif the ðq;UÞ are not stationary solutions. h

This forms an H-Theorem in the canonical ensemble. The freeenergy is always decreasing and it is bounded below, its minimumscorrespond to stationary solutions. Moreover, the existence of thisfunctional indicates that over time, the solutions will converge toone of these stationary solutions.

In order to determine the stability of the minimum values of thefunctional F, we look at the first and second variational derivatives.Taking the first and second variational derivatives of F gives

dF¼Z g

0UsdUsdsþ

Z g

0UdUds�

Z g

0qdUds�

Z g

0dqUdsþ

Z g

0C0ðqÞdqds ð49Þ

d2F¼Z g

0ðdUsÞ2dsþ

Z g

0ðdUÞ2ds�2

Z g

0dqdUdsþ

Z g

0C00ðqÞðdqÞ2ds: ð50Þ

If we find a perturbation dq and dU such that at a fixed point,d2Fðdq; dUÞ > 0, then the fixed point is unstable under this pertur-bation. With this goal in mind, we consider perturbations that pre-serve the total amount of chemical density and that satisfy theboundary conditions. Assuming that

dqðsÞ ¼Xk>0

Ak cosðkps=gÞ ð51Þ

dUðsÞ ¼Xl>0

Cl cosðlps=gÞ; ð52Þ

it is possible to seeZ g

0ðdUsÞ2ds ¼ g

2

Xl>0

l2l C2

l ð53ÞZ g

0ðdUÞ2ds ¼ g

2

Xl>0

C2l ð54Þ

� 2Z g

0dqdUdx ¼ �

Xl;k>0

dk;lgAkCk; ð55Þ

where dk;l is the Kronecker delta and lk ¼ ðpkg Þ. So ðd2FÞðdq; dUÞ < 0

when

0 >1g

Z g

0

Xk>0

Ak cosðlksÞ !2

C 00ds�Xk;l>0

AkCldk;l

þXl>0

l2l

2þ 1

2

� �C2

l : ð56Þ

Since Ak and Cl are constant for given perturbation, we can mini-mize the right hand side of the equation with respect to each Cl.This has the effect of choosing the most unstable choice of dU forgiven choice of dq. The critical points that occur must be minimumssince evaluating the second derivative with respect to an arbitraryCl yields

l2l þ 1 > 0:

Therefore, the minimums occur when

�Xk>0

Akdk;l þ ðl2l þ 1ÞCl ¼ 0

or

Cl ¼Al

l2l þ 1

:

Substituting this choice of Cl in for the unstable case gives

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx 7

0 >1g

Z g

0ðdqÞ2C 00ds�

Xk>0

A2k

l2l þ 1

þXk>0

A2k

2ðl2l þ 1Þ

ð57Þ

¼ 1g

Z g

0ðdqÞ2C00ds�

Xk>0

A2k

2ðl2k þ 1Þ : ð58Þ

If instability will occur for the prescribed perturbations, it will occurwhen the stationary solution satisfies the general instabilityconditionZ g

0C 00ðqðsÞÞdqðsÞ2ds <

Xk>0

A2kg

2ðl2k þ 1Þ : ð59Þ

Since the instability condition holds if and only if the prescribedperturbation is unstable with respect to the stationary solution, iffor any concentration preserving perturbation that is the limit ofsums of cosines, i.e., functions in L2ð½0;g�Þ with zero mean that sat-isfy the boundary conditions, we have

0 6 supdq

Xk>0

A2kg

2ðl2k þ 1Þ �

Z g

0C 00ðqðsÞÞdq2ds; ð60Þ

then the stationary solution is stable. This condition is equivalent toall n� n matrices of the form

R� Q

having non-positive eigenvalues, where R is a diagonal matrix with

Rii ¼g

2ðl2i þ 1Þ

and

Qi;j ¼Z g

0C00ðqðsÞÞ cosðlisÞ cosðljsÞds:

