pattern formation in partial differential equations

Pattern Formation in Partial DifferentialEquations

Natalya M. St. Clair

Professor Jon Jacobsen, AdvisorProfessor Chris Towse, Reader

Submitted to Scripps College in Partial Fulfillmentof the Degree of Bachelor of Arts

April 7, 2006

Department of Mathematics

Abstract

Based on the works of Murray [15] and Plaza, et al. [16] we focus on anapproach to pattern formation via systems of reaction-diffusion equations.In particular, we examine how such patterns evolve and vary with respectto different geometric surfaces. Of interest is how shape and curvature im-pact the selected patterns, and how these compare with the linear stabilityanalyses for the same partial differential equations on flat domains.

There are many uses and applications for pattern formation, making thetopic particularly valuable when considering different geometric surfaces.First, it can be seen as a model for how stripes and spots form on animalhides. Second, it is useful for modeling pattern formations on butterflywings, in alligators’ teeth, and even on tree branching.

Some open questions to explore include finding sufficient reaction ki-netics under which patterns form on different surfaces, geometric analysisfor objects such as the sphere, torus, and cone, and performing numericalsimulations in an attempt to produce models closely resembling real lifespecimens.

Contents

Abstract iii

Acknowledgments ix

1 Introduction to Reaction-Diffusion Mechanisms 11.1 Notes on Pattern Formation . . . . . . . . . . . . . . . . . . . 11.2 Three Classic Kinetics . . . . . . . . . . . . . . . . . . . . . . . 31.3 Linear Stability Analysis and Evolution of Spatial Patterns . 10

2 Turing Spaces 192.1 Determining Turing Spaces . . . . . . . . . . . . . . . . . . . 192.2 Numerical Methods for Solving Turing Spaces . . . . . . . . 25

3 Curvature and Geometry Effects on Reaction-Diffusion Mecha-nisms 273.1 Curvature on Pattern Formation . . . . . . . . . . . . . . . . 273.2 Pattern Formation on Tori and Spheres . . . . . . . . . . . . . 333.3 Experimenting With Pattern Formation on Greater Surfaces 37

4 Conclusion 394.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . 39

A Parameterizations Used For Different Geometries 43A.1 Clover . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.2 Cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.3 Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.4 Sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45A.5 Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

Bibliography 47

List of Figures

1.1 Illustration of the Field Without Grasshoppers Analogy . . . 31.2 Illustration of the Field With Grasshoppers Analogy . . . . . 31.3 Sample pattern for Nondimensionalized Reaction-Diffusion

System with the Schnakenberg Kinetics . . . . . . . . . . . . 81.4 Sample pattern for Nondimensionalized Reaction-Diffusion

System with the Gierer and Meinhardt Kinetics . . . . . . . . 91.5 Sample pattern for Nondimensionalized Reaction-Diffusion

System with the Thomas Kinetics . . . . . . . . . . . . . . . . 9

2.1 Some Typical Spatial Plots For Wavenumbers in the UnstableRange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 Illustration of the Glass in an Earthquake Analogy . . . . . . 242.3 Slices of Turing Space for the Schnakenberg, Gierer and Mein-

hardt, and Thomas kinetics . . . . . . . . . . . . . . . . . . . 25

3.1 Simulated Pattern on a Cone Using Thomas kinetics, Viewedin xyz-space and rs-space . . . . . . . . . . . . . . . . . . . . 31

3.2 Three Tapering Cylinders Simulated Under Thomas kineticsWith Different Values of γ . . . . . . . . . . . . . . . . . . . . 31

3.3 Simulations for Cylinders with the Same Approximate Sur-face Area and Different Shape . . . . . . . . . . . . . . . . . . 32

3.4 Illustration of the Activator-Inhibitor System on a LeopardReposing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.5 Schnakenberg Kinetics on Three Different-Sized Tori Rings . 343.6 Torus Simulations for γ = 400, 800, 1200, Respectively With

Schnakenberg Kinetics and Gierer and Meinhardt Kinetics. . 343.7 A Wide Range of Sphere Simulations for Prototypical Kinet-

ics Such as the Schnakenberg Kinetics. . . . . . . . . . . . . . 35

viii List of Figures

3.8 Comparison of the Sphere and Torus With Proportional Sur-face Area Sizes . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.9 Simulations of The Gierer and Meinhardt Kinetics With aLogarithmically-Profiled Cone . . . . . . . . . . . . . . . . . 37

3.10 Schnakenberg and Thomas Kinetics Simulated on a Logarithmica-lly-Profiled Clover and Exponentially-Profiled Cone . . . . . 38

3.11 Clover Simulations for Reaction-Diffusion Mechanisms WithSystems Such as the Schnakenberg Kinetics . . . . . . . . . . 38

Acknowledgments

I would first like to thank my advisor Professor Jon Jacobsen for not onlyintroducing me to to this beautiful field of mathematics but also for his en-thusiasm and support throughout the entire project. I would also like tothank my second reader, Professor Chris Towse, for his support. Researchfor this thesis was presented at the 2006 AMS/MAA Joint MathematicsMeetings in San Antonio. Without the advice from Professor Arthur Ben-jamin to fund the presentation, this would not have happened. I would liketo thank the members of my research group for their dedication, enthusi-asm, and camaraderie–Julijana Gjorgjieva and Simon Stump. Along thoselines, I am very appreciative towards Jeff Hellrung for releasing his Matlabprograms to use for research purposes. Finally, to the individuals who readmy thesis twice and spotted those last minute corrections, Kevin Zielnickiand Jeremy Mejia, I am incredibly grateful towards both of you.

I am eternally ungrateful to Priya V. Prasad for not reading my thesisand for making me help with her poster and thesis instead.

Chapter 1

Introduction toReaction-DiffusionMechanisms

1.1 Notes on Pattern Formation

Pattern formation is a topic in mathematical biology that studies how struc-tures and patterns in nature evolve over time. One of the mainstream topicsin pattern formation involves the reaction-diffusion mechanisms of two chem-icals, originally proposed by Alan Turing [21] in 1952.

Turing’s idea was that diffusion of two chemical species (or morphogens)could lead to patterns in concentrations that break symmetry [1]. Normally,diffusion acts as a steadying mechanism—for example, diffusion governshow the temperature in a room distributes, as modeled by the heat equa-tion. However, under Turing’s model, diffusion serves as a competitor foranother autocatalytic chemical reaction. He suggested that, under certainconditions, chemicals can react and diffuse in such a way that steady-statepatterns emerge. These patterns are known as Turing patterns.

In 1972, Gierer and Meinhardt [3] and Segel and Jackson [19] indepen-dently showed that two features play an important role in pattern forma-tion: local self-enhancement and long-range inhibition. An activator a is saidto be self-enhancing (or autocatalytic) if small increases of concentration ofa over a homogeneous steady-state concentration causes a further increasein a. Self-enhancement alone does not generate sufficient kinetics for stablepatterns to form because once a begins to increase at a given position, itsown feedback would lead to an overall activation. Thus, self-enhancement

2 Introduction to Reaction-Diffusion Mechanisms

of a must be complemented by the action of a fast diffusing antagonist [8].Namely, an inhibitor b is said to have long-range inhibition if b diffuses muchfaster than its controlling activator a. In other words, the strong short-rangepositive feedback of a must be supplemented by a longer negative feedbackof b [11]. Several types of antagonistic reactions are available, which will bediscussed in Section 1.2.

