Question

A particular region has a rabbit population of 1600. Two foxes are introduced to control the population of rabbits. Following this, the number of rabbits decreases according to the formula

R(t)=1700-A e^{k t} where A and k are constants, and R(t) is the number of rabbits in the region t years after the introduction of the foxes. (a) Given that the population of rabbits drops by one quarter after 5 years, find the values of A and k. (b) Following this model, how long will it take for the rabbits to become extinct? Give your answer to two decimal places. (c) Let F(t) be the number of foxes in the region t years after their introduction. If \frac{d F}{d t}=0.7 F(t) find the time at which the rate of decrease of the rabbit population is equal to the rate of increase of the fox population, correct to two decimal places. Hint. Note that dR/dt and dF/dt represent the rates of change of the rabbit and fox populations respectively. (d) Identify any problems with this model.

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