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Discrete & Continuous Dynamical Systems - B
June 2016 , Volume 21 , Issue 4
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2016, 21(4): 1027-1050
doi: 10.3934/dcdsb.2016.21.1027
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We study mathematical and physical properties of a family of recently introduced, reduced-order approximate deconvolution models. We first show a connection between these models and the Navier-Stokes-Voigt model, and also that Navier-Stokes-Voigt can be re-derived in the approximate deconvolution framework. We then study the energy balance and spectra of the model, and provide results of some turbulent-flow computations that backs up the theory. Analysis of global attractors for the model is also provided, as is a detailed analysis of the Voigt model's treatment of pulsatile flow.
We study mathematical and physical properties of a family of recently introduced, reduced-order approximate deconvolution models. We first show a connection between these models and the Navier-Stokes-Voigt model, and also that Navier-Stokes-Voigt can be re-derived in the approximate deconvolution framework. We then study the energy balance and spectra of the model, and provide results of some turbulent-flow computations that backs up the theory. Analysis of global attractors for the model is also provided, as is a detailed analysis of the Voigt model's treatment of pulsatile flow.
2016, 21(4): 1051-1077
doi: 10.3934/dcdsb.2016.21.1051
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The main goal of this paper is to generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on $L^p$-space with $p>4$. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on $L^p$-space with $p>4$.
The main goal of this paper is to generalize to Banach spaces the well-known results for diffusions on Hilbert spaces obtained in [Peszat, S. and Zabczyk, J. (1995). Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 23(1): 157-172.]. More precisely, we are aiming to prove the strong Feller property and irreducibility of the solutions to a stochastic evolution equations (SEEs) in Banach spaces. We give sufficient conditions on the path space and the coefficients of the SEEs for these aforementioned properties to hold. We apply our result to investigate the long-time behavior of a stochastic nonlinear heat equations on $L^p$-space with $p>4$. Our result implies the uniqueness of the invariant measure, if it exists, for the stochastic nonlinear heat equations on $L^p$-space with $p>4$.
2016, 21(4): 1079-1099
doi: 10.3934/dcdsb.2016.21.1079
+[Abstract](1754)
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Abstract:
We consider the spectral stability problem for a family of standing pulse and wave front solutions to the one-dimensional version of the $M^5$-model formulated by Hillen [T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585--616], to describe the mesenchymal cell motion inside tissue. The stability analysis requires the definition of spectrum, which is divided into two disjoint sets: the point spectrum and the essential spectrum. Under this partition the eigenvalue zero belongs to the essential spectrum and not to the point spectrum. By excluding the eigenvalue zero we can bring the spectral problem into an equivalent scalar quadratic eigenvalue problem. This leads, naturally, to deduce the existence of a negative eigenvalue which also turns out to belong to the essential spectrum. Beyond this result, the scalar formulation enables us to use the integrated equation technique to establish, via energy methods, that the point spectrum is empty. Our main result is that the family of standing waves is spectrally stable. To prove it, we go back to the original scalar problem and show that the rest of the essential spectrum is a subset of the open left-half complex plane.
We consider the spectral stability problem for a family of standing pulse and wave front solutions to the one-dimensional version of the $M^5$-model formulated by Hillen [T. Hillen, $M^5$ mesoscopic and macroscopic models for mesenchymal motion, J. Math. Biol., 53 (2006), 585--616], to describe the mesenchymal cell motion inside tissue. The stability analysis requires the definition of spectrum, which is divided into two disjoint sets: the point spectrum and the essential spectrum. Under this partition the eigenvalue zero belongs to the essential spectrum and not to the point spectrum. By excluding the eigenvalue zero we can bring the spectral problem into an equivalent scalar quadratic eigenvalue problem. This leads, naturally, to deduce the existence of a negative eigenvalue which also turns out to belong to the essential spectrum. Beyond this result, the scalar formulation enables us to use the integrated equation technique to establish, via energy methods, that the point spectrum is empty. Our main result is that the family of standing waves is spectrally stable. To prove it, we go back to the original scalar problem and show that the rest of the essential spectrum is a subset of the open left-half complex plane.
