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Unformatted text preview: Math 334 Lecture #38 9.4: Competing Species Outcome A: Apply Phase Plane Analysis to Population Dynamics . Suppose in a closed environment, there are two species whose populations at time t are x and y . In the absence of competition, the two populations are assumed to be governed by logistic growth (Section 2.5): dx/dt = x ( 1- 1 x ) , dy/dt = y ( 2- 2 y ) , where 1 , are the intrinsic growth rates and 1 / 1 , 2 / 2 ) the saturation levels. When the two species compete for the available food, they reduce each others growth rates and saturations levels. A modification of the uncoupled logistic equations above that accounts for this competi- tion is dx/dt = x ( 1- 1 x- 1 y ) , dy/dt = y ( 2- 2 y- 2 x ) , where 1 , 2 are quantities describing the interference the two species with each other. The parameters 1 , 2 , 1 , 2 , 1 , and 2 depend on the particular species and are estimated from empirical studies. Since we are dealing with populations, we restrict attention to those solutions that lie in the first quandrant, i.e., x 0 and y 0. The competing species model is an autonomous nonlinear system that is generally not solvable for explicit solutions x ( t ), y ( t )....
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This note was uploaded on 03/08/2011 for the course MATH 334 taught by Professor Smith during the Spring '11 term at Vanderbilt.
- Spring '11