A nonsmooth two-sex population model

This paper considers a two-dimensional logistic model to study populations with two genders. The growth behavior of a population is guided by two coupled ordinary differential equations given by a non-differentiable vector field whose parameters are the secondary sex ratio (the ratio of males to females at time of birth), inter-, intra- and outer-gender competitions, fertility and mortality rates and a mating function. For the case where there is no inter-gender competition and the mortality rates are negligible with respect to the density-dependent mortality, using geometrical techniques, we analyze the singularities and the basin of attraction of the system, determining the relationships between the parameters for which the system presents an equilibrium point. In particular, we describe conditions on the secondary sex ratio and discuss the role of the average number of female sexual partners of each male for the conservation of a two-sex species.Comment: 18 pages, 6 figures. Section 2, in which the model is presented, was rewritten to better explain the elements of the proposed model. The description of parameter "r" was correcte

Average sex ratio and population maintenance cost

The ratio of males to females in a population is a meaningful characteristic of sexual species. The reason for this biological property to be available to the observers of nature seems to be a question never asked. Introducing the notion of historically adapted populations as global minimizers of maintenance cost functions, we propose a theoretical explanation for the reported stability of this feature. This mathematical formulation suggests that sex ratio could be considered as an indirect result shaped by the antagonism between the size of the population and the finiteness of resources.Comment: 18 pages. A revised new version, where all the text was improved to become more clear for the reade

Weak KAM methods and ergodic optimal problems for countable Markov shifts

Let $\sigma:\boldsymbol{\Sigma}\to\boldsymbol{\Sigma}$ be the left shift acting on $\boldsymbol{\Sigma}$, a one-sided Markov subshift on a countable alphabet. Our intention is to guarantee the existence of $\sigma$-invariant Borel probabilities that maximize the integral of a given locally H\"older continuous potential $A : \boldsymbol{\Sigma} \to \mathbb R$. Under certain conditions, we are able to show not only that $A$-maximizing probabilities do exist, but also that they are characterized by the fact their support lies actually in a particular Markov subshift on a finite alphabet. To that end, we make use of objects dual to maximizing measures, the so-called sub-actions (concept analogous to subsolutions of the Hamilton-Jacobi equation), and specially the calibrated sub-actions (notion similar to weak KAM solutions).Comment: 15 pages. To appear in Bulletin of the Brazilian Mathematical Society

Exponential rate of decay of correlations of equilibrium states associated with non-uniformly expanding circle maps

In the context of expanding maps of the circle with an indifferent fixed point, understanding the joint behavior of dynamics and pairs of moduli of continuity $(\omega, \Omega)$ may be a useful element for the development of equilibrium theory. Here we identify a particular feature of modulus $\Omega$ (precisely $\lim_{x \to 0^+} \sup_{\mathsf d} \Omega\big({\mathsf d} x \big) / \Omega(\mathsf d) = 0$) as a sufficient condition for the system to exhibit exponential decay of correlations with respect to the unique equilibrium state associated with a potential having $\omega$ as modulus of continuity. This result is derived from obtaining the spectral gap property for the transfer operator acting on the space of observables with $\Omega$ as modulus of continuity, a property that, as is well known, also ensures the Central Limit Theorem. Examples of application of our results include the Manneville-Pomeau famil