# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

March 2020 , Volume 25 , Issue 3

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2020, 25(3): 815-839 doi: 10.3934/dcdsb.2019191 +[Abstract](1313) +[HTML](246) +[PDF](416.65KB)
Abstract:

We derive a one-dimensional cable model for the electric potential propagation along an axon. Since the typical thickness of an axon is much smaller than its length, and the myelin sheath is distributed periodically along the neuron, we simplify the problem geometry to a thin cylinder with alternating myelinated and unmyelinated parts. Both the microstructure period and the cylinder thickness are assumed to be of order \begin{document}$\varepsilon$\end{document}, a small positive parameter. Assuming a nonzero conductivity of the myelin sheath, we find a critical scaling with respect to \begin{document}$\varepsilon$\end{document} which leads to the appearance of an additional potential in the homogenized nonlinear cable equation. This potential contains information about the geometry of the myelin sheath in the original three-dimensional model.

2020, 25(3): 841-857 doi: 10.3934/dcdsb.2019192 +[Abstract](1796) +[HTML](218) +[PDF](482.36KB)
Abstract:

In this paper, input-to-state stability (ISS) of continuous-time systems is analyzed via finite-time Lyapunov functions. ISS of a continuous-time system is first proved via finite-time robust Lyapunov functions for an introduced auxiliary system of the considered system. It is then obtained that the existence of a finite-time ISS Lyapunov function implies that the continuous-time system is ISS. The converse finite-time ISS Lyapunov theorem is proposed. Furthermore, we explore the properties of finite-time ISS Lyapunov functions for the continuous-time system on a bounded and compact set without a small neighborhood of the origin. The effectiveness of our results is illustrated by four examples.

2020, 25(3): 859-875 doi: 10.3934/dcdsb.2019193 +[Abstract](1328) +[HTML](252) +[PDF](347.77KB)
Abstract:

In this paper, we study a shadow system of a two species Lotka-Volterra competition-diffusion-advection system, where the ratio of diffusion and advection rates are supposed to be a positive constant. We show that for any given migration, if the product of interspecific competition coefficients of competitors is small, then the shadow system has coexistence state; otherwise we can always find some migration such that it has no coexistence state. Moreover, these findings can be applied to steady state of the two-species Lotka-Volterra competition-diffusion-advection model. Particularly, we show that if the interspecific competition coefficient of the invader is sufficiently small, then rapid diffusion of the invader can drive to coexistence state.

2020, 25(3): 877-901 doi: 10.3934/dcdsb.2019194 +[Abstract](1168) +[HTML](259) +[PDF](964.75KB)
Abstract:

This paper presents a reaction-diffusion system modeling interactions of the intraguild predator and prey in an unstirred chemostat, in which the predator can also compete with its prey for one single nutrient resource that can be stored within individuals. Under suitable conditions, we first show that there are at least three steady-state solutions for the full system, a trivial steady-state solution with neither species present, and two semitrivial steady-state solutions with just one of the species. Then we establish that coexistence of the intraguild predator and prey can occur if both of the semitrivial steady-state solutions are invasible by the missing species. Comparing with the system without predation, our numerical simulations show that the introduction of predation in an ecosystem can enhance the coexistence of species. Our mathematical arguments also work for the linear food chain model (top-down predation), in which the top-down predator only feeds on the prey but does not compete for nutrient resource with the prey. In our numerical studies, we also do a comparison of intraguild predation and top-down predation.

2020, 25(3): 903-915 doi: 10.3934/dcdsb.2019195 +[Abstract](1405) +[HTML](261) +[PDF](750.7KB)
Abstract:

We consider a discrete non-autonomous semi-dynamical system generated by a family of continuous maps defined on a locally compact metric space. It is assumed that this family of maps uniformly converges to a continuous map. Such a non-autonomous system is called an asymptotically autonomous system. We extend the dynamical system to the metric one-point compactification of the phase space. This is done via the construction of an associated skew-product dynamical system. We prove, among other things, that the omega limit sets are invariant and invariantly connected. We apply our results to two populations models, the Ricker model with no Allee effect and Elaydi-Sacker model with the Allee effect, where it is assumed that the reproduction rate changes with time due to habitat fluctuation.

