# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

November 2020 , Volume 25 , Issue 11

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2020, 25(11): 4119-4126 doi: 10.3934/dcdsb.2020090 +[Abstract](899) +[HTML](246) +[PDF](321.25KB)
Abstract:

In this article we analyze a recently proposed model for boundary layer flow of a nanofluid past a permeable stretching/shrinking sheet. The boundary value problem (BVP) resulting from this model is governed by two physical parameters; \begin{document}${{\lambda}}$\end{document}, which controls the stretching (\begin{document}${{\lambda}} >0$\end{document}) or shrinking (\begin{document}${{\lambda}} < 0$\end{document}) of the sheet, and \begin{document}$S$\end{document}, which controls the suction (\begin{document}$S>0$\end{document}) or injection (\begin{document}$S<0$\end{document}) of fluid through the sheet. For \begin{document}${{\lambda}} \ge 0$\end{document} and \begin{document}$S\in \mathbb{R}$\end{document}, we present a closed-form solution to the BVP and prove that this solution is unique. For \begin{document}${{\lambda}} < 0$\end{document} and \begin{document}$S< 2\sqrt{-{{\lambda}}}$\end{document} we prove no solution exists. For \begin{document}${{\lambda}} < 0$\end{document} and \begin{document}$S = 2\sqrt{-{{\lambda}}}$\end{document} we present a closed-form solution to the BVP and prove that it is unique. For \begin{document}${{\lambda}} < 0$\end{document} and \begin{document}$S> 2\sqrt{-{{\lambda}}}$\end{document} we present two closed-form solutions to the BVP and prove the existence of an infinite number of solutions in this parameter range. The analytical results proved here differ from the numerical results reported in the literature. We discuss the mathematical aspects of the problem that lead to the difficulty in obtaining accurate numerical approximations to the solutions.

2020, 25(11): 4127-4164 doi: 10.3934/dcdsb.2020091 +[Abstract](792) +[HTML](249) +[PDF](655.27KB)
Abstract:

We study the distribution of autonomously replicating genetic elements, so-called plasmids, in a bacterial population. When a bacterium divides, the plasmids are segregated between the two daughter cells. We analyze a model for a bacterial population structured by their plasmid content. The model contains reproduction of both plasmids and bacteria, death of bacteria, and the distribution of plasmids at cell division. The model equation is a growth-fragmentation-death equation with an integral term containing a singular kernel. As we are interested in the long-term distribution of the plasmids, we consider the associated eigenproblem. Due to the singularity of the integral kernel, we do not have compactness. Thus, standard approaches to show the existence of an eigensolution like the Theorem of Krein-Rutman cannot be applied. We show the existence of an eigensolution using a fixed point theorem and the Laplace transform. The long-term dynamics of the model is analyzed using the Generalized Relative Entropy method.

2020, 25(11): 4165-4188 doi: 10.3934/dcdsb.2020092 +[Abstract](1256) +[HTML](321) +[PDF](5508.01KB)
Abstract:

Pattern formation in various biological systems has been attributed to Turing instabilities in systems of reaction-diffusion equations. In this paper, a rigorous mathematical description for the pattern dynamics of aggregating regions of biological individuals possessing the property of chemotaxis is presented. We identify a generalized nonlinear degenerate chemotaxis model where a destabilization mechanism may lead to spatially non homogeneous solutions. Given any general perturbation of the solution nearby an homogenous steady state, we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along the finite number of fastest growing modes. The theoretical results are tested against two different numerical results in two dimensions showing an excellent qualitative agreement.

2020, 25(11): 4189-4210 doi: 10.3934/dcdsb.2020093 +[Abstract](1067) +[HTML](292) +[PDF](459.5KB)
Abstract:

In this paper, we are concerned with a diffusive Leslie-Gower predator-prey model in heterogeneous environment. The global existence and boundedness of solutions are shown. By analyzing the sign of the principal eigenvalue corresponding to each semi-trivial solution, we obtain the linear stability and global stability of semi-trivial solutions. The existence of positive steady state solution bifurcating from semi-trivial solutions is obtained by using local bifurcation theory. The stability analysis of the positive steady state solution is investigated in detail. In addition, we explore the asymptotic profiles of the steady state solution for small and large diffusion rates.

