Discrete models and Fisher's maximum principle in ecology

Fisher's (1930) maximum principle in ecology states that \Any net advantage gained by an organism will be conserved in the form of an increase in population, rather than in an increase in the average Malthusian parameter, which is kept by this adjustment always near zero." We know today that we cannot make such general statements. Nevertheless, several ecologists, including Nicholson (1960), have stressed this principle as a general ecological principle. Based on a number of theoretical counterexamples, we cannot conclude that this principle is not supported by any essential biological facts. This paper examines simple examples that illustrate when the principle is valid. We use a discrete modeling approach to account for the fact that several boreal populations are constrained to reproduce at well-defined discrete moments. Several authors have pointed out that the above maximum principle ceases to be valid when predation is present. With reference to the Ricker competition case, we suggest how the principle could be reformulated so as to cover that case.