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1531-3492
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Discrete & Continuous Dynamical Systems - B
May 2005 , Volume 5 , Issue 2
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2005, 5(2): 175-188
doi: 10.3934/dcdsb.2005.5.175
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Abstract:
We study SIR and SIS epidemic models with multiple pathogen strains. In our models we assume total cross immunity, standard incidence, and density-dependent host mortality. We derive conditions on the models parameters which guarantee competitive exclusion between the n strains. An example is given to show that if these conditions are not satisfied then coexistence between the strains is possible. Furthermore, numerical results are presented to indicate that our conditions on the parameters are sufficient but not necessary for competitive exclusion.
We study SIR and SIS epidemic models with multiple pathogen strains. In our models we assume total cross immunity, standard incidence, and density-dependent host mortality. We derive conditions on the models parameters which guarantee competitive exclusion between the n strains. An example is given to show that if these conditions are not satisfied then coexistence between the strains is possible. Furthermore, numerical results are presented to indicate that our conditions on the parameters are sufficient but not necessary for competitive exclusion.
2005, 5(2): 189-214
doi: 10.3934/dcdsb.2005.5.189
+[Abstract](1330)
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Abstract:
We introduce a new variant of the multilayer Saint-Venant system. The classical Saint-Venant system is a well-known approximation of the incompressible Navier-Stokes equations for shallow water flows with free moving boundary. Its efficiency, robustness and low computational cost make it very commonly used. Nevertheless its range of application is limited and it does not allow to access to the vertical profile of the horizontal velocity. Hence and thanks to a precise analysis of the shallow water assumption we propose here a new approximation of the Navier-Stokes equations which consists in a set of coupled Saint-Venant systems, extends the range of validity and gives a precise description of the vertical profile of the horizontal velocity while preserving the computational efficiency of the classical Saint-Venant system. We validate the model through some numerical examples.
We introduce a new variant of the multilayer Saint-Venant system. The classical Saint-Venant system is a well-known approximation of the incompressible Navier-Stokes equations for shallow water flows with free moving boundary. Its efficiency, robustness and low computational cost make it very commonly used. Nevertheless its range of application is limited and it does not allow to access to the vertical profile of the horizontal velocity. Hence and thanks to a precise analysis of the shallow water assumption we propose here a new approximation of the Navier-Stokes equations which consists in a set of coupled Saint-Venant systems, extends the range of validity and gives a precise description of the vertical profile of the horizontal velocity while preserving the computational efficiency of the classical Saint-Venant system. We validate the model through some numerical examples.
2005, 5(2): 215-238
doi: 10.3934/dcdsb.2005.5.215
+[Abstract](1076)
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Abstract:
This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new type of attractor which includes some classes of noncompact attractors such as unbounded unstable manifolds. We then adapt two cell mapping algorithms to the nonautonomous setting and use the computer program GAIO for the analysis of an explicit example, a two-dimensional system of nonautonomous difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffing-van der Pol oscillator.
This paper is devoted to the numerical approximation of attractors. For general nonautonomous dynamical systems we first introduce a new type of attractor which includes some classes of noncompact attractors such as unbounded unstable manifolds. We then adapt two cell mapping algorithms to the nonautonomous setting and use the computer program GAIO for the analysis of an explicit example, a two-dimensional system of nonautonomous difference equations. Finally we present numerical data which indicate a bifurcation of nonautonomous attractors in the Duffing-van der Pol oscillator.
2005, 5(2): 239-264
doi: 10.3934/dcdsb.2005.5.239
+[Abstract](979)
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Abstract:
We consider a finite horizon optimal control problem for an ODE system, with trajectories presenting a delayed two-values switching along a fixed direction. In particular the system exhibits hysteresis. Due to the presence of the switching component of the trajectories, several definitions of value functions are possible. None of these value functions is in general continuous. We prove that, under general hypotheses, the "least value function", i.e. the value function of the more relaxed problem, is the unique lower semicontinuous viscosity solution of two suitably coupled Hamilton-Jacobi-Bellman equations. Such a coupling involves boundary conditions in the viscosity sense.
