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
Phase Plot of Predator vs. Prey Set parameters a=b=c=d=1 Set initial conditions: x=2 (prey), y=2 (predators) Equilibrium Point: x=(c/d), y= (a/b) Counter-clockwise motion Equilibrium point (1,1)
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
Phase Plot: Case 2 When initial conditions equal the equilibrium point: Parameters: a=b=c=d=1 Initial conditions: x=1 (prey), y=1 (predators)
Phase Plot: Case 3 When parameters are changed: Parameters: a=c=d=1, b=2 *Increase the death rate due to predation Initial Conditions: x=2 (prey), y=2 (predator)
Phase Plot: Case 4 When a species dies out: Parameters: a=1, b=c=d=1 Initial Conditions: x=50 (prey), y=500 (predator) Prey dies, therefore predator dies too.
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
Phase Plot of Prey vs. Predator vs. Super- predator Parameters: a=b=c=d=1, e=0.5, f=0.01, g=0.02 Initial Condition: X=1, Y=1, Z=1
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
Thank you Thank you to Anatoly for helping us with this presentation and helping us to make programs in MATLAB.