Math 211 Math 211 Lecture #28 December 7, 2000 2 Predator-Prey - - PDF document

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Math 211 Math 211

Lecture #28 December 7, 2000

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Predator-Prey Predator-Prey

Lotka-Volterra system x′ = (a − by)x (prey – fish) y′ = (−c + dx)y (predator – sharks)

• Equilbrium points

⋄ (0, 0) is a saddle. ⋄ (x0, y0) = (c/d, a/b) is a linear center.

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• The axes are invariant.
• The positive quadrant is invariant.
• The solution curves appear to be closed. Is

this actually true?

1 John C. Polking

System Return

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Along the solution curve y = y(x) we have dy dx = y(−c + dx) x(a − by) . The solution is H(x, y) = by − a ln y + dx − c ln x = C

• This is an implicit equation for the solution
• curve. ⇒ All solution curves are closed, and

represent periodic solutions.

System Return

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Why Fishing Leads to More Fish Why Fishing Leads to More Fish

Compute the average of the fish & shark populations. d dt ln x(t) = x′ x = a − by 0 = 1 T T d dt ln x(t) dt = a − by. So y = a/b = y0. Similarly x = x0 = c/d.

System Averages

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The effect of fishing that does not distinquish between fish and sharks is the system x′ = (a − by)x − ex y′ = (−c + dx)y − ey This is the same system with a replaced by a − e and c replaced by c + e.

2 John C. Polking

Averages

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The average populations are x1 = c + e d and y1 = a − e b Fishing causes the average fish population to increase and the average shark population to decrease.

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Cottony Cushion Scale Insect & the Ladybird Beetle Cottony Cushion Scale Insect & the Ladybird Beetle

• Cottony cushion scale insect accidentally

introduced from Australia in 1868. ⋄ Threatened the citrus industry.

• Ladybird beetle imported from Australia

⋄ Natural predator – reduced the insects to manageable low.

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DDT kills the scale insect.

• Massive spraying ordered.

⋄ Despite the warnings of mathematicians and biologists.

• The scale insect increased in numbers, as

predicted by Volterra.

3 John C. Polking

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Model of the Immune System in Action Model of the Immune System in Action

• How does the immune system develops

immunity to virus caused diseases? ⋄ Diseases such as flu, the cold, mumps, . . .

• Infectious Diseases of Humans - Roy M.

Anderson & Robert M. May, Oxford University Press 1992

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The model includes the interactions between virus cells and lymphocytes generated by the immune system.

• V (t) = number of virus cells
• Two types of lymphocytes, E1(t) & E2(t).

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Lymphocytes Lymphocytes

Both types:

• Are recruited from bone marrow at a

constant rate

• Die at a rate proportional to their numbers
• Proliferate due to contact with each other

4 John C. Polking

E cells Return

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Model With No Virus Present Model With No Virus Present

E′

1 = Λ1 − µ1E1 + a1

E1E2 1 + b1E1E2 E′

2 = Λ2 − µ2E2 + a2

E1E2 1 + b2E1E2

• For pplane5 use parameters Λ1 = Λ1 = 1,

µ1 = µ1 = 1.25, a1 = a2 = 0.252, and b1 = b2 = 0.008.

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Dynamics of the Lymphocytes Dynamics of the Lymphocytes

E1 ’ = 1 − 1.25 E1 + 0.252 E1 E2/(1 + 0.008 E1 E2) E2 ’ = 1 − 1.25 E2 + 0.252 E1 E2/(1 + 0.008 E1 E2) 5 10 15 20 25 30 5 10 15 20 25 30 E1 E2 Virgin State Immune State

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Interactions with the Virus Interactions with the Virus

• Virus cells have an intrinsic growth rate r.
• Lymphocytes of type E1:

⋄ kill virus because of contacts with them ⋄ proliferate because of contacts with virus

• Lymphocytes of type E2:

⋄ do not directly interact with the virus ⋄ regulate cells of type E1

5 John C. Polking

No virus Interactions Return

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Model With Virus Present Model With Virus Present

E′

1 = Λ1 − µ1E1 + a1

E1E2 1 + b1E1E2 + KV E1 E′

2 = Λ2 − µ2E2 + a2

E1E2 1 + b2E1E2 V ′ = rV − kV E1

• For ode45 use K = 0.5, k = 0.01 and

r = 0.1.

System Dynamics No virus

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Equilibrium Points Equilibrium Points

• There are three realistic equilibrium points

  E1 E2 V   =   1 1   ,   5 5   , &   20 20  

• The first two are unstable. The third is

asymptotically stable.

• What is the global behavior? The best we

can do is to check with ode45.

6 John C. Polking