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7/2/04
Philip J. McDonnell on Evolution
The mosquito  bug spray example cited by Mr. Park strikes me as a classic predator prey model. In this example the mosquito is the prey and the humans with spray are the predator. The crucial difference is that as the mosquito population diminishes the human incentive to buy bug spray will decline. After an appropriate lag the mosquito population will bloom. At which point humans will make a note to get more bug spray on their next trip to the store. After the expected delay they will quickly reduce the blooming population.
This predator prey interaction can be modeled by the LotkaVolterra equations. Suppose we have the following: x = prey population, y = predator population a = growth rate of the prey b = rate at which predators destroy prey
Then the net change in the prey population for the next period is given by: chg.x = ax  bxy
Then if we let: c = death rate of predators d = growth rate of predators given prey
The formula for the growth of the predator population is given by: chg.y = cy + dxy
We thus have a pair of differential or difference equations in the case of discrete time periods. Each equation is interdependent and produces a change amount which determines the new levels of predator and prey in the next period. The pair of equations form a coupled recurrence relation which helps determine the future oscillations of both the predator and prey populations.