Course:MATH102/Question Challenge/1997 December Q6

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Question

"Stuart Pimm, a professor of ecology and environmental biology at the University of Tennessee, told a recent Washington summit on biodiversity that he and other researchers have calculated the effect of shrinking forest area on birds. They predict "50 percent of the world's bird species will go extinct in the next half-century and 75 percent in the next century, just based on what we [humans] have already done."

(a) Assuming this to be the case, suggest a simple differential equation that would fit this projection.

(b) What is the half-life of the number of species?

(c) If there are $10^{5}$ species now, how many will there be in $t$ years from now?

Hints

[show]Hint
You can start out by dividing the time into equal intervals, note that it is 50% in half a century and 75% in a full century..therefore, at what rate is it decaying in every half-century interval?

Solutions

[show]Solution

Part A) We know that after 50 years the population has halved. Therefore we can represent the bird population with the following equation: $0.5=e^{-k(50)}$ $ln0.5=-50k$ $-k={\frac {ln0.5}{50}}$ Exponential functions have differential equations of the form: ${\frac {dy}{dx}}=-ky$ Therefore a differential equation which would fit Stuart Pimm's projection is given by: ${\frac {dy}{dx}}={\frac {-ln0.5}{50}}y$ Part B) The half-life of the bird species is the period it takes for their population to decrease to half of its initial value. This occurs every half-century so the bird population have a half-life of 50 years. Part C) $P(t)=10^{5}e^{-kt}$