# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

August 2016 , Volume 21 , Issue 6

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2016, 21(6): 1671-1687 doi: 10.3934/dcdsb.2016017 +[Abstract](1895) +[PDF](1008.2KB)
Abstract:
In this paper, we are interested in the mathematical and numerical study of a variational model derived as Reaction-Diffusion System for image denoising. We use a nonlinear regularization of total variation (TV) operator's, combined with a decomposition approach of $H^{-1}$ norm suggested by Guo and al. ([19],[20]). Based on Galerkin's method, we prove the existence and uniqueness of the solution on Orlicz space for the proposed model. At last, compared experimental results distinctly demonstrate the superiority of our model, in term of removing noise while preserving the edges and reducing staircase effect.
2016, 21(6): 1689-1711 doi: 10.3934/dcdsb.2016018 +[Abstract](1906) +[PDF](547.1KB)
Abstract:
In this paper, we prove the existence and uniqueness of a Gevrey regularity solution for a class of nonlinear bistable gradient flows, where with the energy may be decomposed into purely convex and concave parts. Example equations include certain epitaxial thin film growth models and phase field crystal models. The energy dissipation law implies a bound in the leading Sobolev norm. The polynomial structure of the nonlinear terms in the chemical potential enables us to derive a local-in-time solution with Gevrey regularity, with the existence time interval length dependent on a certain $H^m$ norm of the initial data. A detailed Sobolev estimate for the gradient equations results in a uniform-in-time-bound of that $H^m$ norm, which in turn establishes the existence of a global-in-time solution with Gevrey regularity.
2016, 21(6): 1713-1728 doi: 10.3934/dcdsb.2016019 +[Abstract](1737) +[PDF](271.6KB)
Abstract:
In this work, the effect of a host refuge on a host-parasitoid inter- action is investigated. The model is built upon a modi ed Nicholson-Bailey system by assuming that in each generation a constant proportion of the host is free from parasitism. We derive a sucient condition based on the model parameters for both populations to coexist. We prove that it is possible for the system to undergo a supercritical and then a subcritical Neimark-Sacker bifurcation or for the system only to exhibit a supercritical Neimark-Sacker bifurcation. It is illustrated numerically that a constant proportion of host refuge can stabilize the host-parasitoid interaction.
2016, 21(6): 1729-1755 doi: 10.3934/dcdsb.2016020 +[Abstract](1897) +[PDF](3096.6KB)
Abstract:
We study the spatial propagating dynamics in a neural network of excitatory and inhibitory populations. Our study demonstrates the existence and nonexistence of traveling pulse solutions with a nonsaturating piecewise linear gain function. We prove that traveling pulse solutions do not exist for such neural field models with even (symmetric) couplings. The neural field models only support traveling pulse solutions with asymmetric couplings. We also show that such neural field models with asymmetric couplings will lead to a system of delay differential equations. We further compute traveling 1--bump solutions using the system of delay differential equations. Finally, we develop Evans functions to assess the stability of traveling 1--bump solutions.
2016, 21(6): 1757-1774 doi: 10.3934/dcdsb.2016021 +[Abstract](2115) +[PDF](410.7KB)
Abstract:
Let a fourth and a second order evolution equations be coupled via the interface by transmission conditions, and suppose that the first one is stabilized by a localized distributed feedback. What will then be the effect of such a partial stabilization on the decay of solutions at infinity? Is the behavior of the first component sufficient to stabilize the second one? The answer given in this paper is that sufficiently smooth solutions decay logarithmically at infinity even the feedback dissipation affects an arbitrarily small open subset of the interior. The method used, in this case, is based on a frequency method, and this by combining a contradiction argument with the Carleman estimates technique to carry out a special analysis for the resolvent.
2016, 21(6): 1775-1802 doi: 10.3934/dcdsb.2016022 +[Abstract](2079) +[PDF](572.7KB)
Abstract:
We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows through membrane channels with fixed boundary ion concentrations and electric potentials. We consider two ion species, one positively charged and one negatively charged, and assume zero permanent charge. Bikerman's local hard-sphere potential is included in the model to account for ion size effects on the ionic flow. The model problem is treated as a boundary value problem of a singularly perturbed differential system. Our analysis is based on the geometric singular perturbation theory but, most importantly, on specific structures of this concrete model. The existence of solutions to the boundary value problem for small ion sizes is established and, treating the ion sizes as small parameters, we also derive approximations of individual fluxes and I-V (current-voltage) relations, from which qualitative properties of ionic flows related to ion sizes are studied. A detailed characterization of complicated interactions among multiple and physically crucial parameters for ionic flows, such as boundary concentrations and potentials, diffusion coefficients and ion sizes, is provided.
