1. MA354 Building Systems of Difference Equations T H 2:30pm– 3:45 pm Dr. Audi Byrne

2. Recall: Models for Population Growth Very generally,  P =  (increases in the population) –  (decreases in the population) Classically,  P = “births” – “deaths” “Conservation Equation”

3. Practice: Writing Down Difference Equations

4. Problem 1: Rabbit Population Model Birth rate: Suppose that the birth rate of a rabbit population is 0.5. (For every two bunnies alive in year t approximately 1 bunny is born in year t+1). Death rate: Suppose that an individual rabbit has a 25% chance of dying each year. Write down a difference equation that describes the given dynamics of the bunny population.

5. Problem 1: Rabbit Population Model  R n+1 = R n+1 = What is the long time behavior of the chicken population?

6. Problem 2: Chicken Population Model Birth rate: Suppose that 2000 chickens are born per year on a chicken farm. Death rate: Suppose that the chicken death rate is 10% per year. Write down a difference equation that describes the given dynamics of the chicken population.

7. Problem 2: Chicken Population Model  C n+1 = C n+1 = What is the long time behavior of the chicken population?

8. Practice: Writing Down Systems of Difference Equations

9. Systems of Difference Equations Two or more populations interact with one another through birth or death terms. Each population is given their own difference equation. To find an equilibrium value for the system, all populations must simultaneously be in equilibrium.

10. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction: W(t+1) = (w b -w d ) W(t) + k 1 W(t)R(t) R(t+1) = (r b -r d ) R(t) – k 2 W(t)R(t)

11. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction: W(t+1) = (w b -w d ) W(t) + k 1 W(t)R(t) R(t+1) = (r b -r d ) R(t) – k 2 W(t)R(t)

12. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction: W(t+1) = (w b -w d ) W(t) + k 1 W(t)R(t) R(t+1) = (r b -r d ) R(t) – k 2 W(t)R(t) Just depends on independent births and deaths…

13. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction:

14. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction: W(t+1) = (w b -w d ) W(t) + k 1 W(t)R(t)

15. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction: W(t+1) = (w b -w d ) W(t) + k 1 W(t)R(t) R(t+1) = (r b -r d ) R(t) – k 2 W(t)R(t)

16. Example: Predator Prey Model W(t) – wolf population R(t) – rabbit population Without interaction: W(t+1) = (w b -w d ) W(t) R(t+1) = (r b -r d ) R(t) With predator/prey interaction: W(t+1) = (w b -w d ) W(t) + k 1 W(t)R(t) R(t+1) = (r b -r d ) R(t) – k 2 W(t)R(t) “mass-action” type interaction k 1 W(t)R(t)

17. Example: Predator Predator Model W(t) – wolf population H(t) – hawk population

18. Example: Predator Predator Model W(t) – wolf population H(t) – hawk population Without interaction: W(t+1) = (w b -w d ) W(t) H(t+1) = (h b -h d ) H(t)

19. Example: Predator Predator Model W(t) – wolf population H(t) – hawk population Without interaction: W(t+1) = (w b -w d ) W(t) H(t+1) = (h b -h d ) H(t) With predator/predator interaction: W(t+1) = (w b -w d ) W(t) - k 1 W(t)H(t) H(t+1) = (h b -h d ) H(t) – k 2 W(t)H(t)

20. Example: Predator/Predator/Prey Model W(t) – wolf population H(t) – hawk population R(r) – rabbit population System with interactions: W(t+1) = (w b -w d ) W(t) – k 1 W(t)H(t) + k 3 W(t)R(t) H(t+1) = (h b -h d ) H(t) – k 2 W(t)H(t) + k 4 W(t)R(t) R(t+1) = (h r -h r ) R(t) – k 5 W(t)R(t) – k 6 H(t)R(t)

21. Example: Predator/Predator/Prey Model W(t) – wolf population H(t) – hawk population R(r) – rabbit population System with interactions: W(t+1) = (w b -w d ) W(t) – k 1 W(t)H(t) + k 3 W(t)R(t) H(t+1) = (h b -h d ) H(t) – k 2 W(t)H(t) + k 4 W(t)R(t) R(t+1) = (h r -h r ) R(t) – k 5 W(t)R(t) – k 6 H(t)R(t)

22. Finding Equilibrium Values of Systems of Difference Equations

23. Example: Predator Predator Model W(t) – wolf population H(t) – hawk population Dynamical System: W(t+1) = (w b -w d ) W(t) - k 1 W(t)H(t) H(t+1) = (h b -h d ) H(t) – k 2 W(t)H(t) W(t+1) = 1.2 W(t) – 0.001 W(t)H(t) H(t+1) = 1.3 H(t) – 0.002 W(t)H(t)

24. Problem 3: Predator-Predator System What is the long time behavior of the two populations?

25. A Car Rental Company A rental company rents cars in Orlando and Tampa. It is found that 60% of cars rented in Orlando are returned to Orlando, but 40% end up in Tampa. Of the cars rented in Tampa, 70% are returned to Tampa and 30% are returned to Orlando. Write down a system of difference equations to describe this scenario and decide how many cars should be kept in each city if there are 7000 cars in the fleet.

26. Problem 4: Rental Car Companies What is the dynamical system describing this scenario? What is the long time behavior of the two populations?