Reference no: EM131433255
Question 1. A non-dimensional model describing the early stages of Dictyostelium discoideum mound formation is given by
ut = uxx - Χ(uvx)x + ru(1 - u)
vt = dvxx + au - bv,
where u(x, t) is the density of the Dictyostehum cells, v(x, t) describes the con-centration of the chemical chemoattractant (cAMP), and the model parameters Χ, r, d, a, b are all positive constants. Show that there is only one spatially-uniform steady state (us, vs) for which us > 0 and vs > 0. By carrying out a linear stability analysis at this steady state, establish that aggregation patterns can only develop if X > Χ*, where Χ* is a constant depending on the other parameters. Give a biological interpretation of this result.
Question 2. The following model describes chemorepulsion, the movement of a cell population down the gradient of a chemical concentration:
∂u/∂t = d.∂2u/∂x2 + α.∂/∂x(u∂u/∂x) + f(u)
∂u/∂t = ∂2u/∂x2 + g(u, v).
The model parameters a and d are both positive constants.
(a) Consider f (u) = u(1 - u) and g(u, v) = a - v, where a is a positive constant. Determine the spatially-uniform steady state (us, vs) for which us > 0 and vs > 0. Show that this steady state is stable to homogeneous perturbations. Is it possible for the steady state to be unstable to an inhomogeneous perturbation of the form (Uk(s, t), Vk(x, t)) = (U'k(t), V'k(t))eikx?
(b) Now consider f(u) = u(1 - u) and g(u, v) = a - uv, where a is a positive constant. Again, determine the positive, spatially-uniform steady state and show that it is stable to homogeneous perturbations. Calculate a condition on the parameters d, a and a such that the steady state is unstable to an inhomogeneous perturbation of the form (Uk(x, t), Vk(x, t)) = (U'k(t), V'k(t))eikx.
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