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This thesis examines the construction of static solutions of (3+l)-dimensional SU{N) Skyrme models, usual and alternative, and pure massive SU(N) Yang-Mills theories. In particular, the application of harmonic maps from S(^2) into the subspace of fields configuration space M. Here, the harmonic maps are used as an ansatz to factoring out the angular dependence part of the solutions from the field equations. In this thesis, we consider the harmonic maps S(^2) → Gr(n, N), where Gr(n, N) is the Grassmann manifold of n-dimensional planes passing through the origin in C(^N). Using the harmonic map ansatz of S(^2) → Gr(2, N) to study the usual SU(N) Skyrme models, we have found that our approximate solutions have marginally higher energies in comparison to the corresponding results previously obtained using CP(^N-1) as target space M. For exact spherically symmetric solutions, we present arguments which suggest that the only solutions obtained this way are embeddings. For the alternative SU(N) Skyrme models, using the harmonic map ansatz of S(^2) → CP(^N-1), we have found that our results for the energies of the exact spherically symmetric solutions are higher than in the usual models. When considering the pure massive SU(N) Yang-Mills theories, we have shown that by choosing the gauge potential to be of almost pure gauge form, the theories reduce to the usual SU(N) Skyrme models. This observation has suggested to us to consider the harmonic map ansatz of S(^2) → CP(^N-1) previously applied to monopole theories. Using this ansatz, we have constructed some bounded spherically symmetric solutions of the theories having finite energies.

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 excersize 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!

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