# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

February 2019 , Volume 24 , Issue 2

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2019, 24(2): 415-421 doi: 10.3934/dcdsb.2018179 +[Abstract](3109) +[HTML](398) +[PDF](328.44KB)
Abstract:

In this short paper we study the existence and uniqueness of solutions of free boundary problems coming from ecology in heterogeneous environments.

2019, 24(2): 423-447 doi: 10.3934/dcdsb.2018180 +[Abstract](2906) +[HTML](443) +[PDF](582.6KB)
Abstract:

We consider an attraction-repulsion chemotaxis model coupled with the Navier-Stokes system. This model describes the interaction between a type of cells (e.g., bacteria), which proliferate following a logistic law, and two chemical signals produced by the cells themselves that degraded at a constant rate. Also, it is considered that the chemoattractant is consumed with a rate proportional to the amount of organisms. The cells and chemical substances are transported by a viscous incompressible fluid under the influence of a force due to the aggregation of cells. We prove the existence of global mild solutions in bounded domains of \begin{document}$\mathbb{R}^N,$\end{document} \begin{document}$N = 2, 3,$\end{document} for small initial data in \begin{document}$L^p$\end{document}-spaces.

2019, 24(2): 449-465 doi: 10.3934/dcdsb.2018181 +[Abstract](3010) +[HTML](407) +[PDF](438.88KB)
Abstract:

In this paper, we investigate the asymptotic behavior for non-autonomous stochastic complex Ginzburg-Landau equations with multiplicative noise on thin domains. For this aim, we first show that the existence and uniqueness of random attractors for the considered equations and the limit equations. Then, we establish the upper semicontinuity of these attractors when the thin domains collapse onto an interval.

2019, 24(2): 467-486 doi: 10.3934/dcdsb.2018182 +[Abstract](3794) +[HTML](417) +[PDF](1083.04KB)
Abstract:

In this paper, we study the Hopf bifurcation and spatiotemporal pattern formation of a delayed diffusive logistic model under Neumann boundary condition with spatial heterogeneity. It is shown that for large diffusion coefficient, a supercritical Hopf bifurcation occurs near the non-homogeneous positive steady state at a critical time delay value, and the dependence of corresponding spatiotemporal patterns on the heterogeneous resource function is demonstrated via numerical simulations. Moreover, it is proved that the heterogeneous resource supply contributes to the increase of the temporal average of total biomass of the population even though the total biomass oscillates periodically in time.

2019, 24(2): 487-510 doi: 10.3934/dcdsb.2018183 +[Abstract](3109) +[HTML](548) +[PDF](2468.96KB)
Abstract:

We study the Turing-Hopf bifurcation and give a simple and explicit calculation formula of the normal forms for a general two-components system of reaction-diffusion equation with time delays. We declare that our formula can be automated by Matlab. At first, we extend the normal forms method given by Faria in 2000 to Hopf-zero singularity. Then, an explicit formula of the normal forms for Turing-Hopf bifurcation are given. Finally, we obtain the possible attractors of the original system near the Turing-Hopf singularity by the further analysis of the normal forms, which mainly include, the spatially non-homogeneous steady-state solutions, periodic solutions and quasi-periodic solutions.

2019, 24(2): 511-528 doi: 10.3934/dcdsb.2018184 +[Abstract](3167) +[HTML](422) +[PDF](441.4KB)
Abstract:

This paper is devoted to the existence, uniqueness and stability of bistable traveling waves for a recursive system, which is defined by the iterations of the Ponicaré map of a yearly periodic age-structured population model derived in the companion paper [8]. The existence of the wave is established by appealing to a monotone dynamical system theory, and the uniqueness and stability are obtained by employing a squeezing method.

2019, 24(2): 529-546 doi: 10.3934/dcdsb.2018194 +[Abstract](3108) +[HTML](407) +[PDF](471.06KB)
Abstract:

In this paper, we investigate the asymptotic regularity of the minimal pullback attractor of a non-autonomous quasi-linear parabolic \begin{document}$p$\end{document}-Laplacian equation with dynamical boundary condition. First, we establish the higher-order integrability of the difference of solutions near the initial time. Then we show that, under the assumption that the time-depending forcing terms only satisfy some \begin{document}$L^2$\end{document} integrability, the \begin{document}$L^2(Ω)× L^2(\partialΩ)$\end{document} pullback \begin{document}$\mathscr{D}$\end{document}-attractor can actually attract the \begin{document}$L^2(Ω)× L^2(\partialΩ)$\end{document}-bounded set in \begin{document}$L^{2+δ}(Ω)× L^{2+δ}(\partialΩ)$\end{document}-norm for any \begin{document}$δ∈[0,∞)$\end{document}.