The general form of the reaction-diffusion system is

∂~c

∂t= D∆~c + ~f(~c), (1.1)

where ~c is a vector of morphogen concentrations, ∆ is the Laplace operator,~f is a function of reaction kinetics, and D is a diagonal matrix of positiveconstant diffusion coefficients. Note that (1.1) is a system of nonlinear par-tial differential equations for the chemical concentrations. Intuitively, thespecial form of (1.1) allows us to think of it as a system of ordinary differen-tial equations and partial differential equations. For this reason, our studyof (1.1) will involve techniques from partial differential equations and or-dinary differential equations to determine when patterns form.

In this thesis, we will consider two-dimensional systems for two species,say A and B, acting on a two dimensional domain. In this case the system(1.1) takes the form:

∂A

∂t= DA∆A + F (A,B), (1.2)

∂B

∂t= DB∆B + G(A,B), (1.3)

where F and G are the reaction kinetics, which will always be nonlinear.The basic idea of Turing, captured in the system (1.2)–(1.3), is the fol-

lowing. Without diffusion, the chemicals would simply grow to a linearsteady-state. On the other hand, under certain conditions, the Turing insta-bility will drive spatially inhomogeneous solutions to give rise to Turingpatterns. These patterns can look like spots, stripes, spirals, and more on atwo-dimensional surface.

In 1989 J.D. Murray [15] cited an intuitive analogy to think about sys-tems of reaction-diffusion equations. Consider a field of grass so that if afire passed through this field, we would simply get the charred remains ofa burned down countryside (see Figure 1.1). Now consider a case wherethere are grasshoppers in the field, and when they get hot, these grasshop-pers sweat enough to prevent patches of grass from burning when the fire

Three Classic Kinetics 3

Figure 1.1: In a field without grasshoppers (i.e., no inhibition), when a firesweeps through, the end result would be simply a charred field.

Figure 1.2: In a field with sweating grasshoppers, their combined efforts toprevent the fire from spreading would cause patches of grass in a charredarea once the process has finished.

sweeps through the area. The grasshoppers would save the grass fromcharring in the following way. The fire (activator) starts to spread, say witha diffusion coefficient DF . Sensing the fire, the grasshoppers (inhibitor)move quickly ahead of the flame front; i.e., they have a diffusion coefficient,say DG, which is much larger than DF . The grasshoppers then sweat pro-fusely, enough to prevent areas of the field from getting charred. If, insteadof a single fire there was a random scattering of flames, we can imaginehow this would result in spatially homogeneous steady state distributionand uncharred regions with a spatial distribution of grasshoppers. If thegrasshoppers and flame front ”diffused” at the same rate, then no spatialpattern would evolve. Other analogies have been proposed; a more con-temporary model constructed with firefighters working together to put outa forest fire [14]. Although whimsical, the grasshopper analogy capturesthe essence of the morphogen dynamics behind pattern formation.

1.2 Three Classic Kinetics

In reaction-diffusion systems, kinetics ultimately determine the types ofpatterns that will emerge. Several sufficient models have been proposedsince 1972 for modeling pattern formation, some originating from real-


world models in other fields of study such as chemistry. Here we aregoing to discuss three well-known models: activator-substrate, activator-inhibitor, and substrate-inhibition.

First, the activator-substrate model occurs when lateral inhibition is a-chieved via substrate consumption in autocatalysis. Lateral inhibition isa signal produced by one reactant that prevents the other from synthesiz-ing [2]. This depletion lowers the rate of the self-enhancing system. Theprototypical example is the Schnakenberg kinetics [17]. With the system(1.2)–(1.3), this set of kinetics takes the form

F (A,B) = k1 − k2A + k3A2B, (1.4)

G(A,B) = k4 − k3A2B, (1.5)

where k1 is a source of A, −k2A is exponential decay, and k3A2B repre-

sents the autocatalysis creating A, but B acts as the substrate, necessaryto facilitate the autocalytic reaction. Like most activator-substrate models,the Schnakenberg system tends to develop wider, less sharp wavefronts incontrast to its counterpart activator-inhibitor system.

Second, the activator-inhibitor model occurs when an inhibitory sub-stance is produced by the activator that, in turn, slows down the activationproduction. The classic example is the Gierer and Meinhardt kinetics:

F (A,B) = k1 − k2A +k3A

2

B, (1.6)

G(A,B) = k4A2 − k5B, (1.7)

which has been widely studied and used since their mechanism was pro-posed in 1972. Notice that in the autocatalytic term k3A2

B , B now acts asan inhibitor to the creation of more A in the F -term. Koch and Meinhardtgive an excellent review of the applications of the Gierer-Meinhardt modelin biological pattern formation in different structures [8]. They offer a widerange of applications to the system (1.6)–(1.7) and variations it can take.

Finally, the activator-substrate model occurs when uncompetitive inhibi-tion is achieved via more than one substrate binding to an active site at thesame time.


The prototypical example is the Thomas kinetics:

F (A,B) = k1 − k2A− k5AB

k6 + k7A + k8A2, (1.8)

G(A,B) = k3 − k4B − k5AB

k6 + k7A + k8A2, (1.9)

which was derived experimentally in 1975 (see Murray [15]). In the sys-tem (1.8)–(1.9), A and B are specifically the concentrations of substrate oxy-gen and enzyme uricase, respectively. Substrate inhibition is present in therightmost terms, via k8A

2. Note that the final terms in F (A,B) and G(A,B)are negative, thus they contribute to the reduction of A and B, and this ratebecomes inhibited with sufficiently large A.

Now a worthwhile endeavor is to nondimensionalize systems such as(1.4)–(1.5) with the reaction-diffusion system (1.2)–(1.3). For beginners tothis technique, Lin and Segel [10] offer an excellent introduction to dimen-sional analysis. To illustrate the technique we will nondimensionalize thesystem (1.2)–(1.3) with the Schnakenberg kinetics (1.4)–(1.5). FollowingMurray [15], we introduce a characteristic length scale L and choose di-mensionless quantities:

~x∗ = ~xL , γ = L2k2

DA, d = DB

DA, u = A

(k3k2

) 12 ,

t∗ = DAtL2 , a = k1

k2

(k3k2

) 12 , b = k4

k2

(k3k2

) 12 , v = B

(k3k2

) 12 .

(1.10)

The chain rule will be used to rewrite the system (1.2)–(1.3) with kinetics(1.4)–(1.5), so we start by taking derivatives of ~x∗ and t∗ with respect to A.The partial derivative of A with respect to t∗ is

∂A

∂t∗=

∂A

∂t

∂t

∂t∗=

L2

DA

∂A

∂t, (1.11)

since ∂t∂t∗ = L2

DA.


Taking the partial derivative of A with respect to ~x∗ gives

∂A

∂~x∗=

∂A

∂~x

∂~x

∂~x∗=

∂A

∂~xL. (1.12)

Using equation (1.12), computing the second partial derivative of A withrespect to ~x∗ yields

∂2A

∂~x∗2=

∂2A

∂x2L2

which implies

1L2

∆~x∗A = ∆~xA. (1.13)

Now rewrite the left and right hand sides in equation (1.2). Recall thatequation (1.2) is

∂A

∂t= DA∆~xA + F (A,B).

