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

Classical understanding of the outcome of the struggle for existence results\nin the Darwinian survival of the fittest. Here we show that the situation may\nbe different, more complex and arguably more interesting. Specifically, we show\nthat different versions of non-homogeneous logistic-like models with a\ndistributed Malthusian parameter imply non-Darwinian survival of everybody. In\ncontrast, the non-homogeneous logistic equation with distributed carrying\ncapacity shows Darwinian survival of the fittest. We also consider an\nnon-homogeneous birth-and-death equation and give a simple proof that this\nequation results in the survival of the fittest. In addition to this known\nresult, we find an exact limit distribution of the parameters of this equation.\nWe also consider frequency-dependent non-homogeneous models and show that\nalthough some of these models show Darwinian survival of the fittest, there is\nnot enough time for selection of the fittest species. We discuss the well-known\nGauze Competitive exclusion principle that states that Complete competitors\ncannot coexist. While this principle is often considered as a direct\nconsequence of the Darwinian survival of the fittest, we show that from the\npoint of view of developed mathematical theory complete competitors can in fact\ncoexist indefinitely.\n