Ecology models and newton vector fields

We generate phase plane portraits of autonomous differential equations by constructing integrals of motion-even in nonconservative systems, where no integrals produced by quadrature exist. These phase plane portraits are produced with no propagated error. Phase portraits do not usually display the parametrization along orbits. The use of high resolution color graphics allows us to display this parametrization conveniently. We examine three ecology models-the logistic equation in the complex domain, and the predator-prey and competing species equations in the real domain.