2 = 0 ~ 9X







The dependence of the results on the initial conditions is discussed below.

Here we investigate the process of tumor growth and progression in relation to the degree of inhibition and promotion. First we will present a mathematical analysis and then biological insights and results of simulations.

9.3.1 Turing stability analysis

Again, we are going to assume that promoters adjust instantaneously to their equilibrium level. By replacing P with C defined by P ~ jrC, we

Op can rewrite equation (9.7) as i-b^ttttH-^ <->

This equation together with equation (9.9) gives a Turing model.

Let us go back to the system of ODEs, (9.4-9.5), and assume that solution (C, I) is a stable equilibrium. Of course, this solution also satisfies the system of PDEs, (9.10,9.9). Let us consider a wave-like deviation from this spatially uniform solution:

Here, the amplitudes of the perturbation, A and B, are small compared to the amplitude of the spatially uniform solution, and we assume an infinitely large space. The equation for the new eigenvalue, A, is

. ^ f a-Dcuj2 -A -3 \ , det( ; -bj-D^- a)=°' (9-n>

where we define

Cra„ „ „ C2rav a =-^-- >0, p=-r—^-— > 0.

Equation (9.11) can be written as

A2 + \{bi -a + (Dc + Di)uj2) + atf - (b7 + J9,w2)(a - Dcto2) = 0. (9.12)

This is the dispersion relation which connects the growth-rate, A, with the spatial frequency of the perturbation, w. The stability conditions are now given by bI-a+{Dc + DI)w2 >0, (9.13)

Note that the stability conditions for solution (C, I) of the system of ODEs, (9.4-9.5), are obtained automatically from the conditions above by setting w = 0:

Inequality (9.13) is always satisfied because of inequality (9.15). Let us derive conditions under which the spatially uniform solution is unstable. This requires that condition (9.14) is reversed. This can be expressed as follows:

where we denoted for simplicity,

This is a fourth order polynomial, symmetrical with respect to the line lo = 0, with a positive leading term. The points, ±|w|, satisfying

correspond to the two minima of the left hand side of inequality (9.17). Let us call these values of u>, ±wc. The condition F(ujc) < 0 defines that the uniform solution (C, I) is unstable.

Let us plot the function F(u) for different values of aj, see Figure 9.5. For small values of aj, F(lj) is strictly positive, and the spatially uniform

0 0

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