Question

In at tropical forest, the populations of falcons and mice are related to each other. The mice eat everything they can while the falcons eat the mice. The population sizes of falcons and mice evolve according to the rule (*) Fₖ₊₁=0.7Fₖ+0.1Mₖ and Mₖ₊₁=−0.5Fₖ+1.3Mₖ where Fₖ and Mₖ are the numbers of falcons and mice in the forest, respectively, k years after 2023. At the beginning of 2023 , there were 10 falcons and 40 mice. (a) Approximately how many falcons and mice will there be at the beginning of 2068 ?

Answer 1

Approximately there will be about 27,429 falcons and 137,147 mice

To approximate the populations of falcons and mice at the beginning of 2068 using the given recursive rules, we follow these steps:

Step 1: Define the Recursive Rules

The populations of falcons F and mice M evolve according to the following rules:

`\[ F_(k+1) = 0.7F_k + 0.1M_k \]`

`\[ M_(k+1) = -0.5F_k + 1.3M_k \]`

where
`\( F_k \) and \( M_k \)`represent the populations of falcons and mice, respectively, in the k -th year after 2023.

Step 2: Initial Conditions

At the beginning of 2023 (which we'll consider as year k = 0 ), the populations are given as:

-
`\( F_0 = 10 \) falcons`

-
`\( M_0 = 40 \) mice`

Step 3: Calculate the Population for Each Year

We will calculate the populations of falcons and mice for each year from 2023 up to 2068. The year 2068 is 45 years after 2023, so we need to find
`\( F_(45) \) and \( M_(45) \).`

Step 4: Perform the Calculations

We will iteratively calculate the populations for each subsequent year using the given recursive formulas until we reach the year 2068.

Let's perform these calculations

Approximately at the beginning of 2068, there will be about 27,429 falcons and 137,147 mice in the tropical forest, based on the given population evolution rules.