2020, 25(11): 4211-4220 doi: 10.3934/dcdsb.2020094 +[Abstract](952) +[HTML](288) +[PDF](3706.91KB)
Abstract:

In this paper, the dynamics of the celebrated \begin{document}$p-$\end{document}periodic one-dimensional logistic map is explored. A result on the global stability of the origin is provided and, under certain conditions on the parameters, the local stability condition of the \begin{document}$p-$\end{document}periodic orbit is shown to imply its global stability.

2020, 25(11): 4221-4255 doi: 10.3934/dcdsb.2020095 +[Abstract](803) +[HTML](257) +[PDF](554.3KB)
Abstract:

In this paper we study the abstract semilinear parabolic problem of the form

as the limit of the corresponding fractional approximations

in a Banach space \begin{document}$X$\end{document}, where the operator \begin{document}$A:D(A) \subset X \to X$\end{document} is a sectorial operator in the sense of Henry [22]. Under suitable assumptions on nonlinearities \begin{document}$f:X^\alpha\to X$\end{document} (\begin{document}$X^\alpha: = D(A^\alpha$\end{document})), we prove the continuity with rate (with respect to the parameter \begin{document}$\alpha$\end{document}) for the global attractors (as seen in Babin and Vishik [4] Chapter 8, Theorem 2.1). As an application of our analysis we consider a fractional approximation of the strongly damped wave equations and we study the convergence with rate of solutions of such approximations.

2020, 25(11): 4257-4276 doi: 10.3934/dcdsb.2020096 +[Abstract](740) +[HTML](256) +[PDF](521.49KB)
Abstract:

We consider the general second order difference equation \begin{document}$x_{n+1} = F(x_n, x_{n-1})$\end{document} in which \begin{document}$F$\end{document} is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that \begin{document}$F$\end{document} has a semi-convex compact invariant domain, then make an extension of \begin{document}$F$\end{document} on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of \begin{document}$F.$\end{document} Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.

2020, 25(11): 4277-4293 doi: 10.3934/dcdsb.2020097 +[Abstract](739) +[HTML](246) +[PDF](462.51KB)
Abstract:

By exploring the smooth effect of the heat flows and the weighted-Chemin-Lerner technique, we obtain the global solutions of large energy to the viscous shallow water equations with initial data in the critical Besov spaces, which improves the previous small energy type arguments [5], [13]. Moreover, the method used here is quiet different from [5], [13].

2020, 25(11): 4295-4316 doi: 10.3934/dcdsb.2020098 +[Abstract](673) +[HTML](238) +[PDF](801.73KB)
Abstract:

We estimate the time that a point or set, respectively, requires to approach the attractor of a radially symmetric gradient type stochastic differential equation driven by small noise. Here, both of these times tend to infinity as the noise gets small. However, the rates at which they go to infinity differ significantly. In the case of a set approaching the attractor, we use large deviation techniques to show that this time increases exponentially. In the case of a point approaching the attractor, we apply a time change and compare the accelerated process to a process on the sphere and obtain that this time increases merely linearly.

2020, 25(11): 4317-4333 doi: 10.3934/dcdsb.2020099 +[Abstract](857) +[HTML](240) +[PDF](402.76KB)
Abstract:

In this paper, we prove the unique global strong solution for the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows when the initial density can contain vacuum states, as long as the initial data satisfies some compatibility condition. Furthermore, our main result improves all the previous results where the initial density is strictly positive. The main ingredient of the proof is to use some critical Sobolev inequality of logarithmic type, which were originally due to Brezis-Gallouet in [3] and Brezis-Wainger in [4], some regularity properties of Stokes system and some delicate energy estimates for nonhomogeneous incompressible heat conducting flows.

