On the determination of evolutionary outcomes directly from the population dynamics of the resident

In order to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary in an ab initio calculation to find the fitness and its derivatives at a singular point. It has been suggested that the possible evolutionary behaviour can be predicted directly from the resident population dynamics, without the need for calculation, by applying three criteria-one based on the form of the density dependent rates and two on the role played by the evolving parameters. The existing arguments for these criteria are rather limited: they apply to systems in which individuals enter an initial class and can then move through any number of other population classes sequentially. (Extensions are included but only apply for systems of two and three classes.) Additionally, many of the arguments depend on the use of a phenomenologically motivated fitness (shown equivalent to the standard form but in a rather long and indirect manner). The present paper removes all these flaws-the criteria are established directly from the standard definition of fitness and individuals can enter any class and move through the classes non-sequentially without restriction on their number. The criteria thus established underlie a geometric description of the singular behaviour in adaptive dynamics which allows direct inferences to be made from population dynamics to the possible singular behaviour depending on which of the criteria apply and on the nature of the trade-off between evolving parameters. The method has the great advantage of leaving the trade-off explicit but unspecified.