Linear Stability at Critical Points of the Model

1. Linear Stability at Critical Points of the Model

2. 4 Critical Points • (0,0) • (0,1) • (1,0) • (n1 *,n2 *) • n1 * = (1-alpha2/beta)/ (1-alpha1alpha2) • n2 * = (1 – alpha1beta(1 – alpha2beta/(1- alpha1alpha2)))

3. Linear Stability • We notice that similar to a scalar ODE • dx/dt = Ax ,x(0) = x0 where denotes vector Has solution x(t) = x0 exp(At), where A is the Jacobian matrix

4. Decomposing A • By writing • A = SDS-1 • Exp(At) = exp[(SDS-1)t] • then taylor expanding the following • sum{ (SDS-1 t)n / n! } from 0…inf • we can see that the eigenvalues of A determine the behavior of the solution. • If Eig(A(criticalpt)) = both neg. then the point is stable • If Eig(A(criticalpt)) = both pos. then the point is unstable • If Eig(A(criticalpt)) = pos/ neg. then it is a saddle point

5. Tedious details of Analysis • This needs to be typed in latex • Show all A matrices evaluated at each critical point • Eigen values of each matrix A • Phase plane behavior determined by above. A couple plots for different cases of alphas, betas, etc. would be nice