MA354

Building Systems of Difference Equations

T H 2:30pm– 3:45 pm

Dr. Audi Byrne

Recall: Models for Population Growth

• Very generally, P = (increases in the population)

– (decreases in the population)

• Classically,

P = “births” – “deaths”

“Conservation Equation”

Practice: Writing Down Difference Equations

Problem 1: Rabbit Population Model

• Birth rate: Suppose that the birth rate of a rabbit population is 0.5*.

• 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.

* E.g., for every two bunnies alive in year t approximately 1 bunny is born in year t+1.

Problem 1: Rabbit Population Model

• Rn+1 =

• Rn+1 =

• What is the long time behavior of the chicken population?

Problem 3: 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.

Problem 2: Chicken Population Model

• Cn+1 =

• Cn+1 =

• What is the long time behavior of the chicken population?

Problem 2: Chicken Population Model

• Cn+1 =

• Cn+1 =

• What is the long time behavior of the chicken population?

Practice: Writing Down Systems of Difference Equations

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.

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb-wd) W(t)

R(t+1) = (rb-rd) R(t)

With predator/prey interaction:

W(t+1) = (wb-wd) W(t) + k1 W(t)R(t)

R(t+1) = (rb-rd) R(t) – k2 W(t)R(t)

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb-wd) W(t)

R(t+1) = (rb-rd) R(t)

With predator/prey interaction:

W(t+1) = (wb-wd) W(t) + k1 W(t)R(t)

R(t+1) = (rb-rd) R(t) – k2 W(t)R(t)

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb-wd) W(t)

R(t+1) = (rb-rd) R(t)

With predator/prey interaction:

W(t+1) = (wb-wd) W(t) + k1 W(t)R(t)

R(t+1) = (rb-rd) R(t) – k2 W(t)R(t)

Population size just depends on independentbirths and deaths…

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb-wd) W(t)

R(t+1) = (rb-rd) R(t)

With predator/prey interaction:

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb-wd) W(t)

R(t+1) = (rb-rd) R(t)

With predator/prey interaction:

W(t+1) = (wb-wd) W(t) + k1 W(t)R(t)

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb-wd) W(t)

R(t+1) = (rb-rd) R(t)

With predator/prey interaction:

W(t+1) = (wb-wd) W(t) + k1 W(t)R(t)

R(t+1) = (rb-rd) R(t) – k2 W(t)R(t)

Example: Predator Prey Model

W(t) – wolf population

R(t) – rabbit population

Without interaction:

W(t+1) = (wb – wd) W(t)

R(t+1) = (rb – rd) R(t)

With predator/prey interaction:

W(t+1) = (wb – wd) W(t) + k1 W(t)R(t)

R(t+1) = (rb – rd) R(t) – k2 W(t)R(t)

“mass-action”type interaction

k1 W(t)R(t)

Example: Predator Predator Model

W(t) – wolf population

H(t) – hawk population

Example: Predator Predator Model

W(t) – wolf population

H(t) – hawk population

Without interaction:

W(t+1) = (wb – wd) W(t)

H(t+1) = (hb – hd) H(t)

Example: Predator Predator Model

W(t) – wolf population

H(t) – hawk population

Without interaction:

W(t+1) = (wb – wd) W(t)

H(t+1) = (hb – hd) H(t)

With predator/predator interaction:

W(t+1) = (wb – wd) W(t) – k1 W(t)H(t)

H(t+1) = (hb – hd) H(t) – k2 W(t)H(t)

Example: Predator/Predator/Prey Model

W(t) – wolf population

H(t) – hawk population

R(r) – rabbit population

System with interactions:W(t+1) = (wb-wd) W(t) – k1 W(t)H(t) + k3 W(t)R(t)

H(t+1) = (hb-hd) H(t) – k2 W(t)H(t) + k4 W(t)R(t)

R(t+1) = (hr-hr) R(t) – k5 W(t)R(t) – k6 H(t)R(t)

Example: Predator/Predator/Prey Model

W(t) – wolf population

H(t) – hawk population

R(r) – rabbit population

System with interactions:W(t+1) = (wb – wd) W(t) – k1 W(t)H(t) + k3 W(t)R(t)

H(t+1) = (hb – hd) H(t) – k2 W(t)H(t) + k4 W(t)R(t)

R(t+1) = (hr – hr) R(t) – k5 W(t)R(t) – k6 H(t)R(t)

Finding Equilibrium Valuesof Systems of Difference

Equations

Example: Predator Predator Model

W(t) – wolf population

H(t) – hawk population

Dynamical System:

W(t+1) = (wb-wd) W(t) - k1 W(t)H(t)

H(t+1) = (hb-hd) H(t) – k2 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)

• What is the long time (equilibrium) behavior of the two populations?

(Solution will be demonstrated

on the board.)

Example: Predator Predator Model

Example: 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.

• What is the dynamical system describing this scenario?

• What is the long time behavior of the two populations?

Example: A Car Rental Company