Video: Finding the Solution of Logistic Differential Equations

Suppose a population’s growth is governed by the logistic equation d𝑃/d𝑑 = 0.07𝑃(1 βˆ’ (𝑃/900)), where 𝑃(0) = 50. Write the formula for 𝑃(𝑑).

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Video Transcript

Suppose a population’s growth is governed by the logistic equation d𝑃 by d𝑑 equals 0.07𝑃 multiplied by one minus 𝑃 over 900, where 𝑃 of zero is equal to 50. Write the formula for 𝑃 of 𝑑.

Writing 𝑃 of 𝑑 means that we need to find the solution to this logistic equation. We can begin by writing down its general form. We know that for the logistic equation d𝑃 by d𝑑 equals π‘˜π‘ƒ multiplied by one minus 𝑃 over 𝐿 that its solution is given by 𝑃 equals 𝐿 over one plus 𝐴𝑒 to the negative π‘˜π‘‘, where 𝐴 is equal to 𝐿 minus 𝑃 nought over 𝑃 nought. Here π‘˜ represents the growth rate of the population. 𝐿 is the carrying capacity. And 𝑃 nought is the initial population. We can identify each of these values from the information given in the question.

First, we see that π‘˜ is equal to 0.07 and 𝐿 is equal to 900. We’re also told that 𝑃 zero is equal to 50. So we can fill in each of the values in the general solution. Let’s work out 𝐴 first of all. 𝐴 is equal to 𝐿 minus 𝑃 nought over 𝑃 nought. That’s 900 minus 50 over 50 or 850 over 50, which is equal to 17.

Now we can substitute into the general form of the solution. 𝑃 is equal to 𝐿 β€” that’s 900 β€” over one plus 𝐴 β€” that’s 17 β€” 𝑒 to the power of negative π‘˜ β€” that’s negative 0.7 β€” 𝑑. So we have our solution for 𝑃 or 𝑃 of 𝑑. It’s equal to 900 over one plus 17𝑒 to the power of negative 0.07𝑑.

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