The rate of changedtdP 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.dtdP=□ Submit Answer
Q. The rate of change dtdP 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.dtdP=□ Submit Answer
Logistic Differential Equation: The logistic differential equation is generally given by the formula: dtdP=rP(1−KP)where:- dtdP is the rate of change of the population P with respect to time t,- r is the intrinsic growth rate of the population,- P is the current population size,- K is the carrying capacity of the environment (the maximum population size that the environment can sustain indefinitely).
Given Information: We are given the following information:- The carrying capacity K is 955 wolves.- The current population size P is 218 wolves.- The rate of change of the population dtdP is 26 wolves per day.We need to find the intrinsic growth rate r.
Calculate Intrinsic Growth Rate: To find the intrinsic growth rate r, we can use the given rate of change dtdP and the current population size P to solve for r in the logistic equation:26=r×218×(1−955218)
Solve for r: Now we solve for r:26=r×218×(1−955218)26=r×218×(955737)26=r×218×0.77173 (approximately)26=r×168.29734 (approximately)r≈168.2973426r≈0.1545 (approximately)
Final Differential Equation: We can now write the differential equation with the value of r we found: dtdP=0.1545×P×(1−955P) 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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