Page - 98 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python

98 4 SolvingOrdinaryDifferentialEquations condition. Numerically,wemore literally need an initial condition: weneed to knowa starting value at the left endof the interval in order to get the computa- tional formulagoing. In fact,wedonotneedacomputer sincewesee a repetitivepatternwhendoing handcalculations,which leadsus toamathematical formula forN..kC1/ t/, : N..kC1/ t/DN.k t/C trN.k t/DN.k t/.1C tr/ DN..k 1/ t/.1C tr/2 ::: DN0.1C tr/kC1 : Rather thanusing (4.2) as a computationalmodel directly, there is a strong tra- dition forderivingadifferential equation fromthisdifferenceequation. The idea is to consider a very small time interval t and look at the instantaneous growth as this time interval is shrunk to an infinitesimally small size. Inmathematical terms, itmeans thatwe let t ! 0. As (4.2) stands, letting t ! 0will just producean equation0D0, sowehave todivideby t and then take the limit: lim t!0 N.tC t/ N.t/ t D rN.t/: The termon the left-hand side is actually the definition of the derivativeN 0.t/, so wehave N 0.t/D rN.t/; which is thecorrespondingdifferential equation. There is nothing in our derivation that forces the parameter r to be constant – it can changewith timedue to, e.g., seasonal changesormorepermanent environ- mental changes. Detour:Exactmathematicalsolution If you have taken a course onmathematical solution methods for differential equations,youmaywanttorecaphowanequationlikeN 0 D rN orN 0 D r.t/N is solved.Themethodof separationofvariables is themostconvenientsolution strategy in this case: N 0 D rN dN dt D rN dN N D rdt NZ N0 dN N D tZ 0 rdt