2016, 21(6): 1803-1812 doi: 10.3934/dcdsb.2016023 +[Abstract](1572) +[PDF](392.0KB)
Abstract:
In this paper we study the approximate controllability of the following semilinear difference equation $z(n+1)=A(n)z(n)+B(n)u(n)+f(n,z(n),u(n)), \quad n\in \mathbb{N}^*,$ $z(n)\in Z$, $u(n)\in U$, where $Z$, $U$ are Hilbert spaces, $A\in l^{\infty}(\mathbb{N},L(Z))$, $B\in l^{\infty}(\mathbb{N},L(U,Z))$, $u\in l^2(\mathbb{N},U)$ and the nonlinear term $f:\mathbb{N} \times Z\times U\longrightarrow Z$ is a suitable function. We prove that, under some conditions on the nonlinear term $f$, the approximate controllability of the linear equation is preserved. Finally, we apply this result to a discrete version of the semilinear wave equation.
2016, 21(6): 1813-1837 doi: 10.3934/dcdsb.2016024 +[Abstract](1333) +[PDF](1238.1KB)
Abstract:
Systems of ODE's with forms related to the discrete nonlinear Schrödinger equation arise in a variety of semi-classical molecular models, such as models of polymers with coupling between nearby excitable states, and reductions of quantum many-body systems. Similar systems also arise as models of arrays of nonlinear optical wave-guides, which can involve new features such as binary alternation of coefficients: so-called Binary Exciton Chain Systems.
Solutions of such systems are often seen to develop regions of slow variation even when the initial data are impulsive: in particular the emergence of a slowly varying leading pulse that propagates in an approximately traveling wave form, and in stationary oscillations near the endpoints. This has motivated the search for long-wave approximations by PDEs.
In this article it is observed that the patterns of slow variation are substantially different from those assumed in some previously-considered long-wave approximations, and several new PDE approximations are presented: third order systems describing leading pulses of approximately traveling wave form (related to the Airy PDE in the linearized case), and a quite different system describing stationary oscillations near an endpoint with zero boundary conditions. Numerical solutions of these PDE models and some linear analysis confirm that they provide a good agreement with the long-wave phenomena observed in the ODE systems.
2016, 21(6): 1839-1858 doi: 10.3934/dcdsb.2016025 +[Abstract](2222) +[PDF](438.0KB)
Abstract:
This paper concerns the Neumann problem of a reaction-diffusion system, which has a variable exponent Laplacian term and could be applied to image denoising. It is shown that the problem admits a unique renormalized solution for each integrable initial datum.
2016, 21(6): 1859-1867 doi: 10.3934/dcdsb.2016026 +[Abstract](1897) +[PDF](330.4KB)
Abstract:
A ratio--dependent predator-prey model with stage structure for prey was investigated in [8]. There the authors mentioned that they were unable to show if such a model admits limit cycles when the unique equilibrium point $E^*$ at the positive octant is unstable.
Here we characterize the existence of Hopf bifurcations for the systems. In particular we provide a positive answer to the above question showing for such models the existence of small--amplitude Hopf limit cycles being the equilibrium point $E^*$ unstable.
2016, 21(6): 1869-1893 doi: 10.3934/dcdsb.2016027 +[Abstract](1616) +[PDF](698.9KB)
Abstract:
As the practice of aquaculture has increased the interplay between large fish farms and wild fisheries in close proximity has become ever more pressing. Infectious salmon anemia virus (ISAv) is a flu-like virus affecting a variety of finfish. In this article, we adapt the standard deterministic within host model of a viral infection to each patch of a two patch system and couple the patches via linear diffusion of the virus. We determine the basic reproductive ratio $\mathcal{R}^0$ for the full system as well as invariant subsystems. We show the existence of unique positive equilibrium in the full system and subsystems and relate the existence of the equilibrium to the $\mathcal{R}^0$ values. In particular, we show that if $\mathcal{R}^0>1$, the virus persists in the environment and is enzootic in the host population; if $\mathcal{R}^0\leq 1$, the virus is cleared and the system asymptotically approaches the disease free equilibrium. We also show that, with positive diffusivity, it is possible for the virus to be excluded when there is a susceptible host population in only one patch, but to persist if there are susceptible host populations in both patches. We analyze the local stability of the equilibria and show the existence of Hopf bifurcations.