2020, 25(3): 917-933 doi: 10.3934/dcdsb.2019196 +[Abstract](1125) +[HTML](376) +[PDF](503.46KB)
Abstract:

It is well known that CTL (cytotoxic T lymphocyte) immune response could be broadly classified into lytic and nonlytic components, nonlinear functions can better reproduce saturated effects in the interaction processes between cell and viral populations, and distributed intracellular delay can realistically reflect the stochastic element in the delay effects. For these reasons, we develop an HTLV-I (Human T-cell leukemia virus type I) infection model with nonlinear lytic and nonlytic CTL immune responses, nonlinear incidence rate, distributed intracellular delay and immune impairment. Through conducting complete analysis, it is revealed that all these factors influence the concentration level of infected T-cells at the chronic-infection equilibrium, whereas intracellular distributed delay and nonlinear incidence rate may change the expression of the basic reproduction number \begin{document}$\mathfrak{R}_0$\end{document} in the context where the model proposed still preserves the threshold dynamics. Our analysis results obtained may improve several existing works by comparison. We also perform global sensitivity analysis for \begin{document}$\mathfrak{R}_0$\end{document} in order to explore the effective strategies of lowering the concentration level of infected T-cells.

2020, 25(3): 935-955 doi: 10.3934/dcdsb.2019197 +[Abstract](1111) +[HTML](215) +[PDF](382.19KB)
Abstract:

The goal of this paper is to study the long-time behavior of a class of extensible beams equation with the nonlocal weak damping

on a bounded smooth domain \begin{document}$\Omega\subset\mathbb{R}^{n}$\end{document} with hinged (clamped) boundary condition. Under some suitable conditions on the Kirchhoff coefficient \begin{document}$m(\|\nabla u\|^2)$\end{document} and the nonlinear term \begin{document}$f(u)$\end{document}, the well-posedness is established by means of the monotone operator theory and the existence of a global attractor is obtained in the subcritical case, where the asymptotic smooothness of the semigroup is verified by the energy reconstruction method.

2020, 25(3): 957-979 doi: 10.3934/dcdsb.2019198 +[Abstract](1056) +[HTML](215) +[PDF](470.1KB)
Abstract:

In this work we study a degenerate pseudo-parabolic system with cross diffusion describing the evolution of the densities of an unsaturated two-phase flow mixture with dynamic capillary pressure in porous medium with saturation-dependent relaxation parameter and hypocoercive diffusion operator modeling cross diffusion. The equations are derived in a thermodynamically correct way from mass conservation laws. Global-in-time existence of weak solutions to the system in a bounded domain with equilibrium boundary conditions is shown. The main tools of the analysis are an entropy inequality and a crucial apriori bound which allows for controlling the degeneracy.

2020, 25(3): 981-999 doi: 10.3934/dcdsb.2019199 +[Abstract](1420) +[HTML](248) +[PDF](359.03KB)
Abstract:

This paper deals with the spreading speed of the disease described by a partially degenerate and cooperative epidemic model with free boundaries. We show that the spreading speed is determined by a semi-wave problem. To find such a semi-wave solution, we prove the existence of a monotone solution to a reduced ODE by an upper and lower solution approach. And then we establish the uniqueness of the semi-wave solution via the sliding method. It is demonstrated that the precise asymptotic spreading speed is less than the minimal speed of traveling waves.

2020, 25(3): 1001-1013 doi: 10.3934/dcdsb.2019205 +[Abstract](1262) +[HTML](273) +[PDF](5647.89KB)
Abstract:

There are many works on self-excited and hidden attractors. However the relationship between them is less investigated. In this study we present a system which can have both hidden self-excited attractors. Dynamical properties of the chaotic system are studied using the equilibrium points and Eigenvalues analysis, Lyapunov exponents and bifurcation plots. Since fractional order models are more interesting in engineering applications, the fractional order version of the proposed system is derived using Adomian decomposition method. Bifurcation and stability analysis of the fractional order model shows the existence of chaotic oscillations. To demonstrate the engineering importance of the fractional order model, we have designed a digital communication system with SCSK (Symmetric Chaos Shift Keying) modulation method separately for both self-excited attractor and hidden attractor, the bit error rate performance was compared.

