Question

A model for population growth is given by the following differential equation (i.e. the "logistic" equation) $\frac{dP}{dt} = kP(1 - \frac{P}{c})$. Where $P(t)$ is the population at time $t$, 'k' is a constant and 'c' is also a constant, called the carrying capacity. The carrying capacity 'c' represents the maximum population that the environment is capable of sustaining. Biologists stocked a lake with 400 fish and estimated the carrying capacity for fish of that species in that lake to be 10 000. The number of fish tripled in the first year. (a) Assuming that the size of the fish population satisfies the logistic equation, find an expression for the size of the population after t years. (b) How long will it take the population to grow to 5000? Answer: 2.68 years.

          A model for population growth is given by the following differential equation (i.e. the
"logistic" equation)
$\frac{dP}{dt} = kP(1 - \frac{P}{c})$.
Where $P(t)$ is the population at time $t$, 'k' is a constant and 'c' is also a constant, called
the carrying capacity. The carrying capacity 'c' represents the maximum population that
the environment is capable of sustaining.
Biologists stocked a lake with 400 fish and estimated the carrying capacity for fish of that
species in that lake to be 10 000. The number of fish tripled in the first year.
(a) Assuming that the size of the fish population satisfies the logistic equation, find
an expression for the size of the population after t years.
(b) How long will it take the population to grow to 5000?
Answer: 2.68 years.

A model for population growth is given by the following differential equation (i.e. the
"logistic" equation)
(dP)/(dt) = kP(1 - (P)/(c)).
Where P(t) is the population at time t, 'k' is a constant and 'c' is also a constant, called
the carrying capacity. The carrying capacity 'c' represents the maximum population that
the environment is capable of sustaining.
Biologists stocked a lake with 400 fish and estimated the carrying capacity for fish of that
species in that lake to be 10 000. The number of fish tripled in the first year.
(a) Assuming that the size of the fish population satisfies the logistic equation, find
an expression for the size of the population after t years.
(b) How long will it take the population to grow to 5000?
Answer: 2.68 years.

Added by Gregory J.

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