Simplify the logistic model with constant harvesting

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Unformatted text preview: stant harvesting by centering the bifurcation at the origin. Let and . Then [2] becomes . [3] 3 Ch8Lecs This is the normal form of the saddle-node bifurcation. For there are two hyperbolic equilibrium points, for there is a single nonhyperbolic equilibrium, and for there are no equilibria. ■ The saddle-node bifurcation requires 3 conditions on the vector field: Singularity condition (the equilibrium point [3] is nonhyperbolic at ). Non-degeneracy condition (the coefficient of in [3] is nonzero). Transversality condition (that guarantees that the parameter perturbs the nonhyperbolic equilibrium point in a transverse way: in [3]). Local Bifurcations of Limit Cycles To further illustrate the meaning of “local bifurcation” let’s briefly describe local bifurcations of limit cycles. sketch limit cycle , section , intersection Consider a Poincaré map . corresponds to a fixed point of : . The eigenvalues of are the Floquet multipliers (not including the trivial unit multiplier). Real and imaginary axes, region of evls within the unit circle for , arrows showing that evls can traverse the circle though , , or a complex conjugate pair can pass out of the unit circle off the real axis. Example A saddle-node bifurcation of limit cycles. Suppose that for there is a semistable limit cycle and an associated Poincaré map: sketch semistable limit cycle in 2D and nearby trajectories sketch -, -, the line , the Poincaré map is concave down and tangent to the line at the origin, cobweb paths inside and outside the limit cycle sketch pair of limit cycles. The outer one is stable and the inner one unstable. Nearby trajectories. sketch -, -, the line , the Poincaré map has shifted so there are now two intersections with the line. Three cobweb paths. 4 Ch8Lecs sketch spiral flow without limit cycle no intersection between the Poincare map with the line . The saddle node bifurcation corresponds to a single eigenvalue reaching the unit circle at and then disappearing.■ Global Bifurcations of Limit Cycles Global bifurcations cannot be described by a local analysis. Example Homoclinic bifurcation in sketch like Meiss Fig. 8.18. the limit cycle for ■ . and without the point and axis. Notice 8.2 Preservation of Equilibria Consider . An equilibrium point satisfies . When we linearize about the equilibrium point, the Jacobian matrix is . If the Jacobian matrix has a zero eigenvalue then it is singular and we say the equilibrium point is degenerate. Otherwise the Jacobian matrix is nonsingular and the equilibrium point is nondegenerate. Contrast this with the definition of a nonhyperbolic equilibrium point, where only the real part of an eigenvalue need be zero. For example, if a Jacobian matrix has...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .

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