Questions: A population of 30 deer are introduced into a wildlife sanctuary. It is estimated that the sanctuary can sustain up to 300 deer. Absent constraints, the population would grow by 70% per year. Estimate the population after one year p1= Estimate the population after two years p2=

A population of 30 deer are introduced into a wildlife sanctuary. It is estimated that the sanctuary can sustain up to 300 deer. Absent constraints, the population would grow by 70% per year.

Estimate the population after one year
p1=

Estimate the population after two years
p2=

Transcript text: A population of 30 deer are introduced into a wildlife sanctuary. It is estimated that the sanctuary can sustain up to 300 deer. Absent constraints, the population would grow by $70 \%$ per year. Estimate the population after one year \[ p_{1}= \] $\square$ Estimate the population after two years \[ p_{2}= \] $\square$ Question Help: Video 1 Video 2 Message instructor Submit Question

Solution

Solution Steps

To estimate the population after each year, we can use the logistic growth model. This model accounts for the carrying capacity of the environment. The formula for the population at the next time step is:

\[ p_{t+1} = p_t + r \cdot p_t \cdot \left(1 - \frac{p_t}{K}\right) \]

where:

$ p_t $ is the current population,
$ r $ is the growth rate (0.7 for 70%),
$ K $ is the carrying capacity (300 deer).

We will calculate the population for each year using this formula.

Step 1: Initial Parameters

We start with an initial population of $ p_0 = 30 $ deer. The growth rate is $ r = 0.7 $ and the carrying capacity of the sanctuary is $ K = 300 $.

Step 2: Population After One Year

Using the logistic growth formula: \[ p_1 = p_0 + r \cdot p_0 \cdot \left(1 - \frac{p_0}{K}\right) \] Substituting the values: \[ p_1 = 30 + 0.7 \cdot 30 \cdot \left(1 - \frac{30}{300}\right) = 48.9 \]

Step 3: Population After Two Years

Using the same formula for the next year: \[ p_2 = p_1 + r \cdot p_1 \cdot \left(1 - \frac{p_1}{K}\right) \] Substituting the values: \[ p_2 = 48.9 + 0.7 \cdot 48.9 \cdot \left(1 - \frac{48.9}{300}\right) \approx 77.55 \]