Please help me with the following:

a.) Prove this by solving for the equilibrium solutions for the nonlinear system.

b.)Find a linear system that approximates the nonlinear system near (A, 0) by letting (N, P ) = (A + u, v), plugging into the nonlinear system, and assuming that u and v are small enough that nonlinear terms (u2, v2, uv) are negligible.

c.)Show that if you do the same thing for the critical point (1, A ? 1) (that is, let (N, P ) = (1+u, A?1+v)) and simplify, you get the linear system that we used in the prelab, u? = ??(Au + v), v? = (A ? 1)u.

d.)Rewrite the system you got in (c) as a second order equation in v, as you did in the prelab exercise 4a.

e.)Solve the linear system from (c) and the second order equation from (d) and show that you get the same characteristic equation and same solution. (Assume that A > 1 and ?

In lab 3 we consider the nonlinear system N' = 7(A N (1 + P)), P' = P(N 1). (a) In the prelab we stated that the critical points of this system are (N, P) = (A, 0) and (N, P) 2 (LA 1). Prove this by solving for the equilibrium solutions for the nonlinear system. (b) Find a linear system that approximates the nonlinear system near (A, 0) by letting (N, P) = (A + um), plugging into the nonlinear system, and assuming that u and 'U are small enough that nonlinear terms (M, '02, no) are negligible. (c) Show that if you do the same thing for the critical point (1, A 1) (that is, let (N, P) = (1+u, A1+v)) and simplify, you get the linear system that we used in the prelab, u' = 'y(Au + 'U), 11' = (A 1)u. (d) Rewrite the system you got in (c) as a second order equation in 'U, as you did in the prelab exercise 4a. e Solve the linear system from C and the second order e uation from q d and ShOW that you et the same characteristic e uation and same g q solution. (Assume that A > 1 and "y