In general, it seems that this expression has to be evaluated numer-ically in order to determine the stability of the solution. However,we can state a few estimates about this equation. If we chooseA1 ¼ 1 and Ak ¼ 0 for k – 1, then using Holder’s inequality we seeinstability will occur if

jjC00jjL1 <1

ðpg Þ2 þ 1

: ð61Þ

For the stability condition, we note that

supdq

Xk>0

A2k

g2ðl2

k þ 1Þ �Z g

0C 00ðqðsÞÞdq2ds

6 supdq

Xk>0

A2k

g2ðl2

1 þ 1Þ �Z g

0C 00ðqðsÞÞdq2ds ð62Þ

6 supdq

g2ðl2

1 þ 1ÞXk>0

A2k �

Z g

0C 00ðqðsÞÞdq2ds ð63Þ

6 supdq

1l2

1 þ 1jjdqjj2L2 �

Z g

0C 00ðqðsÞÞdq2ds ð64Þ

6 supdq

1l2

1 þ 1jjdqjj2L2 � inf C 00ðqðsÞÞ

Z g

0dq2ds ð65Þ

6 supdqjjdqjj2L2

1l2

1 þ 1� inf C 00ðqðsÞÞ

� �ð66Þ

where the first inequality follows from lk > l1, and the third andfifth lines from the observation that

jjdqjj2L2 ¼Z g

0dq2ds ¼

Z g

0

Xk>0

Ak cosðlksÞ !2

ds ¼ g2

Xk>0

A2k :

Using (66) and (60), we have stability if

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

1l2

1 þ 1< inf

sC 00ðqðsÞÞ ð67Þ

These results lead to the following Theorem.

Theorem 3.2. Assuming the hypothesis found in Theorem 3.1,stationary solutions to the system (35, 36) are stable if they satisfy

inf C00 P1

ðpg Þ2 þ 1

ð68Þ

and are unstable if

jjC 00jjL1 <1

ðpg Þ2 þ 1

: ð69Þ

Notice that inequality (68) is in agreement with the linear sta-bility condition, which can be obtained by computing the eigen-values of the linearization matrix by assuming the perturbationsof the components of the solution ðM;MÞ proportional tocosðl1xÞ (see Ref. [2,12]). These eigenvalues are non-positive if

and only if hðMÞgðMÞ P

1ðpgÞ

2þ1. If this condition does not hold (for example,

for large gðMÞ, which corresponds to large x in the case of system(3, 4)), a bifurcation occurs and the constant solution becomesunstable, converging in time towards an inhomogeneous steadystate solution. In particular, the constant solution ðM;MÞ is linearly

unstable for the perturbations cosðlkxÞ, if hðMÞgðMÞ <

1

ðpkg Þ

2 þ 1.

3.2. Limiting behaviour of domain length

For this section, we will consider the model system describedby (6), (7), and (8) and analyze the behavior as the domain lengthgrows unbounded. Using Corollary 2.4 along with the definition ofC00,

jjC 00jjL1 ¼bx

max1þ q2

min

qminð1� qminÞ2 ;

1þ q2max

qmaxð1� qmaxÞ2

" #: ð70Þ

When a new non-homogeneous solution emerges from the fixedpoint, x ¼ xk þ �, we can asymptotically approximate the solutionsq ¼ eM þ �q1 þ Oð�2Þ and U ¼ eM þ �U1 þ Oð�2Þ. Substituting theseexpansions into (34) and (59), we notice that

1 <l2

l þ 1l2

k þ 1þ Oð�Þ ð71Þ

when l < k, we get that l > 1 implies dq ¼ cosðl1xÞ is an unstableperturbation. So, all non-single step perturbations start outunstable.

Note that an interesting phenomenon happens if we allow thedomain length, g, to increase. For a given K, as g approaches infin-ity, the points ðUmin;0Þ and ðUmax;0Þ approach the seperatrix atS1; S3, or both. As a result, jjC00jjL1 stays bounded for large g.

Looking at (34), we see that as g becomes unbounded every mul-ti-step solution will eventually emerge. Previous phase plane anal-ysis and (60) indicates that if a solution with k-step emerges and isstable in the limit, then all l-step solutions, with l < k, will also be-come stable. In particular the single-step solutions will always stayor become stable if a given k-step solution becomes stable. In prac-tice though, this may require incredibly large values of g.

