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Classical understanding of the outcome of the struggle for existence resultsin the Darwinian survival of the fittest. Here we show that the situation maybe different, more complex and arguably more interesting. Specifically, we showthat different versions of non-homogeneous logistic-like models with adistributed Malthusian parameter imply non-Darwinian survival of everybody. Incontrast, the non-homogeneous logistic equation with distributed carryingcapacity shows Darwinian survival of the fittest. We also consider annon-homogeneous birth-and-death equation and give a simple proof that thisequation results in the survival of the fittest. In addition to this knownresult, we find an exact limit distribution of the parameters of this equation.We also consider frequency-dependent non-homogeneous models and show thatalthough some of these models show Darwinian survival of the fittest, there isnot enough time for selection of the fittest species. We discuss the well-knownGauze Competitive exclusion principle that states that Complete competitorscannot coexist. While this principle is often considered as a directconsequence of the Darwinian survival of the fittest, we show that from thepoint of view of developed mathematical theory complete competitors can in factcoexist indefinitely.

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