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Highlights It is proposed a model based on non-local operators capable of reproduce several of the equations traditionally used in research on population dynamics, as well as having additional features, useful for describing the phenomenon of pattern formation. The combination of two of the parameters present in the model, namely, α and β , determines the existence of a phase space, characterized by regions with and without patterns. The patterns formed will have M peaks and M valleys, with M ≥ 2. There are ( α, β ) combinations which lead to the formation of degenerate states, where the system seems not to forget its initial conditions. The model allows the study of real systems, such is the case of bacterial populations subjected to inhomogeneous growth conditions. The parameter β represents a macroscopic measure of changes that occur at an individual level in the concentrations of proteins in bacteria, as a product of significant variations in the environment.

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This work studies the compound nucleus' de-excitation during spallation nuclear reactions, with focus on the behavior of the Coulomb barrier for emission of charged particles and the level density parameter at the fission saddle point, establishing for the first time a relationship between these magnitudes and the level density. This relationship manifests itself through two differential equations explicitly dependent on the excitation energy and the level density parameter of the nucleus after the emission of light particles. These equations were implemented into the CRISP code and used to simulate proton induced spallation reactions on Pb 208 and 238 U, respectively, comparing the results with data coming from GSI experiments as well as from other codes and obtaining satisfactory results that confirm the reliability of the new models.

This work presents new approximate analytical solutions for the Riccati equation (RE) resulting from the application of the method of variation of parameters. The original equation is solved using another RE explicitly dependent on the independent variable. The solutions obtained are easy to implement and highly applicable, which is confirmed by solving several examples corresponding to REs whose solution is known, as well as optimizing the method to determine the density of the members that make up a population. In this way, new perspectives are open in the study of the phenomenon of pattern formation.