Also note that 1þq2

qð1�qÞ2has a minimum 7:41375 at q ¼ 0:295598.

Thus, if

7:41375 >x

bððpg Þ2 þ 1Þ

;

then all stationary solutions are stable by Theorem 3.2. Since Lemma2.5 is not a strict if and only if statement, we cannot exclude the

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

8 R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx

possible existence of stable non-constant solutions in this region. Fur-thermore, the constant solution, ðq;UÞ ¼ ðM;MÞ, is unstable if andonly if 1þM2

Mð1�MÞ2< x

bððpgÞ2þ1Þ

using either Theorem 3.2 or condition (60).

3.3. Plateaus vs. Spikes

In this section, we consider the system

qs ¼ ðhðqÞqsÞs � ðgðqÞv sÞs ð72ÞUs ¼ Uss �Uþ q ð73Þqsð0Þ ¼ Usð0Þ ¼ qsðgÞ ¼ UsðgÞ ¼ 0: ð74Þ

and assume that h and g are chosen such that unique solutions existand qðUÞ can be well defined as a monotonically increasing functionin a similar way as in the case corresponding to (11). As described in[7], the classification of one dimensional local maxima into spikesor plateaus is based on the observations concerning the nonlocalgradient of the first derivative at maxima. In particular, for manyqualitative spikes (plateaus) it was observed that the non-local gra-dient of the first derivative was larger (smaller) than the secondderivative.

In this section, we will use the equivalent definition to classifythe maximum of stationary solutions of (72, 73) as a spike (pla-teau) if the fourth derivative of U or q is positive (negative). Fora stationary solution, one can integrate the right hand side of(72) with respect to s, and set the integration constant to zero tosatisfy the boundary condition as with (9), to see that criticalpoints of qðs0Þ :¼ qðs0; �Þ correspond to critical pointsUðs0Þ :¼ Uðs0; �Þ. Note that if either q or U is maximum at s0, thenthe assumption that qðUÞ exists, and is monotonically increasing,causes the other to be a maximum, since at a critical point

@ssqðUðs0ÞÞ ¼ q0ðUðs0ÞÞ@ssUðs0Þ:

The equilibrium equations yield

0 ¼ ðhðqðsÞÞqðsÞ0Þ0 � ðgðqðsÞÞUðsÞ0Þ0 ð75Þ0 ¼ UðsÞ00 þ qðsÞ �UðsÞ: ð76Þ

Taking the two derivatives on the top and bottom gives

0 ¼ ðhq0Þ000 � ðgðqÞU0Þ000 ð77Þ0 ¼ UIV þ q00 �U00: ð78Þ

Observe that evaluating (75) and (76) at s0 we get

q00ðs0Þ ¼ gU00=h ð79ÞU00ðs0Þ ¼ �qþU: ð80Þ

Similarly, evaluating (77) and (78) at s0 allows us to get

qIV ðs0Þ ¼ ð�3@qhðq00Þ2 þ 3@qgq00U00 þ gUIV Þ=h ð81ÞUIV ðs0Þ ¼ �q00 þU00: ð82Þ

Substituting (79) and (82) into (81), we get

qIV ðs0Þ ¼ 3ðU00Þ2 ghðh@qg � g@qhÞ

h2 þ gh

UIV ð83Þ

¼ ðU00Þ gh

3U00@qgh

� �� g

hþ 1

h ið84Þ

since U00ðs0Þ < 0; g=h > 0, and by (80) we have qðs0Þ is a spike if andonly if

�3ðq�UÞ@qgh

� �� g

h� 1

� �< 0 ð85Þ

Recall, C00 ¼ h=g and we can multiply both sides by C00 to rewrite thespike condition as

0 < 3ðq�UÞC 00@q1C 00

� �þ 1� C 00

ð86Þ

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

Notice that q�U > 0 since we are at maximum. This allows us towrite the spike condition for q as

3C 000

C 00<

1� C 00

q�U

� �ð87Þ

Similarly, the plateau condition is the same inequality, but with thesign reversed.

