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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: D’Ancona ► 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.d ir/predprey.dir/predprey.html http://www.stolaf.edu/people/mckelvey/envision.d ir/predprey.dir/predprey.html http://www.stolaf.edu/people/mckelvey/envision.d ir/predprey.dir/predprey.html ► http://www.shodor.org/scsi/handouts/twosp.html http://www.shodor.org/scsi/handouts/twosp.html ► http://www.math.duke.edu/education/ccp/materia ls/diffeq/predprey/pred2.html http://www.math.duke.edu/education/ccp/materia ls/diffeq/predprey/pred2.html http://www.math.duke.edu/education/ccp/materia ls/diffeq/predprey/pred2.html ► http://www.biology.mcgill.ca/undergrad/c571/artic les/Lecture09-PredPrey.pdf http://www.biology.mcgill.ca/undergrad/c571/artic les/Lecture09-PredPrey.pdf http://www.biology.mcgill.ca/undergrad/c571/artic les/Lecture09-PredPrey.pdf

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