Let us now consider the general system (12.5.4) defined on a bounded domain. In the case of one-space dimension we consider the domain to be the interval v To complete the formulation of the problem we require boundary as well as initial conditions. That is, in addition to (12.5.4) we assume c1(x, t), c2(x, t) satisfy
(12.6.1) where f1(x) and f2(x) are given concentrations. Also we impose the “no flux” boundary conditions
(12.6.2)
The reason for choosing zero flux boundary conditions is that we are primarily interested in self organisation of patterns. No flux boundary conditions mean that there is to be no external input of morphogens.
To develop an analysis of Turing driven instabilities in this case we proceed in precisely the same way as we did above for an infinite domain. The crucial point is that the perturbations d1 and d2, which solve the system (12.5.11), must now satisfy the conditions
A little thought shows that d1, d2 must be of the form (12.5.13) where (12.6.4)
for x=0, l. Consequently, rather than being arbitrary, the parameter k must take the discrete values kn, n=1, …, where
i.e.,
(12.6.5)
This observation is crucial; it is fundamental to deciding which patterns are selected. Proceeding precisely as before we find that the critical wave number kc is given by
(12.6.6)
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which may or may not be satisfied for a given l. However if one is allowed to vary l then it is possible to choose a mode characterised by n so that (12.6.6) is satisfied. In other words a critical pattern of a certain form is dependent on the size of the domain.
More generally we see that spatial patterns evolve if we can select integer values of n so that . This can only happen if we can choose n so that
(12.6.7)
where , i=1, 2…, satisfy That is,
(12.6.8) where A=(a11a22−a12a21)>0. The number of integers n for which (12.6.8) is satisfied determines the modes of pattern selection. To see this suppose the domain size l is such that (12.6.8) is satisfied only for n=1. The only unstable mode is cos and morphogen concentration
(12.6.9) where is the positive root of (12.5.6). This unstable mode is the dominant one which emerges as t increases. If we say that black corresponds to a concentration above the steady state c1,0 and white corresponds to a concentration below c1,0 then we have the pattern shown in Figure 12.6.1.
FIGURE 12.6.1: Morphogen pattern for n=1.
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FIGURE 12.6.2: Morphogen pattern for n=2.
Similarly if n=2 is the only value of n for which the inequality (12.6.8) holds then (12.6.9) becomes (12.6.10) leading to the pattern shown in Figure 12.6.2.
Which mode or combination of modes and hence patterns are selected depends on initial conditions (12.6.1). Now consider the two-dimensional domain Ω defined by 0≤x≤l, 0≤ y≤h, with rectangular boundary ∂Ω on which no-flux boundary conditions are imposed.
Once again the theory developed above in the one-dimensional case is followed with only minor modifications. Most importantly we seek solutions of the linearised problem (12.5.11) of the form
(12.6.11) where the wave numbers k1 and k2 are chosen so that d1 and d2 satisfy the boundary conditions:
(12.6.12) i=1, 2.
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By the method of separation of variables, or otherwise, we find that
(12.6.13) for integers m, n=1, 2….v
Now proceed precisely as before to see that if we define
(12.6.14)
then we again arrive at equations (12.5.16) and (12.5.22) with k2 simply replaced by K2. The critical wave number is given by
(12.6.15) and modes characterised by m and n exist if they satisfy the inequalities
(12.6.16) To illustrate possible modes, suppose the domain size is sufficiently large so that (12.6.16) holds for m=3, n=2. Then the pattern shown in Figure 12.6.3 is possible where the shaded areas indicate regions in which the morphogen concentration is above the steady state.
FIGURE 12.6.3: Morphogen pattern for m=3, n=2.
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The fundamental assumption of pattern formation via Turing diffusion driven instabilities is that the linearly unstable modes that grow exponentially in time will eventually be bounded by the non-linear kinetic terms in (12.5.4). This is indeed the case. To prove mathematically that this happens one has to show that in the positive quadrant of morphogen space there is a compact region about the uniform steady state (c1,0, c2,0) to which c1 and c2 are always confined. Such a region is called a confined set or a contracting set. The determination of such sets is often difficult and is beyond the scope of this book. The interested reader is encouraged to consult the literature cited in the notes at the end of this chapter.