2020, 25(11): 4335-4359 doi: 10.3934/dcdsb.2020100 +[Abstract](912) +[HTML](266) +[PDF](581.05KB)
Abstract:

In this paper we discuss the Wong-Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for stochastic Ginzburg-Landau equations driven by a white noise. We first apply the Galerkin method and compactness argument to prove the existence and uniqueness of weak solutions. Consequently, we show that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. At last, when the stochastic Ginzburg-Landau equation is driven by a linear multiplicative noise, we establish the convergence of solutions of Wong-Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.

2020, 25(11): 4361-4382 doi: 10.3934/dcdsb.2020101 +[Abstract](678) +[HTML](239) +[PDF](433.96KB)
Abstract:

This paper is devoted to the existence of at least one positive periodic solution for generalized Basener-Ross model with time-dependent coefficients. Our proof is based on Manásevich-Mawhin continuation theorem, Leray-Schauder alternative principle, fixed point theorem in cones. Moreover, we obtain that there are at least two positive periodic solutions for this model.

2020, 25(11): 4383-4396 doi: 10.3934/dcdsb.2020102 +[Abstract](820) +[HTML](256) +[PDF](418.96KB)
Abstract:

In a bounded domain \begin{document}$\Omega\subset \mathbb{R}^n$\end{document} with smooth boundary, this work considers the indirect pursuit-evasion model

with positive parameters \begin{document}$\chi, \xi, \lambda, \mu$\end{document}, \begin{document}$a$\end{document} and \begin{document}$b$\end{document}.

It is firstly asserted that when \begin{document}$n\le 3$\end{document}, for any given suitably regular initial data the corresponding homogeneous Neumann initial-boundary problem admits a global and bounded smooth solution. Moreover, it is shown that if \begin{document}$b\lambda<\mu$\end{document} and under some explicit smallness conditions on \begin{document}$\chi$\end{document} and \begin{document}$\xi$\end{document}, any nontrival bounded classical solution converges to the spatially homogeneous coexistence state in the large time limit; if \begin{document}$b\lambda>\mu$\end{document}, however, then under an explicit smallness assumption on \begin{document}$\chi$\end{document} but without any restriction on \begin{document}$\xi$\end{document}, any bounded classical solution \begin{document}$(u, v)$\end{document} with \begin{document}$u\not\equiv 0$\end{document} stabilizes to \begin{document}$(\lambda, 0)$\end{document} as \begin{document}$t\to \infty$\end{document}.

2020, 25(11): 4397-4410 doi: 10.3934/dcdsb.2020103 +[Abstract](800) +[HTML](254) +[PDF](469.07KB)
Abstract:

In this paper we study existence of nonradial stationary solutions of a free boundary problem modeling the growth of nonnecrotic tumors. Unlike the models studied in existing literatures on this topic where boundary value condition for the nutrient concentration \begin{document}$\sigma$\end{document} is linear, in this model this is a nonlinear boundary condition. By using the bifurcation method, we prove that nonradial stationary solutions do exist when the surface tension coefficient \begin{document}$\gamma$\end{document} takes values in small neighborhoods of certain eigenvalues of the linearized problem at the radial stationary solution.

2020, 25(11): 4411-4426 doi: 10.3934/dcdsb.2020104 +[Abstract](1290) +[HTML](475) +[PDF](3195.69KB)
Abstract:

Filamentous fungi contribute to ecosystem and human-induced processes such as primary production, bioremediation, biogeochemical cycling and biocontrol. Predicting the dynamics of fungal communities can hence improve our forecasts of ecological processes which depend on fungal community structure. In this work, we aimed to develop simple theoretical models of fungal interactions with ordinary and partial differential equations, and to validate model predictions against community dynamics of a three species empirical system. We found that space is an important factor for the prediction of community dynamics, since the performance was poor for models of ordinary differential equations assuming well-mixed nutrient substrate. The models of partial differential equations could satisfactorily predict the dynamics of a single species, but exhibited limitations which prevented the prediction of empirical community dynamics. One such limitation is the arbitrary choice of a threshold local density above which a fungal mycelium is considered present in the model. In conclusion, spatially explicit simulation models, able to incorporate different factors influencing interaction outcomes and hence dynamics, appear as a more promising direction towards prediction of fungal community dynamics.

