MATH-M 344 Chapter Notes - Chapter 9: Phase Portrait, Nonlinear System, Curve
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Rate of change of populations (system known as the lokta-volterra equation): Number of interactions between species proportional to product of populations: each encounter tends to promote growth of predator, hinder growth of prey. With no predators, prey grows at rate proportional to current population; when (cid:1877)=0, (cid:3031)(cid:3051)(cid:3031)(cid:3047)=(cid:1876)>0 and (cid:3031)(cid:3052)(cid:3031)(cid:3047)=0. Because ,(cid:1855)>0, e-values have opposite signs- unstable saddle point: near (cid:4672) (cid:3030),(cid:4673): (cid:3031)(cid:3031)(cid:3047)[(cid:1873)(cid:1874)]=[0. Pure imaginary- stable center for linear system; cannot use theorem 9. 3. 2 (no conclusion about stability comes from it for locally linear system w/ pure imaginary e-values: construct linear system around each point, determine e-values. Stability of critical points: focus on second linear system around (cid:4672)(cid:3030),(cid:4673): (cid:3031)(cid:3031)(cid:3047)[(cid:1873)(cid:1874)]=[0. {(cid:3031)(cid:3048)(cid:3031)(cid:3047)= (cid:3030) (cid:1874) (cid:3031)(cid:3049)(cid:3031)(cid:3047)=(cid:1873) (cid:3031)(cid:3049)(cid:3031)(cid:3048)= (cid:4672)(cid:4673)(cid:3048) (cid:4672) (cid:4673)(cid:3049) (cid:3031)(cid:3049)(cid:3031)(cid:3048)=2(cid:3048)2(cid:3030)(cid:3049) (cid:2870)(cid:1873) (cid:1856)(cid:1873)= (cid:2870)(cid:1855)(cid:1874) (cid:1856)(cid:1874) (cid:2870) (cid:1873)(cid:1856)(cid:1873)= (cid:2870)(cid:1855) (cid:1874)(cid:1856)(cid:1874) 0 constant of integration system of this form increases, meaning parametric direction goes in direction shown.
