Question

The number of individuals in a population is modeled by the following discrete-difference equation and initial condition: x n + 1 = 0.80 x n + 30 , x 0 = 100 In this model, x n represents the population at year n . Find the solution to this problem. Your answer should be an expression for xn that depends only on n. Write x_n for xn. What happens to the population in the long-run, as n→[infinity]?

Answer 1

The solution to the discrete-difference equation
`\(x_(n+1) = 0.80x_n + 30\)`, with the initial condition
`\(x_0 = 100\)`, is given by
`\(x_n = 50 * 0.8^n + 50\)`. In the long run, as
`\(n \rightarrow \infty\)`, the population approaches a stable value of 50.

Step-by-step explanation:

The given difference equation represents a discrete dynamical system modeling population growth. To find a solution, we can use the initial condition
`\(x_0 = 100\)` and recursively apply the difference equation. The formula
`\(x_n\)` is derived by recognizing the pattern in the recurrence relation. In this case,
`\(x_n\)` follows a geometric sequence with a common ratio of 0.8 and an initial term of 50, resulting in the expression
`\(x_n = 50 * 0.8^n + 50\)`.

In the long run, as
`\(n \rightarrow \infty\)`, the behavior of the population becomes apparent. The common ratio of 0.8 is less than 1, indicating that the population experiences exponential decay. The term
`\(50 * 0.8^n\)` approaches zero as n becomes infinitely large,
`\(x_n\)` converging to the stable value of 50. This suggests that over time, the population settles into a steady state, and further changes become negligible.

Understanding the behavior of the population model, in the long run, is crucial for predicting the system's stability and making informed decisions about resource management or intervention strategies. In this case, the population stabilizes at 50, reflecting a balance between growth and the limiting effect of the difference equation.