2016, 21(4): 1101-1117
doi: 10.3934/dcdsb.2016.21.1101
+[Abstract](1688)
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Abstract:
We prove the global asymptotic stability of the disease-free and the endemic equilibrium for general SIR and SIRS models with nonlinear incidence. Instead of the popular Volterra-type Lyapunov functions, we use the method of Dulac functions, which allows us to extend the previous global stability results to a wider class of SIR and SIRS systems, including nonlinear (density-dependent) removal terms as well. We show that this method is useful in cases that cannot be covered by Lyapunov functions, such as bistable situations. We completely describe the global attractor even in the scenario of a backward bifurcation, when multiple endemic equilibria coexist.
We prove the global asymptotic stability of the disease-free and the endemic equilibrium for general SIR and SIRS models with nonlinear incidence. Instead of the popular Volterra-type Lyapunov functions, we use the method of Dulac functions, which allows us to extend the previous global stability results to a wider class of SIR and SIRS systems, including nonlinear (density-dependent) removal terms as well. We show that this method is useful in cases that cannot be covered by Lyapunov functions, such as bistable situations. We completely describe the global attractor even in the scenario of a backward bifurcation, when multiple endemic equilibria coexist.
2016, 21(4): 1119-1148
doi: 10.3934/dcdsb.2016.21.1119
+[Abstract](1765)
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Abstract:
In the present study, we investigate the dynamics of shunting inhibitory cellular neural networks (SICNNs) with impulsive effects. We give a mathematical description of the chaos for the multidimensional dynamics of impulsive SICNNs, and prove its existence rigorously by taking advantage of the external inputs. The Li-Yorke definition of chaos is used in our theoretical discussions. In the considered model, the impacts satisfy the cell and shunting principles. This enriches the applications of SICNNs and makes the analysis of impulsive neural networks deeper. The technique is exceptionally useful for SICNNs with arbitrary number of cells. We make benefit of unidirectionally coupled SICNNs to exemplify our results. Moreover, the appearance of cyclic irregular behavior observed in neuroscience is numerically demonstrated for discontinuous dynamics of impulsive SICNNs.
In the present study, we investigate the dynamics of shunting inhibitory cellular neural networks (SICNNs) with impulsive effects. We give a mathematical description of the chaos for the multidimensional dynamics of impulsive SICNNs, and prove its existence rigorously by taking advantage of the external inputs. The Li-Yorke definition of chaos is used in our theoretical discussions. In the considered model, the impacts satisfy the cell and shunting principles. This enriches the applications of SICNNs and makes the analysis of impulsive neural networks deeper. The technique is exceptionally useful for SICNNs with arbitrary number of cells. We make benefit of unidirectionally coupled SICNNs to exemplify our results. Moreover, the appearance of cyclic irregular behavior observed in neuroscience is numerically demonstrated for discontinuous dynamics of impulsive SICNNs.
2016, 21(4): 1149-1166
doi: 10.3934/dcdsb.2016.21.1149
+[Abstract](2038)
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Abstract:
In this paper we discuss an empirical phenomena known as the 20-60-20 rule. It says that if we split the population into three groups, according to some arbitrary benchmark criterion, then this particular ratio often implies some sort of balance. From practical point of view, this feature leads to efficient management or control. We provide a mathematical illustration, justifying the occurrence of this rule in many real world situations. We show that for any population, which could be described using multivariate normal vector, this fixed ratio leads to a global equilibrium state, when dispersion and linear dependance measurement is considered.
In this paper we discuss an empirical phenomena known as the 20-60-20 rule. It says that if we split the population into three groups, according to some arbitrary benchmark criterion, then this particular ratio often implies some sort of balance. From practical point of view, this feature leads to efficient management or control. We provide a mathematical illustration, justifying the occurrence of this rule in many real world situations. We show that for any population, which could be described using multivariate normal vector, this fixed ratio leads to a global equilibrium state, when dispersion and linear dependance measurement is considered.
2016, 21(4): 1167-1187
doi: 10.3934/dcdsb.2016.21.1167
+[Abstract](2144)
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Abstract:
When describing the anisotropic evolution of microstructures in solids using phase-field models, the anisotropy of the crystalline phases is usually introduced into the interfacial energy by directional dependencies of the gradient energy coefficients. We consider an alternative approach based on a wavelet analogue of the Laplace operator that is intrinsically anisotropic and linear. The paper focuses on the classical coupled temperature/Ginzburg--Landau type phase-field model for dendritic growth. For the model based on the wavelet analogue, existence, uniqueness and continuous dependence on initial data are proved for weak solutions. Numerical studies of the wavelet based phase-field model show dendritic growth similar to the results obtained for classical phase-field models.
