Assume there is a certain population of deer in a forest whose growth is described by the logistic equation. The constant of proportionality for this type of deer is k=0.59, and the carrying capacity of the forest is 10,000 deer.
If the starting population is 8800 deer, then after one breeding season the population of the forest is?
We can use the logistic equation to model the growth of the deer population:
dP/dt = k * P * (1 - P/K)
where P is the population of deer, t is time, k is the constant of proportionality, and K is the carrying capacity of the forest.
Using the given values, we have:
dP/dt = 0.59 * 8800 * (1 - 8800/10000)
dP/dt = 0.59 * 8800 * 0.12
dP/dt = 628.32
This means that the population of the forest will increase by 628.32 deer after one breeding season. Therefore, the new population will be:
8800 + 628.32 = 9428 (rounded to the nearest whole number)
So, the population of the forest after one breeding season will be about 9428 deer.
I solved it! I got 9297.2222 so no need to help
To calculate the population of the forest after one breeding season using the logistic equation, we need to use the formula:
P(t) = K / (1 + A * e^(-k * t))
Where:
- P(t) is the population at time t
- K is the carrying capacity of the forest
- A is the initial population size relative to the carrying capacity (A = P(0) / K)
- k is the constant of proportionality
- e is the base of the natural logarithm
Given:
- k = 0.59
- K = 10,000 (carrying capacity)
- P(0) = 8800 (initial population)
First, calculate A:
A = P(0) / K
A = 8800 / 10000
A = 0.88
Now, substitute the values into the formula:
P(t) = 10000 / (1 + 0.88 * e^(-0.59 * t))
Since we are interested in after one breeding season, we can set t = 1.
P(1) = 10000 / (1 + 0.88 * e^(-0.59 * 1))
Now, let's calculate P(1):
P(1) = 10000 / (1 + 0.88 * e^(-0.59))
P(1) ≈ 10000 / (1 + 0.88 * 0.5547) (using the value of e as approximately 0.5547)
P(1) ≈ 10000 / (1 + 0.4861)
P(1) ≈ 10000 / 1.4861
P(1) ≈ 6726.02
Therefore, after one breeding season, the population of the forest is approximately 6726 deer.
To calculate the population of the deer after one breeding season using the logistic equation, we need to use the following formula:
P(t) = (K * P₀ * e^(k*t)) / (K + P₀ * (e^(k*t) - 1))
Where:
- P(t) is the population after time t
- K is the carrying capacity of the forest
- P₀ is the initial population
- k is the constant of proportionality
- e is the base of the natural logarithm (approximately 2.71828)
- t is the time period (in this case, one breeding season)
Given:
- K = 10,000
- P₀ = 8800
- k = 0.59
- t = 1
Let's plug these values into the formula and calculate the population:
P(t) = (10,000 * 8800 * e^(0.59*1)) / (10,000 + 8800 * (e^(0.59*1) - 1))
First, calculate e^(0.59*1) using a calculator:
e^(0.59*1) ≈ 1.80074
Next, substitute the values into the formula and simplify:
P(t) = (10,000 * 8800 * 1.80074) / (10,000 + 8800 * (1.80074 - 1))
P(t) = (79,207,200) / (10,000 + 8800 * 0.80074)
P(t) = (79,207,200) / (10,000 + 7045.67272)
P(t) ≈ 79,207,200 / 17,045.67272
P(t) ≈ 4643.239
Therefore, after one breeding season, the population of the forest is approximately 4643 deer.