We investigate global dynamics of the following systems of difference equations , , , where the parameters , , , , , and are positive numbers and the initial conditions and are arbitrary nonnegative numbers. This system is a version of the Leslie-Gower competition model for two species. We show that this system has rich dynamics which depends on the part of parametric space.

In this paper will be observed the population dynamics of a three-species Lotka-Volterra model: two predators and their prey. This simplified model yields a more complicated dynamical system than classic Lotka-Volterra model. We will give the conditions under which one of the predators becomes extinct and when the coexistence between predators is possible. Given will be sufficient conditions for the existence of solutions for certain classes of Cauchy’s solutions of Lotka-Volterra model. The behavior of integral curves in the neighborhoods of an arbitrary integral curves will be considered.

The proven safe operation and the high availability of aerial ropeways is mainly thanks to their design. This is based on the manufacturer’s extensive experience as well as the strict application of the relevant rules and norms. In that regard, this paper describes an analysis of the haulage ropes on ropeways in case of accidental loads. In solving this problem, analyzed ropeway system with sufficient accuracy was modeled as a system with three degrees of freedom. For solving differential equations we have used the software “Wolfram Mathematica”. At the end of the paper, we discuss the results of the dynamic forces in the haulage ropes in a certain time interval, and the results of the safety factors which in this case are sufficient to ensure reliable operation of the system.

This paper presents sufficient conditions for the existence of solutions for certain classes of Cauchy’s solutions of the Lagerstrom equation as well as their behavior. Behavior of integral curves in the neighborhoods of an arbitrary or integral curve are considered. The obtained results contain the answer to the question on approximation of solutions whose existence is established. The errors of the approximation are defined by functions that can be sufficiently small. The theory of qualitative analysis of differential equations and topological retraction method are used.

In this paper, the method of measuring the bending properties of wood using spruce (Picea abies Karsten) by elementary bending theory has been examined. Three-point bending tests (EN 310) were conducted under various span/depth (L/h) ratios. Load-deflection diagram was simultaneously measured. Modulus of rapture MOR, bending strength in proportional limits SPL and Modulus of elasticity in bending MOE, were conducted under 3.4 to 30 span/depth ratios. The results suggest that for testing solid wood spruce the ratio should be L/h ≥ 20. This principle can be used as an aid to engineering purposes, particularly in controlling the production process. 1. BACKGRAUND Application of wood and composite materials based on wood are treated as unloaded elements in most of the wood industry. In accordance with that and with the European standard EN 310 [2], bending strength and modulus of elasticity in bending are tested by method of three point.The adopted method for flexural strength test ignores the effect of transverse forces or shear stresses caused by loading and reduces the calculation of bending strength of the Euler-Bernoulli theory to pure bending. Many factors influence the mechanical properties of wood, as non-homogeneous and orthotropic material: moisture contents, slope of grain, temperature, annual ring width, density, etc. Given the adopted method of testing, bending depends on the specimen, span/depth ratio, load, loading speed etc.