Adaptive dynamics for an age-structured population model with a Shepherd recruitment function

English: In this study the evolution of the genetic composition of certain species will be replaced by the evolution of the traits that represent these genetic compositions. Depending on the nature of the trait of interest, a scalar valued parameter called the strategy parameter will be assigned to this trait making the simulation of strategy evolution possible. The trait of interest, and therefore the strategy associated, will be the ability of a population to keep its densities within the carrying capacity of the environment they find themselves in. The Shepherd function, on account of its wide use in population simulations as well as composing of exactly such a density parameter, will be the density curbing mechanism of choice in the age-structured population model designed here. An algorithm will be designed to simulate strategy evolution towards an evolutionary stable strategy or ESS that will ensure not only an optimal fit for this environment but also render the population immune against future invasion by other members of the population practising slight variations of this strategy. There are two ways to come by such an optimal strategy without directly involving genetics. The first is game theory, allowing strategists to compete for this position, and the second is with the use of adaptive dynamics, converting winning and loosing instead into tangible mathematics. Combining these two classics will show that the quest is an exercise in strategy optimization, not only from the point of view of an already established population but also from the point of view of an initially small one. It will be interesting!