Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Explore our Q&A platform to find in-depth answers from a wide range of experts in different fields. Get immediate and reliable solutions to your questions from a community of experienced professionals on our platform.

Assume there is a certain population of fish in a pond whose growth is described by the logistic equation. It is estimated that the carrying capacity for the pond is 1200 fish. Absent constraints, the population would grow by [tex]$180\%$[/tex] per year.

If the starting population is given by [tex]$p_0 = 400$[/tex], then after one breeding season the population of the pond is given by
[tex]
p_1 =
[/tex]

After two breeding seasons the population of the pond is given by
[tex]
p_2 =
[/tex]
[tex]\square[/tex]

Sagot :

To calculate the population of fish in the pond after one and two breeding seasons using the logistic growth model, we will follow these steps:

1. Determine the carrying capacity (K), which is 1200 fish.
2. Set the growth rate (r) at 180%, or as a decimal, 1.8.
3. Initial population (p0) of the pond is given as 400 fish.

Step-by-Step Solution:

### After One Breeding Season:

Using the logistic growth model, we calculate the population after one breeding season.

The logistic equation for one breeding season is:
[tex]\[ p_1 = p_0 + r \cdot p_0 \left(1 - \frac{p_0}{K}\right) \][/tex]

Plugging in the values:
[tex]\[ p_1 = 400 + 1.8 \cdot 400 \left(1 - \frac{400}{1200}\right) \][/tex]

First, simplify the fraction:
[tex]\[ 1 - \frac{400}{1200} = 1 - \frac{1}{3} = \frac{2}{3} \][/tex]

Now, multiply and calculate:
[tex]\[ 1.8 \cdot 400 \cdot \frac{2}{3} = 1.8 \cdot 400 \cdot 0.6667 \approx 480 \][/tex]

Add this to the initial population:
[tex]\[ p_1 = 400 + 480 = 880 \][/tex]

So, the population after one breeding season [tex]\( p_1 \)[/tex] is:
[tex]\[ p_1 = 880 \][/tex]

### After Two Breeding Seasons:

We use the new population [tex]\( p_1 \)[/tex] to find [tex]\( p_2 \)[/tex]:

The logistic equation for the next breeding season is:
[tex]\[ p_2 = p_1 + r \cdot p_1 \left(1 - \frac{p_1}{K}\right) \][/tex]

Plugging in the values:
[tex]\[ p_2 = 880 + 1.8 \cdot 880 \left(1 - \frac{880}{1200}\right) \][/tex]

First, simplify the fraction:
[tex]\[ 1 - \frac{880}{1200} = 1 - \frac{22}{30} = \frac{8}{30} = \frac{4}{15} \][/tex]

Now, multiply and calculate:
[tex]\[ 1.8 \cdot 880 \cdot \frac{4}{15} = 1.8 \cdot 880 \cdot 0.2667 \approx 422.4 \][/tex]

Add this to the population after one breeding season:
[tex]\[ p_2 = 880 + 422.4 = 1302.4 \][/tex]

So, the population after two breeding seasons [tex]\( p_2 \)[/tex] is:
[tex]\[ p_2 = 1302.4 \][/tex]

Therefore, the population of the pond after one breeding season is:
[tex]\[ p_1 = 880 \][/tex]

And after two breeding seasons the population is:
[tex]\[ p_2 = 1302.4 \][/tex]
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.