When describing the anisotropic evolution of microstructures in solids using phase-field models, the anisotropy of the crystalline phases is usually introduced into the interfacial energy by directional dependencies of the gradient energy coefficients. We consider an alternative approach based on a wavelet analogue of the Laplace operator that is intrinsically anisotropic and linear. The paper focuses on the classical coupled temperature/Ginzburg--Landau type phase-field model for dendritic growth. For the model based on the wavelet analogue, existence, uniqueness and continuous dependence on initial data are proved for weak solutions. Numerical studies of the wavelet based phase-field model show dendritic growth similar to the results obtained for classical phase-field models.
2016, 21(4): 1189-1202
doi: 10.3934/dcdsb.2016.21.1189
+[Abstract](1539)
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Abstract:
This article is concerned with a mutualism ecological model with Lévy noise. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistence in mean are also established.
This article is concerned with a mutualism ecological model with Lévy noise. The local existence and uniqueness of a positive solution are obtained with positive initial value, and the asymptotic behavior to the problem is studied. Moreover, we show that the solution is stochastically bounded and stochastic permanence. The sufficient conditions for the system to be extinct are given and the conditions for the system to be persistence in mean are also established.
2016, 21(4): 1203-1223
doi: 10.3934/dcdsb.2016.21.1203
+[Abstract](1906)
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Abstract:
A bi-spatial pullback attractor is obtained for non-autonomous and stochastic FitzHugh-Nagumo equations when the initial space is $L^2(\mathbb{R}^n)^2$ and the terminate space is $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$. Some new techniques of positive and negative truncations are used to investigate the regularity of attractors for coupling equations and to correct the essential mistake in [T. Q. Bao, Discrete Cont. Dyn. Syst. 35(2015), 441-466]. A counterexample is given for an important lemma for $H^1$-attractor in several literatures included above.
A bi-spatial pullback attractor is obtained for non-autonomous and stochastic FitzHugh-Nagumo equations when the initial space is $L^2(\mathbb{R}^n)^2$ and the terminate space is $H^1(\mathbb{R}^n)\times L^2(\mathbb{R}^n)$. Some new techniques of positive and negative truncations are used to investigate the regularity of attractors for coupling equations and to correct the essential mistake in [T. Q. Bao, Discrete Cont. Dyn. Syst. 35(2015), 441-466]. A counterexample is given for an important lemma for $H^1$-attractor in several literatures included above.
2016, 21(4): 1225-1236
doi: 10.3934/dcdsb.2016.21.1225
+[Abstract](1514)
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Abstract:
In this article, we study the dynamic transition for the one dimensional generalized Kuramoto-Sivashinsky equation with periodic condition. It is shown that if the value of the dispersive parameter $\nu$ is strictly greater than $\nu^{\ast}$, then the transition is Type-I (continuous) and the bifurcated periodic orbit is an attractor as the control parameter $\lambda$ crosses the critical value $\lambda_0$. In the case where $\nu$ is strictly less than $\nu^{\ast}$, then the transition is Type-II (jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter $\lambda$ crosses the critical value $\lambda_0$. The value of $\nu^{\ast}$ is also calculated in this paper.
In this article, we study the dynamic transition for the one dimensional generalized Kuramoto-Sivashinsky equation with periodic condition. It is shown that if the value of the dispersive parameter $\nu$ is strictly greater than $\nu^{\ast}$, then the transition is Type-I (continuous) and the bifurcated periodic orbit is an attractor as the control parameter $\lambda$ crosses the critical value $\lambda_0$. In the case where $\nu$ is strictly less than $\nu^{\ast}$, then the transition is Type-II (jump) and the trivial solution bifurcates to a unique unstable periodic orbit as the control parameter $\lambda$ crosses the critical value $\lambda_0$. The value of $\nu^{\ast}$ is also calculated in this paper.
