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Find and solve a differential equation governing population growth with given conditions

by RoRi

The population of a town at time t = 0 is 365. The population growth factor is e, and the town experiences a death rate of one percent of the population per day. Find a differential equation modeling the population of the town as a function of time and find

  1. the actual population of the town after t years,
  2. the cumulative total of the fatalities from the town’s death rate.

Incomplete.

2 comments

  1. Evangelos says:

    Apologies for the terse comment here, but my previous solution got spam-filtered. WordPress needs to get a damn clue.

    Growth rate

        \begin{align*} P(t) &= 365 e^{t} \\ \frac{dP}{dt} &= P(t) \\ &= 365 e^{t} \end{align*}

    D**th rate

        \begin{align*} D' &= \frac{365}{100} P(t) \end{align*}

    Differential equation for population change

        \begin{align*} \frac{dP}{dt} &= P(t) - D'\\ &= -2.65P(t) \end{align*}

    Solve for P

        \begin{align*} P(t) &= 365e^{-2.65t} \\ \\ \frac{dP}{dt} &= (-2.65)(365)e^{-2.65t} \end{align*}

    Net change in population at time x

        \begin{align*} \int_{0}^{x} (-2.65)(365)e^{-2.65t} dt &= [365e^{-2.65t}]_{x}^{0} \\ &= 365(1 - e^{-2.65x}) \end{align*}

  2. Evangelos says:

    This problem was poorly-worded all around. The solution will reflect the back-of-book answers, and not necessarily the answers requested in the exercise.

    From the given information, the initial population of Coyote Gulch is 365. If there were no “accidental” deaths in the town, the population would multiply by a factor of e each year. We can model this growth as follows

        \begin{align*} P(t) &= 365 e^{t} \\ \frac{dP}{dt} &= P(t) \\ &= 365 e^{t} \end{align*}

    The rate of “accidental” deaths is 1 death per 100 people each day, or 365 deaths per year. If we let D’ denote the number of deaths per year, we have

        \begin{align*} D' &= \frac{365}{100} P(t) \end{align*}

    Subtracting this from our population growth rate gives us a differential equation to model the actual population change over time

        \begin{align*} \frac{dP}{dt} &= P(t) - D'\\ &= -2.65P(t) \end{align*}

    Separating variables and integrating the equation, with P(0) = 365, we get

        \begin{align*} P(t) &= 365e^{-2.65t} \\ \\ \frac{dP}{dt} &= (-2.65)(365)e^{-2.65t} \end{align*}

    Now, to find the net population change at a given time x, we simply take an indefinite integral from time 0 to time x of the derivative dP/dt

        \begin{align*} \int_{0}^{x} (-2.65)(365)e^{-2.65t} dt &= [365e^{-2.65t}]_{x}^{0} \\ &= 365(1 - e^{-2.65x}) \end{align*}

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