2019, 24(2): 547-561 doi: 10.3934/dcdsb.2018196 +[Abstract](2853) +[HTML](276) +[PDF](776.58KB)
Abstract:

We explore an approach based on the theory of cosymmetry to model interaction of predators and prey in a two-dimensional habitat. The model under consideration is formulated as a system of nonlinear parabolic equations with spatial heterogeneity of resources and species. Firstly, we analytically determine system parameters, for which the problem has a nontrivial cosymmetry. To this end, we formulate cosymmetry relations. Next, we employ numerical computations to reveal that under said cosymmetry relations, a one-parameter family of steady states is formed, which may be characterized by different proportions of predators and prey. The numerical analysis is based on the finite difference method (FDM) and staggered grids. It allows to follow the transformation of spatial patterns with time. Eventually, the destruction of the continuous family of equilibria due to mistuned parameters is analyzed. To this end, we derive the so-called cosymmetric selective equation. Investigation of the selective equation gives an insight into scenarios of local competition and coexistence of species, together with their connection to the cosymmetry relations. When the cosymmetry relation is only slightly violated, an effect we call 'memory on the lost family' may be observed. Indeed, in this case, a slow evolution takes place in the vicinity of the lost states of equilibrium.

2019, 24(2): 563-586 doi: 10.3934/dcdsb.2018197 +[Abstract](3051) +[HTML](455) +[PDF](10760.32KB)
Abstract:

This paper deals with a methodology for defining and computing an analytical Fourier-Taylor series parameterisation of local invariant manifolds associated to periodic orbits of polynomial vector fields. Following the Parameterisation Method, the functions involved in the series result by solving some linear non autonomous differential equations. Exploiting the Floquet normal form decomposition, the time dependency is removed and the differential problem is rephrased as an algebraic system to be solved for the Fourier coefficients of the unknown periodic functions. The procedure leads to an efficient and fast computational algorithm. Motivated by mission design purposes, the technique is applied in the framework of the Circular Restricted Three Body problem and the parameterisation of local invariant manifolds for several halo orbits is computed and discussed.

2019, 24(2): 587-613 doi: 10.3934/dcdsb.2018198 +[Abstract](3089) +[HTML](340) +[PDF](487.02KB)
Abstract:

In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the \begin{document}$p$\end{document}th moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.

2019, 24(2): 615-635 doi: 10.3934/dcdsb.2018199 +[Abstract](3200) +[HTML](365) +[PDF](455.19KB)
Abstract:

In this paper, we consider the numerical approximation for a class of fractional stochastic partial differential equations driven by infinite dimensional fractional Brownian motion with hurst index \begin{document}$H∈ (\frac{1}{2}, 1)$\end{document}. By using spectral Galerkin method, we analyze the spatial discretization, and we give the temporal discretization by using the piecewise constant, discontinuous Galerkin method and a Laplace transform convolution quadrature. Under some suitable assumptions, we prove the sharp regularity properties and the optimal strong convergence error estimates for both semi-discrete and fully discrete schemes.

2019, 24(2): 637-655 doi: 10.3934/dcdsb.2018200 +[Abstract](2576) +[HTML](373) +[PDF](529.13KB)
Abstract:

In this paper we study a one-dimensional linear advection-diffusion equation on a half-line driven by a Lévy boundary noise. The problem is motivated by the study of contaminant transport models under random sources (P. P. Wang and C. Zheng, Ground water, 43 (2005), [34]). We determine the closed form formulae for mild solutions of this equation with Dirichlet and Neumann noise and study approximations of these solutions by classical solutions obtained with the help of Wong-Zakai approximations of the driving Lévy process.

