The rate of changedtdP 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.dtdP=□
Q. The rate of change dtdP 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.dtdP=□
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).In this case, we need to find the value of r since we have the carrying capacity K and the current rate of change (dtdP) at a specific population size P.
Given Parameters: We are given:- The carrying capacity K=888 foxes,- The current population size P=210 foxes,- The current rate of change dtdP=36 foxes per day.We can use the given rate of change to find the intrinsic growth rate r by rearranging the logistic equation:36=r×210×(1−888210)
Calculate Intrinsic Growth Rate: Now we solve for r:36=r×210×(1−210/888)36=r×210×(678/888)36=r×210×(0.7635)36=r×160.335r=36/160.335r≈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:dtdP=0.2245×P×(1−888P)
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