After substituting equations (1.11)–(1.13) into equation (1.2) and collectingterms, we get

∂A

∂t∗= DA∆~x∗A +

L2

DAF (A,B). (1.14)

Our last task is to express equation (1.14) with (1.4) in dimensionless

forms using the terms in (1.10). First, note that since u = A(

k3k2

) 12 , this

implies A =(

k2k3

) 12 u. Hence

∂A

∂t∗=(

k2

k3

) 12

ut∗, (1.15)

which, after writing equation (1.14) with (1.4), implies

(k2

k3

) 12

ut∗ =(

k2

k3

) 12

∆~x∗u +L2

DA

(k1 − k2A + k3A

2B)

. (1.16)

Now put each righthand term of (1.10) into equation (1.16). We get

ut∗ = ∆~x∗u +L2

DA

(k3

k2

) 12

k1 − k2

(k2

k3

) 12

u + k3

((k2

k3

) 12

)2

u2(

k2

k3

) 12

v

,


which reduces to

ut∗ = ∆~x∗u +L2

DAk1

(k3

k2

) 12

−(

L2

DAk2

)u +

L2

DAk2u

2v. (1.17)

Finally, replacing the terms in equation (1.17) with each lefthand termin (1.10) produces the nondimensionalized formula for equation (1.4) of theSchnakenberg system:

ut∗ = ∆~x∗u + γa− γu− γu2v,

or

ut∗ = ∆~x∗u + γ(a− u− u2v

)where

γ =L2k2

DA. (1.18)

Similarly, for equation (1.3), we get

vt∗ = d∆~x∗v + γ(b− u2v

), (1.19)

where γ is defined in (1.18).giving the desired nondimensionalized equations for the Schnakenberg sys-tem:

ut = ∆u + γ(a− u + u2v

), (1.20)

vt = d∆v + γ(b− u2v

), (1.21)

after dropping the asterisks for convenience. Note that the seven parame-ters in (1.4)–(1.5) are reduced to four parameters (a, b, γ, d). This is a greatreduction, since to study all systems (1.4)–(1.5) is equivalent to exploringthe four dimensional (a, b, γ, d) parameter space. In the system (1.20)–(1.21)a and b are positive parameters and u and v are the activator and inhibitorterms, respectively.


Figure 1.3: Sample pattern for the system (1.26)–(1.27) with the Schnaken-berg kinetics (1.20)–(1.21) with γ = 400, viewed in three dimensions andfrom overhead, respectively. The z-axis represents the concentration of theactivator u.

Similar to how we obtained the system (1.20)–(1.21), nondimensional-izing of (1.6)–(1.7) gives

f(u, v) = a− bu +u2

v, (1.22)

g(u, v) = u2 − v, (1.23)

where a and b are positive parameters.Finally, nondimensionalizing of (1.8)–(1.9) yields

f(u, v) = a− u− ρuv

1 + u + Ku2, (1.24)

g(u, v) = α(b− v)− ρuv

1 + u + Ku2, (1.25)

where a, b, α, ρ, and K are positive parameters.In general the system (1.2)–(1.3) can be nondimensionalized and scaled

to take the form

ut = ∆u + γf(u, v), (1.26)vt = d∆v + γg(u, v), (1.27)

where d = DvDu

is the ratio of the diffusion coefficients of u and v, f andg are the reaction kinetics, and γ is a key parameter. For example, in the


Figure 1.4: Sample pattern for the system (1.26)–(1.27) with the Gierer andMeinhardt kinetics (1.22)–(1.23) simulated on a plane with γ = 400 , viewedin three dimensions and from overhead, respectively. Notice that thesewavesolutions are more isolated and sharp in contrast to those in Figure1.3.

Figure 1.5: Sample pattern for the system (1.26)–(1.27) with the Thomaskinetics (1.24)–(1.25) with γ = 400, viewed in three dimensions and fromoverhead, respectively.


Schnakenberg system (1.20)–(1.21), γ = L2(

k2DA

), so γ is related to the fol-

lowing:

(1) In two dimensions, γ is proportional to the area, L2. We will see theimportance of this in Section 2.1.

(2) γ represents a relative strength of reaction terms.

(3) An increase in γ can also be thought of as a decrease in DA, the diffu-sion coefficient in equation (1.2).

The advantage of writing γ and d in this way allow a wide interpre-tation in biological modeling as opposed to the dimensionalized systems(1.4)–(1.5) through (1.8)–(1.9). When we look at the domains where spatialpatterns appear, we can also conveniently look at the parameters in (γ, d)space. Later it will be clear why we wrote mechanisms such as (1.20)–(1.21)this way once we investigate how patterns form across different-size struc-tures.

Finally, whether the systems (1.2)–(1.3) can even generate Turing pat-terns depends on the reaction kinetics of f and g, and the values of γ andd. We explore this idea in the next section.

1.3 Linear Stability Analysis and Evolution of SpatialPatterns

We conclude this chapter by introducing an important technique knownas linear stability analysis, and deriving sufficient conditions for when pat-terns form with the system (1.26)–(1.27). Many ideas from this section fol-low Murray [15], although we go a little more in depth here.

In particular, we are interested in when the reaction-diffusion mecha-nism (1.26)–(1.27) exhibits a diffusion-driven instability, or Turing instabil-ity. Counter-intuitively, the driving force of this instability is diffusion; thismechanism also determines the type of pattern that forms. Naturally, all ofthis depends on values of γ, d, and reaction kinetics of f and g.

We consider the system on a rectangular domain Ω = (0, L) × (0,H).We also add to the partial differential equation system boundary condi-tions. It is natural to assume that nothing enters the system and nothingexits the system. This makes sense intuitively because none of the mor-phogens would, for example, spontaneously leave the animal hide they

Linear Stability Analysis and Evolution of Spatial Patterns 11

occupy. Thus, we will take zero flux (Neumann) boundary conditions forthe flat domain. The system (1.26)–(1.27) becomes

ut = ∆u + γf(u, v) on Ω,

vt = d∆v + γg(u, v) on Ω,

∇u · ν = ∇v · ν = 0 on ∂Ω,

(1.28)

where Ω = (0, L)× (0,H) ⊆ R2 and ν is the unit outer normal on ∂Ω.We first consider the case of spatially homogeneous solutions. In this

case the partial differential equation system (1.28) is equivalent to the ordi-nary differential equation

ut = γf(u, v),vt = γg(u, v).

(1.29)

We derive conditions so that the equilibrium solution of this system is lin-early stable. This steady-state (u0, v0) solves

f(u0, v0) = 0, g(u0, v0) = 0.

Consider a perturbation ~w from the steady-state

~w(t) =

(u(t)− u0

v(t)− v0

)=

(w1

w2

). (1.30)

Since

~w ′(t) =

(u′(t)v′(t)

),

we may substitute u′ = γf(u0 + w1, v0 + w1) and v′ = γg(u0 + w2, v0 + w2).The Taylor series expansion from multivariable calculus gives

w′1 = γ

(f(u0, v0) + ∂f

∂uw1 + ∂f∂v w2 + O(w2

1, w22, w1w2)

),

w′2 = γ

(g(u0, v0) + ∂g

∂uw1 + ∂g∂vw2 + O(w2

1, w22, w1w2)

),


and since f(u0, v0) = 0 and g(u0, v0) = 0,

w′1 = γ

(∂f

∂uw1 +

∂f

∂vw2 + O(w2

1, w22, w1w2)

),

w′2 = γ

(∂g

∂uw1 +

∂g

∂vw2 + O(w2

1, w22, w1w2)

),

For ||~w|| small, u and v are small, and the quadratic terms O(w21, w

22, w1w2)

in u and v are extremely small. Hence, the perturbation is locally governedby the equation

~wt = γA~w, (1.31)

where

γA = γ

(∂f∂u

∂f∂v

∂g∂u

∂f∂v

) ∣∣∣(u0,v0) ,

is the Jacobian matrix evaluated at (u0, v0). This linear system is stablewhen all eigenvalues of A have Re(λ) < 0. Since the characteristic polyno-mial of A is

det(γA− λI) =γfu − λ γfv

γgu γgv − λ= 0,

or

λ2 − γ(fu + gv)λ + γ2(fugv − fvgu) = 0, (1.32)

it follows that the eigenvalues are

λ1,2 =γ(fu + gv)±

√(fu + gu)2 − 4(fugv − fvgu)2

2. (1.33)

Therefore, linear stability (Re(λ) < 0) is guaranteed if

tr(A) = fu + gv < 0, (1.34)det(A) = fugv − fvgu > 0. (1.35)

Equations (1.34)–(1.35) are the two key conditions for the homogeneoussolution of (1.29) to be linearly stable. The conditions will play an impor-tant role in establishing when Turing instability can form.


