Homework SourseSuppose a population Pt satisfies dPdt 4P 001P2 P0 50 Whe
Suppose a population P(t) satisfies
dP/dt = .4P - .001P^2 P(0) = 50
Where t is measured in years
What is the carrying capacity?
What is P\'(0)?
When will the population reach 50% of the carrying capacity?
dP/dt = .4P - .001P^2 P(0) = 50
Where t is measured in years
What is the carrying capacity?
What is P\'(0)?
When will the population reach 50% of the carrying capacity?
Solution
Carrying capacity refers to the maximum of the population. To maximize the function, we must set dP/dt = 0 1) 0.4P -0.001P^2 = 0 ---> P(0.4 - 0.001P) = 0, P = 0 is the trivial solution 0.4 - 0.001P = 0 ---> P = 0.4 /0.001 = 400 2) P\'(0) = 0.4*P(0) - 0.001*P(0)^2, but P(0) = 50, so P\'(0) = 0.4*50 - 0.001*50^2 = 17.5