2019, 24(2): 657-670 doi: 10.3934/dcdsb.2018201 +[Abstract](2537) +[HTML](301) +[PDF](427.79KB)
Abstract:

We characterize the attractors for a two-parameter class of two-dimensional piecewise affine maps. These attractors are strange attractors, probably having finitely many pieces, and coincide with the support of an ergodic absolutely invariant probability measure. Moreover, we demonstrate that every compact invariant set with non-empty interior contains one of these attractors. We also prove the existence, for each natural number \begin{document}$n,$\end{document} of an open set of parameters in which the respective transformation exhibits at least \begin{document}$2^n$\end{document} non connected two-dimensional strange attractors each one of them formed by \begin{document}$4^n$\end{document} pieces.

2019, 24(2): 671-694 doi: 10.3934/dcdsb.2018202 +[Abstract](2475) +[HTML](351) +[PDF](999.67KB)
Abstract:

We study the dynamics of the Fisher-KPP equation on the infinite homogeneous tree and Erdős-Réyni random graphs. We assume initial data that is zero everywhere except at a single node. For the case of the homogeneous tree, the solution will either form a traveling front or converge pointwise to zero. This dichotomy is determined by the linear spreading speed and we compute critical values of the diffusion parameter for which the spreading speed is zero and maximal and prove that the system is linearly determined. We also study the growth of the total population in the network and identify the exponential growth rate as a function of the diffusion coefficient, α. Finally, we make predictions for the Fisher-KPP equation on Erdős-Rényi random graphs based upon the results on the homogeneous tree. When α is small we observe via numerical simulations that mean arrival times are linearly related to distance from the initial node and the speed of invasion is well approximated by the linear spreading speed on the tree. Furthermore, we observe that exponential growth rates of the total population on the random network can be bounded by growth rates on the homogeneous tree and provide an explanation for the sub-linear exponential growth rates that occur for small diffusion.

2019, 24(2): 695-717 doi: 10.3934/dcdsb.2018203 +[Abstract](3331) +[HTML](364) +[PDF](568.6KB)
Abstract:

In this paper, we investigate the strong convergence rate of the split-step theta (SST) method for a kind of stochastic differential equations with piecewise continuous arguments (SDEPCAs) under some polynomially growing conditions. It is shown that the SST method with \begin{document}$θ∈[\frac{1}{2},1]$ \end{document} is strongly convergent with order \begin{document}$\frac{1}{2}$ \end{document} in \begin{document}$p$ \end{document}th(\begin{document}$p≥ 2$ \end{document}) moment if both drift and diffusion coefficients are polynomially growing with regard to the delay terms, while the diffusion coefficients are globally Lipschitz continuous in non-delay arguments. The exponential mean square stability of the improved split-step theta (ISST) method is also studied without the linear growth condition. With some relaxed restrictions on the step-size, it is proved that the ISST method with \begin{document}$θ∈(\frac{1}{2},1]$ \end{document} is exponentially mean square stable under the monotone condition. Without any restriction on the step-size, there exists \begin{document}$θ^*∈(\frac{1}{2},1]$ \end{document} such that the ISST method with \begin{document}$θ∈(θ^*,1]$ \end{document} is exponentially stable in mean square. Some numerical simulations are presented to illustrate the analytical theory.

2019, 24(2): 719-735 doi: 10.3934/dcdsb.2018204 +[Abstract](2539) +[HTML](429) +[PDF](398.58KB)
Abstract:

In this paper we study a class of multi-term time fractional integral diffusion equations. Results on existence, uniqueness and regularity of a strong solution are provided through the Rothe method. Several examples are given to illustrate the applicability of main results.

2019, 24(2): 737-754 doi: 10.3934/dcdsb.2018205 +[Abstract](2440) +[HTML](325) +[PDF](607.79KB)
Abstract:

Nonlocal Lotka-Volterra models have the property that solutions concentrate as Dirac masses in the limit of small diffusion. Motivated by the existence of moving Dirac-concentrations in the time-dependent problem, we study the qualitative properties of steady states in the limit of small diffusion. Under different conditions on the growth rate and interaction kernel as motivated by the framework of adaptive dynamics, we will show that as the diffusion rate tends to zero the steady state concentrates (ⅰ) at a single location; (ⅱ) at two locations simultaneously; or (ⅲ) at one of two alternative locations. The third result in particular shows that solutions need not be unique. This marks an important difference of the non-local equation with its local counterpart.

