1. Predator-Prey Models Sarah Jenson Stacy Randolph

2. Outline ► Basic Theory of Lotka-Volterra Model ► Predator-Prey Model Demonstration ► Refinements of Lotka-Volterra Model

3. 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

4. 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

5. 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

6. 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.

7. 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

8. 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

9. NetLogo Predator-Prey Model

10. 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.

11. 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.

12. 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.

13. Questions?

14. 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