During the last 30 years considerable attention was paid to open systems far from thermal equilibrium. Under certain conditions these dissipative systems show a qualitatively new behavior on macroscopic length scales, which are known as spatiotemporal structures. These new structures arise as a feature of collective behavior of a many-body systems. One particular example of dissipative systems considered in the present Thesis is the systems with reactant birth and death. Such systems arise, e.g., in description of the population growth or the kinetics of chemical reactions. To describe the systems with a large number of particles, one has to impose some restrictions. So, it is assumed that individual properties of particles are not important, only their interaction and interaction result (reaction) are taken into account. A number of rules, which describe the behavior of particles on the microscopic level, are known as a mathematical model. There exist two methods to analyze properties of a mathematical model. The first is analysis based on the master equation. In general, this method fails to describe the properties of spatiotemporal structures. There are no analytical approximations taking into account the effect of long-range particle correlation, which is important for description of the changes on a macroscopic range. The second approach are Monte Carlo (MC) computer simulations, which actually is alternative to experiments. The MC method takes into account long-range reactant correlations. They arise as a result of microscopical model. MC has disadvantages typical for all numerical methods, e.g., a large simulation time. In the present Thesis the Lotka-type and the A+B->0 models are considered in detail. These reactions are commonly found as one of a component in many chemical reactions. The emphasis is made on understanding the basic properties of these models. Further, several physically important modifications of the Lotka-type and the A+B->0 models are made. Firstly, in Chapter 1. the Lotka-type model is extended to investigate the resonance properties. Secondly, the effect of reactant diffusion and interaction is incorporated into Lotka-type model in Chapter 2. Thirdly, the standard A+B->0 reaction is extended to the case of surface reconstruction in Chapter 3. General conclusion is presented at the end of the Thesis, which is ended by four Appendices.