Predator-Prey Models

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Predator-Prey Models

• Sarah Jenson
• Stacy Randolph

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Outline

• Basic Theory of Lotka-Volterra Model
• Predator-Prey Model Demonstration
• Refinements of Lotka-Volterra Model

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Lotka-Volterra Model

• Vito Volterra
• (1860-1940)
• famous Italian mathematician
• Retired from pure mathematics in 1920
• Son-in-law DAncona

• Alfred J. Lotka
• (1880-1949)
• American mathematical biologist
• primary example plant population/herbivorous
  animal dependent on that plant for food

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Lotka-Volterra Model cont.

• The Lotka-Volterra equations are a pair of first
  order, non-linear, differential equations that
  describe the dynamics of biological systems in
  which two species interact.
• Earliest predator-prey model based on sound
  mathematical principles
• Forms the basis of many models used today in the
  analysis of population dynamics
• Original form has problems

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Lotka-Volterra Model cont.

• Describes interactions between two species in an
  ecosystem a predator and a prey
• Consists of two differential equations
• dF/dt F(a-bS)
• dS/dt S(cF-d)
• F Initial fish population
• S Initial shark population
• a reproduction rate of the small fish
• b shark consumption rate
• c small fish nutritional value
• d death rate of the sharks
• dt time step increment

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Prey Equation

• dF/dt F(a-bS)
• The small-fish population will grow exponentially
  in the absence of sharks
• Will decrease by an amount proportional to the
  chance that a a shark and a small fish bump into
  one another.

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Predator Equation

• dS/dt S(cF-d)
• Shark population can increase only proportionally
  to the number of small fish
• Sharks are simultaneously faced with decay due to
  constant death rate

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Experimental Evidence for Lotka-Volterra

• Georgii Frantsevich Gause (1910 1986)
• Competitive exclusion
• Predator-Prey System
• Two ciliates
• Results
• 1 Extinction of both prey and predator
• 2 With prey refuge extinction of predator
• 3 with immigration of predator and prey
  sustained oscillations

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NetLogo Predator-Prey Model
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Issues with Lotka-Volterra Model

• Will always contain a fixed point
• Example managing an ecosystem of small fish
  and sharks
• Will always have an infinite number of limit
  cycles that appear to orbit around the embedded
  fixed point.

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Refinement of Theory

• 1930s Competition in the Prey
• 1950s Leslie
• removed the prey dependency in the birth of the
  predators
• changed the death term for the predator to have
  both the number of predators and the ratio of
  predators to prey.
• 1960s May
• Discovered that predators are never not hungry.
  He fixed this by adding a piece to the prey death
  that would control this term.

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Conclusions

• The simplest models of population dynamics reveal
  the delicate balance that exists in almost all
  ecological systems.
• Refined Lotka-Volterra models appear to be the
  appropriate level of mathematical sophistication
  to describe simple predator-prey models.

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Questions?
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Sources

• Flake, G.W. The Computational Beauty of
  Nature,1998
• http//www.stolaf.edu/people/mckelvey/envision.dir
  /predprey.dir/predprey.html
• http//www.shodor.org/scsi/handouts/twosp.html
• http//www.math.duke.edu/education/ccp/materials/d
  iffeq/predprey/pred2.html
• http//www.biology.mcgill.ca/undergrad/c571/articl
  es/Lecture09-PredPrey.pdf