Spreading speeds and traveling wave solutions of diffusive vector-borne disease models without monotonicity

Vector-borne diseases, such as chikungunya, dengue, malaria, West Nile virus, yellow fever and Zika, pose a major global public health problem worldwide. In this paper we investigate the propagation dynamics of diffusive vector-borne disease models in the whole space, which characterize the spatial expansion of the infected hosts and infected vectors. Due to the lack of monotonicity, the comparison principle cannot be applied directly to this system. We determine the spreading speed and minimal wave speed when the basic reproduction number of the corresponding kinetic system is larger than one. The spreading speed is mainly estimated by the uniform persistence argument and generalized principal eigenvalue. We also show that solutions converge locally uniformly to the positive equilibrium by employing two auxiliary monotone systems. Moreover, it is proven that the spreading speed is the minimal wave speed of travelling wave solutions. In particular, the uniqueness and monotonicity of travelling waves are obtained. When the basic reproduction number of the corresponding kinetic system is not larger than one, it is shown that solutions approach to the disease-free equilibrium uniformly and there is no travelling wave solutions. Finally, numerical simulations are presented to illustrate the analytical results.

Spatiotemporal patterns induced by four mechanisms in a tussock sedge model with discrete time and space variables

In this paper, we investigate the spatiotemporal patterns of a freshwater tussock sedge model with discrete time and space variables. We first analyze the kinetic system and show the parametric conditions for flip and Neimark–Sacker bifurcations respectively. With spatial diffusion, we then show that the obtained stable homogeneous solutions can experience Turing instability under certain conditions. Through numerical simulations, we find periodic doubling cascade, periodic window, invariant cycles, chaotic behaviors, and some interesting spatial patterns, which are induced by four mechanisms: pure-Turing instability, flip-Turing instability, Neimark–Sacker–Turing instability, and chaos.

Spatiotemporal dynamics for a diffusive mussel–algae model near a Hopf bifurcation point

In this paper, the Hopf bifurcation and Turing instability for a mussel–algae model are investigated. Through analysis of the corresponding kinetic system, the existence and stability conditions of the equilibrium and the type of Hopf bifurcation are studied. Via the center manifold and Hopf bifurcation theorem, sufficient conditions for Turing instability in equilibrium and limit cycles are obtained, respectively. In addition, we find that the strip patterns are mainly induced by Turing instability in equilibrium and spot patterns are mainly induced by Turing instability in limit cycles by numerical simulations. These provide a comprehension on the complex pattern formation of a mussel–algae system.

A simple model of time-dependent ionization in Type IIP supernova envelopes

ABSTRACT We propose a model kinetic system of the hydrogen atom (two levels plus continuum) under the conditions typical for atmospheres of Type IIP supernovae in the plateau stage. Despite the simplicity of this system, it describes realistically the basic properties of the complete system. Analysis shows that the ionization ‘freeze-out’ effect is always manifest at large times. We give a simple criterion for checking the statistical equilibrium of a system under the given conditions at any time. It is shown that if the system is in non-equilibrium at early times, the time-dependent effect of ionization necessarily exists. We also generalize this criterion to the case of arbitrary multilevel systems. We discuss various factors that affect the strength of the time-dependent effect in the kinetics during the photospheric phase of a supernova explosion.

Competitiveness of nonstationary states in linear kinetic systems

The master equation formalism is used to describe the possibility for peak population amplitudes of two nonstationary states in a 3-stage linear kinetic system to be endowed with an untraditional physical quantity — competitiveness — established in regard to the differences for the degree of the peak responses to a change in the input rate constants. Calculated coefficients of competitiveness are found to agree with observations of performance for the three optical materials with respect to their reliability in different operating windows. It is concluded that, for a non-equilibrium linear kinetic system, the competitiveness constitutes a common dynamic property of its nonstationary states and, in the case of their directed irreversible evolution, comprises the property of a system’s anti-cooperativity.

Simulation of the Building Materials Production Process

We study characteristics and properties of modified Godunov-Sultangazin kinetic system solution allows us to describe the nature and time of technological processes for the production of building materials based on autocatalysis reaction. This system is nonlinear, which explains the difficulty of its analytical solution. So, we use the numerical method of its investigation. The numerical results obtained and analyzed.