Question

Suppose that poaching reduces the population of an endangered animal by 1% per year. Further suppose that when the population of this animal falls below 40 ​, its extinction is inevitable​ (owing to the lack of reproductive options without severe​ in-breeding). If the current population of the animal is ​1600, when will it face​ extinction? Comment on the validity of this exponential model.

It will take about (how many) years for the animal to face extinction?

Answer 1

Using an exponential decay model, it will take approximately 161 years for the population to fall below 40 and face extinction, assuming a constant rate of decline of 1% per year from an initial population of 1600.

Step-by-step explanation:

Extinction Timeline Calculation

If poaching reduces the population of an endangered animal by 1% per year, we can model this with an exponential decay function. The formula for the population after t years is:

P(t) =
`P0 * (0.99)^t`

where P0 is the initial population and P(t) is the population after t years. We are given that P0 is 1600 and we want to find the number of years (t) it takes for the population to drop below 40. Setting P(t) = 40 and solving for t:

40 =
`1600 * (0.99)^t`

Divide both sides by 1600:

0.025 =
`(0.99)^t`

Apply the logarithm:

ln(0.025) = t * ln(0.99)

Solve for t:

t = ln(0.025) / ln(0.99) ≈ 160.67

The animal population will face extinction in approximately 161 years.

As for the validity of this exponential model, it is important to note that it assumes a constant rate of decline and does not account for factors such as increased mortality due to lower numbers, changes in poaching rates, or conservation efforts. In reality, populations are subject to complex dynamics and the model should ideally be adjusted for these factors to improve its accuracy.