We consider a finite horizon optimal control problem for an ODE system, with trajectories presenting a delayed two-values switching along a fixed direction. In particular the system exhibits hysteresis. Due to the presence of the switching component of the trajectories, several definitions of value functions are possible. None of these value functions is in general continuous. We prove that, under general hypotheses, the "least value function", i.e. the value function of the more relaxed problem, is the unique lower semicontinuous viscosity solution of two suitably coupled Hamilton-Jacobi-Bellman equations. Such a coupling involves boundary conditions in the viscosity sense.
2005, 5(2): 265-276
doi: 10.3934/dcdsb.2005.5.265
+[Abstract](834)
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Abstract:
In this work we study the asymptotic behavior of the solutions of some population equations with diffusion in unbounded domains by using the notion of critical spectrum introduced recently by R. Nagel and J. Poland [9]. To do this, we extend the abstract results of Brendle-Nagel-Poland [4], concerning the persistence under perturbations of the critical spectrum of a semigroup, to Hille-Yosida operators.
In this work we study the asymptotic behavior of the solutions of some population equations with diffusion in unbounded domains by using the notion of critical spectrum introduced recently by R. Nagel and J. Poland [9]. To do this, we extend the abstract results of Brendle-Nagel-Poland [4], concerning the persistence under perturbations of the critical spectrum of a semigroup, to Hille-Yosida operators.
2005, 5(2): 277-288
doi: 10.3934/dcdsb.2005.5.277
+[Abstract](1058)
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Abstract:
In this paper, we propose a delay differential equation with continuous periodic parameters to model the circadian pacemaker in a periodic environment. First, we show the existence of a positive periodic solution by using the theory of coincidence degree. Then we establish the global attractivity of the periodic solution under two sufficient conditions. These conditions are easily verifiable and are independent of each other. Some numerical simulations are also performed to demonstrate the main results.
In this paper, we propose a delay differential equation with continuous periodic parameters to model the circadian pacemaker in a periodic environment. First, we show the existence of a positive periodic solution by using the theory of coincidence degree. Then we establish the global attractivity of the periodic solution under two sufficient conditions. These conditions are easily verifiable and are independent of each other. Some numerical simulations are also performed to demonstrate the main results.
2005, 5(2): 289-298
doi: 10.3934/dcdsb.2005.5.289
+[Abstract](1003)
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The dynamics of the Poincaré map, associated with a periodic tridiagonal system modeling cooperative-competitive ecological interactions, is investigated. It is shown that the limit-set is either a fixed point or is contained in the boundary of the positive cone and itself contains a cycle of fixed points. Furthermore, the dynamics is trivial if the number of interactive species is not greater than 4.
The dynamics of the Poincaré map, associated with a periodic tridiagonal system modeling cooperative-competitive ecological interactions, is investigated. It is shown that the limit-set is either a fixed point or is contained in the boundary of the positive cone and itself contains a cycle of fixed points. Furthermore, the dynamics is trivial if the number of interactive species is not greater than 4.
2005, 5(2): 299-318
doi: 10.3934/dcdsb.2005.5.299
+[Abstract](1015)
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Abstract:
Transport equations are intensively used in Mathematical Biology. In this article the moment closure for transport equations for an arbitrary finite number of moments is presented. With use of a variational principle the closure can be obtained by minimizing the $L^2(V)$-norm with constraints. An $H$-Theorem for the negative $L^2$-norm is shown and the existence of Lagrange multipliers is proven. The Cattaneo closure is a special case for two moments and was studied in Part I (Hillen 2003). Here the general theory is given and the three moment closure for two space dimensions is calculated explicitly. It turns out that the steady states of the two and three moment systems are determined by the steady states of a corresponding diffusion problem.
Transport equations are intensively used in Mathematical Biology. In this article the moment closure for transport equations for an arbitrary finite number of moments is presented. With use of a variational principle the closure can be obtained by minimizing the $L^2(V)$-norm with constraints. An $H$-Theorem for the negative $L^2$-norm is shown and the existence of Lagrange multipliers is proven. The Cattaneo closure is a special case for two moments and was studied in Part I (Hillen 2003). Here the general theory is given and the three moment closure for two space dimensions is calculated explicitly. It turns out that the steady states of the two and three moment systems are determined by the steady states of a corresponding diffusion problem.
