In a time before Dorothy Gale, the Wicked Witch of the East invades Munchkin Country, forcing the Munchkins to emigrate to the Emerald City at a rate proportional to the population p

In a time before Dorothy Gale, the Wicked Witch of the East invades Munchkin Country, forcing the Munchkins to emigrate to the Emerald City at a rate proportional to the population p.

The equation for modelling the population of Munchkin Country is now given as

dp/dt = rp(1 – p²/N²) – kp,

Where k > 0 is the rate of emigration.

(a) What are the dimensions of k?

(b) Non-dimensionalise equation (2), reducing the number of parameters to one variable.

(c) Determine the critical points of your non-dimensional emigration model.

(d) Sketch the phase lines of your non-dimensional emigration model for

i. k < r,

ii. k > r.

(e) Using the linear stability criteria, determine the stability of the critical points in your non-dimensional emigration model for

i. k < r,

ii. k > r.

(f) Sketch the bifurcation diagram for your non-dimensional emigration model.

(g) Using your above analysis and setting r = 0.05g and k = 0.03d, what is the population of Munchkin Country with emigration for very large time t? You can assume that the initial population is greater than zero.

Let g and d be the respective seventh and eighth digits of your student number, i.e. if g = 3 and d = 5, then r = 0.053 and k = 0.035.