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Unformatted text preview: Math 334 Lecture #38 Â§ 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 Ïƒ 1 x ) , dy/dt = y ( 2 Ïƒ 2 y ) , where 1 , 2 are the intrinsic growth rates and 1 /Ïƒ 1 , 2 /Ïƒ 2 the saturation levels. When the two species compete for the available food, they reduce each othersâ€™ growth rates and saturations levels. A modification of the uncoupled logistic equations above that accounts for this competi tion is dx/dt = x ( 1 Ïƒ 1 x Î± 1 y ) , dy/dt = y ( 2 Ïƒ 2 y Î± 2 x ) , where Î± 1 , Î± 2 are quantities describing the interference the two species with each other. The parameters 1 , 2 , Ïƒ 1 , Ïƒ 2 , Î± 1 , and Î± 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 â‰¥ 0 and y â‰¥ 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 11/29/2011 for the course MATH 334 taught by Professor Dallon during the Fall '08 term at BYU.
 Fall '08
 DALLON
 Math, Differential Equations, Equations

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