2005, 5(2): 319-334
doi: 10.3934/dcdsb.2005.5.319
+[Abstract](1036)
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Abstract:
We study a kinetic model for chemotaxis introduced by Othmer, Dunbar, and Alt [23], which was motivated by earlier results of Alt, presented in [1], [2]. In two papers by Chalub, Markowich, Perthame and Schmeiser, it was rigorously shown that, in three dimensions, this kinetic model leads to the classical Keller-Segel model as its drift-diffusion limit when the equation of the chemo-attractant is of elliptic type [4], [5]. As an extension of these works we prove that such kinetic models have a macroscopic diffusion limit in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type, which is the original version of the chemotaxis model.
We study a kinetic model for chemotaxis introduced by Othmer, Dunbar, and Alt [23], which was motivated by earlier results of Alt, presented in [1], [2]. In two papers by Chalub, Markowich, Perthame and Schmeiser, it was rigorously shown that, in three dimensions, this kinetic model leads to the classical Keller-Segel model as its drift-diffusion limit when the equation of the chemo-attractant is of elliptic type [4], [5]. As an extension of these works we prove that such kinetic models have a macroscopic diffusion limit in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type, which is the original version of the chemotaxis model.
2005, 5(2): 335-352
doi: 10.3934/dcdsb.2005.5.335
+[Abstract](814)
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Abstract:
A model of interaction between nutrient, prey, and predator with intratrophic predation of the predator and a limiting periodic nutrient input is proposed and studied. Dynamics of the system are shown to depend on two thresholds. These thresholds are expressed in terms of certain periodic solutions of the system. Intratrophic predation can have impact on the model only if both thresholds are greater than zero. In this case positive periodic solutions exist. Numerical techniques are then used to explore the effect of intratrophic predation by examining the mean value and stability of these positive periodic solutions. It is demonstrated numerically that intratrophic predation can increase the stability region of the positive periodic solutions. It can also elevate the mean values of prey population and decrease the mean values of nutrient concentration for stable positive periodic solutions. Moreover, intratrophic predation can eliminate the chaotic behavior of the system when the degree of intratrophic predation is large enough.
A model of interaction between nutrient, prey, and predator with intratrophic predation of the predator and a limiting periodic nutrient input is proposed and studied. Dynamics of the system are shown to depend on two thresholds. These thresholds are expressed in terms of certain periodic solutions of the system. Intratrophic predation can have impact on the model only if both thresholds are greater than zero. In this case positive periodic solutions exist. Numerical techniques are then used to explore the effect of intratrophic predation by examining the mean value and stability of these positive periodic solutions. It is demonstrated numerically that intratrophic predation can increase the stability region of the positive periodic solutions. It can also elevate the mean values of prey population and decrease the mean values of nutrient concentration for stable positive periodic solutions. Moreover, intratrophic predation can eliminate the chaotic behavior of the system when the degree of intratrophic predation is large enough.
2005, 5(2): 353-364
doi: 10.3934/dcdsb.2005.5.353
+[Abstract](961)
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We derive simple conditions for the stability or instability of the synchronized oscillation of a class of networks of coupled phase-oscillators, which includes many of the systems used in neural modelling.
We derive simple conditions for the stability or instability of the synchronized oscillation of a class of networks of coupled phase-oscillators, which includes many of the systems used in neural modelling.
2005, 5(2): 365-384
doi: 10.3934/dcdsb.2005.5.365
+[Abstract](1083)
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In this paper, we examine the problems of stochastic stability and stabilization for a class of interconnected systems with Markovian jump parameters. The jumping parameters are treated as continuous-time, discrete- state Markov process. The purpose is to design a decentralized state feedback controller such that stochastic stability and a prescribed $H_\infty$-performance are guaranteed. Next, the robust $H_\infty$-control problem for linear interconnected systems with Markovian jump parameters and parametric uncertainties is studied. The parametric uncertainties are assumed to be real, time-varying and norm-bounded that appear in the state matrix. Both cases of finite-horizon and infinite-horizon are analyzed. We establish that the decentralized control problem for interconnected Markovian jump systems with and without uncertain parameters can be essentially solved in terms of the solutions of a finite set of coupled differential (or algebraic) Riccati equations. Extension of the developed results to the case of uncertain jumping rates is provided.