2016, 21(4): 1237-1257
doi: 10.3934/dcdsb.2016.21.1237
+[Abstract](1593)
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Abstract:
We consider a spatially-heterogeneous generalization of a well-established model for the dynamics of the Human Immunodeficiency Virus-type 1 (HIV) within a susceptible host. The model consists of a nonlinear system of three coupled reaction-diffusion equations with parameters that may vary spatially. Upon formulating the model, we prove that it preserves the positivity of initial data and construct global-in-time solutions that are both bounded and smooth. Finally, additional results concerning the local and global asymptotic behavior of these solutions are also provided.
We consider a spatially-heterogeneous generalization of a well-established model for the dynamics of the Human Immunodeficiency Virus-type 1 (HIV) within a susceptible host. The model consists of a nonlinear system of three coupled reaction-diffusion equations with parameters that may vary spatially. Upon formulating the model, we prove that it preserves the positivity of initial data and construct global-in-time solutions that are both bounded and smooth. Finally, additional results concerning the local and global asymptotic behavior of these solutions are also provided.
2016, 21(4): 1259-1277
doi: 10.3934/dcdsb.2016.21.1259
+[Abstract](1819)
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A nonlinear reaction-diffusion problem with a general, both spatially and delay distributed reaction term is considered in an unbounded domain $\mathbb{R}^N$. The existence of a unique weak solution is proved. A locally compact attractor together with entropy bound is also established.
A nonlinear reaction-diffusion problem with a general, both spatially and delay distributed reaction term is considered in an unbounded domain $\mathbb{R}^N$. The existence of a unique weak solution is proved. A locally compact attractor together with entropy bound is also established.
2016, 21(4): 1279-1295
doi: 10.3934/dcdsb.2016.21.1279
+[Abstract](2536)
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In this study, a previously published mathematical model of mixed immunotherapy and chemotherapy of tumors is considered. The stability analysis of the tumor-free equilibrium obtained in the previous study of this model is flawed. In this paper, a suitable analysis is performed to correct this error, and the parameter conditions for the stability of the tumor-free equilibrium are obtained. The stability condition gives an indicator of the host's ability to fight a cancer. The parameter conditions are examined using experimental data from clinical studies to show that the immune system is able to control a small tumor, and the host's ability to fight a cancer depends on individual variation. A numerical method based on the continuation technique is employed for one-parameter bifurcation analysis of the mathematical model with periodically pulsed therapies. The unstable fixed point curve provides a good approximation of the maximum tumor burden as a function of the dosage. Chemotherapy-induced lymphocyte damage, which may cause treatment failure, is observed in the numerical simulation. The numerical method also produces a set of combined chemotherapy and immunotherapy dosages from which an efficient and safe combination of dosages can be determined.
In this study, a previously published mathematical model of mixed immunotherapy and chemotherapy of tumors is considered. The stability analysis of the tumor-free equilibrium obtained in the previous study of this model is flawed. In this paper, a suitable analysis is performed to correct this error, and the parameter conditions for the stability of the tumor-free equilibrium are obtained. The stability condition gives an indicator of the host's ability to fight a cancer. The parameter conditions are examined using experimental data from clinical studies to show that the immune system is able to control a small tumor, and the host's ability to fight a cancer depends on individual variation. A numerical method based on the continuation technique is employed for one-parameter bifurcation analysis of the mathematical model with periodically pulsed therapies. The unstable fixed point curve provides a good approximation of the maximum tumor burden as a function of the dosage. Chemotherapy-induced lymphocyte damage, which may cause treatment failure, is observed in the numerical simulation. The numerical method also produces a set of combined chemotherapy and immunotherapy dosages from which an efficient and safe combination of dosages can be determined.
2016, 21(4): 1297-1316
doi: 10.3934/dcdsb.2016.21.1297
+[Abstract](1865)
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In this paper, we study the initial boundary value problem of a reaction-convection-diffusion epidemic model for cholera dynamics, which was developed in [38], named susceptible-infected-recovered-susceptible-bacteria (SIRS-B) epidemic PDE model. First, a local well-posedness result relying on the theory of cooperative dynamics systems is obtained. Via a priori estimates making use of the special structure of the system and continuation of local theory argument, we show that in fact this problem is globally well-posed. Secondly, we analyze the local asymptotic stability of the solutions based on the basic reproduction number associated with this model.
