= Area-preserving seasonal predator-prey map
{title2=$du'\,dv'=du\,dv,\quad u=\log X,\ v=\log Y$}
The <annual pulse-breeding predator-prey model> is an <area-preserving map> in logarithmic population coordinates: $u'=u+\log r-\kappa e^v$, then $v'=v+\log s+\log(1+be^{u'})$. Each step is a shear of <determinant> one. At a positive <fixed point>, the ordinary-population <Jacobian matrix> has <determinant> one and <trace> $2-(1-s)\log r$. <Eigenvalues> lie on the <unit circle> for $0<(1-s)\log r<4$, but this linear neutrality does not prove nonlinear stability at every resonance. Area preservation excludes an isolated attracting positive periodic cycle.
Back to article page
