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The rate of change (dP)/(dt) of the number of wolves at a national park is modeled by a logistic differential equation. The maximum capacity of the park is 955 wolves. At 8 PM, the number of wolves at the national park is 218 and is increasing at a rate of 26 wolves per day. Write a differential equation to describe the situation. (dP)/(dt)=◻" Submit Answer "

The rate of change dPdt \frac{d P}{d t} of the number of wolves at a national park is modeled by a logistic differential equation. The maximum capacity of the park is 955955 wolves. At 88 PM, the number of wolves at the national park is 218218 and is increasing at a rate of 2626 wolves per day. Write a differential equation to describe the situation.\newlinedPdt=□ Submit Answer  \frac{d P}{d t}=\square \text { Submit Answer }

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Q. The rate of change dPdt \frac{d P}{d t} of the number of wolves at a national park is modeled by a logistic differential equation. The maximum capacity of the park is 955955 wolves. At 88 PM, the number of wolves at the national park is 218218 and is increasing at a rate of 2626 wolves per day. Write a differential equation to describe the situation.\newlinedPdt=□ Submit Answer  \frac{d P}{d t}=\square \text { Submit Answer }
  1. Logistic Differential Equation: The logistic differential equation is generally given by the formula: \newlinedPdt=rP(1−PK)\frac{dP}{dt} = rP(1 - \frac{P}{K})\newlinewhere:\newline- dPdt\frac{dP}{dt} is the rate of change of the population PP with respect to time tt,\newline- rr is the intrinsic growth rate of the population,\newline- PP is the current population size,\newline- KK is the carrying capacity of the environment (the maximum population size that the environment can sustain indefinitely).
  2. Given Information: We are given the following information:\newline- The carrying capacity KK is 955955 wolves.\newline- The current population size PP is 218218 wolves.\newline- The rate of change of the population dPdt\frac{dP}{dt} is 2626 wolves per day.\newlineWe need to find the intrinsic growth rate rr.
  3. Calculate Intrinsic Growth Rate: To find the intrinsic growth rate rr, we can use the given rate of change dPdt\frac{dP}{dt} and the current population size PP to solve for rr in the logistic equation:\newline26=r×218×(1−218955)26 = r \times 218 \times (1 - \frac{218}{955})
  4. Solve for r: Now we solve for r:\newline26=r×218×(1−218955)26 = r \times 218 \times (1 - \frac{218}{955})\newline26=r×218×(737955)26 = r \times 218 \times (\frac{737}{955})\newline26=r×218×0.7717326 = r \times 218 \times 0.77173 (approximately)\newline26=r×168.2973426 = r \times 168.29734 (approximately)\newliner≈26168.29734r \approx \frac{26}{168.29734}\newliner≈0.1545r \approx 0.1545 (approximately)
  5. Final Differential Equation: We can now write the differential equation with the value of rr we found: dPdt=0.1545×P×(1−P955)\frac{dP}{dt} = 0.1545 \times P \times (1 - \frac{P}{955}) This is the logistic differential equation that models the rate of change of the number of wolves at the national park.

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