ru (1-3) G-1). Recall the strong Allee effect from the previous homework. The diffusion equation in this case is ди 224

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ru (1-3) G-1). Recall the strong Allee effect from the previous homework. The diffusion equation in this case is ди 224 D +ru at 0,2 To make the computations a little easier, use the numbers given in the last homework as well: ди au D at Pəx2 +ru(1 – 2u) (u – 1). (a) Let u(x, t) = U(x - c), and V = U'. Derive a 2-dimensional system of ODE's for U and V. (1) Sketch the nullclines and direction arrows in the UV-plane. (e) Find the critical points. From the direction arrows you should see that two are saddles. For the other you'll need to compute the Jacobian. Show from the Jacobian that this is a sink. (d) Compute the minimal wave speed c such that the sink has real eigenvalues. (e) Sketch the general solution in the UV-plane for the above value of c. (1) There are two possible traveling wave solutions. Sketch both of them for t = 0, 1 = 1, t = 2 (s) One traveling wave solution could happen if i < (10) > 5 for all 1. The other if 0 <u(1, 0) < 1 for all 1. Explain the meaning of both. (Note: For general u(,0) with regions where u is above and below 1, the solution u(s, t) will not look like a traveling wave, and could be quite a bit different. One would have to solve for it numerically.)