Now to include spatial effects we must consider the full reaction-diffusionequation (i.e., with the diffusion term), and with the linearized ~w, the sys-tem of partial differential equations (1.28) becomes

~wt = D∆~w + γA~w, D =

(1 00 d

). (1.36)

Using separation of variables we look for solutions to (1.36) of the form

~w =∑k

~ckeλktΦk,

where Φk solves the Helmholtz equation∆Φk + k2Φk = 0 on Ω,∂Φk∂ν = 0 on ∂Ω.

(1.37)

Here k2 > 0 is a positive constant. Substituting a test function ~ckeλktΦk into

(1.36) yields

λΦk

(~cke

λt)

= k2DΦk

(~cke

λt)

+ γAΦk

(~cke

λt)

,

which reduces to

λΦk = Dk2Φk + γAΦk,

or,

0 = Φk

[γA−Dk2 − λI

]≡[λI − γA + Dk2

]Φk. (1.38)

For convenience, call[λI − γA + Dk2

]the matrix M. This system will have

nontrivial solutions if det(M ) = 0, i.e.,∣∣∣λI − γA + Dk2∣∣∣ = 0,

which is ∣∣∣∣∣ λ− γfu + k2 γfv

−γgu λ− γgv + dk2

∣∣∣∣∣ = 0.

Expanding the determinant yields a quadratic in λ:

λ2 + λ(dk2 − γgv − γfu + k2

)+ dk4 − γk2 (dfu + gv) + γ2|A| = 0. (1.39)


Equation (1.39) has the form

λ2 + αλ + β = 0

with

α = dk2 − γgv − γfu + k2, (1.40)β = h(k2) = dk4 − γk2 (dfu + gv) + γ2|A|. (1.41)

Recall that steady-state solutions are linearly unstable if Re(λ) > 0, and ifMΦk = 0, we are looking for the null space of M.

Now, a solution ~w = ~ckeλtΦk will be unstable to spatial disturbances

if Re(λ) > 0. In this case the quadratic is simply equation (1.32), and thatquadratic yielded conditions (1.34)–(1.35). Now we want the steady stateunstable to spatial perturbations, so we require Re(λ(k)) > 0 for some k 6=0. This occurs if either equation (1.40) is negative, or when h(k2) < 0 forsome k 6= 0. We already know that when k2 = 0, we get the condition(1.34). In particular, since we require fu + gu < 0 from condition (1.34), andk2(1 + d) > 0 for all k 6= 0, we require equation (1.40) has the form

k2(1 + d)− γ(fu + gv) > 0,

so the only way to get Re(λ) > 0 is when h(k2) < 0 for some k. We canverify this condition from the solutions of the polynomial (1.39), namely

λ1,2 =−(k2(1 + d)− γ(gv + fu)

)±√

[k2(1 + d)− γ(gv + fu)]2 − 4h(k2)2

.

Because we required det(A) > 0 from condition (1.35), we can see thath(k2) < 0 only occurs when

dfu + gv > 0. (1.42)

The inequality (1.42) is a needed but insufficient condition forRe(λ) > 0. To get h(k2) < 0 for some k 6= 0, we require that the minimumhmin < 0. From equation (1.41), differentiation with respect to k2 is

2dk2m − γ(dfu + gv) = 0, (1.43)


thus

k2 = k2m = γ

dfu + gv

2d, (1.44)

which yields

hmin =

[|A| − (dfu + gv)2

4d

]γ2.

Thus, the condition h(k2) < 0 is

|A| < γ(dfu + gv)2

4d

for k 6= 0, and at bifurcation, when hmin = 0,

|A| = (dfu + gv)2

4d.

For fixed kinetics parameters (think of any of the kinetics in Section 1.2),we define a critical diffusion coefficient dc. We know that this dc > 1, andthat

0 = (γdcfu + gv)2 .

Expanding terms gives

0 = γ2d2cf

2u + 2γdcfugv + g2

v .

Consider the case when hmin = 0, at bifurcation. Then we require

0 = |A| = fugv − fvgu =(dfu + gv)2

4d,

which implies

0 = dcf2u + 2dc(2fvgu + fugv) + g2

v .

Hence, the critical wavenumber kc is given by

k2c = γ

dcfu + gv

2dc= γ

√|A|dc

= γ

√fugv − fvgu

dc.

When h(k2) < 0, equation (1.39) has a solution λ > 0 for the same rangeof wavenumbers that make h < 0. From equation (1.39), with d > dc the


range of unstable wavenumbers is k21 < k2 < k2

2, which is obtained fromthe quadratic

k2 = γdfu + gv ±

√(dfu + gv)

2 − 4d|A|2d

.

Thus,

k1 =γ

2d

[dfu + gv −

√(dfu + gv)

2 − 4d|A|]

(1.45)

< k2 (1.46)

<γ

2d

[dfu + gv +

√(dfu + gv)

2 − 4d|A|]

(1.47)

= k2. (1.48)

The expression λ = λ(k2) is called a dispersion relation. In an unstable range,Reλ(k2) > 0 has maximum kmax with d > dc. This implies that there existsa fastest growing mode in the summation of ~w. Thus, we have

~w ∝k2∑k1

~ckeλ(k2)tΦk for large t.

To recap our results, we state the following theorem.

Theorem 1 ([15]). Consider the system of partial differential equations (1.26)–(1.27) with the equilibrium point (u0, v0). The conditions for Turing patterns toform are

fu + gv < 0, (1.49)fugv − fvgu > 0, (1.50)

dfu + gv > 0, (1.51)

(dfu + gv)2 − 4d(fugv − fvgu) > 0, (1.52)

where all partial derivatives are evaluated at (u0, v0). If these conditions are sat-isfied, then there exists a range of patterns depending on γ, with wavenumbersthat are linearly unstable given in (1.45)–(1.48). These spatial patterns are theeigenfunctions Φk with wavenumbers k1 and k2.


Note that it follows from (1.49) and (1.51) that d 6= 1, and fu and gv

must have opposite signs. For example, if fu > 0 and gv < 0, then d > 1.Thinking in terms of reaction-diffusion, this means that the inhibitor of thesystem (1.26)–(1.27) must diffuse faster than the activator, as noted before.This is the case in the activator-inhibitor model such as the Schnakenbergkinetics (1.20)–(1.21).

