MacQuarie Island is a small island about half-way between

Problem 11

MacQuarie Island is a small island about half-way between Antarctica and New Zealand. Between 2000 and 2006, the population of rabbits on the island rose from 4,000 to 130,000. Model the growth in the rabbit population R(t) at time t using an exponential growth model d R dt = k R, where t = 0 corresponds to the year 2000. What is an appropriate value for the growth-rate parameter k, and what does this model predict for the population in the year 2010. (For more information on why the population of rabbits exploded, see Review Exercise 22 in Chapter 2.)

                                                              Step-step by step solution

Step 1 of 2

Consider the following model the growth in the rabbit population  at time

Here   corresponds to the year 2000.

The population of rabbits on the island rose from 4,000 to 130,000 between 2000 and 2006.

That is,  and .

Solve the differential equation   ,

                               [Separate the variables]

                     [ Integrate on both sides]

                Here  is an integrating constant.

                        Take exponential on both sides.

                  Here