2020, 25(3): 1015-1041 doi: 10.3934/dcdsb.2019206 +[Abstract](1329) +[HTML](248) +[PDF](384.5KB)
Abstract:

We consider an equilibrium model of the Limit Order Book in a stock market, where a large number of competing agents post "buy" or "sell" orders. For the "one-shot" game, it is shown that the two sides of the LOB are determined by the distribution of the random size of the incoming order, and by the maximum price accepted by external buyers (or the minimum price accepted by external sellers). We then consider an iterated game, where more agents come to the market, posting both market orders and limit orders. Equilibrium strategies are found by backward induction, in terms of a value function which depends on the current sizes of the two portions of the LOB. The existence of a unique Nash equilibrium is proved under a natural assumption, namely: the probability that the external order is so large that it wipes out the entire LOB should be sufficiently small.

2020, 25(3): 1043-1058 doi: 10.3934/dcdsb.2019207 +[Abstract](1166) +[HTML](264) +[PDF](397.74KB)
Abstract:

In this paper, valuation of a defaultable corporate bond with credit rating migration risk is considered under the structure framework by using a free boundary model. The existence, uniqueness and regularity of the solution are obtained. Furthermore, we analyze the solution's asymptotic behavior and prove that the solution is convergent to an closed form solution. In addition, numerical examples are also shown.

2020, 25(3): 1059-1081 doi: 10.3934/dcdsb.2019208 +[Abstract](1383) +[HTML](247) +[PDF](406.44KB)
Abstract:

We study the collective dynamics of Kuramoto ensemble under uncertain coupling strength. For a finite ensemble, we can model the dynamics of the Kuramoto ensemble by the stochastic Kuramoto system with multiplicative noise. In contrast, for an infinite ensemble, the dynamics is effectively described by the Kuramoto-Sakaguchi-Fokker-Planck(KS-FP) equation with state dependent degenerate diffusion. We present emergent synchronization estimates for the stochastic and kinetic models, which yield the stability of the phase-locked state for identical Kuramoto ensemble with the same natural frequencies. We also provide a brief explanation on the mean-field limit between two models.

2020, 25(3): 1083-1096 doi: 10.3934/dcdsb.2019209 +[Abstract](984) +[HTML](219) +[PDF](379.92KB)
Abstract:

We are concerned with the singularity formation of strong solutions to the two-dimensional (2D) non-barotropic non-resistive compressible magnetohydrodynamic equations with zero heat conduction in a bounded domain. It is showed that the strong solution exists globally if the density and the magnetic field as well as the pressure are bounded from above. Our method relies on critical Sobolev inequalities of logarithmic type.

2020, 25(3): 1097-1108 doi: 10.3934/dcdsb.2019210 +[Abstract](1107) +[HTML](258) +[PDF](346.33KB)
Abstract:

This paper investigates the homoclinic orbits and chaos in the generalized Lorenz system. Using center manifold theory and Lyapunov functions, we get non-existence conditions of homoclinic orbits associated with the origin. The existence conditions of the homoclinic orbits are obtained by Fishing Principle. Therefore, sufficient and necessary conditions of existence of homoclinic orbits associated with the origin are given. Furthermore, with the broken of the homoclinic orbits, we show that the chaos is in the sense generalized Shil'nikov homoclinic criterion.

