Ivan Jordanov

On nonlinear population waves

We discuss a model system of partial differential equations for description of the spatio-temporal dynamics of interacting populations. We are interested in the waves caused by migration of the populations. We assume that the migration is a diffusion process influenced by the changing values of the birth rates and coefficients of interaction among the populations. For the particular case of one population and one spatial dimension the general model is reduced to analytically tractable PDE with polynomial nonlinearity up to 4th order. We investigate this particular case and obtain two kinds of solutions: (i) approximate solution for small value of the ratio between the coefficient of diffusion and the wave velocity and (ii) exact solutions which describe nonlinear kink and solitary waves. In an appropriate phase space the kinks correspond to a connection between two states represented by a saddle point and a stable node. Finally we derive conditions for the asymptotic stability of the obtained solutions.

Analysis of a Japan government intervention on the domestic agriculture market

We investigate an economic system in which one large agent—the Japan government changes the environment of numerous smaller agents—the Japan agriculture producers by indirect regulation of prices of agriculture goods. The reason for this intervention was that before the oil crisis in 1974 Japan agriculture production prices exhibited irregular and large amplitude changes. By means of analysis of correlations and a combination of singular spectrum analysis (SSA), principal component analysis (PCA), and time delay phase space construction (TDPSC) we study the influence of the government measures on the domestic piglet prices and production in Japan. We show that the government regulation politics was successful and lead: (i) to a decrease of the nonstationarities and to increase of predictability of the piglet price; (ii) to a coupling of the price and production cycles; and (iii) to increase of determinism of the dynamics of the fluctuations of piglet price around the year average price. The investigated case is an example confirming the thesis that a large agent can change in a significant way the environment of the small agents in complex (economic or financial) systems which can be crucial for their survival or extinction.

ON NONLINEAR WAVES IN THE SPATIO-TEMPORAL DYNAMICS OF INTERACTING POPULATIONS

Abstract In this paper the spatial-temporal dynamics of the members of interacting populations is described by means of nonlinear partial differential equations. We consider the migration as a diffusion process influenced by the changing values of the birth rates and the coefficients of interaction between the populations. The general model is reduced to analytically tractable partial differential equations (PDE) with polynomial nonlinearity up to third order for the particular case of one population and one spatial dimension. We obtain an analytical solution which describes nonlinear kink and solitary waves in the population dynamics by applying the modified method of simplest equation to the described model.

Numerical modeling of dynamics of a population system with time delay: DUSHKOV et al.

Mathematical models of interacting populations are often constructed as systems of differential equations, which describe how populations change with time. Below we study one such model connected to the nonlinear dynamics of a system of populations in presence of time delay. The consequence of the presence of the time delay is that the nonlinear dynamics of the studied system become more rich, e.g., new orbits in the phase space of the system arise which are dependent on the time-delay parameters. In more detail we introduce a time delay and generalize the model system of differential equations for the interaction of three populations based on generalized Volterra equations in which the growth rates and competition coefficients of populations depend on the number of members of all populations \cite{Dimitrova2001a},\cite{Dimitrova2001b} and then numerically solve the system with and without time delay. We use a modification of the method of Adams for the numerical solution of the system of model equations with time delay. By appropriate selection of the parameters and initial conditions we show the impact of the delay time on the dynamics of the studied population system.

On Nonlinear Waves in the Space-Temporal Dynamics of the ERK Signaling Protein

In this paper we model the spatial and temporal dynamics of ERK and STAT protein interaction by means of nonlinear partial differential equations. We show that the diffusion together with the corresponding biochemical reactions is likely to play a critical role in governing the dynamical behavior of the ERK and STAT interaction system. We reduce the above mentioned system to analytically tractable PDE with polynomial nonlinearity up to third order. By applying the modified method of simplest equation to the described model we obtain an analytical solution which describes drop and jump propagation of the ERK protein concentration.