ma354 building systems of difference equations t h 2:30pm– 3:45 pm dr. audi byrne
TRANSCRIPT
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MA354
Building Systems of Difference Equations
T H 2:30pm– 3:45 pm
Dr. Audi Byrne
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Recall: Models for Population Growth
• Very generally, P = (increases in the population)
– (decreases in the population)
• Classically,
P = “births” – “deaths”
“Conservation Equation”
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Practice: Writing Down Difference Equations
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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.
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Problem 1: Rabbit Population Model
• Rn+1 =
• Rn+1 =
• What is the long time behavior of the chicken population?
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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.
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Problem 2: Chicken Population Model
• Cn+1 =
• Cn+1 =
• What is the long time behavior of the chicken population?
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Problem 2: Chicken Population Model
• Cn+1 =
• Cn+1 =
• What is the long time behavior of the chicken population?
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Practice: Writing Down Systems of Difference Equations
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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.
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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)
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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)
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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…
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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:
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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)
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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)
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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)
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Example: Predator Predator Model
W(t) – wolf population
H(t) – hawk population
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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)
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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)
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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)
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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)
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Finding Equilibrium Valuesof Systems of Difference
Equations
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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)
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• What is the long time (equilibrium) behavior of the two populations?
(Solution will be demonstrated
on the board.)
Example: Predator Predator Model
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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.
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• What is the dynamical system describing this scenario?
• What is the long time behavior of the two populations?
Example: A Car Rental Company