For a given system (1.26)–(1.27) these equations determine which val-ues of the parameters will yield patterns. For a given model the set ofparameters in which patterns exists is known as Turing Space. In the nextchapter we take a closer look at these parameters.

Chapter 2

Turing Spaces

2.1 Determining Turing Spaces

In this chapter, we further investigate how different kinetics affect Turingspace (or pattern formation space). In the case of the Schnakenberg kinetics(1.20)–(1.21) there exist three parameters (a, b, d) under which the mecha-nism (1.26)–(1.27) is unstable. We start by giving a detailed analysis of theSchnakenberg kinetics (1.20)–(1.21) in order to gain a better understandingof when spatial patterns form.

Recall that the Schnakenberg kinetics (1.20)–(1.21) is of the form

f(u, v) = a− u + u2v,

g(u, v) = b− u2v.

The equilibrium points are obtained by setting f(u, v) = g(u, v) = 0 :

0 = a− u + u2v,

0 = b− u2v.

It follows u0 = a + b, and since u20v0 = b, this implies v0 = b

u20, and the

equilibrium point is

(u0, v0) =(

a + b,b

a + b2

).

20 Turing Spaces

The Jacobian of the system at (u0, v0) yields:

A =

(fu fv

gu gv

)|(u0,v0)

=

(−1 + 2uv u2

−2uv −u2

)|(u0,v0)

=

(b−aa+b

)(a + b)2

−2(

ba+b

)− (a + b)2

,

and using the condition (1.49) of Theorem 1, we get:

tr(A) =b− a

a + b− (a + b)2 < 0

therefore

0 < b− a < (a + b)3 .

Similarly, the condition (1.50) of Theorem 1 gives

|A| = −((

b− a

a + b

)(a + b)2

)−(

(a + b)2(− 2b

a + b

))= b2 + a2 + 2ba

= (a + b)2

> 0.

Recall that the condition (1.51) of Theorem 1 is

dfu + gv > 0,

therefore

d(b− a) > (a + b)3.

Finally, the condition (1.52) of Theorem 1 is

(dfu + gv)2 − 4d(fugv − fvgu) > 0,

Determining Turing Spaces 21

thus [d(b− a)− (a + b)3

]2> 4d(a + b)4.

In summary, we obtain the following conditions for the Schnakenberg ki-netics under which a range of Turing patterns form:

0 < b− a < (a + b)3, (2.1)

(a + b)2 > 0, (2.2)

d(b− a) > (a + b)3, (2.3)[d(b− a)− (a + b)3

]2> 4d(a + b)4. (2.4)

In this way, we have determined the Turing space for the Schnakenbergkinetics. The homogeneous equilibrium solution of

ut = ∆u + a− u + u2v, (2.5)

vt = d∆v + b− u2v (2.6)

will grow unstable under the slightest spatial perturbations where the con-ditions (2.1)–(2.4) hold.

For example, consider the eigenproblem (1.37) and choose the planardomain Ω = (0,H)× (0, L) for some positive constants H,L. Using separa-tion of variables, the general solution is

w(x, y) =∞∑

n=1

∞∑m=1

An cos(

nπx

L

)Bm cos

(mπy

H

). (2.7)

Consider equation (2.7) with equation (1.36). The entire solution to thespatially patterned partial differential equation becomes:

w(x, y) =∞∑

n=1

∞∑m=1

eλ(k2)tAn cos(

nπx

L

)Bm cos

(mπy

H

). (2.8)

where

k2 = π2

(n2

L2+

m2

H2

)m,n ∈ Z.

22 Turing Spaces

and each k lives in the parameter

γP (a, b, d) = k21 < k2 = π2

(n2

L2+

m2

H2

)< k2

2 = γQ(a, b, d). (2.9)

Using the inequalities (1.45)–(1.48) with the k’s in the parameter (2.9),we get for k1,

P =dfu + gv −

√(dfu + gv)

2 − 4d|A|2d

,

which with the Schnakenberg kinetics (1.20)–(1.21) is

P =d(b− a)− (a + b)3 −

√[d(b− a)− (a + b)3]2 − 4d(a + b)4

2d(a + b), (2.10)

and for k2,

Q =d(b− a)− (a + b)3 +

√[d(b− a)− (a + b)3]2 − 4d(a + b)4

2d(a + b). (2.11)

Now, thinking of the parameters (2.10)–(2.11) in terms of the wavelengthω = 2π

k , the wavelengths are bounded by unstable modes with an upperand lower bound ω1 and ω2, which we now know are

ω1 =(

2π

k1

)2

=4π2

γP (a, b, d)> ω2 > ω2

2 =4π2

γQ(a, b, d). (2.12)

The parameters in (2.12) explain what sort of role γ plays in the reaction-diffusion system (1.26)–(1.27). With γ very small, the parameters (2.10)–(2.11) tells us that the range of wavenumbers and hence Turing patternswill not form (i.e., there is no unstable region in the Turing space). On theother hand, with γ sufficiently large, this ensures that the system (1.26)–(1.27) is spatially unstable and therefore patterns will form.

In summary, the best way to think of the solution is in terms of oscillat-ing waves across a plane. Depending on values of m and n, at each unstablesolution waves will form across the Laplacian, with the amount of wavescorresponding to values of m and n. So more waves correspond to largernumbers for m and n, meaning a denser plane of patterns viewed from thetop.

Determining Turing Spaces 23

Figure 2.1: Some two-dimensional spatial pattern simulations plotted withcliff plots and three-dimensional views. White regions are when u > u0, theuniform steady-state. Three-dimensional plots are included on the right forreference.

24 Turing Spaces

Figure 2.2: In a hypothetical apartment in downtown San Francisco, a seriesof earthquakes disturb a glass of water resting on a table. When viewedorthogonally, the quantity of waves would increase as the magnitude ofthe earthquakes increase.

Numerical Methods for Solving Turing Spaces 25

Figure 2.3: Slices of Turing space for the Schnakenberg (1.20)–(1.21), Giererand Meinhardt (1.22)–(1.23), and Thomas (1.24)–(1.25) kinetics, respec-tively.

To drive home this argument, consider the following analogy: think ofthe two-dimensional domain as a glass of water sitting on a surface, say adining room table in a second floor apartment in downtown San Francisco.If a minor earthquake vibrates the table surface, the water surface in theglass becomes perturbed, if only slightly. A few waves stream across thesurface when viewed from the top, but the overall number is quite small.Now say a strong earthquake hits the building, vibrating the surface to thepoint of a fissure on the tabletop. As the earthquake occurs, the numberof waves would in this case be numerous, much more than the first. Thisanalogy, along with sketches of the overhead view of the glass in each case,are illustrated in Figure 2.2.

2.2 Numerical Methods for Solving Turing Spaces

Finding the equilibrium points (u0, v0) of the system (1.26)–(1.27) withSchnakenberg kinetics (1.20)–(1.21) in Section 2.1 involved relatively simplealgebra and clean steps. Fortunately, it is also possible to similarly find theequilibrium points of the system (1.26)–(1.27) with Gierer and Meinhardtkinetics (1.22)–(1.23). Eventually, however, one can see that the methodused in Section 2.1 grows complicated with mechanisms such as the Thomaskinetics (1.24)–(1.25). Solving numerically is a more practical way to findthe Turing Space with different kinetics. The algebra grows complex, butfor any kinetics, it is straightforward to program a computer to scan throughthe parameters and determine if the Turing conditions hold or not.