Now we examine the case when U is a plateau. Substituting(79) into (82), we get

UIV ðs0Þ ¼ � ghþ 1

� �U00: ð88Þ

Since U00ðs0Þ < 0, we have Uðs0Þ is a spike if and only if

gðqðs0ÞÞhðqðs0ÞÞ

> 1 ð89Þ

For (6) and (7), this is equivalent to

ð1þ qðs0Þ2ÞqðsoÞð1� qðs0ÞÞ2

<xb:

Also, the plateau condition becomes

C00ðqðs0ÞÞ > 1:

The plateau versus spike condition for U looks similar to the condi-tion for stability versus in-stability, except we evaluate at a singlepoint rather than over an integral against a perturbation. Also, wehave the right hand side of the plateau condition is larger thanthe right hand side of the stability condition since

1 P jjdq2jj2L2 ð90Þ

Pg2

Xk>0

A2k ð91Þ

>Xk>0

A2k

g2ðl2

k þ 1Þ ð92Þ

The spike condition

C00ðqðs0ÞÞ < 1;

is similar to the condition that the solution becomes unstable as ggoes to infinity. In fact, it can be seen that if a periodic solution is aspike, then on a long enough domain it will become unstable. Whilewe cannot state for certain that all unstable solutions are spikes, assome may be plateaus, it is possible to use Theorem 3.2 to see

Corollary 3.3. All non-constant solutions to the system described in(72, 73, 74) that satisfy the instability conditions in Theorem 3.2 havea spike in U.

Proof. Use Theorem 3.2 and the fact that the condition on jjC00jjL1is easier to satisfy than the inequality of (89) to get the result. h

Notice for our application Eq. (6), C000ðqÞ > 0 is minimum atq� ¼ 0:295598. For qðs0Þ < q�, the spike condition for q clearlyholds when 1 > C00ðqðs0ÞÞ. Likewise, if uðs0Þ is larger than the loca-tion of the minimum, e.g., if M > q�, the plateau condition holdswhen 1 < C00ðqðs0ÞÞ. Otherwise the equation will have to be evalu-ated to determine if q is a plateau or a spike.

Fortunately, in many of the cases that are dealt with numeri-cally, qðs0Þ > q� and so plateaus for Uðs0Þ coincide with plateausfor qðs0Þ.

4. Numerics

To demonstrate the existence of stable multi-step solutions aswell as analyze the properties of such solutions, we use numericsto evaluate the various expressions as well as solve the PDE.

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx 9

Specifically, we use MATLAB’s pdepe code which uses Skeel andBerzin’s method for discretizing the spatial domain [16] in orderto apply the method of lines coupled with MATLAB’s stiff variableorder ODE solver, ode15s. For this section, we examine the rescaledsystem (6, 8) for x ¼ 1 and b ¼ 0:1.

First, we wish to examine what C00ðqÞ looks like. Graphing it, wesee a concave up function with two singularities, with the steepersingularity at q ¼ 0 (see Fig. 3a).

If we assume that Si ¼ ðU;wÞ ¼ ðM;0Þ, for M ¼ 0:25 and somei ¼ 1;2;3, then one question we can ask is, what are the other Si

values, for given K, as we vary bx. For such fixed points, Si, we have

w ¼ 0; Ui ¼ qi, and can use K ¼ QðMÞ �M and (11, 12) to solve forvalid constant Ui. We can then graph the roots of QðUÞ � K �U ¼ 0for different values of b

x (see Fig. 3(b)). For large ratios of diffusion,b, to chemotactic strength, x, only one constant solution is al-lowed. As the ratio decreases, we see the emergence of two newsteady solutions, and the line for U ¼ 0:25 becomes U2. As the ratiodecreases further, the line for U ¼ 0:25 switches from being U1 to

U2. This point coincides with the location whenbx

1þM2

Mð1�MÞ2

equals 1. Past this point, for sufficiently large domains, U2 ¼ Mbecomes unstable.

Note that the different seperatrix values are included to give anintuition to the Hamiltonian relationship between solutions withthe same K value, not necessarily to indicate a bifurcation in thedynamics as one normally sees in saddle node and transcriticalbifurcations. In fact, since the different values of Ui correspond todifferent values of M, it is not possible to dynamically move fromone of these constant steady states to another.