2019, 24(2): 755-781 doi: 10.3934/dcdsb.2018206 +[Abstract](2935) +[HTML](342) +[PDF](567.23KB)
Abstract:

In this paper, we investigate a parabolic variational inequality problem associated with the American strangle option pricing. We obtain the existence and uniqueness of \begin{document}$W^{2, 1}_{p, \rm{loc}}$\end{document} solution to the problem. Also, we analyze the smoothness and monotonicity of two free boundaries. Finally, numerical results of the model based on this problem are described and used to show the boundary properties and the price behavior.

2019, 24(2): 783-800 doi: 10.3934/dcdsb.2018207 +[Abstract](3822) +[HTML](352) +[PDF](1128.48KB)
Abstract:

In this paper, we propose a latent HIV infection model with general incidence function and multiple delays. We derive the positivity and boundedness of solutions, as well as the existence and local stability of the infection-free and infected equilibria. By constructing Lyapunov functionals, we establish the global stability of the equilibria based on the basic reproduction number. We further study the global dynamics of this model with Holling type-Ⅱ incidence function through numerical simulations. Our results improve and generalize some existing ones. The results show that the prolonged time delay period of the maturation of the newly produced viruses may lead to the elimination of the viruses.

2019, 24(2): 801-829 doi: 10.3934/dcdsb.2018208 +[Abstract](2642) +[HTML](359) +[PDF](536.11KB)
Abstract:

In this paper, we investigate a two-species weak competition system of reaction-diffusion-advection with double free boundaries that represent the expanding front in a one-dimensional habitat, where a combination of random movement and advection is adopted by two competing species. The main goal is to understand the effect of small advection environment and dynamics of the two species through double free boundaries. We provide a spreading-vanishing dichotomy, which means that both of the two species either spread to the entire space successfully and survive in the new environment as time goes to infinity, or vanish and become extinct in the long run. Furthermore, if the spreading or vanishing of the two species occurs, some sufficient conditions via the initial data are established. When spreading of the two species happens, the long time behavior of solutions and estimates of spreading speed of both free boundaries are obtained.

2019, 24(2): 831-849 doi: 10.3934/dcdsb.2018209 +[Abstract](2803) +[HTML](316) +[PDF](539.06KB)
Abstract:

This paper deals with a boundary-value problem for a coupled chemotaxis-Stokes system with logistic source

in three-dimensional smoothly bounded domains, where the parameters $ξ\ge0$, $μ>0$ and $φ∈ W^{1, ∞}(Ω)$, $D$ is a given function satisfying $D(n)\ge C_{D}n^{m-1}$ for all $n>0$ with $m>0$ and $C_{D}>0$. $\mathcal{S}$ is a given function with values in $\mathbb{R}^{3×3}$ which fulfills

with some $C_{\mathcal{S}}>0$ and $α>0$. It is proved that for all reasonably regular initial data, global weak solutions exist whenever $m+2α>\frac{6}{5}$. This extends a recent result by Liu el at. (J. Diff. Eqns, 261 (2016) 967-999) which asserts global existence of weak solutions under the constraints $m+α>\frac{6}{5}$ and $m\ge\frac{1}{3}$.

2019, 24(2): 851-879 doi: 10.3934/dcdsb.2018210 +[Abstract](2523) +[HTML](312) +[PDF](5876.6KB)
Abstract:

We consider the Czirók model for collective motion of locusts along a one-dimensional torus. In the model, each agent's velocity locally interacts with other agents' velocities in the system, and there is also exogenous randomness to each agent's velocity. The interaction tends to create the alignment of collectivemotion. By analyzing the associated nonlinear Fokker-Planck equation, we obtain the condition for the existence of stationary order states and the conditions for their linear stability. These conditions depend on the noise level, which should be strong enough, and on the interaction between the agent's velocities, which should be neither too small, nor too strong. We carry out the fluctuation analysis of the interacting system and describe the large deviation principle to calculate the transition probability from one order state to the other. Numerical simulations confirm our analytical findings.

