# Mr Nichols PHHS. History Alfred Lotka Vito Volterra -American biophysicist -Proposed the predator- prey model in 1925 -Italian mathematician -Proposed.

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Mr Nichols PHHS

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

2-Species Models

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

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

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/works hop/2DS.html

3-Species Model (Super-predator)

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

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

Sources: http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectu res_files/lotka-volterra.pdf http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectu res_files/lotka-volterra.pdf http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/work shop/2DS.html http://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/work shop/2DS.html http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equati on http://en.wikipedia.org/wiki/Lotka%E2%80%93Volterra_equati on http://isolatium.uhh.hawaii.edu/m206L/lab8/predator/predator.htm http://isolatium.uhh.hawaii.edu/m206L/lab8/predator/predator.htm http://www4.ncsu.edu/eos/users/w/white/www/white/ma302/l ess10.PDF http://www4.ncsu.edu/eos/users/w/white/www/white/ma302/l ess10.PDF http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectu res_files/lotka-volterra.pdf http://www.cs.unm.edu/~forrest/classes/cs365/CS%20365/Lectu res_files/lotka-volterra.pdf http://www.stolaf.edu/people/mckelvey/envision.dir/lotka- volt.html http://www.stolaf.edu/people/mckelvey/envision.dir/lotka- volt.html

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