Question 3 (20 Marks) A population of giant tortoises is modelled by the delay differential equation dN (t) = bN(t T) - c[N(t T)] m[N(t)], dt - where m, b and c are positive parameters and T is the time delay between egg laying and hatching. (a) Give a biological interpretation for each of the three terms in the right hand side of the equation. Explain how we can see that the tortoises have a naturally long lifespan when the population is small. (b) Show that there is a unique positive equilibrium N., and find a linear differential equation for h(t) = N(t)- N that approximately holds when h is small. Find the characteristic equation. (c) Using parameter continuation, show that as T is increased from zero, instability of the equilibrium N. will first be observed (if at all) when the following system has a real solution for wand T: (c-m) cos(wT) = -2m (cm) sin(wT) = (c+m)w/b. (d) What is the meaning of the parameter w? Find an expression for w in terms of b, c and m only. Show that N is in fact locally stable for all T> 0 if c/m is not too large.