Question

A biologist studies the population development of black bears in the US state of Pennsylvania. The study begins in 2000, when there were 15,000 bears in the state. Let p(t) denote the number of bears in Pennsylvania t years after the beginning of the survey.

The birth rate and the death rate for the population are constant 0, 16 and
0, 15 respectively. In other words, after t years (when the number of bears is p(t)), 0.16p(t) bears are born and 0.15p(t) bears die. There is also an estimated constant net influx of 50 bears/year from the surrounding states.
(a) Set up an initial value problem (i.e., a differential equation with initial conditions) for p(t).
(b) Solve the initial value problem in (a) and estimate the number of bears in Pennsylvania in 2022.​​​​​​​

Answer 1

Step-by-step explanation:

(a) The change in population over time is the net result of births, deaths, and immigration. We have been given the birth rate as 0.16 and the death rate as 0.15, while the net influx is a constant of 50 bears/year. Therefore, the equation for the population growth can be described as:

dp/dt = 0.01p + 50, where p(0)=15,000

(b) Separating variables, we obtain:

dp/(0.01p+50) = dt

Integrating both sides, we get:

ln(0.01p+50) = t + C

where C is the constant of integration. Applying the initial condition, we find:

ln(0.01*15000+50) = 0 + C

C = ln(200)

Therefore, the equation governing population growth is:

ln(0.01p+50) = t + ln(200)

Solving for p, we have:

0.01p+50 = 200e^t

p = 20000e^(0.01t) - 5000

Using the equation, we can estimate p(22) as follows:

p(22) = 20000e^(0.01*22) - 5000 = 31,348.5

Therefore, we can estimate that there will be approximately 31,348 bears in Pennsylvania in the year 2022.