1SolutionsSection 1.11.The rate of change in the populationP(t)is the derivativeP0(t). TheMalthusian Growth Law states that the rate of change in the population isproportional toP(t). ThusP0(t) =kP(t), wherekis the proportionalityconstant. Without reference to thetvariable, the differential equationbecomesP0=kP2.a. This statement mathematically isb(t) =b0P(t)where we have usedb0to represent the proportionality constant.b. This statement translates asd(t) =d0P2(t)where we have usedd0torepresent the proportionality constant.c. The overall growth rate isP0(t). Thus the Logistic Growth Law isP0(t) =b(t)d(t)=b0P(t)d0P2(t)= (b0d0P(t))P(t):3.Torricelli’s law states that the change in height,
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