2019, 24(2): 881-905 doi: 10.3934/dcdsb.2018211 +[Abstract](2843) +[HTML](318) +[PDF](362.69KB)
Abstract:

In this paper we consider an $n$ dimensional piecewise smooth dynamical system. This system has a co-dimension 2 switching manifold Σ which is an intersection of two co-dimension one switching manifolds Σ1 and Σ2. We investigate the relation of periodic orbit of PWS between periodic orbit of its regularized system. If this PWS system has an asymptotically stable crossing periodic orbit γ or sliding periodic orbit, we establish conditions to ensure that also a regularization of the given system has a unique, asymptotically stable, limit cycle in a neighbourhood of γ, converging to γ as the regularization parameter goes to 0.

2019, 24(2): 907-929 doi: 10.3934/dcdsb.2018212 +[Abstract](2887) +[HTML](361) +[PDF](534.24KB)
Abstract:

This paper focuses on the mathematical analysis of a self-suggested model arising from biology, consisting of dynamics of oxygen diffusion and consumption, chemotaxis process and viscous incompressible non-Newtonian fluid in a bounded domain \begin{document}$Ω \subset \mathbb{R}^d$\end{document}, with \begin{document}$d = 2, 3.$\end{document} The viscosity of the studied fluid is supposed to be non constant and depends on the shear-rate \begin{document}$|{\bf{D}}\boldsymbol{v}|$\end{document} as well as the cell density \begin{document}$m$\end{document} and the oxygen concentration \begin{document}$c$\end{document}. Nonlinearities are also considered in the diffusion terms for the convection-diffusion equations corresponding to \begin{document}$m$\end{document} and \begin{document}$c$\end{document}. Under the choice of suitable structures and convenient assumptions for the nonlinear fluxes, we prove global existence of weak solutions, in the case of a smooth bounded domain subject to Navier's slip conditions at the boundary and for large range of initial data.

2019, 24(2): 931-940 doi: 10.3934/dcdsb.2018213 +[Abstract](2798) +[HTML](439) +[PDF](221.08KB)
Abstract:

In 2011, Liu et. al. proposed a three-component reaction-diffusion system to model the spread of bacteria and its signaling molecules (AHL) in an expanding cell population. At high AHL levels the bacteria are immotile, but diffuse with a positive diffusion constant at low distributions of AHL. In 2012, Fu et. al. studied a reduced system without considering nutrition and made heuristic arguments about the existence of traveling wave solutions. In this paper we provide rigorous proofs of the existence of traveling wave solutions for the reduced system under some simple conditions of the model parameters.

2019, 24(2): 941-964 doi: 10.3934/dcdsb.2018255 +[Abstract](2308) +[HTML](176) +[PDF](459.67KB)
Abstract:

We are concerned with the initial boundary value problem of the Long-Short wave equations on the whole line. A fully discrete spectral approximation scheme is structured by means of Hermite functions in space and central difference in time. A priori estimates are established which are crucial to study the numerical stability and convergence of the fully discrete scheme. Then, unconditionally numerical stability is proved in a space of $H^1({\Bbb R})$ for the envelope of the short wave and in a space of $L^2({\Bbb R})$ for the amplitude of the long wave. Convergence of the fully discrete scheme is shown by the method of error estimates. Finally, numerical experiments are presented and numerical results are illustrated to agree well with the convergence order of the discrete scheme.

Xianbo Sun and
2019, 24(2): 965-987 doi: 10.3934/dcdsb.2018341 +[Abstract](2130) +[HTML](136) +[PDF](522.07KB)
Abstract:

In this paper, we consider a generalized BBM equation with weak backward diffusion, dissipation and Marangoni effects, and study the existence of periodic and solitary waves. Main attention is focused on periodic and solitary waves on a manifold via studying the number of zeros of some linear combination of Abelian integrals. The uniqueness of the periodic waves is established when the equation contains one coefficient in backward diffusion and dissipation terms, by showing that the Abelian integrals form a Chebyshev set. The monotonicity of the wave speed is proved, and moreover the upper and lower bounds of the limiting wave speeds are obtained. Especially, when the equation involves Marangoni effect due to imposed weak thermal gradients, it is shown that at most two periodic waves can exist. The exact conditions are obtained for the existence of one and two periodic waves as well as for the co-existence of one solitary and one periodic waves. The analysis is mainly based on Chebyshev criteria and asymptotic expansions of Abelian integrals near the solitary and singularity.

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