Struggle for Existence: the models for Darwinian and non-Darwinian selection

Classical understanding of the outcome of the struggle for existence results in the Darwinian survival of the fittest. Here we show that the situation may be different, more complex and arguably more interesting. Specifically, we show that different versions of non-homogeneous logistic-like models with a distributed Malthusian parameter imply non-Darwinian survival of everybody. In contrast, the non-homogeneous logistic equation with distributed carrying capacity shows Darwinian survival of the fittest. We also consider an non-homogeneous birth-and-death equation and give a simple proof that this equation results in the survival of the fittest. In addition to this known result, we find an exact limit distribution of the parameters of this equation. We also consider frequency-dependent non-homogeneous models and show that although some of these models show Darwinian survival of the fittest, there is not enough time for selection of the fittest species. We discuss the well-known Gauze Competitive exclusion principle that states that Complete competitors cannot coexist. While this principle is often considered as a direct consequence of the Darwinian survival of the fittest, we show that from the point of view of developed mathematical theory complete competitors can in fact coexist indefinitely.