The logistic equation practice problems
Logistic growth: equilibria, the solution P(t) = K / (1 + Ae^(−rt)), and when the population reaches a level. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Solve \( \displaystyle P' = \frac{1}{10}P\left(1 - \frac{P}{500}\right) \), \( \displaystyle P(0) = 100 \). Find the equilibrium solutions, \( \displaystyle P(7) \), and the time when the population reaches half its carrying capacity.
Solve \( \displaystyle P' = \frac{1}{5}P\left(1 - \frac{P}{5000}\right) \), \( \displaystyle P(0) = 1000 \). Find the equilibrium solutions, \( \displaystyle P(9) \), and the time when the population reaches half its carrying capacity.
A population follows the logistic model with growth rate \( \displaystyle r = \frac{1}{2} \) per year, carrying capacity 5000, and initial population 1250. Find \( \displaystyle P(t) \), \( \displaystyle P(2) \), and when the population reaches 2500.
Solve \( \displaystyle P' = \frac{1}{5}P\left(1 - \frac{P}{5000}\right) \), \( \displaystyle P(0) = 1250 \). Find the equilibrium solutions, \( \displaystyle P(1) \), and the time when the population reaches half its carrying capacity.
Solve \( \displaystyle P' = \frac{1}{5}P\left(1 - \frac{P}{10000}\right) \), \( \displaystyle P(0) = 500 \). Find the equilibrium solutions, \( \displaystyle P(1) \), and the time when the population reaches half its carrying capacity.
Solve \( \displaystyle P' = \frac{1}{10}P\left(1 - \frac{P}{100}\right) \), \( \displaystyle P(0) = 20 \). Find the equilibrium solutions, \( \displaystyle P(7) \), and the time when the population reaches half its carrying capacity.
Solve \( \displaystyle P' = \frac{3}{10}P\left(1 - \frac{P}{500}\right) \), \( \displaystyle P(0) = 10 \). Find the equilibrium solutions, \( \displaystyle P(3) \), and the time when the population reaches half its carrying capacity.
Solve \( \displaystyle P' = \frac{1}{2}P\left(1 - \frac{P}{1000}\right) \), \( \displaystyle P(0) = 100 \). Find the equilibrium solutions, \( \displaystyle P(4) \), and the time when the population reaches half its carrying capacity.
A population follows the logistic model with growth rate \( \displaystyle r = 1 \) per year, carrying capacity 500, and initial population 10. Find \( \displaystyle P(t) \), \( \displaystyle P(4) \), and when the population reaches 250.
Solve \( \displaystyle P' = \frac{1}{5}P\left(1 - \frac{P}{1000}\right) \), \( \displaystyle P(0) = 250 \). Find the equilibrium solutions, \( \displaystyle P(6) \), and the time when the population reaches half its carrying capacity.