Evolutionary invasion analysis

Evolutionary invasion analysis, also known as adaptive dynamics, is a set of mathematical modeling techniques that use differential equations to study the long-term evolution of traits in asexually reproducing populations. It rests on the following four assumptions about mutation and natural selection in the population under study: Individuals reproduce clonally. Mutations are infrequent, and natural selection acts quickly. The population can be assumed to be at equilibrium when a new mutant arises. The number of individuals with the mutant trait is initially negligible in the large, established resident population. Phenotypic mutations occur in small but not infinitesimal steps. Evolutionary invasion analysis makes it possible to identify conditions on model parameters for which the mutant population dies out, replaces the resident population, and/or coexists with the resident population. Long-term coexistence (on the evolutionary timescale) is known as evolutionary branching. When branching occurs, the mutant establishes itself as a second resident in the environment. Central to evolutionary invasion analysis is the mutant's invasion fitness. This is a mathematical expression for the mutant's long-term exponential growth rate when it is introduced into the resident population in small numbers. If the invasion fitness is positive, the mutant population can grow in the environment set by the resident organism. If the invasion fitness is negative, the mutant population swiftly goes extinct. The basic principle of evolution via natural selection was outlined by Charles Darwin in his 1859 book, On the Origin of Species. Though controversial at the time, the central ideas remain largely unchanged to this date, even though much more is now known about the biological basis of inheritance. Darwin expressed his arguments verbally, but many attempts have since then been made to formalise the theory of evolution.