In this paper, we examine the problems of stochastic stability and stabilization for a class of interconnected systems with Markovian jump parameters. The jumping parameters are treated as continuous-time, discrete- state Markov process. The purpose is to design a decentralized state feedback controller such that stochastic stability and a prescribed $H_\infty$-performance are guaranteed. Next, the robust $H_\infty$-control problem for linear interconnected systems with Markovian jump parameters and parametric uncertainties is studied. The parametric uncertainties are assumed to be real, time-varying and norm-bounded that appear in the state matrix. Both cases of finite-horizon and infinite-horizon are analyzed. We establish that the decentralized control problem for interconnected Markovian jump systems with and without uncertain parameters can be essentially solved in terms of the solutions of a finite set of coupled differential (or algebraic) Riccati equations. Extension of the developed results to the case of uncertain jumping rates is provided.
2005, 5(2): 385-410
doi: 10.3934/dcdsb.2005.5.385
+[Abstract](934)
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We make precise the sense in which spatial homogenization to a constant function in space is attained in a linear parabolic problem when large diffusion in all parts of the domain is assumed. Also interaction between diffusion and boundary flux terms is considered. Our starting point is a detailed analysis of the large diffusion effects on the associated elliptic and eigenvalue problems. Here convergence is shown in the energy space $H^1(\Omega)$ and in the space of continuous functions $C(\overline{\Omega})$. In the parabolic case we prove convergence in the functional space $L^\infty ((0,T),L^2(\Omega)) \bigcap L^2 ((0,T),H^1(\Omega)).
We make precise the sense in which spatial homogenization to a constant function in space is attained in a linear parabolic problem when large diffusion in all parts of the domain is assumed. Also interaction between diffusion and boundary flux terms is considered. Our starting point is a detailed analysis of the large diffusion effects on the associated elliptic and eigenvalue problems. Here convergence is shown in the energy space $H^1(\Omega)$ and in the space of continuous functions $C(\overline{\Omega})$. In the parabolic case we prove convergence in the functional space $L^\infty ((0,T),L^2(\Omega)) \bigcap L^2 ((0,T),H^1(\Omega)).
2005, 5(2): 411-422
doi: 10.3934/dcdsb.2005.5.411
+[Abstract](1096)
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In the past few decades, efforts have been made to clean sites polluted by heavy metals such as chromium. One of the new innovative methods of eradicating metals from soil is phytoremediation. Phytoremediation uses plants to pull metals from the soil through the roots. This article develops a system of differential equations to model the plant metal interaction of phytoremediation. We prove there exists a threshold time, $t$*, where the amount of metals in the environment meet a prescribed EPA criteria. The cost of phytoremediating up to time $t$* is computed. The cost function can be used to estimate the feasibility of clearing a polluted site through phytoremediation as opposed to alternate techniques such as brown filling.
In the past few decades, efforts have been made to clean sites polluted by heavy metals such as chromium. One of the new innovative methods of eradicating metals from soil is phytoremediation. Phytoremediation uses plants to pull metals from the soil through the roots. This article develops a system of differential equations to model the plant metal interaction of phytoremediation. We prove there exists a threshold time, $t$*, where the amount of metals in the environment meet a prescribed EPA criteria. The cost of phytoremediating up to time $t$* is computed. The cost function can be used to estimate the feasibility of clearing a polluted site through phytoremediation as opposed to alternate techniques such as brown filling.
2005, 5(2): 423-460
doi: 10.3934/dcdsb.2005.5.423
+[Abstract](868)
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This article presents a numerical and dynamical study of a system of partial differential equations, describing the motion of a lamellar phase in a solution of surfactants in a Couette-Taylor system. It has been shown that, at high shear rate, a stabilization of the system occurs. We show, under a hypothesis on the spectrum, that this system has a local center manifold. This hypothesis on the spectrum is verified numerically, by using a finite difference method. The numerical results show that a Hopf bifurcation occurs at some shear rate. The velocity of the layers at the Hopf bifurcation corresponds to the one when the layers break themselves in the physical case. In addition, an instability result at low shear rate is proved.