2020, 25(3): 1109-1128 doi: 10.3934/dcdsb.2019211 +[Abstract](1199) +[HTML](243) +[PDF](456.63KB)
Abstract:

For \begin{document}$n\geq2$\end{document} let \begin{document}$\mathit{\Omega }\subset {\mathbb{R}}^n$\end{document} be a bounded domain with smooth boundary as well as some nonnegative functions \begin{document}$0\not \equiv u_0\in W^{1, \infty}(\mathit{\Omega })$\end{document} and \begin{document}$v_0\in W^{1, \infty}(\mathit{\Omega })$\end{document}. With \begin{document}$\varepsilon\in(0, 1)$\end{document} we want to know in which sense (if any!) solutions to the parabolic-parabolic system

converge to those of the system where \begin{document}$\varepsilon = 0$\end{document} and where the initial condition for \begin{document}$v$\end{document} has been removed. We will see in our theorem that indeed the solutions of these systems converge in a meaningful way if \begin{document}$m>1+\frac{n-2}{n}$\end{document} without the need for further conditions, e. g. on the size of \begin{document}$\left\|{{u_0}}\right\|_{L^p(\mathit{\Omega })}$\end{document} for some \begin{document}$p\in[1, \infty]$\end{document}.

2020, 25(3): 1129-1139 doi: 10.3934/dcdsb.2019212 +[Abstract](1172) +[HTML](258) +[PDF](688.28KB)
Abstract:

Building upon the technique that we developed earlier for perturbed sweeping processes with convex moving constraints and monotone vector fields (Kamenskii et al, Nonlinear Anal. Hybrid Syst. 30, 2018), the present paper establishes the conditions for global asymptotic stability of global and periodic solutions to perturbed sweeping processes with prox-regular moving constraints. Our conclusion can be formulated as follows: closer the constraint to a convex one, weaker monotonicity is required to keep the sweeping process globally asymptotically stable. We explain why the proposed technique is not capable to prove global asymptotic stability of a periodic regime in a crowd motion model (Cao-Mordukhovich, DCDS-B 22, 2017). We introduce and analyze a toy model which clarifies the extent of applicability of our result.

2020, 25(3): 1141-1158 doi: 10.3934/dcdsb.2019213 +[Abstract](1822) +[HTML](295) +[PDF](376.13KB)
Abstract:

In this paper, we focus on fast-slow stochastic partial differential equations in which the slow variable is driven by a fractional Brownian motion and the fast variable is driven by an additive Brownian motion. We establish an averaging principle in which the fast-varying diffusion process will be averaged out with respect to its stationary measure in the limit process. It is shown that the slow-varying process \begin{document}$L^{p}(p \geq 2)$\end{document} converges to the solution of the corresponding averaging equation. To reduce the complexity, one can concentrate on the limit process instead of studying the original full fast-slow system.

2020, 25(3): 1159-1167 doi: 10.3934/dcdsb.2019214 +[Abstract](1247) +[HTML](265) +[PDF](282.39KB)
Abstract:

A predator-prey model with Sigmoid functional response is studied. The main purpose is to investigate the global stability of a positive (co-existence) equilibrium, whenever it exists. A recent developed approach shows that, associated with the model, there is an implicitly defined function which plays an important rule in determining the global stability of the positive equilibrium. By performing an analytic and geometrical analysis we demonstrate that a crucial property of this implicitly defined function is governed by the local stability of the positive equilibrium. With this crucial property we are able to show that the global and local stability of the positive equilibrium, whenever it exists, is equivalent. We believe that our approach can be extended to study the global stability of the positive equilibrium for predator-prey models with some other types of functional response.

2020, 25(3): 1169-1192 doi: 10.3934/dcdsb.2019215 +[Abstract](1053) +[HTML](243) +[PDF](426.73KB)
Abstract:

This paper is concerned with the Cauchy problem for a fractal Burgers equation in two dimensions. When \begin{document}$\alpha\in (0, 1)$\end{document}, the same problem has been studied in one dimensions, we can refer to [1, 17, 24]. In this paper, we study well-posedness of solutions to the Burgers equation with supercritical dissipation. We prove the local existence with large initial data and global existence with small initial data in critical Besov space by energy method. Furthermore, we show that solutions can blow up in finite time if initial data is not small by contradiction method.

2019  Impact Factor: 1.27