Suppose a population’s growth is
governed by the logistic equation d𝑃 by d𝑡 equals 0.07𝑃 multiplied by one minus
𝑃 over 900, where 𝑃 of zero is equal to 50. Write the formula for 𝑃 of 𝑡.
Writing 𝑃 of 𝑡 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 d𝑡 equals 𝑘𝑃 multiplied by one minus 𝑃 over 𝐿 that its solution
is given by 𝑃 equals 𝐿 over one plus 𝐴𝑒 to the negative 𝑘𝑡, where 𝐴 is equal
to 𝐿 minus 𝑃 nought over 𝑃 nought. Here 𝑘 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 𝑘 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 — that’s 900 —
over one plus 𝐴 — that’s 17 — 𝑒 to the power of negative 𝑘 — that’s negative 0.7
— 𝑡. So we have our solution for 𝑃 or
𝑃 of 𝑡. It’s equal to 900 over one plus
17𝑒 to the power of negative 0.07𝑡.