ESPM-EEP%202010%20Lecture%2013

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ESPM 104/EEP 115 Fall 2010: Lecture 13 1. Population interactions: consider the pair of models in which the per-capita growth rate f of a species x is affected by the density of species y and the per-capita growth rate g of a species y is affected by the density of species x—that is: dx dt = xf ( x , y ) dy dt = yg ( x , y ) a. Competition: intraspecific f x < 0 and g y < 0 interspecific f y < 0 and g x < 0 b. Resource x , consumer y : f y < 0 and g x > 0 c. Mutualism: more complicated. 2. More generally, this can also be written in terms of the total growth rates F and G. dx dt = F ( x , y ) dy dt = G ( x , y ) 3. The solutions to the equations F ( x , y ) = 0 G ( x , y ) = 0 are called the null-isoclines, or simply the nullclines. 4. The equilibrium solutions to the equations modeling the temporal dynamics of two interacting populations are where the nullclines intersect. In the case of the per-capita model, the

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Unformatted text preview: nullclines are: (Download SSG8.7: 2D Nonlinear , this document has not been properly proofread) x-nullclines : x =0 and y = h ( x ) where f ( x , h ( x ) ) =0 y-nullclines : y =0 and y = k ( x ) where g ( x , k ( x ) ) =0 5. Phase space diagram in x-y plane. Plot regions where dx dt > ( < )0 and dy dt > ( < )0 by looking at the signs of f ( x , h ( x ) ) and g ( x , k ( x )) in different zones mapped out by the plots of the nullclines in the positive quadrant of the x-y plane. Arbitrary example an equilibrium center and an attractor at (0,0). The separatrix (dotted line) is approximate 6. Prey-predator equations. See various texts including Case 2000. Also see http://www.scholarpedia.org/article/Predator-prey_model...
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