Question 1. A non-dimensional model describing the early stages of Dictyostelium discoideum mound formation is given by

u_t = u_xx - Χ(uv_x)_x + ru(1 - u)

v_t = dv_xx + 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 (u_s, v_s) for which u_s > 0 and v_s > 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 (u_s, v_s) for which u_s > 0 and v_s > 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 (U_k(s, t), V_k(x, t)) = (U'_k(t), V'_k(t))e^ikx?

(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 (U_k(x, t), V_k(x, t)) = (U'_k(t), V'_k(t))e^ikx.