Predator-Prey Models Sarah Jenson Stacy Randolph
Outline • Basic Theory of Lotka-Volterra Model • Predator-Prey Model Demonstration • Refinements of 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 Lotka-Volterra Model
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
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
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.
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
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
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.
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.
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.
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/diffeq/predprey/pred2.html • http://www.biology.mcgill.ca/undergrad/c571/articles/Lecture09-PredPrey.pdf