popmodels.pdf - Population Models Basic Assumption:...

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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 birthsβPΔtTotal deathsδPΔtChange in population is the difference:ΔP(β-δ)PΔt=ΔPΔt(β-δ)P=dPdt= (β-δ)P(take limitΔt0)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
Growth rate proportional to populationSupposeδ=0 andβ=kPwithk>0 constant: no death, while birth rate increaseswith population. ThendPdt= (β-δ)P=kP2=ZPP0˜P-2d˜P=Zt0kd˜t=⇒ -P-1+P-10=kt=P(t) =P01-P0ktP(t)ast1P0k(in finite time!)Ridiculous model for long term: typically large population re-duces birth rate in real-world situations.P0tP01P0kThe Logistic equationAssume growth rateβ-δalinearfunction ofP:β-δ=k(M-P(t))whereM,kare constant.Definition.The Logistic Differential Equation isdPdt=kP(M-P).The logistic equation is separable, henceZt0kd˜t=ZPP01˜P(M-˜P)d˜P=1MZPP01˜P+1M-˜Pd˜P(partial fractions)=Mkt=ln(M-P0)P

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