This article presents a numerical and dynamical study of a system of partial differential equations, describing the motion of a lamellar phase in a solution of surfactants in a Couette-Taylor system. It has been shown that, at high shear rate, a stabilization of the system occurs. We show, under a hypothesis on the spectrum, that this system has a local center manifold. This hypothesis on the spectrum is verified numerically, by using a finite difference method. The numerical results show that a Hopf bifurcation occurs at some shear rate. The velocity of the layers at the Hopf bifurcation corresponds to the one when the layers break themselves in the physical case. In addition, an instability result at low shear rate is proved.
2005, 5(2): 461-468
doi: 10.3934/dcdsb.2005.5.461
+[Abstract](930)
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We investigate global stability of the regulated logistic growth model (RLG) $n'(t)=rn(t)(1-n(t-h)/K-cu(t))$, $u'(t)=-au(t)+bn(t-h)$. It was proposed by Gopalsamy and Weng [1, 2] and studied recently in [4, 5, 6, 9]. Compared with the previous results, our stability condition is of diff erent kind and has the asymptotical form. Namely, we prove that for the fixed parameters $K$ and $\mu=bcK/a$ (which determine the levels of steady states in the delayed logistic equation $n'(t)=rn(t)(1-n(t-h)/K)$ and in RLG) and for every $hr < \sqrt{2}$ the regulated logistic growth model is globally stable if we take the dissipation parameter a sufficiently large. On the other hand, studying the local stability of the positive steady state, we observe the improvement of stability for the small values of a: in this case, the inequality $rh<\pi (1+\mu)/2$ guarantees such a stability.
We investigate global stability of the regulated logistic growth model (RLG) $n'(t)=rn(t)(1-n(t-h)/K-cu(t))$, $u'(t)=-au(t)+bn(t-h)$. It was proposed by Gopalsamy and Weng [1, 2] and studied recently in [4, 5, 6, 9]. Compared with the previous results, our stability condition is of diff erent kind and has the asymptotical form. Namely, we prove that for the fixed parameters $K$ and $\mu=bcK/a$ (which determine the levels of steady states in the delayed logistic equation $n'(t)=rn(t)(1-n(t-h)/K)$ and in RLG) and for every $hr < \sqrt{2}$ the regulated logistic growth model is globally stable if we take the dissipation parameter a sufficiently large. On the other hand, studying the local stability of the positive steady state, we observe the improvement of stability for the small values of a: in this case, the inequality $rh<\pi (1+\mu)/2$ guarantees such a stability.
2005, 5(2): 469-488
doi: 10.3934/dcdsb.2005.5.469
+[Abstract](922)
+[PDF](193.2KB)
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In this paper, we use the Chebyshev rational pseudospectral method to analyze the long-time behavior of Cauchy problem for nonlinear Schrödinger equation (NSE) with weak damping. We obtain the error estimate of approximate solution and prove the existence and weak upper semicontinuity of approximate attractor.
In this paper, we use the Chebyshev rational pseudospectral method to analyze the long-time behavior of Cauchy problem for nonlinear Schrödinger equation (NSE) with weak damping. We obtain the error estimate of approximate solution and prove the existence and weak upper semicontinuity of approximate attractor.
2005, 5(2): 489-512
doi: 10.3934/dcdsb.2005.5.489
+[Abstract](1270)
+[PDF](437.5KB)
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Solutions of the complex KdV equation and the complex KdV- Burgers equation are studied theoretically and numerically. Attention is focused on whether their solutions are regular for all time. This is a difficult issue partially because the conservation laws of the KdV equation no longer yield a priori bounds for its complex-valued solutions in the $L^2$-space. The problem is tackled here on several fronts including investigating how the regularity of the real part is related to that of the imaginary part, studying blow-up of series solutions, and assessing the impact of dissipation. Systematic numerical simulations are performed to complement the theoretical results.
Solutions of the complex KdV equation and the complex KdV- Burgers equation are studied theoretically and numerically. Attention is focused on whether their solutions are regular for all time. This is a difficult issue partially because the conservation laws of the KdV equation no longer yield a priori bounds for its complex-valued solutions in the $L^2$-space. The problem is tackled here on several fronts including investigating how the regularity of the real part is related to that of the imaginary part, studying blow-up of series solutions, and assessing the impact of dissipation. Systematic numerical simulations are performed to complement the theoretical results.
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