12.7 Notes
Diffusion through membranes
Most of the material here is based on the results of S.I.Rubinow and we recommend his book Introduction to Mathematical Biology, John Wiley & Sons, New York, 1975 for futher developments and a host of biological applications. See also J.D.Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
Global behaviour of nerve impulse transmissions
For a fuller background to the material here, we refer to the survey article of S.P.Hastings, Some
mathematical problems from neurobiology, Am. Math. Monthly, 82, 881–895, 1975 and the detailed results in J.Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, Berlin, 1983.
Global behaviour in chemical reactions
An excellent treatment of the Belousov-Zhabotinskii reaction and many other diffusion problems in biology are to be found in J.D.Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
Diffusion driven instability and pattern formation; Finite pattern forming domains
The interested reader is recommended to read the fundamental papers: A.M.Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Lond, B237, 37–72, 1952; and L.Wolpert, Positional information and the spatial pattern of cellular differentiation, J. Theor. Biol., 25, 1–47, 1969. See also the treatment in J.D.Murray, Mathematical Biology, Springer-Verlag, Berlin, 1993.
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Exercises
12.1A cell of solute concentration c and characteristic length δ is placed in a large bathing solution of solute with fixed concentration c0. If the region 0≤x≤δ represents the cell while the regions x<0, x>δ represent the cell exterior, show that the solution concentration of the cell is given by
where D is the diffusion constant.
12.2In one dimension, let there be a slab of solute of uniform concentration c0 and of thickness 2a. At t=0 Find the concentration c(x, t) for all x and t.
12.3A stationary spherical cell of radius a is metabolising a nutrient that is at uniform concentration c0 in the surrounding medium initially. Assume that the cell instantaneously metabolises any nutrient molecules
that enter it, so that the nutrient concentration at the cell wall is zero at all times. Solve the diffusion equation in spherical polar co-ordinates and find the nutrient concentration c=c(r, t) in the surrounding medium as a function of the radial postion r and the time t.
12.4By considering the energy function
show that the ordinary differential equation
where , cannot have a periodic solution of period T. 12.5Consider the reaction-diffusion system
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defined for , and where u, υ satisfy the following initial and boundary conditions
Define the energy function
and show that
where d=min(d1, d2) and
Here, maxu,υ means the maximum over the solution values of u and υ. Deduce that if m<2π2d then E(t)→0 as t→∞. Interpret this result.
12.6From the general theory of diffusion driven instability and using the notation of Section 12.5 derive the inequalities
where
12.7Consider the reaction-diffusion system
where u and υ represent concentrations of morphogens and α, β and δ are positive constants. Find the non-zero equilibrium point and determine conditions on α and β for a Turing instability to occur.
Calculate the values of δ for which the Turing instability can take place.
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12.8A model capable of diffusion driven instability is
Determine the homogeneous steady state and show that it is stable provided
By linearising the system about the homogeneous steady state show that diffusion instability can occur if 12.9Derive diffusion driven instability and pattern forming properties of the two dimensional reaction-diffusion
system
where a, b are positive parameters and d is a positive diffusion coefficient. The system is defined on the rectangular region 0≤x≤A, 0≤y≤B and the morphogens u and υ satisfy the boundary conditions
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Page 301 Chapter 13
Bifurcation and Chaos
13.1 Bifurcation
Equilibrium points have been considered in Chapter 5. The position of an equilibrium point may depend upon a parameter. For instance, the numbers infected when an epidemic reaches equilibrium might be different for different rates of infection. How the variation of a parameter can affect equilibrium is the topic to be discussed here. Such a parameter will be called a control parameter to distinguish it from any parameters which remain fixed.
We begin with an example to indicate the sort of thing that can happen. Example 13.1.1
Consider the non-linear conservative system in which μ≥0 and The equilibrium points are given by y=0 and solutions of
(13.1.1)
One solution of (13.1.1) is x=0. To see if there are any other solutions we examine the derivative with respect to x of the left-hand side of (13.1.1), namely μ cos x−1. The derivative is always negative if 0≤μ<1. Therefore, for this range of μ, the left-hand side of (13.1.1) decreases steadily as x increases from 0. Since it is zero when x=0 it cannot vanish subsequently and there is no other solution of (13.1.1) for 0≤μ<1.