2016, 21(6): 1895-1915 doi: 10.3934/dcdsb.2016028 +[Abstract](2397) +[PDF](2062.8KB)
Abstract:
We discuss the optimization of chemotherapy treatment for low-grade gliomas using a mathematical model. We analyze the dynamics of the model and study the stability of solutions. The dynamical model is incorporated into an optimal control problem for which different objective functionals are considered. We establish the existence of optimal controls and give a detailed discussion of the necessary optimality conditions. Since the control variable appears linearly in the control problem, optimal controls are concatenations of bang-bang and singular arcs. We derive a formula of the singular control in terms of state and adjoint variables. Using discretization and optimization methods we compute optimal drug protocols in a number of scenarios. For small treatment periods, the optimal control is bang-bang, whereas for larger treatment periods we obtain both bang-bang and singular arcs. In particular, singular controls illustrate the metronomic chemotherapy.
2016, 21(6): 1917-1936 doi: 10.3934/dcdsb.2016029 +[Abstract](1554) +[PDF](9063.3KB)
Abstract:
In 1977 Robe considered a modification of the Restricted Three Body Problem, where one of the primaries is a shell filled with an incompressible liquid. The motion of the small body of negligible mass takes place inside this sphere and is therefore affected by the buoyancy force of the liquid. We investigate the existence and stability of the equilibrium points in the planar circular problem and discuss the range of the parameters for which the problem has a physical meaning.
Our main contribution is to establish the Lyapunov stability for the equilibrium point at the center of the shell. We achieve this by putting the Hamiltonian function of Robe's problem into its normal form and then use the theorems of Arnol'd, Markeev and Sokol'skii. Resonance cases and some exceptional cases require special treatment.
2016, 21(6): 1937-1951 doi: 10.3934/dcdsb.2016030 +[Abstract](1522) +[PDF](463.3KB)
Abstract:
We provide qualitative predictions on the rheology of mucus of healthy individuals (Wild Type or WT-mucus) versus those infected with Cystic Fibrosis (CF-mucus) using an experimentally guided, multi-phase, multi-species ionic gel model. The theory which accounts for mucus (as polymer phase), water (as solvent phase) and ions, H$^+$, Na$^+$ and Ca$^{2+}$, is linearized to study the hydration of spherically symmetric mucus gels and calibrated against the experimental data of mucus diffusivities. Near equilibrium, the linearized form of the solution reduces to an expression similar to the well known kinetic theory of neutral gels. Numerical studies reveal that the Donnan potential is the dominating mechanism driving the mucus swelling/deswelling transition. However, the altered swelling kinetics of the Cystic Fibrosis infected mucus is not merely governed by the hydroelectric composition of the swelling media, but also due to the altered movement of electrolytes as well as due to the defective properties of the mucin polymer network.
2016, 21(6): 1953-1973 doi: 10.3934/dcdsb.2016031 +[Abstract](1908) +[PDF](517.7KB)
Abstract:
In this paper, we consider the following quasilinear attraction-repulsion chemotaxis system of parabolic-parabolic type \begin{equation*} \left\{ \begin{split} &u_t=\nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w),\qquad & x\in\Omega,\,\, t>0,\\ &v_t=\Delta v+\alpha u-\beta v,\qquad &x\in\Omega, \,\,t>0,\\ &w_t=\Delta w+\gamma u-\delta w,\qquad &x\in\Omega,\,\, t>0 \end{split} \right. \end{equation*} under homogeneous Neumann boundary conditions, where $D(u)\geq c_D (u+\varepsilon)^{m-1}$ and $\Omega\subset\mathbb{R}^2$ is a bounded domain with smooth boundary. It is shown that whenever $m>1$, for any sufficiently smooth nonnegative initial data, the system admits a global bounded classical solution for the case of non-degenerate diffusion (i.e., $\varepsilon>0$), while the system possesses a global bounded weak solution for the case of degenerate diffusion (i.e., $\varepsilon=0$).
2016, 21(6): 1975-1998 doi: 10.3934/dcdsb.2016032 +[Abstract](1825) +[PDF](670.1KB)
Abstract:
In this paper, we study a size-structured cannibalism model with environment feedback and delayed birth process. Our focus is on the asymptotic behavior of the system, particularly on the effect of cannibalism and time lag on the long-term dynamics. To this end, we formally linearize the system around a steady state and study the linearized system by $C_0$-semigroup framework and spectral analysis methods. These analytical results allow us to achieve linearized stability, instability and asynchronous exponential growth results under some conditions. Finally, some examples are presented and simulated to illustrate the obtained stability conclusions.
