A model for the spruce budworm population u(t) is governed by

du/dt = ru (1-u/q)-u²/(1+u²),           (1)

where q and r are positive dimensionless parameters. The non-zero steady states are thus given by the intersection of the two curves

U(u) = r(1-u/q), V(u) - u/(1+u²)

(a) Show, using the conditions for a double root, that the curve in q, r space which divides it into regions where there are 1 or 3 positive steady states is given parametrically by

q(a) = 2a³/(a²-1), r(a) =2a³/(1+a²)²

[Hint: find a cubic H(u) such that at the double root u = a, H(a) = H'(a) = 0 and solve these equations for q and r.]

(b) Show that the two curves meet in a cusp, that is where q'(a) = r'(a) = 0, at a =√3

(c) Use MAPLE to plot the curves in (q(a),r(a)) space, for 1.01 < a < 20 noting in particular the behaviours of q(a) and r(a) as a→∞ and a→1.

(d) Consider equation (1) in the case when r = 0.55. Use MAPLE to plot the solution for various choices of the initial condition u(0) in the cases when q = 6, q = 10 and q = 20. Discuss the change in behaviour as q increases.