A spatio-temporal model of megacity development that treats the megacity as an active medium is presented. From our point of view, it is advisable to consider the process of urban ecosystem development from the standpoint of the theory of autowave self-organization in active media. According to this concept, the urban ecosystem is considered as interacting with each other’s natural and anthropogenic subsystems with significant heterogeneity of areas affected by human intervention and urban geobiocoenoses. The model is based on the general principles of active medium dynamics; therefore, it is universal for any object to be considered an active medium. The only difference when using the model to predict the development of urban ecosystems in countries with different socio-economic and political prerequisites is the variety of parameters included in the model, i.e., the activation parameter, the autowave process inhibitors, and the characteristic scales of the activator and inhibitor. The model was tested on the example of Moscow expansion in the period of 1952–1968 and showed good agreement with the map data. By means of the model, a prediction of Shanghai and surrounding territory development until 2030 was made.

A two-dimensional singularly perturbed elliptic equation referred to in applications as the reaction–diffusion equation is considered. The nonlinearity describing the reaction is assumed to be discontinuous on a certain closed curve. On the basis of the generalized asymptotic comparison principle, the existence of smooth solution is proven and the accuracy of the asymptotic approximation is estimated.

Ecosystem carbon dioxide, energy, and water fluxes were measured using eddy covariance in a fresh clear-cut surrounded by a mixed spruce-birch-aspen forest in the boreal zone of European Russia. Measurements were initiated in spring 2016 following timber harvest and continued for five months. The influence of surrounding forest on air flow and turbulent fluxes within the clear-cut were examined using a process-based two-dimensional (2D) hydrodynamic turbulent exchange model. The clear-cut was a source of CO₂ to the atmosphere prior to onset of vegetation growth during early spring. During this period the mean daily latent (LE) and sensible (H) heat fluxes were very similar and the Bowen ratio (β = H/LE) averaged about 1.0. Daily net ecosystem exchange of CO₂ (NEE) was around 0 gC m⁻² d⁻¹ following onset of vegetation growth from mid-spring through summer, while β declined to 0.6–0.7. There was strong diurnal variability in NEE, LE and H over the measurement period that was governed by solar radiation and temperature as well as the leaf area index (LAI) of regrown vegetation. Modeled vertical CO₂ and H₂O fluxes along a transect that crossed the clear-cut and coincided with the dominate wind direction showed that the clear-cut strongly influenced turbulent fluxes within the atmospheric surface layer. Furthermore, modeled atmospheric dynamics suggested that the clear-cut had a large influence on turbulent fluxes in the downwind forest, but little impact on the upwind side. An aggregated approach including field measurements and process-based models can be a useful approach to estimate energy, water and carbon dioxide fluxes in non-uniform forest landscapes.

A singularly perturbed initial boundary value problem is considered for a parabolic equation that is known in application as the reaction-diffusion-advection equation. An asymptotic expansion of solutions with a moving front is constructed. This asymptotics is proved by the method of differential inequalities, which is based on well-known comparison theorems and develops the ideas of formal asymptotics for constructing upper and lower solutions in singularly perturbed problems with internal and boundary layers.

A singularly perturbed boundary value problem for a system of equations with different powers of a small parameter is considered in the one-dimensional case. The asymptotic behavior and existence of a solution with an internal transition layer are analyzed. The asymptotics are substantiated using the asymptotic method of differential inequalities.

A two-dimensional reaction–diffusion equation in a medium with discontinuous characteristics is considered; the existence, local uniqueness, and asymptotic stability of its stationary...