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The rate of change
(dP)/(dt) of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is 888 fox. At 8 PM, the number of fox at the national park is 210 and is increasing at a rate of 36 fox per day. Write a differential equation to describe the situation.

(dP)/(dt)=◻

The rate of change $\frac{d P}{d t}$ of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is $888$ fox. At $8$ PM, the number of fox at the national park is $210$ and is increasing at a rate of $36$ fox per day. Write a differential equation to describe the situation. $\newline$ $\frac{d P}{d t}=\square$

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Evaluate two-variable equations: word problems

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Q. The rate of change

\frac{d P}{d t}

of the number of fox at a national park is modeled by a logistic differential equation. The maximum capacity of the park is

888

fox. At

8

PM, the number of fox at the national park is

210

and is increasing at a rate of

36

fox per day. Write a differential equation to describe the situation.

\newline

\frac{d P}{d t}=\square

Logistic Differential Equation: The logistic differential equation is generally given by the formula: $\newline$ $(\frac{dP}{dt}) = rP(1 - \frac{P}{K})$ $\newline$ where: $\newline$ - $(\frac{dP}{dt})$ is the rate of change of the population $P$ with respect to time $t$ , $\newline$ - $r$ is the intrinsic growth rate of the population, $\newline$ - $P$ is the current population size, $\newline$ - $K$ is the carrying capacity of the environment (the maximum population size that the environment can sustain indefinitely). $\newline$ In this case, we need to find the value of $r$ since we have the carrying capacity $K$ and the current rate of change $(\frac{dP}{dt})$ at a specific population size $P$ .
Given Parameters: We are given: $\newline$ - The carrying capacity $K = 888$ foxes, $\newline$ - The current population size $P = 210$ foxes, $\newline$ - The current rate of change $\frac{dP}{dt} = 36$ foxes per day. $\newline$ We can use the given rate of change to find the intrinsic growth rate $r$ by rearranging the logistic equation: $\newline$ $36 = r \times 210 \times (1 - \frac{210}{888})$
Calculate Intrinsic Growth Rate: Now we solve for $r$ : $36 = r \times 210 \times (1 - 210/888)$ $36 = r \times 210 \times (678/888)$ $36 = r \times 210 \times (0.7635)$ $36 = r \times 160.335$ $r = 36 / 160.335$ $r \approx 0.2245$ per day
Write Logistic Differential Equation: Now that we have the value of $r$ , we can write the logistic differential equation for this situation: $\newline$ $\frac{dP}{dt} = 0.2245 \times P \times (1 - \frac{P}{888})$

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