Page - 211 - in Programming for Computations – Python - A Gentle Introduction to Numerical Simulations with Python 3.6, Volume Second Edition

8.2 PopulationGrowth:AFirstOrderODE 211 easily deal with almost any differential equation! This is exactly the topic of the presentchapter. 8.2.1 DerivationoftheModel It can be instructive to show how an equation like (8.1) arises. Consider some population of an animal species and let N(t) be the number of individuals in a certain spatial region, e.g. an island. We are not concerned with the spatial distributionof theanimals, just thenumberof theminsomeregionwherethereisno exchangeof individualswithotherregions.DuringatimeintervalΔt, someanimals willdieandsomewillbeborn.Thenumbersofdeathsandbirthsareexpected tobe proportional toN. For example, if there are twice as many individuals, we expect them to get twice as many newborns. In a time intervalΔt, the net growth of the populationwill thenbe N(t+Δt)−N(t)= b¯N(t)− d¯N(t), where b¯N(t) is the number of newborns and d¯N(t) is the number of deaths. If we double Δt, we expect the proportionality constants b¯ and d¯ to double too, so it makes sense to think of b¯ and d¯ as proportional to Δt and “factor out” Δt. That is, we introduce b = b¯/Δt and d = d¯/Δt to be proportionality constants for newborns and deaths independent of Δt. Also, we introduce r = b − d, which is the net rate of growth of the population per time unit. Our model then becomes N(t+Δt)−N(t)=ΔtrN(t). (8.2) Equation (8.2) is actually a computational model. GivenN(t), we can advance thepopulationsize by N(t+Δt)=N(t)+ΔtrN(t). This is called a difference equation. If we knowN(t) for some t, e.g.,N(0)=N0, wecancompute N(Δt)=N0+ΔtrN0, N(2Δt)=N(Δt)+ΔtrN(Δt), N(3Δt)=N(2Δt)+ΔtrN(2Δt), ... N((k+1)Δt)=N(kΔt)+ΔtrN(kΔt),