In this paper, we study the initial boundary value problem of a reaction-convection-diffusion epidemic model for cholera dynamics, which was developed in [38], named susceptible-infected-recovered-susceptible-bacteria (SIRS-B) epidemic PDE model. First, a local well-posedness result relying on the theory of cooperative dynamics systems is obtained. Via a priori estimates making use of the special structure of the system and continuation of local theory argument, we show that in fact this problem is globally well-posed. Secondly, we analyze the local asymptotic stability of the solutions based on the basic reproduction number associated with this model.
2016, 21(4): 1317-1327
doi: 10.3934/dcdsb.2016.21.1317
+[Abstract](1745)
+[PDF](436.6KB)
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This paper considers the parabolic-parabolic Keller-Segel system with nonlinear sensitivity $u_t=\Delta u-\nabla (u^{\alpha}\nabla v)$, $v_t=\Delta v-v+u$, subject to homogeneous Neumann boundary conditions with smooth and bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq1$. It is proved that if $\alpha\geq\max\{1,\frac{2}{n}\}$, then the solutions are globally bounded, and both the components $u$ and $v$ decay to the same constant steady state $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u_0(x) dx$ exponentially in the $L^\infty$-norm provided both $\|u_0\|_{L^{q^{\ast}}(\Omega)}$ and $\|\nabla v_0\|_{L^{p^{\ast}}(\Omega)}$ small enough with $q^{\ast}=\frac{n\alpha K}{n+K}$, $p^{\ast}=\frac{n\alpha K}{n+K-n\alpha}$, $K\in[n,2n\alpha-n]\cap ((\alpha-1)n,\infty)$.
This paper considers the parabolic-parabolic Keller-Segel system with nonlinear sensitivity $u_t=\Delta u-\nabla (u^{\alpha}\nabla v)$, $v_t=\Delta v-v+u$, subject to homogeneous Neumann boundary conditions with smooth and bounded domain $\Omega\subset\mathbb{R}^{n}$, $n\geq1$. It is proved that if $\alpha\geq\max\{1,\frac{2}{n}\}$, then the solutions are globally bounded, and both the components $u$ and $v$ decay to the same constant steady state $\bar{u}_0=\frac{1}{|\Omega|}\int_\Omega u_0(x) dx$ exponentially in the $L^\infty$-norm provided both $\|u_0\|_{L^{q^{\ast}}(\Omega)}$ and $\|\nabla v_0\|_{L^{p^{\ast}}(\Omega)}$ small enough with $q^{\ast}=\frac{n\alpha K}{n+K}$, $p^{\ast}=\frac{n\alpha K}{n+K-n\alpha}$, $K\in[n,2n\alpha-n]\cap ((\alpha-1)n,\infty)$.
2016, 21(4): 1329-1346
doi: 10.3934/dcdsb.2016.21.1329
+[Abstract](1887)
+[PDF](434.4KB)
Abstract:
A model with age of infection is formulated to study the possible effects of variable infectivity on HBV transmission dynamics. The stability of equilibria and persistence of the model are analyzed. The results show that if the basic reproductive number $\mathcal{R}_0<1$, then the disease-free equilibrium is globally asymptotically stable. For $\mathcal{R}_0>1$, the disease is uniformly persistent, and a Lyapunov function is used to show that the unique endemic equilibrium is globally stable in a special case.
A model with age of infection is formulated to study the possible effects of variable infectivity on HBV transmission dynamics. The stability of equilibria and persistence of the model are analyzed. The results show that if the basic reproductive number $\mathcal{R}_0<1$, then the disease-free equilibrium is globally asymptotically stable. For $\mathcal{R}_0>1$, the disease is uniformly persistent, and a Lyapunov function is used to show that the unique endemic equilibrium is globally stable in a special case.
2016, 21(4): 1347-1388
doi: 10.3934/dcdsb.2016.21.1347
+[Abstract](1531)
+[PDF](1013.7KB)
Abstract:
In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the terminal values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.
In this paper, we study the minimum time planar tilting maneuver of a spacecraft, from the theoretical as well as from the numerical point of view, with a particular focus on the chattering phenomenon. We prove that there exist optimal chattering arcs when a singular junction occurs. Our study is based on the Pontryagin Maximum Principle and on results by M.I. Zelikin and V.F. Borisov. We give sufficient conditions on the terminal values under which the optimal solutions do not contain any singular arc, and are bang-bang with a finite number of switchings. Moreover, we implement sub-optimal strategies by replacing the chattering control with a fixed number of piecewise constant controls. Numerical simulations illustrate our results.
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