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We consider an ensemble of mass collisionless particles, which interact mutually either by an attraction of Newton's law of gravitation or by an electrostatic repulsion of Coulomb's law, under a background downward gravity in a…
Standard diffusion equation is based on Brownian motion of the dispersing species without considering persistence in the movement of the individuals. This description allows for the instantaneous spreading of the transported species over an…
This paper proposes a unified approach for studying global exponential stability of a general class of switched systems described by time-varying nonlinear functional differential equations. Some new delay-independent criteria of global…
In this paper we consider the diffusive competition model with free boundary in the heterogeneous time-periodic environment, in which the variable intrinsic growth rates of invasive and native species may change signs and be "very negative"…
We revisit the dynamics of the one-dimensional self-gravitating sheets models. We show that homogeneous and non-homogeneous states have different ergodic properties. The former is non-ergodic and the one-particle distribution function has a…
In this paper, we consider the almost periodic dynamics of an impulsive multispecies Lotka-Volterra competition system with time delays on time scales. By establishing some comparison theorems of dynamic equations with impulses and delays…
Pattern-forming nonequilibrium systems are ubiquitous in nature, from driven quantum matter and biological life forms to atmospheric and interstellar gases. Identifying universal aspects of their far-from-equilibrium dynamics and statistics…
We study dynamics in a neighborhood of a nonhyperbolic fixed point or an irreducible homoclinic tangent point. General type conditions for the existence of infinite sets of periodic points are obtained. A new method, based on the study of…
We analyze under which conditions equilibration between two competing effects, repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction, occurs. This balance leads to continuous compactly supported radially…
Organisms modulate their fitness in heterogeneous environments by dispersing. Prior work shows that there is selection against "unconditional" dispersal in spatially heterogeneous environments. "Unconditional" means individuals disperse at…
We discuss recent work on the static and dynamical properties of the asymmetric exclusion process, generalized to include the effect of disorder. We study in turn: random disorder in the properties of particles; disorder in the spatial…
In theoretical ecology, models describing the spatial dispersal and the temporal evolution of species having non-overlapping generations are often based on integrodifference equations. For various such applications the environment has an…
We construct a continuum model for biological aggregations in which individuals experience long-range social attraction and short range dispersal. For the case of one spatial dimension, we study the steady states analytically and…
To describe population dynamics, it is crucial to take into account jointly evolution mechanisms and spatial motion. However, the models which include these both aspects, are not still well-understood. Can we extend the existing results on…
In this work, we study the dynamics of a spatially heterogeneous single population model with the memory effect and nonlinear boundary condition. By virtue of the implicit function theorem and Lyapunov-Schmidt reduction, spatially…
The current paper is concerned with the asymptotic dynamics of two species competition systems with/without chemotaxis in heterogeneous media. In the previous work \cite{ITBWS17a}, we find conditions on the parameters in such systems for…
In this thesis we investigate cosmological models more general than the isotropic and homogeneous Friedmann-Lemaitre models. We focus on cosmologies with one spatial degree of freedom, whose matter content consists of a perfect fluid and…
In complex systems, the interplay between nonlinear and stochastic dynamics, e.g., J. Monod's necessity and chance, gives rise to an evolutionary process in Darwinian sense, in terms of discrete jumps among attractors, with punctuated…
We show that for a large class of evolutionary nonlinear and nonlocal partial differential equations, symmetry of solutions implies very restrictive properties of the solutions and symmetry axes. These restrictions are formulated in terms…
We study nonlinear stability of spatially homogeneous oscillations in reaction-diffusion systems. Assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase, we prove that spatially localized perturbations…