Ecology models and Newton vector fields (Q1108963)

The paper deals with the visual examination of the global behaviour of solutions to autonomous systems of differential equations. A simple method to do this is to integrate numerically in forward time and then to draw the phase portraits and the Poincaré sections of the corresponding flows by computer. Indeed, the error inherent numerical integration may be propagated which leads to a deviation from the true orbit long before the vicinity of an attractor is reached. The authors propose a new method which has no propagated error. In a second part they draw high resolution computer graphical color phase plane portraits for some differential equations used in mathematical ecology: the logistic equation in the complex domain, the predator-prey and the competing species equation in the real domain. (Unfortunately, the reviewer has not seen these pictures.) The algorithm for constructing integral curves developed by the author is explained in detail for an example in the complex domain.