 Quadratic discrete predator-prey map

This map equals an explicit unit-step update of a logistic predator-prey differential equation. Its coexistence fixed point is $(b,(1-a-b)/a)$, requiring $a+b<1$. Its trace and determinant are $T=2-b/a$ and $D=(1-2b)/a$. The Schur stability criterion gives stability precisely when $b>(1-a)/2$ and $b<a+1/3$. The coexistence flip at $b=a+1/3$ is subcritical on $1/9<a<1/3$, with effective cubic coefficient $-9/[(3a+1{)}^2(9a-1)]$ when the $-1$ eigenvector has first component one. The unit-circle crossing $a+2b=1$ is a supercritical Neimark–Sacker bifurcation away from the strong resonances of a planar map $a=1/7,1/5$; in the eigenvector normalization $q_1=1$ its cubic radial coefficient is $-1/[a^2(1-a)]$. The whole nonnegative quadrant is not forward invariant, because $x^{\prime }$ can become negative. Local fixed-point stability is therefore not a global population-dynamics classification.

// Destination: dynamical-systems.bigb