The rate of change (dP)/(dt) of the number of deer on an island is modeled by a logistic differential equation. The maximum capacity of the island is 593 deer. At 5PM, the number of deer on the island is 210 and is increasing at a rate of 16 deer per day. Write a differential equation to describe the situation. (dP)/(dt)=◻

Question

The rate of change  (dP)/(dt) of the number of deer on an island is modeled by a logistic differential equation. The maximum capacity of the island is 593 deer. At  5PM, the number of deer on the island is 210 and is increasing at a rate of 16 deer per day. Write a differential equation to describe the situation.  (dP)/(dt)=◻

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Logistic Differential Equation: The logistic differential equation is generally given by the formula: $$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$$ where: - $P$ is the population at time $t$, - $r$ is the intrinsic growth rate of the population, - $K$ is the carrying capacity of the environment (maximum population size), - $\frac{dP}{dt}$ is the rate of change of the population with respect to time.Given Carrying Capacity: We are given that the carrying capacity $K$ is 593 deer. This value will be used in our differential equation.Initial Population Data: We are also given that at 5PM, the number of deer $P$ is 210 and is increasing at a rate of 16 deer per day. This information will help us determine the intrinsic growth rate $r$.Calculate Intrinsic Growth Rate: To find the intrinsic growth rate $r$, we use the given rate of change of the population $\frac{dP}{dt} = 16$ when $P = 210$. We plug these values into the logistic equation and solve for $r$: $$16 = r \cdot 210 \left(1 - \frac{210}{593}\right)$$Calculate Fraction: First, calculate the fraction of the carrying capacity that the current population represents: $$\frac{210}{593}$$Subtract Fraction: Perform the division to find the fraction: $$ \frac{210}{593} \approx 0.3541 $$Multiply by Population: Subtract this fraction from 1 to find the term $\left(1 - \frac{P}{K}\right)$: $$ 1 - 0.3541 \approx 0.6459 $$Divide Rate of Change: Now, multiply this term by the current population $P = 210$: $$ 210 \cdot 0.6459 \approx 135.639 $$Calculate Value of r: Finally, divide the given rate of change of the population by this product to solve for $r$: $$ r = \frac{16}{135.639} $$Write Logistic Differential Equation: Perform the division to find the value of $r$: $$ r \approx \frac{16}{135.639} \approx 0.1180 $$Now that we have the value of $r$, we can write the logistic differential equation: $$\frac{dP}{dt} = 0.1180P\left(1 - \frac{P}{593}\right)$$

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