A population of deer live in a forest. They have no natural predator, and are limited only by the resources available to them. (a) The deer population p(t) is governed by the logistic growth equation dp (1-P), dt where r is the growth rate and N is the population carrying capacity. The time variable t is measured in months. (i) If this system is dimensionally consistent, what are the units of r? What does the quantity r mean? =rp (ii) Set r= 0.25, N = 20000, and p(0) = 1000 + 10008, where is the third digit of your MQ student number. Use ode45 to simulate the population over a 15 year period. (b) A nearby village decides to hunt deer in a way that allows the population to be sustainable. They hunt seasonally, so that deer have time to recover after each hunting season. The deer equation is now governed by dp dt = rp 1+ sin 2 (i) Use the same parameters as in the previous question. Set a = 200 and use ode45 to simulate the population over a 15 year period. 2t (ii) How large a value can a take while allowing the deer to reach a sustainable population. What happens if a exceeds this value? Support your answer with example simulations. (iii) Is there a way the villagers could adapt their behaviour to so they can choose larger values of a than the critical value from (ii)? Show this using an example simulation. An overdamped bead on a rotating ring has the following dimensional equation: de fl = -mg sin 0 + mrw sin 0 cos 0. dt de dT Here, 0(t) is the angle of the bead, where angle is a dimensionless quantity in the range 0 (-, T]. The mass of the bead is m and the radius of the ring is r. The ring rotates around. its vertical axis with a constant angular velocity (radians/unit time) of w. The value is the damping coefficient. All constants are positive. (a) If this expression is dimensionally consistent, what are the dimensions of ? (b) By appropriately non-dimensionalizing this equation, obtain = sin 0(k cos 01). (c) What are the critical points of this model for k = 1/2, k = 1, and k = 2? Sketch the phase lines for each case. (d) Using the linear stability criterion, determine the stability of the critical points in the model for k= 1/2 and k = 2. What happens if you try to apply this criterion to the k = 1 case? (e) Show that k = 1 is the bifurcation value. What does this tell us about the required rotational angular velocity required for the bead to settle at a nonzero angle? How does it change if we use a ring with a larger radius? (f) Plot the bifurcation diagram for this model.