1.It is recommended to do Exercise 8.12 prior to the present one. Here we look at the same population growth model N (t) = rN(t), N(0) = N0. The time derivative N (t) can be approximated by various types of finite differences. Exercise 8.12 considers a backward difference (Fig. 8.22), while Sect. 8.2.2 explained the forward difference (Fig. 8.4). A centered difference is more accurate than a backward or forward difference: N (tn + 1 2 Δt) ≈ N(tn + Δt) − N(tn) Δt = Nn+1 − Nn Δt . This type of difference, applied at the point t n+1 2 = tn + 1 2Δt, is illustrated geometrically in Fig. 8.23.
a) Insert the finite difference approximation in the ODE N = rN and solve for the….