Invasion and extermination
a certain country is being infested by rodents that can be modelled using
dp/ dt= rp −k
where p(t) is the population of rodents, t is the time in weeks, r is the relative growth rate of rodents and k is the weekly rate at which exterminators can remove rodents.
the following estimates are available
• the relative growth rate is r = 0.1 per week.
• the extermination rate is k = 200 rodents per week.
• there are currently p(0) = 3,000 rodents in the population.
questions
1. what is the equilibrium rodent population? is this equilibrium stable or unstable?
2. use separation of variables to find the general solution and then also give the particular solution.
3. using the above particular solution, estimate what the rodent population will be after 8 weeks. (give the exact formula and a numerical approximation.)
4. the government goes bankrupt and cannot pay the exterminators, so the extermination rate drops to k = 0. what is the doubling time of the rodent population in this case? (give the exact formula and a numerical approximation.)