Now, we examine solutions to the system. For demonstrationpurposes, we set the initial conditions of the PDE to be constant,with a small cosine wave perturbation:

qðs;0Þ ¼ 0:25þ 0:05 cospg

s� �

;

Uðs;0Þ ¼ 0:0125þ 0:0025 cospg

s� �

:

We allow the function to evolve until time s ¼ 1014. For small g, thePDE converges to a constant steady state solution. For largerlengths, g ¼ 40, multi-step structures emerge during a short transi-tion time. These structures display the characteristic meta-stabilityby remaining almost unchanged for a long period of time. Progres-sively, these structures transition to smaller and smaller numbers ofmeta-stable steps until the solution converges to a stable singlestep. For g ¼ 100, the end convergence results in a double step pat-tern (see Fig. 4 for single and double step plots). Even though a dou-ble step pattern is reached numerically, the solution is not preciselycentered. Since periodicity for reflected solutions is preserved, thisis still an approximate steady state solution, and most likely hap-pened because the initial conditions were biased to one side of

0 0.2 0.4 0.6 0.8 10

10

20

30

40

50

ρ

C’’

(a)Fig. 3. (a) Graph of C00ðqÞ for x ¼ 1; b ¼ 0:1 and (b) Values of cons

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

the system. It is interesting that q and U are almost identical. Thedifferences between them are as small as possible, just to satisfyEq. (7).

Numerically checking the spike versus plateau conditions fornon-stationary solutions qðx; tÞ and Uðx; tÞ at various times indi-cates that both q and U satisfy the plateau conditions near thesame time for both g ¼ 40 and g ¼ 100. Once the solutions satisfythe plateau condition, they continue to satisfy it at later times.

We ran several simulations with varying x and end times ¼ 1014. We then plotted the values of K; eM; sup C00ðuðxÞÞ, andinf C00ðuðxÞÞ for the resulting steady state solutions (see Fig. 5).Changing g between 40 and 100 resulted in little to no qualitativechange in these, so the resulting values are plotted for g ¼ 40. Inthe first plot, K decreases steadily indicating that the difference be-tween U and QðqÞ is increasing (see Fig. 5(a)). For constant solu-tions eM ¼ M, but after the bifurcation that results in q ¼ Mbecoming unstable, the value of eM jumps and starts increasingsub-linearly (see Fig. 5(b)). In the final plot, the value sup C00ðqÞ in-creases exponentially while inf C00ðqÞ remains constant since C00 isbounded below and achieves its minimum as q changes betweenpeaks and troughs (see Fig. 5(c)). This indicates the above stabilityconditions in Theorem 3.2 become less useful as b decreases (e.g. ifx increases), and further methods are needed to simplify the moregeneral stability conditions found in (60).

5. Discussion

Under starvation conditions, the chemotactic amoeba, Dictyos-telium discoideum, is capable of aggregating together and formingslugs. In this paper, we examined a PDE model of the slug forma-tion of the amoeba [11] that does not exhibit blow up in finite time[1,6]. In particular, we examined what patterns may be generatedby the model, when slug formation may occur, and when slug for-mation versus formation of homogeneous densities would bestable.

We have shown the existence of a Lyapunov function for thePDE model as well as convergence of the solutions of the PDE sys-tem to the steady state solutions. The steady state solutions wereclassified as either constants or bounded sequences of steps. Thesteps can be plateaus or spikes. A sequence of plateau steps quali-tatively resembles the formation of several slugs. Stability condi-tions are described for both constant and non-constant steadystate solutions, as well as for the non-constant, steady state, pla-teau solutions. Finally, numerical solutions to the PDE system dem-onstrated transitions in the number of half-steps as either thedomain length, or the ratio of chemotactic sensitivity to cell diffu-sion, increased.

