We present a sufficient condition for robust permanence of ecological (or Kolmogorov) differential equations based on average Liapunov functions. Via the minimax theorem we rederive Schreiber's sufficient condition [S. Schreiber, J. Differential Equations, 162 (2000), pp. 400-426] in terms of Liapunov exponents and give various generalizations. Then we study robustness of permanence criteria against discretizations with fixed and variable stepsizes. Applications to mathematical ecology and evolutionary games are given.

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Background For the understanding of human nature, the evolutionary roots of human moral behaviour are a key precondition. Our question is as follows: Can the altruistic moral rule “Risk your life to save your family members, if you want them to save your life” be evolutionary stable? There are three research approaches to investigate this problem: kin selection, group selection and population genetics modelling. The present study is strictly based on the last approach. Results We consider monogamous and exogamous families, where at an autosomal locus, dominant-recessive alleles determine the phenotypes in a sexual population. Since all individuals’ survival rate is determined by their altruistic family members, we introduce a new population genetics model based on the mating table approach and adapt the verbal definition of evolutionary stability to genotypes. In general, when the resident is recessive, a homozygote is an evolutionarily stable genotype (ESG), if the number of survivors of the resident genotype of the resident homozygote family is greater than that of non-resident heterozygote survivors of the family of the resident homozygote and mutant heterozygote genotypes. Using the introduced genotype dynamics we proved that in the recessive case ESG implies local stability of the altruistic genotype. We apply our general ESG conditions for self-sacrificing life history strategy when the number of new-born offspring does not depend on interactions within the family and the interactions are additive. We find that in this case our ESG conditions give back Hamilton’s rule for evolutionary stability of the self-sacrificing life history strategy. Conclusions In spite of the fact that the kidney transplantations was not a selection factor during the earlier human evolution, nowadays “self-sacrificing” can be observed in the live donor kidney transplantations, when the donor is one of the family members. It seems that selection for self-sacrificing in family produced an innate moral tendency in modulating social cognition in human brain. Electronic supplementary material The online version of this article (10.1186/s12862-019-1478-0) contains supplementary material, which is available to authorized users.

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We report on experiences with an adaptive subdivision method supported by interval arithmetic that enables us to prove subset relations of the form T (W ) ⊂ U and thus to check certain sufficient conditions for chaotic behaviour of dynamical systems in a rigorous way.Our proof of the underlying abstract theorem avoids of referring to any results of applied algebraic topology and relies only on the Brouwer fixed point theorem.The second novelty is that the process of gaining the subset relations to be checked is, to a large extent, also automatized. The promising subset relations come from solving a constrained optimization problem via the penalty function approach.Abstract results and computational methods are demonstrated by finding embedded copies of the standard horseshoe dynamics in iterates of the classical Hénon mapping.