2020, 25(11): 4427-4447 doi: 10.3934/dcdsb.2020106 +[Abstract](770) +[HTML](301) +[PDF](1542.93KB)
Abstract:

The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model.

2020, 25(11): 4449-4477 doi: 10.3934/dcdsb.2020107 +[Abstract](996) +[HTML](271) +[PDF](741.61KB)
Abstract:

This paper is devoted to studying the dynamics of a certain age structured heroin-cocaine epidemic model. More precisely, this model takes into account the following unknown variables: susceptible individuals, heroin users, cocaine users and recovered individuals. Each one of these classes can change or interact with others. In this paper, firstly, we give some results on the existence, uniqueness and positivity of solutions. Next, we obtain a threshold value \begin{document}$r(\Psi'[0])$\end{document} such that an endemic equilibrium exists if \begin{document}$r(\Psi'[0]) > 1$\end{document}. We then show that if \begin{document}$r(\Psi'[0]) < 1$\end{document}, then the disease-free equilibrium is globally asymptotically stable, whereas if \begin{document}$r(\Psi'[0]) > 1$\end{document}, then the system is uniformly persistent. Moreover, for \begin{document}$r(\Psi'[0]) > 1$\end{document}, we show that the endemic equilibrium is globally asymptotically stable under an additional assumption that epidemic parameters for heroin users and cocaine users are same. Finally, some numerical simulations are presented to illustrate our theoretical results.

2020, 25(11): 4479-4492 doi: 10.3934/dcdsb.2020108 +[Abstract](747) +[HTML](244) +[PDF](431.86KB)
Abstract:

The spatial spreading dynamics is considered for a class of convolution differential equation resulting from physical and biological problems. It is shown that this kind of equation with monostable structure admits a spreading speed, even when the nonlinear reaction terms without monotonicity. The upward convergence of spreading speed is also established under appropriate conditions.

2020, 25(11): 4493-4513 doi: 10.3934/dcdsb.2020109 +[Abstract](863) +[HTML](277) +[PDF](793.74KB)
Abstract:

The main aim of this article is to examine almost sure exponential stabilization and suppression of nonlinear systems by periodically intermittent stochastic perturbation with jumps. On the one hand, some sufficient criteria ensure almost sure stabilization of the unstable deterministic system by applying exponential martingale inequality with jumps. On the other hand, sufficient conditions of destabilization are provided under which the system is stable by the well-known strong law of large numbers of local martingale and Poisson process. Both the sample Lyapunov exponents are closely related to the control period \begin{document}$T$\end{document} and noise width \begin{document}$\theta$\end{document}. As for applications, the well-known Lorenz chaotic systems and nonlinear Liénard equation with jumps are discussed. Finally, two simulation examples demonstrating the effectiveness of the results are provided.

2020, 25(11): 4515-4533 doi: 10.3934/dcdsb.2020110 +[Abstract](775) +[HTML](263) +[PDF](456.68KB)
Abstract:

In this paper, we establish a new blowup criterion for the strong solution to the simplified Ericksen-Leslie system modeling compressible nematic liquid crystal flows in a bounded domain \begin{document}$\Omega\subset\mathbb{R}^{3}$\end{document}. Specifically, we obtain the blowup criterion in terms of \begin{document}$\|P\|_{L^\infty_t BMO_{x}}$\end{document} and \begin{document}$\|\nabla d\|_{L^s_t L^\infty_x}$\end{document}, for any \begin{document}$s>3$\end{document}. The appearance of vacuum could be allowed.

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