A Hamiltonian characterization of the steady state solutions isalso given, but this characterization relies on scalars that arecalculated from a given solution, e.g., eM and K. Numerically, wesee that these scalars vary with the parameters, e.g., the chemical

0 0.05 0.1 0.150

0.2

0.4

0.6

0.8

1

β /ω

Φi

(b)tant solution Ui as b

x increases given one Ui ¼ M for M ¼ 0:25

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

Fig. 4. Surface Plots of q and U over time, in log scale, with a stable single step at g ¼ 40 and as Table 2-step at g ¼ 100 with same initial conditions up to scaling in the x-coordinate (a), (b), (d) and (e). Density plots for end time values of q and U at s ¼ 1014 (c), (f)

0.5 1 1.5−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

ω

K

(a)0.5 1 1.5

0.24

0.26

0.28

0.3

0.32

0.34

0.36

ω

M−t

ilde

(b)

0.5 1 1.50

2

4

6

8

10

12

ω

Γ (u

)

inf C" L=40sup C" L=40

(c)Fig. 5. Plots of (a) K, (b) eM , and (c) inf C00ðqÞ and sup C00ðqÞ, for the resulting steady state solution, with g ¼ 40 and varying x.

10 R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx

production rate a, or the chemotactic sensitivity v (entering thereduced system through x). The dependence of these variableson parameters, such as a and v, makes it difficult to establish whenk-step inhomogenous solutions exist. Once we have informationfrom one steady state solution, we can use that information, andknowledge about bifurcating periodic orbits, to conjecture theexistence of other steady state solutions to the system.

The resulting thresholds, derived from the characterization ofthe plateau versus spike solutions and unstable versus stable

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

solutions, are close to each other. In the limit, as the domain lengthapproaches infinity, the thresholds approach each other, suggest-ing that knowledge of whether or not the solution is plateau mightdetermine its stability or visa-versa. However, the condition forstability is global, and requires calculation of an integral over theentire domain, versus the condition for a plateau, which is local.As a result, it is not obvious how to link plateau conditions and sta-bility conditions for an arbitrary inhomogenous solution. Numeri-cally, the resulting approximate steady state solutions, calculated

bal inhomogeneous solutions of a macroscopic model of cell motion, Math.

R. Gejji et al. / Mathematical Biosciences xxx (2012) xxx–xxx 11

over long time runs, satisfy both the stability and the plateau con-ditions, suggesting that stable solutions are plateaus, even if wecannot prove it at this time.

From the biological point of view, it is interesting that, up to nor-malizing factors, the densities of the cells u and of the chemical v arealmost exactly the same. This suggests that not only do cell densityand chemical concentrations influence each other, but that they arestrongly correlated in how they influence each other at steadystates. While future biological experiments will be needed to testthis result, we can now hypothesize that cells that maintain consis-tent chemotactic behavior, when exposed to a constant in time che-motactic pattern, will approximately replicate this pattern.

Based on the results described above, we can predict when a gi-ven parameter set will result in constant steady state solutions ver-sus when it will result in formation of patterns, as determined bydiffusion of the chemical, its decay and production by the cells,as well as by diffusion of the cells in the chemotactic field. It canbe determined whether a given steady state is stable or not, andwhether or not it is a plateau. However, there are still many openquestions related to this system. In particular, when given only ini-tial conditions, we do not currently know how the resulting vari-ables, K and eM , can be calculated. Also, additional study isneeded to determine how to precisely predict when solutions fora given parameter set, and given k, would approach k-step stableplateaus. This would result in gaining an insight into how environ-mental parameters influence the number of slugs that form andhow wide apart these slugs are.

Acknowledgements

The authors thank Chuan Xue and Kun Zhao for helpful discus-sions, and Kathy Phillips for help with editing. R.G. was supported

Please cite this article in press as: R. Gejji et al., Classification and stability of gloBiosci. (2012), http://dx.doi.org/10.1016/j.mbs.2012.03.009

by the University of Notre Dames CAM Fellowship and partiallysupported by the NSF grant DMS 0931642. M.A. was partially sup-ported by the NSF grant DMS 0931642 and NIH grant R01GM100470-01. B.K. was partially supported by MNiSW grant NoNN201548738 and by FPS grant TEAM/2009-3/6.

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