Self-replication and splitting of domain patterns in reaction-diffusion systems with the fast inhibitor

Bump competition and lattice solutions in two-dimensional neural fields

Some forms of competition among activity bumps in a two-dimensional neural field are studied. First, threshold dynamics is included and rivalry evolutions are considered. The relations between parameters and dominance durations can match experimental observations about ageing. Next, the threshold dynamics is omitted from the model and we focus on the properties of the steady-state. From noisy inputs, hexagonal grids are formed by a symmetry-breaking process. Particular issues about solution existence and stability conditions are considered. We speculate that they affect the possibility of producing basis grids which may be combined to form feature maps.

Instabilities of a three-dimensional localized spot

We investigate the behavior of localized spots in three spatial dimensions in a model two-variable system describing the Belousov-Zhabotinsky reaction in water-in-oil microemulsion. We find three types of instabilities: splitting of a single spot (i) into two spots, (ii) into a torus, and (iii) into an unstable shell that splits almost immediately to six or eight new spots.

Excitable population dynamics, biological control failure, and spatiotemporal pattern formation in a model ecosystem

Biological control has been attracting an increasing attention over the last two decades as an environmentally friendly alternative to the more traditional chemical-based control. In this paper, we address robustness of the biological control strategy with respect to fluctuations in the controlling species density. Specifically, we consider a pest being kept under control by its predator. The predator response is assumed to be of Holling type III, which makes the system's kinetics "excitable." The system is studied by means of mathematical modeling and extensive numerical simulations. We show that the system response to perturbations in the predator density can be completely different in spatial and non-spatial systems. In the nonspatial system, an overcritical perturbation of the population density results in a pest outbreak that will eventually decay with time, which can be regarded as a success of the biological control strategy. However, in the spatial system, a similar perturbation can drive the system into a self-sustained regime of spatiotemporal pattern formation with a high pest density, which is clearly a biological control failure. We then identify the parameter range where the biological control can still be successful and describe the corresponding regime of the system dynamics. Finally, we identify the main scenarios of the system response to the population density perturbations and reveal the corresponding structure of the parameter space of the system.

Localized patterns in reaction-diffusion systems

We discuss a variety of experimental and theoretical studies of localized stationary spots, oscillons, and localized oscillatory clusters, moving and breathing spots, and localized waves in reaction-diffusion systems. We also suggest some promising directions for future research in this area.

Spot deformation and replication in the two-dimensional Belousov-Zhabotinski reaction in a water-in-oil microemulsion

In the limit of a large duffusivity ratio, spotlike solutions in the two-dimensional Belousov-Zhabotinski reaction in water-in-oil microemulsion are studied. It is shown analytically that such spots undergo an instability as the diffusivity ratio is decreased. An instability threshold is derived. For spots of small radius, it is shown that this instability leads to a spot splitting into precisely two spots. For larger spots, it leads to deformation, fingering patterns, and space-filling curves. Numerical simulations are shown to be in close agreement with the analytical predictions.

Hydrodynamics of bacterial colonies: a model

We propose a hydrodynamic model for the evolution of bacterial colonies growing on soft agar plates. This model consists of reaction-diffusion equations for the concentrations of nutrients, water, and bacteria, coupled to a single hydrodynamic equation for the velocity field of the bacteria-water mixture. It captures the dynamics inside the colony as well as on its boundary and allows us to identify a mechanism for collective motion towards fresh nutrients, which, in its modeling aspects, is similar to classical chemotaxis. As shown in numerical simulations, our model reproduces both usual colony shapes and typical hydrodynamic motions, such as the whirls and jets recently observed in wet colonies of Bacillus subtilis. The approach presented here could be extended to different experimental situations and provides a general framework for the use of advection-reaction-diffusion equations in modeling bacterial colonies.

Theory of domain patterns in systems with long-range interactions of Coulomb type

We develop a theory of the domain patterns in systems with competing short-range attractive interactions and long-range repulsive Coulomb interactions. We take an energetic approach, in which patterns are considered as critical points of a mean-field free energy functional. Close to the microphase separation transition, this functional takes on a universal form, allowing us to treat a number of diverse physical situations within a unified framework. We use asymptotic analysis to study domain patterns with sharp interfaces. We derive an interfacial representation of the pattern's free energy which remains valid in the fluctuating system, with a suitable renormalization of the Coulomb interaction's coupling constant. We also derive integro-differential equations describing stationary domain patterns of arbitrary shapes and their thermodynamic stability, coming from the first and second variations of the interfacial free energy. We show that the length scale of a stable domain pattern must obey a certain scaling law with the strength of the Coulomb interaction. We analyzed the existence and stability of localized (spots, stripes, annuli) and periodic (lamellar, hexagonal) patterns in two dimensions. We show that these patterns are metastable in certain ranges of the parameters and that they can undergo morphological instabilities leading to the formation of more complex patterns. We discuss nucleation of the domain patterns by thermal fluctuations and pattern formation scenarios for various thermal quenches. We argue that self-induced disorder is an intrinsic property of the domain patterns in the systems under consideration.

Complex patterns predicted in an in vitro experimental model system for the evolution of molecular cooperation

An isothermal biochemical in vitro amplification system with two trans-cooperatively coupled amplifying DNA molecules was investigated homogeneously using a hierarchy of kinetic models and as a simplified reaction-diffusion system. In our model of this recently developed experimental system, no reaction mechanism higher than second order occurs, yet numerical simulations show a variety of complex spatiotemporal patterns which arise in response to finite amplitude perturbations in a flow reactor. In a certain domain of the kinetic parameters the system shows self-replicating spots. These spots can stabilize the cooperative amplification in such evolving systems against emerging parasites. The results are of high relevance for experimental studies on these functional in vitro ecosystems in spatially resolved microstructured reactors.