Using these simulations, we could see how different kinetics affectedthe Turing spaces. To start, from Figure 2.3, notice that the activator-substrate

26 Turing Spaces

kinetics (1.20)–(1.21) has an inverse curve-shape in contrast to that of theactivator-inhibitor kinetics (1.22)–(1.23). On the other hand, the substrate-inhibitor kinetics (1.24)–(1.25) stands alone as looking like a cone with anexponential profile. Note that all the simulations in Figure 2.3 are slices ofa three-dimensional object in (a, b, d)-Turing space.

With the parameter GUI [6], we found that the most useful range ofpatterns live when we set a = .1, b = .9, and d = 10, with γ rangingfrom 400 to 2400 with kinetics (1.20)–(1.21) and (1.22)–(1.23). For kineticssuch as (1.24)–(1.25), since there are more positive parameters to consider,and patterns do not form when a = .1, b = .9 and γ ranges in the 400-1200 range, our approach was to similarly choose values at the tip of wherepatterns form and in doing so found that this generated a wider range ofresults when looking at different curvatures. The results are explored inChapter 3.

Chapter 3

Curvature and GeometryEffects on Reaction-DiffusionMechanisms

3.1 Curvature on Pattern Formation

Since Turing’s pioneering paper in 1952 [21], many applications have beensuggested for reaction-diffusion mechanisms in biology. The original arti-cle does not take into account changes in geometry, yet the roles that patternformation plays in biology are so versatile, demand to study this topic ingreater detail has surfaced in recent years. In Harrison et al. [5] the authorsstudy the branching of different algae using the Gierer-Meinhardt modelon a torus (see Figure 3.6). Harrison et al. particularly examine whorl for-mation in Acetabularia, morphogenesis of branching one-celled chlorophytealgae, and branchings in Micrasterias. In Liaw et al. [9], the authors modelspot formation on the shells of lady beetles using a hemisphere. The rest ofthis thesis considers the topic of the effects of different kinetics on differentsurfaces, our own findings, and poses further questions for study.

Here we are going to introduce our approach to modeling geometryeffects. On a planar domain, we have already determined the scaling ef-fects for the Schnakenberg kinetics (1.20)–(1.21) and how to obtain the Tur-ing space for different mechanisms. However, one question that arises iswhether it is possible to study patterns on different surfaces. Our approachis based on Plaza et al. [16], in which the authors map geometric surfacesto a flat domain. Accordingly, the partial differential equation changes toaccount for different curvatures.

28 Curvature and Geometry Effects on Reaction-Diffusion Mechanisms

Assume there is a surface parameterized by (r, s) modeling the shapeand size of the geometry domain; i.e., the surface in R3 is defined by some(X(r, s), Y (r, s), Z(r, s)). Using a change of variables we study the system(1.26)–(1.27) by parameterizing the domain from regular xyz-space to rs-space. We then map this solution back onto the (X(r, s), Y (r, s), Z(r, s))surface to see the resulting pattern. With this mapping, the nondimensionalmodel (1.26)–(1.27) takes the form:

ut = ∆∗u + γ∗f(u, v), (3.1)vt = d∆∗v + γ∗g(u, v), (3.2)

where ∆∗ and γ∗ are a change of variables for ∆ and γ. For example, aplanar region (0, L)× (0,H) is parameterized by X(r, s)

Y (r, s)Z(r, s)

=

LrHs0

where 0 < r < 1 and 0 < s < 1, with γ∗ = γ and ∆∗ = ∆. When thers-surface wraps around the xyz-coordinates in normal space, the bound-ary conditions may be determined by the geometry. So to complete theproblem, we simply have to appropriate boundary conditions, such as pe-riodic in the case of a sphere. With different geometries, ∆∗ changes be-tween different parameterizations, such as cylindrical or spherical coordi-nates. Appendix A lists the full parameterizations we used for different(X(r, s), Y (r, s), Z(r, s)) surfaces in this thesis, but as a simple example, wewill parameterize a cone and show how our approach differs from paststudies.

3.1.1 Examples: Cones and Cylinders

A cone is parameterized by X(r, s)Y (r, s)Z(r, s)

=

spq cos (2πr)spq sin (2πr)

qs

(3.3)

where p and q are constants. In Figure 3.1 we chose p = .1(

1−4.95

)to taper

off the cone as thin as possible at the tip and q = 1. Compared to the tallercone in Figure 3.2, we took p = .1

(1−25

)and q = 6.

Curvature on Pattern Formation 29

Because of the geometry of the cone on an xyz-axis, we take cylindricalcoordinates instead of the usual cartesian. In cylindrical coordinates ∆ hasthe form:

∆∗ =∂2

∂r2+

1r

(∂2

∂θ2

)+

∂2

∂z2for (r, θ, z).

For a fixed cylinder, r = R is constant, meaning

∂u

∂r=

∂2u

∂r2= 0,

thus,

∆∗ =1

R2

(∂2

∂θ2

)+

∂2

∂z2.

Therefore, the partial differential equation system (3.1)–(3.2) on a cylinderbecomes:

ut =

(1

R2

(∂2

∂θ2

)+

∂2

∂z2

)u + γ∗f(u, v), (3.4)

vt = d

(1

R2

(∂2

∂θ2

)+

∂2

∂z2

)v + γ∗g(u, v). (3.5)

For a cone, we get the full

ut =

(∂2

∂r2+

1r

(∂2

∂θ2

)+

∂2

∂z2

)u + γ∗f(u, v), (3.6)

vt = d

(∂2

∂r2+

1r

(∂2

∂θ2

)+

∂2

∂z2

)v + γ∗g(u, v). (3.7)

The equivalent eigenproblem (1.37) corresponds to

∆∗ ~w + k2 ~w = 0 for ~w(R, θ, z),

with zero flux boundary conditions at z = 0 and z = s, and periodic bound-ary conditions for θ = 0 and θ = 2π.


The equivalent solution to (2.7) becomes

∞∑n

∞∑m

eλ(k2)t cos (nθ) cos(

mπz

s

),

where

k2 =n2

R2+

m2π2

s2

with the equivalent Turing space parameters to (1.45)–(1.48), i.e.,

γ∗P = k21 < k2 < n2

R2 + m2π2

s2 < k22 = γ∗Q. (3.8)

Murray [15] simulates the cone dynamics by comparing the behaviorfor cylinders with different fixed radius R. In contrast, using the parame-terization of the cone which will be defined by the system (3.1)–(3.2) withthe parameterization (3.3), we study the system on the actual cone.

Following Plaza et al. [16], computing

h22 = |Xr|2 ,

= |(pqs (−2π sin(2πr)) , pqs (2π cos(2πr)) , 0)|2 ,

= 4p2q2s2π2,

yields

h2 = 2πpqs, (3.9)

and by a similar calculation,

h21 = q2(1 + p2). (3.10)

The system (3.1)–(3.2) with the parameterization (3.3) becomes:

ut =1

h1h2

[(h2

h1us

)s+(

h1

h2us

)r

]+ γ∗f(u, v), (3.11)

vt =1

h1h2

[(h2

h1vs

)s+(

h1

h2vs

)r

]+ γ∗g(u, v), (3.12)

where h1 and h2 are defined in equations (3.9)–(3.10).Heuristically, Murray [14] used piecewise tapering cylinders to show

that as the size of the radius increases, stripes will evolve into spots. In

Curvature on Pattern Formation 31

Figure 3.1: Pattern on a cone and parameterized rs-domain using Thomaskinetics (1.24)–(1.25). Here we took γ = 1200. The tapering of the cone alsoshows that surface area is a factor in determining spots or stripes.

