M. A. Tsyganov

Pursuit-evasion predator-prey waves in two spatial dimensions

We consider a spatially distributed population dynamics model with excitable predator-prey kinetics, where species propagate in space due to their taxis with respect to each others gradient in addition to, or instead of, their diffusive spread. Earlier, we have described new phenomena in this model in one spatial dimension, not found in analogous systems without taxis: reflecting and self-splitting waves. Here we identify new phenomena in two spatial dimensions: unusual patterns of meander of spirals, partial reflection of waves, swelling wave tips, attachment of free wave ends to wave backs, and as a result, a novel mechanism of self-supporting complicated spatiotemporal activity, unknown in reaction-diffusion population models.

Solitary waves in excitable systems with cross-diffusion

We consider a FitzHugh–Nagumo system of equations where the traditional diffusion terms are replaced with linear cross-diffusion of components. This system describes solitary waves that have unusual form and are capable of quasi-soliton interaction. This is different from the classical FitzHugh–Nagumo system with self-diffusion, but similar to a predator–prey model with taxis of populations on each others gradient which we considered earlier. We study these waves by numerical simulations and also present an analytical theory, based on the asymptotic behaviour which arises when the local dynamics of the inhibitor field are much slower than those of the activator field.

Formation of stationary demarcation zones between population autowaves propagating towards each other

Abstract Two kinds of autowaves are known to exist in the simplest mathematical models of excitable media: Ω and KΩ waves. One of the essential differences between them rests in the fact that the Ω waves collide whereas the KΩ waves do not. In the present paper we show that both Ω- and KΩ-like waves occur in the essentially more complex population expansion mathematical model. The Ω-like waves propagate at higher rates than do the KΩ-like ones. Both types of autowaves can carry out transitions from unstable spatial structures into other, stable ones. The results of computer experiments are supported by our experiments in vivo.

pH track of expanding bacterial populations

A method of pH distribution measurements in agar nutrient media containing expanding bacterial populations is described. It is based on measuring pH microsamples taken at different points of the media. The sample volume was 10 microliters. A pH sensitive field effect transistor was used as a measuring electrode. Acidification was found to occur in glucose media, while alkalization occurred in the media containing peptone.

Oscillatory pulses and wave trains in a bistable reaction-diffusion system with cross diffusion

We study waves with exponentially decaying oscillatory tails in a reaction-diffusion system with linear cross diffusion. To be specific, we consider a piecewise linear approximation of the FitzHugh-Nagumo model, also known as the Bonhoeffer-van der Pol model. We focus on two types of traveling waves, namely solitary pulses that correspond to a homoclinic solution, and sequences of pulses or wave trains, i.e., a periodic solution. The effect of cross diffusion on wave profiles and speed of propagation is analyzed. We find the intriguing result that both pulses and wave trains occur in the bistable cross-diffusive FitzHugh-Nagumo system, whereas only fronts exist in the standard bistable system without cross diffusion.

Bacterial population autowave patterns: spontaneous symmetry bursting

Abstract Bacteria are known to form autowave patterns (population waves) like those formed by propagating nerve impulses, phase transitions, concentration waves in the Belousov-Zhabotinsky reaction, etc. The formation of bacterial waves is due to the ability of bacteria to drift (through chemotaxis) into the regions with higher attractant concentration. As a result, in contrast to other types of autowaves, bacterial population waves have not only a diffusion component of a bacterial flow but a chemotaxis flow as well. We present the experimental results of the study of spontaneous symmetry loss of bacterial autowave patterns. We show that this phenomenon can be simulated with a simple cellular automata model, and symmetry bursting depends on the parameters characterizing chemotactic sensitivity and motility of the cells forming the population wave. In the experiments in vivo we show that the distortion of a bacterial wave shape can be initiated by bacterial density fluctuations in the parent population that the bacterial waves flake off from.

Spontaneous traveling waves in oscillatory systems with cross diffusion.

We identify a type of pattern formation in spatially distributed active systems. We simulate one-dimensional two-component systems with predator-prey local interaction and pursuit-evasion taxis between the components. In a sufficiently large domain, spatially uniform oscillations in such systems are unstable with respect to small perturbations. This instability, through a transient regime appearing as spontaneous focal sources, leads to establishment of periodic traveling waves. The traveling wave regime is established even if boundary conditions do not favor such solutions. The stable wavelength is within a range bounded both from above and from below, and this range does not coincide with instability bands of the spatially uniform oscillations.

Negative refractoriness in excitable systems with cross-diffusion

The results of numerical experiments with mathematical models of excitable systems with cross-diffusion are presented. It was shown that the refractoriness in such systems may be negative. The effects of negative refractoriness on the propagation and interaction of waves are demonstrated.

Quasisolitons in self-diffusive excitable systems, or Why asymmetric diffusivity obeys the Second Law

Solitons, defined as nonlinear waves which can reflect from boundaries or transmit through each other, are found in conservative, fully integrable systems. Similar phenomena, dubbed quasi-solitons, have been observed also in dissipative, “excitable” systems, either at finely tuned parameters (near a bifurcation) or in systems with cross-diffusion. Here we demonstrate that quasi-solitons can be robustly observed in excitable systems with excitable kinetics and with self-diffusion only. This includes quasi-solitons of fixed shape (like KdV solitons) or envelope quasi-solitons (like NLS solitons). This can happen in systems with more than two components, and can be explained by effective cross-diffusion, which emerges via adiabatic elimination of a fast but diffusing component. We describe here a reduction procedure can be used for the search of complicated wave regimes in multi-component, stiff systems by studying simplified, soft systems.

Simulation of growth of colonies of filamentous fungi in a hydrogen peroxide gradient.

Microscopic fungi are potent destructors and utiliz-ers of organic and some inorganic substances. Forexample, microscopic fungi were shown to activelycover and destruct engineering and technological con-structions in the exclusion zone of the ChernobylAtomic Power Plant [1]. The main sources of radioac-tive contamination in this zone are the so-called “hot”particles of 1–100 µm in size, which have various radi-onuclide composition and possess a high specific activ-ity (10–1000 Bq/particle) [2]. Upon interaction withwater, all types of sources radiation generate active rad-icals, primarily hydrogen peroxide as the most stablecompound [3]. These isolated radiation sources createlocal gradients of hydrogen peroxide on moist sub-strates, affecting the growth rate of microscopic fungushyphae.In this work, we used mathematic modeling to studythe dynamics of growth of colonies of microscopicfungi in a hydrogen peroxide gradient.A study of nine strains of four species of filamen-tous fungi [4, 5] revealed two principally different typesof growth responses of filamentous fungi to hydrogenperoxide (Fig. 1). Hypha elongation rate of type A fungigradually decreased as the hydrogen peroxide concen-tration increase and completely stopped at high hydro-gen peroxide concentrations ( 10