Controling delayed transitions with applications to prevent single species extinctions

A new method is proposed to control delayed transitions towards extinction in single population theoretical models with discrete time undergoing saddle-node bifurcations. The control method takes advantage of the delaying properties of the saddle remnant arising after the bifurcation, and allows to sustain populations indefinitely. Our method, which is shown to work for deterministic and stochastic systems, could generally be applied to avoid transitions tied to one-dimensional maps after saddle-node bifurcations.

Dynamical analysis in growth models: Blumberg’s equation

We present a new dynamical approach to the Blumberg's equation, a family of unimodal maps. These maps are proportional to Beta(p, q) probability densities functions. Using the symmetry of the Beta(p, q) distribution and symbolic dynamics techniques, a new concept of mirror symmetry is defined for this family of maps. The kneading theory is used to analyze the effect of such symmetry in the presented models. The main result proves that two mirror symmetric unimodal maps have the same topological entropy. Different population dynamics regimes are identified, when the intrinsic growth rate is modified: extinctions, stabilities, bifurcations, chaos and Allee effect. To illustrate our results, we present a numerical analysis, where are demonstrated: monotonicity of the topological entropy with the variation of the intrinsic growth rate, existence of isentropic sets in the parameters space and mirror symmetry.

Mutation accumulation in Tetrahymena

Background - The rate and fitness effects of mutations are key in understanding the evolution of every species. Traditionally, these parameters are estimated in mutation accumulation experiments where replicate lines are propagated in conditions that allow mutations to randomly accumulate without the purging effect of natural selection. These experiments have been performed with many model organisms but we still lack empirical estimates of the rate and effects of mutation in the protists. Results - We performed a mutation accumulation (MA) experiment in Tetrahymena thermophila, a species that can reproduce sexually and asexually in nature, and measured both the mean decline and variance increase in fitness of 20 lines. The results obtained with T. thermophila were compared with T. pyriformis that is an obligate asexual species. We show that MA lines of T. thermophila go to extinction at a rate of 1.25 clonal extinctions per bottleneck. In contrast, populations of T. pyriformis show a much higher resistance to extinction. Variation in gene copy number is likely to be a key factor in explaining these results, and indeed we show that T. pyriformis has a higher mean copy number per cell than T. thermophila. From fitness measurements during the MA experiment, we infer a rate of mutation to copy number variation of 0.0333 per haploid MAC genome of T. thermophila and a mean effect against copy number variation of 0.16. A strong effect of population size in the rate of fitness decline was also found, consistent with the increased power of natural selection. Conclusions - The rate of clonal extinction measured for T. thermophila is characteristic of a mutational degradation and suggests that this species must undergo sexual reproduction to avoid the deleterious effects detected in the laboratory experiments. We also suggest that an increase in chromosomal copy number associated with the phenotypic assortment of amitotic divisions can provide an alternative mechanism to escape the deleterious effect of random chromosomal copy number variation in species like T. pyriformis that lack the resetting mechanism of sexual reproduction. Our results are relevant to the understanding of cell line longevity and senescence in ciliates.

Scaling law in Saddle-node bifurcations for one-dimensional maps: A complex variable approach

The study of transient dynamical phenomena near bifurcation thresholds has attracted the interest of many researchers due to the relevance of bifurcations in different physical or biological systems. In the context of saddle-node bifurcations, where two or more fixed points collide annihilating each other, it is known that the dynamics can suffer the so-called delayed transition. This phenomenon emerges when the system spends a lot of time before reaching the remaining stable equilibrium, found after the bifurcation, because of the presence of a saddle-remnant in phase space. Some works have analytically tackled this phenomenon, especially in time-continuous dynamical systems, showing that the time delay, tau, scales according to an inverse square-root power law, tau similar to (mu-mu (c) )(-1/2), as the bifurcation parameter mu, is driven further away from its critical value, mu (c) . In this work, we first characterize analytically this scaling law using complex variable techniques for a family of one-dimensional maps, called the normal form for the saddle-node bifurcation. We then apply our general analytic results to a single-species ecological model with harvesting given by a unimodal map, characterizing the delayed transition and the scaling law arising due to the constant of harvesting. For both analyzed systems, we show that the numerical results are in perfect agreement with the analytical solutions we are providing. The procedure presented in this work can be used to characterize the scaling laws of one-dimensional discrete dynamical systems with saddle-node bifurcations.