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Lotka-Volterra Predator-Prey Problems

Lotka-Volterra Predator-Prey Problems. By: Alexandra Silva and Dani Hoover Intro to Systems ESE 251 11/24/09. History. Alfred Lotka. Vito Volterra. -American biophysicist -Proposed the predator-prey model in 1925. -Italian mathematician -Proposed the predator-prey model in 1926.

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Lotka-Volterra Predator-Prey Problems

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  1. Lotka-Volterra Predator-Prey Problems By: Alexandra Silva and Dani Hoover Intro to Systems ESE 251 11/24/09

  2. History • Alfred Lotka • Vito Volterra -American biophysicist -Proposed the predator-prey model in 1925 -Italian mathematician -Proposed the predator-prey model in 1926

  3. 2-Species Models

  4. Equations and Variables • X’ = ax – bxy • Y’ = -cy + dxy • X: the population of prey • Y: the population of predators • a: natural growth rate of prey in the absence of predation • b: death rate due to predation • c: natural death rate of predators in the absence of prey • d: growth rate due to predation

  5. Assumptions • The prey always has an unlimited supply of food and reproduces exponentially • The food supply of the predators depend only on the prey population (predators eat the prey only) • The rate of change of the population is proportional to the size of the population • The environment does not change in favor of one species

  6. Phase Plot of Predator vs. Prey • Set parameters a=b=c=d=1 • Set initial conditions: x=2 (prey), y=2 (predators) • Equilibrium Point: x=(c/d), y= (a/b) • Counter-clockwise motion Equilibrium point (1,1)

  7. Steady-State Orbit explanation A = Too many predators.B = Too few prey.C = Few predator and prey; prey can grow.D= Few predators, ample prey. http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/2DS.html

  8. Phase Plot: Case 2 • When initial conditions equal the equilibrium point: • Parameters: a=b=c=d=1 Initial conditions: x=1 (prey), y=1 (predators)

  9. Phase Plot: Case 3 • When parameters are changed: • Parameters: a=c=d=1, b=2 *Increase the death rate due to predation • Initial Conditions: x=2 (prey), y=2 (predator)

  10. Phase Plot: Case 4 • When a species dies out: • Parameters: a=1, b=c=d=1 • Initial Conditions: x=50 (prey), y=500 (predator) • Prey dies, therefore predator dies too.

  11. 3-Species Model (Super-predator)

  12. Equations and Variables (for 3-species model) • X’= ax-bxy (prey-- mouse) • Y’= -cy+dxy-eyz (predator-- snake) • Z’= -fz+gxz (super-predator-- owl) • a: natural growth rate of prey in the absence of predation • b: death rate due to predation • c: natural death rate of predator • d: growth rate due to predation • e: death rate due to predation (by super-predator) • f: natural death rate of super-predator • g: growth rate due to predation

  13. Phase Plot of Prey vs. Predator vs. Super-predator Parameters: a=b=c=d=1, e=0.5, f=0.01, g=0.02 Initial Condition: X=1, Y=1, Z=1

  14. Problems with Lotka-Volterra Models • The Lotka-Volterra model has infinite cycles that do not settle down quickly. These cycles are not very common in nature. • Must have an ideal predator-prey system. • In reality, predators may eat more than one type of prey • Environmental factors

  15. Thank you • Thank you to Anatoly for helping us with this presentation and helping us to make programs in MATLAB.

  16. Questions?

  17. Sources: • http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectures_files/lotka-volterra.pdf • http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/2DS.html • http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equation • http://isolatium.uhh.hawaii.edu/m206L/lab8/predator/predator.htm • http://www4.ncsu.edu/eos/users/w/white/www/white/ma302/less10.PDF • http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectures_files/lotka-volterra.pdf • http://www.stolaf.edu/people/mckelvey/envision.dir/lotka-volt.html

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