Show That The A B Turing Parameter Space For Diffusion-driven

where b and d are positive constants. Which is the activator and which the inhibitor? Determine the positive steady states and show, by an examination of the eigenvalues in a linear stability analysis of the diffusionless situation, that the reaction kinetics cannot exhibit oscillatory solutions if b < 1.

Determine the conditions for the steady state to be driven unstable by diffusion. Show that the parameter domain for diffusion-driven instability is given by 0 < b < 1, db > 3 + 2V2 and sketch the (b, d) parameter space in which diffusion-driven instability occurs. Further show that at the bifurcation to such an instability the critical wavenumber kc is given by k2 = (1 + V2)/d.

3. An activator-inhibitor reaction diffusion system with activator inhibition is modelled by u2

Vt = u2 - V + dVxx, where K is a measure of the inhibition and a, b and d are constants. Sketch the null clines for positive b, various K > 0 and positive or negative a.

Show that the (a, b) Turing (parameter) space for diffusion-driven instability is defined parametrically by a = bu0 - (1 + Kul)2

combined with b > 2[u(1 + Kul)]-1 - 1' b > b > 2[u(1 + Ku2)]-2 - 1, b < 2[u(1 + Ku2)]-2 - 2^2[du(1 + Ku0)]-1/2 + 1, 0 0 d where the parameter u0 takes on all values in the range (0, to). Sketch the Turing space for (i) K = 0 and (ii) K = 0 for various d (Murray 1982).

4. Determine the relevant axisymmetric eigenfunctions W and eigenvalues k2 for the circular domain bounded by R defined by

Given that the linearly unstable range of wavenumbers k2 for the reaction diffusion mechanism (2.7) is given by

YL(a, b, d) < k2 < yM(a, b, d), where L and M are defined by (2.38), determine the critical radius Rc of the domain below which no spatial pattern can be generated. For R just greater than Rc sketch the spatial pattern you would expect to evolve.

5. Consider the reaction diffusion mechanism given by u2 2

where y , b and d are positive constants. For the domain 0 < x < 1 with zero flux conditions determine the dispersion relation X(k2) as a function of the wave-numbers k of small spatial perturbations about the uniform steady state. Is it possible with this mechanism to isolate successive modes by judicious variation of the parameters? Is there a bound on the excitable modes as d ^ to with b and y fixed?

6. Suppose fishing is regulated within a zone H km from a country's shore (taken to be a straight line) but outside of this zone overfishing is so excessive that the population is effectively zero. Assume that the fish reproduce logistically, disperse by diffusion and within the zone are harvested with an effort E. Justify the following model for the fish population u(x, t).

Ut = ru( 1 - K) - EU + DUxx, u = 0 on x = H, ux = 0 on x = 0, where r, K, E(< r) and D are positive constants.

If the fish stock is not to collapse show that the fishing zone H must be greater than 2 [D/(r — E)]1/2 km. Briefly discuss any ecological implications.

7. Use the approximation method described in Section 2.7 to determine analytically the critical length L 0 as function of r, q and D such that an outbreak can exist in the spruce budworm population model u\ u2

Determine the maximum population um when L = L 0.

8. Consider the Lotka-Volterra predator-prey system (see Chapter 3, Volume I, Section 3.1) with diffusion given by ut = u(1 — v) + Duxx, Vt = av(u — 1) + Dvxx in the domain 0 < x < 1 with zero flux boundary conditions. By multiplying the first equation by a(u — 1) and the second by (v — 1) show that

Determine the minimum S for all u and v. Show that necessarily a ^ 0 as t ^rn by supposing a 2 tends to a nonzero bound, the consequences of which are not possible. Hence deduce that no spatial patterns can be generated by this model in a finite domain with zero flux boundary conditions.

(This result can also be obtained rigorously, using maximum principles; the detailed analysis is given by Murray (1975).)

9. The amoebae of the slime mould Dictyostelium discoideum, with density n(x, t), secrete a chemical attractant, cyclic-AMP, and spatial aggregations of amoebae start to form. One of the models for this process (and discussed in Section 11.4, Volume I) gives rise to the system of equations, which in their one-dimensional form, are nt = Dnnxx — x(nax )x, at = hn — ka + Daaxx, where a is the attractant concentration and h, k, x and the diffusion coefficients Dn and Da are all positive constants. Nondimensionalise the system.

Consider (i) a finite domain with zero flux boundary conditions and (ii) an infinite domain. Examine the linear stability about the steady state (which intro duces a further parameter here), derive the dispersion relation and discuss the role of the various parameter groupings. Hence obtain the conditions on the parameters and domain size for the mechanism to initiate spatially heterogeneous solutions.

Experimentally the chemotactic parameter x increases during the life cycle of the slime mould. Using x as the bifurcation parameter determine the critical wavelength when the system bifurcates to spatially structured solutions in an infinite domain. In the finite domain situation examine the bifurcating instability as the domain is increased.

Briefly describe the physical processes operating and explain intuitively how spatial aggregation takes place.

10. Consider the dimensionless reaction anisotropic diffusion system du , d2u , d2u

In the absence of diffusion the steady state u = (uo, vo) is stable. By carrying out a linear analysis about the steady state by looking for solutions in the form u — uo a ext+i(kxx+kyy), where kx and ky are the wavenumbers, show that if

H(k2x, k2y) = didk + piklk] + d2dk — Yp2k2x — Yk2p3 + Y2( fugv — fvgu), where pi = did4 + d2d3, p2 = d3 fu + di gv, p3 = d4 fu + d2 gv is such that H < 0for some k^k^ = 0 then the system can be diffusionally unstable to spatial perturbations. The maximum linear growth is given by the values k1 and k2 which give the minimum of H (k^, k2 ). Show that the minimum of H is given by

For a spatial pattern to evolve we need real values of kx, ky which requires, from the last equation, that

-d4 fu + d2 gv -di gv + d3 fu
0 0

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