If μ>1, the derivative is positive at x=0. As x increases the derivative decreases, passes through zero and then stays negative up to x=π; for simplicity we limit x to (−π, π). Therefore, as x increases, the left-hand side of (13.1.1) first increases from 0 to some positive maximum where μ cos x=1 and then decreases steadily to x=π. At x=π it is negative and so there is a zero x=x1≠0 of (13.1.1) when μ>1. By the preceding argument μ cos x1<1. Since (13.1.1) is unaffected by changing the sign of x there is also a solution x=−x1. Thus, we have found one equilibrium point for 0≤μ<1 and three for μ>1.
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To examine the nature of the equilibrium we proceed as in Section 5.3. Near the equilibrium point (0, 0) put x=ξ, y=η with ξ and η small. Then
According to Section 5.4 the equation
should be solved now. If μ<1, the two values of λ are imaginary complex conjugates and the equilibrium point is a centre. When μ>1, the two values of λ are real but of opposite sign; the equilibrium point is a saddle-point.
When μ>1 the equilibrium point (x1, 0) can occur. Near it put x=x1+ξ, y=η to obtain In this case
Since μ cos x1−1 is negative, as shown above, the equilibrium point is a centre. The same is true for the equilibrium point (−x1, 0).
Several features of this example should be noted. The position of the equilibrium point x=0 does not change as the control parameter μ is altered but its character switches as μ passes through 1. It could be regarded as going from a stable regime to an unstable one—termed interchange of stabilities by Poincaré. At the switch extra equilibrium points are born; their position varies with the control parameter but not their character. The changing of the character of an equilibrium point and/or the creation of extra ones by alteration of a control parameter is known as bifurcation. The value of μ where bifurcation occurs may be called a bifurcation
point.
An aid to keeping track of what is going on as the control parameter varies is the bifurcation diagram. This is a plot of the positions of the equilibrium points against the control parameter (see Figure 13.1.1). The letters c and s signify which curves represent centres and saddle-points, respectively. Adding information about λ would render the diagram too complicated; so sometimes there is a separate plot of λ against μ. It can be somewhat awkward when λ is complex. To avoid a three-dimensional picture λ can be plotted on the complex λ-plane with values of μ attached to the points.
On account of its appearance a bifurcation like that of Figure 13.1.1 is sometimes known as a pitchfork bifurcation.
If the equation for in Example 13.1.1 is changed to
(13.1.2)
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FIGURE 13.1.2: Bifurcation diagram for (13.1.2).
there are no equilibrium points for μ<0. For μ>0 there are two and . The first of these
corresponds to a centre whereas the second corresponds to a saddle-point. The resulting bifurcation diagram is shown in Figure 13.1.2.
Bifurcation is not restricted to the non-linear conservative systems that have been studied so far. For example, the system
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FIGURE 13.1.3: Bifurcation from node to focus.
with μ≥0 has an equilibrium point at (0,0). It is a node for μ<3 and a focus for μ>3. The bifurcation diagram is displayed in Figure 13.1.3, the letters n and f indicating node and focus, respectively.
The reader should be warned that often the term bifurcation is employed much more strictly than has been used here. It is kept to denote points where the stability of the system changes. A diagram like Figure 13.1.3 would not be regarded as bifurcation in the strict sense because the stability is unaltered although the
behaviour of the system changes radically from non-oscillatory to oscillatory.
It should be stressed that the identification of the type of equilibrium point has been carried out by the approximation of Section 5.3. Thus, the equations are valid only near the equilibrium point. They represent what is occurring locally. How the local scene fits into the global picture can be very difficult to resolve. Of course, there may be more than one control parameter as in
which has three control parameters μ1, μ2 and μ3. Discussion of bifurcation is generally extremely tricky even for the equilibrium point in which x=0, let alone the determination of other equilibrium points. Pictorial
representation is far from straightforward since three dimensions are occupied already by the control parameters. Accordingly, we shall concentrate on cases in which there is a single control parameter.
will become apparent later. Sometimes it will be convenient to use one term in the rest of this chapter and sometimes the other.
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