Math 3260 - Applied Differential Equations Homework #3 due March 31 1. Consider the reaction diffusion system a 2a a2 = Da 2 a + c1 , t x h 2 h h = Dh 2 h + c2 a2 , t x h h (0) = (L) = 0 , x x a a (0) = (L) = 0 , x x where > 0, > 0, c1 > 0 and c2 > 0 are parameters. (a) Find all spatially homogeneous steady-states. There should be one steady-state with a 6= 0 and h 6= 0. We will refer to this equilibrium as (a , h ). (b) Give a condition on the parameter values to ensure the stability of (a , h ) in the absence of diffusion (after setting Da = 0 and Dh = 0). Take = 1, = 2, c1 = c2 = 1 and Dh = 1 for the remainder of the question. (c) Find the critical value of Da , Dac such that for Da > Dac (a , h ) will be stable with respect to diffusive instabilities for all values of L. (d) If we set Da = 0.08, what is the critical value of L = LC such that no pattern forms for L < Lc . 2. The Lotka-Volterra equations with diffusion are given by y 2y = Dy 2 + ry byz , t x z 2z = Dz 2 ez + dyz . t x Where r, b, e and d are positive parameters. Here y(x, t) is the prey population and z(x, t) is the predator population. Show that diffusive instability is not possible for this system. 1