Question

The average population growth rate for whitetail deer is 0.35. Hunting laws are set to limit the time allowed for hunting deer with a goal of achieving about a 35% mortality rate on deer to keep the population in check. Years with a higher than 35% mortality will result in an overall decline in the deer population while years with a lower than 35% mortality rate will result in an increased population. If the growth rate exceeds the mortality rate, and the net effect were a 4% growth rate, how long would it take the population of deer to double

Answer 1

Answer:

The amount of years that it takes the population to double is 17.33 years.

Step-by-step explanation:

We have a model for the population of deer, in we know that the population have a net growth rate of 4%.

That is:

$\dfrac{dP}{dt} =0.04P$

We have to calculate how much time it takes to the population of deers to duplicate its number, given that the growth rate is kept constant.

We can start by solving the first differential equation:

$\dfrac{dP}{dt} =0.04P\\\\\\\int \dfrac{dP}{P}=0.04\int dt\\\\\\ln(P)=0.04t+C_1\\\\\\P(t)=Ce^{0.04t}$

We don't know the initials conditions to calculate C, but we don't need them to solve this problem.

We will estimate the time h that takes the population to double as:

$P(t+h)=2P(t)\\\\Ce^{0.04(t+h)}=2Ce^{0.04t}\\\\e^{0.04t}\cdot e^{0.04h}=2e^{0.04t}\\\\e^{0.04h}=2\\\\0.04h=ln(2)\\\\h=ln(2)/0.04=17.33$

The amount of years that it takes the population to double is 17.33 years.