Figure 3.2: Simulations of tapering cylinders using Thomas kinetics (1.24)–(1.25). Here γ = .5, 3, 19, respectively. This also shows that as the scalefactor γ increases, the likelihood of getting spots increases.


Figure 3.3: The same approximate surface area of the cone in Figure 3.1simulated on different-sized radii and heights for a cylinder.

contrast, our approach is to solve the full system on the cone using theparameterization discussed in the introduction 3.1.

Finally, by simple calculation of a tapering cylinder and cone, we getthat the approximate surface area for the cone in Figure 3.1 is .05π whereasthose in 3.2 are 6.39π. This computation offers insight to how surface areaimpacts resulting patterns as well.

3.1.2 Numerically Solved Patterns on a Cylinder of Same SurfaceArea

We now experiment on a few cylindrical shapes that have the same surfacearea as the cone in Figure 3.1. Creating a cylinder with the same (bottom)radius as the cone in Figure 3.1 causes an adjusted height of about onethird the height of the cone. This results in vertical stripe formation. Weshould expect to get spots when we adjust the cylinder radius and heightto some compromise of the two end shapes in Figure 3.3. Finally, adjustingthe height of the cylinder causes a small radius when the surface areas arethe same, meaning patterns develop horizontally but not vertically. Noticethat in all four cases in Figures 3.1 and 3.3 we get about the same amount ofm’s and n’s, which verifies that both surface area and curvature affect thetypes of patterns that will form.

3.1.3 Biological Applications

As an interesting biological interlude, consider the implications of the conesimulations in this section. In mammals, hair pigmentation is due to cellsknown as melanocytes, which are evenly distributed about the derma.

Pattern Formation on Tori and Spheres 33

Figure 3.4: An illustration of the activator-substrate system (1.20)–(1.21) ona leopard reposing. It has been suggested that reaction-diffusion mecha-nisms can be modeled for spot formation on leopards.

Whether or not these melanocytes produce melanin–what causes hair pig-mentation–is believed to be caused by some undiscovered chemicals laiddown during morphogenesis [8]. The cone shapes in Figures 3.1–3.2 canbe viewed as models for hair pigmentation on, for example, a feline tail ormammalian leg. Indeed, if we look at tapered tails in certain spotted cats,we see spots until just at the tip of the tail, where there are stripes instead.Pattern formation has also been studied with regards to animal hides, suchas snakes or fish.

3.2 Pattern Formation on Tori and Spheres

3.2.1 Torus

The cylinder in Section 3.1.1 was useful for showing when spots and stripesare likely to form. As a second example for this argument, we can look atthe torus viewed from overhead.

For the torus, we are interested in how its curvature compares to thatof the sphere and cone, and also whether there are other ways to view howspots and stripes form with regards to area. Figure 3.5 shows that depend-ing on the width of the ring, we again see this evolution from spots andstripes. Since the torus wraps twice in rs-space, we take periodic condi-tions on each boundary.


Figure 3.5: Schnakenberg kinetics (1.20)–(1.21) on three different-sized torirings. Here we can see how area affects the formation of spots and stripes–in the thickest torus ring, spots form, whereas the thinnest torus ring onlyhas enough surface area for stripes to form.

Figure 3.6: Torus simulations for γ = 400, 800, 1200, respectively withSchnakenberg kinetics (1.20)–(1.21) above and Gierer and Meinhardt kinet-ics (1.22)–(1.23) below.

Note that for the torus, the curvature depends on the ”inside” and ”out-side” of the doughnut-shape. Thus, the directions that the patterns developare going to be different in comparison to a more regularly-curved surface,such as the cone or sphere. In fact, notice the directions that the splittingspots of the torus takes for the Schnakenberg kinetics in Figure 3.6. Ratherthan splitting smoothly, which is the case with the sphere, the torus tendsto shift in directions depending on the location on the shape.

3.2.2 Sphere

The sphere, on the other hand, has a more smooth, constant curvaturethroughout the surface. This allows the patterns to distribute evenly in

Pattern Formation on Tori and Spheres 35

Figure 3.7: A Wide Range of Sphere Simulations. Here we took Schnaken-berg (1.20)–(1.21), Gierer and Meinhardt (1.22)–(1.23), and Thomas (1.24)–(1.25) kinetics in each column, respectively, with γ = 400 and γ = 800 ineach different row.

comparison to the cone, torus, or even cylinder. Thus, aside from the plane,it is easier to see how different kinetics affect the end result on this shape.Figure 3.7 shows some of these effects.

3.2.3 Role of Curvature

In comparing the surfaces between the sphere, plane, and torus, it was es-pecially provocative to view the Schnakenberg simulation for γT = 1200 onthe torus and γS = 800 on the sphere. The surface area of this torus is ap-proximately 67% of the sphere; whereas γS = .67γT . To see if a relationshipmight be present, we looked at simulations of planes with the same surfacearea. Figure 3.8 shows the results.

Figure 3.8 is illustrative of how curvature affects the nondimensional-ized reaction-diffusion system (1.26)–(1.27). With regards to the torus andits corresponding plane, the curvature on the torus almost causes the split-ting of the spots to reverse in location when viewed in rs-space. The sphere,on the other hand, wraps as the plane, most likely because of its constantcurvature.


Figure 3.8: Comparison of the sphere and torus with proportional surfacearea sizes, viewed in xyz-space and rs-space. Simulations of planes withthe same surface area and corresponding γ are also viewed below.

Experimenting With Pattern Formation on Greater Surfaces 37

Figure 3.9: The Gierer and Meinhardt kinetics with a logarithmically-profiled cone, for γ = 2400, 3600, 4000.

3.3 Experimenting With Pattern Formation on GreaterSurfaces

We conclude this chapter with examples of more exotic surfaces. Althoughunconventional, these shapes offer further insight into some deeper realmsof reaction-diffusion systems.

3.3.1 Logarithmic and Exponentially-Profiled Shapes

Some of the cone simulations have limitations when we simply take a lin-ear profile. To further explore the cone we attempted to take exponentialand logarithmic profiles in order gain further insight into how area affectspattern formation. Some of the results obtained are in Figures 3.9–3.10.Perhaps the most intriguing shape is the logarithmically-profiled cloverbecause of its almost asymmetrical shape. Although asymmetrical objectshave not been explored in this context, it might be interesting for furtherstudy to see what sort of patterns form on these types of surfaces.

3.3.2 Clover

Because of its varying curvatures, the clover offered a wide range of pat-terns and thus real insight into how surface area affects the pattern. Inparticular, note that doubling γ in the simulation with the Schnakenbergkinetics (1.20)–(1.21) in Figure 3.11 caused a dramatic variation. We canalso see how with varying curvatures we can get in the same shape a vari-ety of patterns, as shown with equation (1.22)–(1.23) when γ = 800.


Figure 3.10: Schnakenberg kinetics simulated on a logarithmically-profiledclover, and the Thomas kinetics simulated on an exponentially-profiledcone.

Figure 3.11: A wide range of clover simulations. Here, we took kinetics(1.20)–(1.21) through (1.24)–(1.25) in each column, respectively, with γ =400 and γ = 800 in each different row.

Chapter 4

Conclusion

4.1 Concluding Remarks

We have tried to give an introduction to linear stability analysis, Turingspaces, and in particular how geometric surfaces impact the type of pat-terns that will form. Very distinct biological systems can be simulated byassumption of similar mechanisms, which is why we used a wide range ofkinetics and also shapes.

