Population ModelsBasic Assumption: population dynamics of a group controlled by two functions of timeBirth Rateβ(t,P) =average number of births per group member, per unit timeDeath Rateδ(t,P) =average number of deaths per group member, per unit timeIfP(t) =population at timet, then, betweentandt+Δt:•Total births≈βPΔt•Total deaths≈δPΔtChange in population is the difference:ΔP≈(β-δ)PΔt=⇒ΔPΔt≈(β-δ)P=⇒dPdt= (β-δ)P(take limitΔt→0)Different models depend on choices/observations/predictions of birth and death ratesNatural GrowthSupposeβandδconstantNatural growth (β-δ>0) or decay (β-δ<0) equation:dPdt= (β-δ)P=⇒P(t) =P0e(β-δ)twhereP0is the population at timet=0P0tβ>δβ<δβ=δP01