2016, 21(6): 1999-2009 doi: 10.3934/dcdsb.2016033 +[Abstract](2493) +[PDF](375.2KB)
Abstract:
In this paper, we are concerned with the following nonlinear Schrödinger equations with hardy potential and critical Sobolev exponent $$\label{eq0.1} \left\{\begin{array}{ll} -\Delta u+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^{*}-2}u,& \textrm{in}\, \mathbb{R}^N, \\ u>0, & \textrm{in}\,\mathcal{D}^{1,2}(\mathbb{R}^N), （1） \end{array} \right.$$ where $2^{*}=\frac{2N}{N-2}$ is the critical Sobolev exponent, $0\leq \mu<\overline{\mu}=\frac{(N-2)^2}{4}$, $a(x)\in C(\mathbb{R}^N)$. We first use an abstract perturbation method in critical point theory to obtain the existence of positive solutions of （1） for small value of $|\lambda|$. Secondly, we focus on an anisotropic elliptic equation of the form $$\label{eq0.2} -{\rm div}(B_\lambda(x)\nabla u)+\lambda a(x)u^q=\mu\frac{u}{|x|^{2}}+|u|^{2^*-2}u, x\in\mathbb{R}^N. （2）$$ The same abstract method is used to yield existence result of positive solutions of （2） for small value of $|\lambda|$.
2016, 21(6): 2011-2037 doi: 10.3934/dcdsb.2016034 +[Abstract](1554) +[PDF](515.8KB)
Abstract:
We establish a class of intermittent bidirectional dispersal population models with almost periodic parameters and dispersal delays between two patches. The form of dispersal discussed in this paper is different from both continuous and impulsive dispersals, in which the dispersal behavior occurs either in a sustained manner or instantaneously; instead, it is a synthesis of these types. Dynamical properties such as permanence, existence, uniqueness, and globally asymptotic stability of almost periodic solutions are investigated by using Liapunov-Razumikhin type technique, using the comparison theorem, constructing a suitable Lyapunov functional, using almost periodic functional hull theory and analysis approach, etc. Finally, numerical simulations are presented and discussed to illustrate our analytic results, by which we find that intermittent dispersal systems are more complicated than continuous or impulsive dispersal systems.
2016, 21(6): 2039-2056 doi: 10.3934/dcdsb.2016035 +[Abstract](1997) +[PDF](467.1KB)
Abstract:
This paper deals with a parabolic-elliptic-ODE chemotaxis-haptotaxis system with nonlinear diffusion \begin{eqnarray*}\label{1a} \left\{ \begin{split}{} &u_t=\nabla\cdot(\varphi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), \\ &0=\Delta v-v+u, \\ &w_{t}=-vw, \end{split} \right. \end{eqnarray*} under Neumann boundary conditions in a smooth bounded domain $\Omega\subset \mathbb{R}^{n}$ $(n\geq1)$, where $\chi$, $\xi$ and $\mu$ are positive parameters and $\varphi(u)$ is a nonlinear diffusion. Under the non-degenerate diffusion and some suitable assumptions on positive parameters $\chi,\xi,\mu$, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution that is uniformly bounded in $\Omega\times(0,\infty)$. Moreover, under the degenerate diffusion, it is proved that the corresponding problem admits at least one nonnegative global bounded-in-time weak solution. Finally, for the suitably small initial data $w_{0}$, we give the decay estimate of $w$.
2016, 21(6): 2057-2071 doi: 10.3934/dcdsb.2016036 +[Abstract](2034) +[PDF](431.7KB)
Abstract:
In this paper, the Lie symmetry analysis is performed on the KBK equation. By constructing its one-dimensional optimal system, we obtain four classes of reduced equations and corresponding group-invariant solutions. Particularly, the traveling wave equation, as an important reduced equation, is investigated in detail. Treating it as a singular perturbation system in $\mathbb{R}^3$, we study the phase space geometry of its reduced system on a two-dimensional invariant manifold by using the dynamical system methods such as tracking the unstable manifold of the saddle, studying the equilibria at infinity and discussing the homoclinic bifurcation and Poincaré bifurcation. Correspongding wavespeed conditions are determined to guarantee the existence of various bounded traveling waves of the KBK equation.

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