There are several areas we hope to see this project develop for futurestudy. To start, all of the shapes chosen were symmetrical when viewedeither from overhead or a side, but in many ways these shapes only modelthe simplest of real-world applications. Turing’s original question was howsymmetry in nature becomes broken from embryogenesis–why the humanhand, for example, looks completely different when viewed at differentangles. To answer the crucial question that started this study, it might beinteresting to examine asymmetrical shapes, such as a skewed pyramid ora clover with uneven leaf sizes.

Pattern formation on growing domains is another application, one thatis larger in scope than those previously mentioned. This year, J. Gjorgjieva[4] modeled pattern formation on a growing sphere and is using her resultsto show when splitting of spots occurs. Studying growing models–and inparticular, simulating asymmetrical models–would offer insight into vary-ing spots and stripes on animal hides and the like.

One question for biologists is how to refine models such as the Schnaken-berg kinetics (1.20)–(1.21) to get patterns providing a closer model to bio-logical patterns. Another is that the patterns are modeled on very smallsurfaces. To model growth on larger surfaces, cells have to make use of the

40 Conclusion

larger area by activation of certain genes. Koch and Meinhardt [8] attemptto answer both questions by introducing a biological switch to model acti-vating these genes. A simple switch system is given by

∂y

∂t= η

y2

1 + κy2− µy + σext, (4.1)

where η, κ, and µ are constants, and σext describes the external signal. With-out σext y has two stable steady-states, one evaluated at

y0 = 0

and the second at

y0 =η −

√η2 − 4κµ2

2κµ.

Between these two stable steady-states is an unstable steady-state

y0 =η +

√η2 − 4κµ2

2κµ.

So if σext exceeds the threshold in between its stable modes, the systemswitches from a low state to a heavy state.

One system that Koch and Meinhardt proposed using equation (4.1) is

ut = Du∆u + ηu

[u2v

1 + κuu2− u

], (4.2)

vt = Du∆v +σv

1 + κvy− ηvu

2v

1 + κvv2− µvv, (4.3)

yt = ηyy2

1 + κyy2− µyy + σext

y v, (4.4)

which is an activator-inhibitor model with the modified switching-system.The results obtained were more polygonal shapes than those obtained inFigure 1.4. This switching system is useful for modeling more polygonalcoat patterns found on giraffes or leopards. The system (4.2)–(4.4) can evenbe used to model the development of wings on dragonflies or the eye of afruitfly.

Although many examples presented throughout this thesis have beenin biology and chemistry, pattern formation is not reserved for living sys-tems. It has been used to model high sand dunes and sharply contoured

Concluding Remarks 41

rivers, although wind and erosion has presumably redistributed the initialpatterns that formed [13]. Lightning, crystals, and galaxies have similarlybeen used, which Meinhardt discussed in 1982 [11].

These models are just a few of many applications in pattern formation,which is a vast and current topic in mathematical biology. Hopefully, thisthesis introduced the reader to a wide range of kinetics and inspired a fewinteresting future studies. We hope that the reader is at least convincedthat the treatment of pattern formation is feasible and provides essentialinsights into the beautiful processes of nature.

Appendix A

Parameterizations Used ForDifferent Geometries

This appendix includes a short listing of the different geometries we usedand how we parameterized them throughout the thesis. We list each shapealphabetically for convenience. The values we used for the Schnakenbergkinetics (1.20)–(1.21) and Gierer and Meinhardt kinetics (1.22)–(1.23) werea = .1, b = .9, and d = 10, and for Thomas kinetics (1.24)–(1.25) we choseK = .01, α = 1.5, ρ = 18.5, a = 92, b = 64, d = 10. Since γ was a scale factor,we always listed γ in the figure captions or text.

A.1 Clover

The parameterization of the clover in Section 3.11 is given by X(r, s)Y (r, s)Z(r, s)

=

p+cos(q2πr)m cos(2πr)

p+cos(q2πr)m cos(2πr)

s

. (A.1)

Here p, q, and m are constants corresponding to the radius of the clover,the number of ”leaves” we want, and the cylindrical ratio, respectively. Wechose p = 2, q = 4,m = 3.

To add exponential profiling to the clover, one merely needs to multiplythe lefthand side of equation (A.1) by an exponential term, namely exp(s).

44 Parameterizations Used For Different Geometries

A.2 Cone

We already saw the parameterization of the cone in Section 3.1.1 with equa-tion (3.3), but we restate it here: X(r, s)

Y (r, s)Z(r, s)

=

spq cos (2πr)spq sin (2πr)

qs

(A.2)

where p and q are constants. In Figure 3.1 we chose p = .1(

1−4.95

)to taper

off the cone as thin as possible at the tip and q = 1. Compared to the tallercone in Figure 3.2, we took p = .1

(1−25

)and q = 6.

To add exponential profiling to the cone, one merely needs to multi-ply the righthand side of equation (A.1) by an exponential term, namelyexp(s) : X(r, s)

Y (r, s)Z(r, s)

=

exp(s)sp cos (2πr)exp(s)sp cos (2πr)

exp(s)qs

,

and for the logarithmic profile X(r, s)Y (r, s)Z(r, s)

=

1

exp(s)sp cos (2πr)1

exp(s)sp cos (2πr)1

exp(s)

.

In both cases, we took p = .1(

1−4.95

).

A.2.1 Cylinder

In Section 3.1.2, we parameterized each cylinder by X(r, s)Y (r, s)Z(r, s)

=

p cos (2πs)p sin (2πr)

s

.

Plane 45

To match the surface area with that of the cone, we solved for

surface area = 2πp2 + 2πps

.16 ≈ 6.28p

(p + s)

⇒ p ≈ .025p

− p, p > 0,

and then adjusted p and s accordingly. In Figure 3.3 the numbers for p arep = .023, .05, .1, so s = 1, .35, .15.

A.3 Plane

Although we listed a parameterization for the plan in Section 3.1, we restateit here: X(r, s)

Y (r, s)Z(r, s)

=

LrHs0

.

To adjust the width and height, simply multiply r and s by H and L con-stants. Thus to adjust surface area, simply choose area = width× height =HL. In Figure 3.8 we have 3×4 and 4×5 planes to correspond with a torusof surface area ≈ 19.73 and a sphere of surface area ≈ 12.5. The rest ofthe planes were simple 1× 1’s.

A.4 Sphere

In order to prevent NANs during the numerical simulation, we subtractedby two small values to create a fat ”barrel” instead of a sphere: X(r, s)

Y (r, s)Z(r, s)

=

cos (2πr − ε1) · sin (2πs− ε2)cos (2πr − ε1) · sin (2πs− ε2)

cos (πs)

. (A.3)

In our simulations, we chose ε1 = ε2 = .00001. It is also possible to adjustthe sphere size by multiplying the righthand side of the parameterization(A.3) by a positive constant.

46 Parameterizations Used For Different Geometries

A.5 Torus

The normal torus parameter is given by X(r, s)Y (r, s)Z(r, s)

=

(p cos (2πs) + q) cos (2πr)(p cos (2πs) + q) sin (2πr)

p sin (2πs)

where p and q are constants. To adjust the width of the ring, we simplysized down p. In Figure (3.5) we took p = .5, .25, .1, respectively. and fixedq = 1.

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pattern formation in partial differential equations

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