Flexibility of Lyapunov exponents

We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.

Non-smooth Dynamics Emerging from Predator-driven Discontinuous Prey Dispersal

In ecology, the refuge protection of the prey plays a significant role in the dynamics of the interactions between prey and predator. In this paper, we investigate the dynamics of a non-smooth prey-predator mathematical model characterized by density-dependent intermittent refuge protection of the prey. The model assumes the population density of the predator as an index for the prey to decide on when to avail or discontinue refuge protection, representing the level of apprehension of the prey by the predators. We apply Filippov's regularization approach to study the model and obtain the sliding segment of the system. We obtain the criterion for the existence of the regular or virtual equilibria, boundary equilibrium, tangent points, and pseudo-equilibria of the Filippov system. The conditions for the visibility (or invisibility) of the tangent points are derived. We investigate the regular or virtual equilibrium bifurcation, boundary-node bifurcation and pseudo-saddle-node bifurcation. Further, we examine the effects of dispersal delay on the Filippov system associated with prey vigilance in identifying the predator population density. We observe that the hysteresis in the Filippov system produces stable limit cycles around the predator population density threshold in some bounded region in the phase plane. Moreover, we find that the level of apprehension and vigilance of the prey play a significant role in their refuge-dispersion dynamics.

Evaluation of Hormone-Induced Stress Responses Using Endogenous Cortisol in Carp (Cyprinus carpio)

With stress studies in fish it is often difficult to determine the degree of response to various stressors, and the interpretation of this influence is usually based on an increase in endogenous cortisol levels. Simulation of stress with synthetic corticosteroids is widely used in ichthyological practice, which raises the question of whether endogenous cortisol is the most appropriate parameter for measuring stress levels in such studies. This work presents the dynamics of the plasma cortisol level in simulating acute and chronic stress in 24 carps by a single injection of dexamethasone in the first experimental group and betamethasone in the second experimental group, in comparison with the control group (without injection) for 21 days. The analysis was performed before injection, as well as after 7, 14 and 21 days of treatment. The hormonal response was compared with that of fish stressed by natural factors (hypoxia). It was found that betamethasone inactivates the production of endogenous cortisol during all subsequent days of the experiment after injection from 353.68±66.39 ng/ml to 7.28±1.27 ng/ml by day 21, while the effect of dexamethasone caused multidirectional fluctuations in its level: from 346.25±43.16 ng/ml to 242.25±58.49 ng/ml on the 7th day, 388.25±37.51 ng/ml on the 14th day and 264.25±21.21 ng/ml on day 21 compared with smooth dynamics in control fish: 376.25±44.04 ng/ml, 366.75±42.82 ng/ml, 335.33±8.57 ng/ml and 366.00±89.22 ng/ml, respectively. It was concluded that measuring the level of endogenous cortisol is not recommended when assessing the degree of stress imitation by these hormones, and in studies of this type it is necessary to search for other indicators. Keywords: carp, cortisol, stress, dexamethasone, betamethasone, hormone