МОДЕЛЮВАННЯ ЧИСЕЛЬНОСТІ ГРИЗУНІВ У ЛІСОВИХ БІОТОПАХ ЗАХІДНОГО ПОДІЛЛЯ (НА ПРИКЛАДІ MYODES GLAREOLUS)

Monitoring and predicting the dynamics of abundance of species living in natural habitats is an important component stability analysis of ecosystem as well as dynamics and direction of change of biotic communities under global climate change and pressure of the alien species. The aim of the work was to build a matrix model and study the state of stabilisation of the dynamics of the bank vole population within the Leslie model. The object of the study was the population dynamics of Myodes glareolus Schreber, 1780 = Clethrionomys glareolus auct. The study is based on materials obtained during 2017–2019. This period covered one phase of the long-term population dynamics of the bank vole, named “population growth”. The research was carried out according to generally accepted methods. A total of 6400 trap-days were processed, and 358 forest fistulas were collected and studied. The intensity of harmful activity of rodents is due to the variability of the number of animals in the population. The quantitative population changes are the result of three factors: births, deaths, and migrations. The main condition for the existence of the species is the stability of the population, which is determined by the action of thecompensatory mechanisms. The growth phase of the bank vole lasted all three years of the research, the quantitative indicators were respectively: 2017 – 1.8 individuals per 100 trap-days; 2018 – 2.0 individuals per 100 trap-days; 2019 – 2.7 individuals per 100 trap-days. Low levels of the abundance in the spring of each year of the study, namely at the beginning of the breeding season (3.7 – 2.6 – 8.9 individuals per 100 trap-days). Favourable for the abundance growth was the sex ratio of the population (approximately 1:1), with some rise in the share of females, which decreases on the period of spring 2018 to autumn 2019). Some decrease in the share of immature individuals (4.5 – 3.9 – 3.1 %) is an indirect confirmation of the stability of puberty of animals with subsequent replenishment of the "stock", which led to accelerated reproduction and, consequently, provided prerequisites for further population growth. The causal mechanisms of population control established by us, without a doubt, can serve as a basis for further prognosis, of the number of pests in natural habitats. To predict population changes, the Leslie model, which is widely used in mathematical analyses of the abundance of both plant and animal groups, was chosen. The algorithm for building a matrix model, detailed in the article, has five following steps. The exponential nature of the actual and projected growth of the bank vole population during the five-year cycle (2017–2019 with a prognosis until 2023) revealed in the analysis can be explained not so much by the power of the species' reproductive potential as by the lack of the significant changes in habitat, caused by constant weather conditions, low individual mortality from predators and non-communicable diseases or other accidents. The application of the matrix model allowed to confirm the key role of the main compensatory mechanisms of population dynamics, as they contribute to the stabilisation of the population and, as a consequence, are an important condition for the existence of the species.

From kinetic to fluid models of liquid crystals by the moment method

<p style='text-indent:20px;'>This paper deals with the convergence of the Doi-Navier-Stokes model of liquid crystals to the Ericksen-Leslie model in the limit of the Deborah number tending to zero. While the literature has investigated this problem by means of the Hilbert expansion method, we develop the moment method, i.e. a method that exploits conservation relations obeyed by the collision operator. These are non-classical conservation relations which are associated with a new concept, that of Generalized Collision Invariant (GCI). In this paper, we develop the GCI concept and relate it to geometrical and analytical structures of the collision operator. Then, the derivation of the limit model using the GCI is performed in an arbitrary number of spatial dimensions and with non-constant and non-uniform polymer density. This non-uniformity generates new terms in the Ericksen-Leslie model.</p>

Solution landscapes of the simplified Ericksen–Leslie model and its comparisonwith the reduced Landau–deGennes model

We investigate the solution landscapes of a simplified Ericksen–Leslie (sEL) vector model for nematic liquid crystals, confined in a two-dimensional square domain with tangent boundary conditions. An efficient numerical algorithm is developed to construct the solution landscapes by utilizing the symmetry properties of the model and the domain. Since the sEL model and the reduced Landau–de Gennes (rLdG) models can be viewed as Ginzburg–Landau functionals, we systematically compute the solution landscapes of the sEL model, for different domain sizes, and compare them with the solution landscapes of the corresponding rLdG model. There are many similarities, including the stable diagonal and rotated states, bifurcation behaviours and sub-solution landscapes with low-index saddle solutions. Significant disparities also exist between the two models. The sEL vector model exhibits the stable solution C ± with interior defects, high-index ‘fake defect’ solutions, novel tessellating solutions and certain types of distinctive dynamical pathways. The solution landscape approach provides a comprehensive and efficient way for model comparison and is applicable to a wide range of mathematical models in physics.

The Uniaxial Limit of the Non-Inertial Qian–Sheng Model for Liquid Crystals

In this article, we consider the Qian–Sheng model in the Landau–de Gennes framework describing nematic liquid crystal flows when the inertial effect is neglected. By taking the limit of elastic constant to zero (also called the uniaxial limit) and utilizing the so-called Hilbert expansion method, we provide a rigorous derivation from the non-inertial Qian–Sheng model to the Ericksen–Leslie model.

Concentration-cancellation in the Ericksen–Leslie model

We establish the subconvergence of weak solutions to the Ginzburg–Landau approximation to global-in-time weak solutions of the Ericksen–Leslie model for nematic liquid crystals on the torus $${\mathbb {T}^2}$$ T 2 . The key argument is a variation of concentration-cancellation methods originally introduced by DiPerna and Majda to investigate the weak stability of solutions to the (steady-state) Euler equations.