The slope of the contour line near the singular point determines the answers to the following questions. When near a singular point two neighbouring phenotypes can invade each other, the resulting dimorphism will contract over evolutionary time only if the singular point is uninvadable, otherwise it will expand. Near singular points that monomorphically attract and dimorphically repel, the evolutionary trajectory will branch. The rightmost figure shows repeated instances of such branching events in a simulated evolutionary trajectory. These simplifications allow rapid scientific progress, while the analyses of special cases suggests that the predictions obtained usually aggree with those from more sophisticated models.
Evolution proceeds by the continual replacement of resident types by novel mutants.
The latter originate by chance but their evolutionary fate depends on their fitness, i.e. on their capacity to increase in numbers. This fitness necessarily depends the current environment E which is set by the composition of the resident population. By eliminating E we get the invasion function, i.e. the fitness of potential mutants as a function of the type of the mutant and of the types constituting the resident population. In order to calculate this function we start with describing the dependence of individual population-dynamical behaviour on supposedly heriditary traits. From this we obtain the initial exponential growth rate of a mutant population in the ergodic environment set by the resident population-dynamical attractor.