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1531-3492
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Discrete & Continuous Dynamical Systems - B
January 2015 , Volume 20 , Issue 1
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2015, 20(1): 1-21
doi: 10.3934/dcdsb.2015.20.1
+[Abstract](1486)
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Abstract:
By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
By constructing an invariant set in the three dimensional space, we establish the existence of traveling wave solutions to a reaction-diffusion-chemotaxis model describing biological processes such as the bacterial chemotactic movement in response to oxygen and the initiation of angiogenesis. The minimal wave speed is shown to exist and the role of each process of reaction, diffusion and chemotaxis in the wave propagation is investigated. Our results reveal three essential biological implications: (1) the cell growth increases the wave speed; (2) the chemotaxis must be strong enough to make a contribution to the increment of the wave speed; (3) the diffusion rate plays a role in increasing the wave speed only when the cell growth is present.
2015, 20(1): 23-38
doi: 10.3934/dcdsb.2015.20.23
+[Abstract](1564)
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Abstract:
In this note, we establish under mild smoothness assumptions the pathwise convergence rate of an Euler-type method with projection for delay stochastic differential equations on unbounded domains.
In this note, we establish under mild smoothness assumptions the pathwise convergence rate of an Euler-type method with projection for delay stochastic differential equations on unbounded domains.
2015, 20(1): 39-58
doi: 10.3934/dcdsb.2015.20.39
+[Abstract](1136)
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Abstract:
We study a minimal time control problem under the presence of a saturation point on the singular locus. The system describes a fed-batch reactor with one species and one substrate. Our aim is to find an optimal feedback control steering the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc which is non-necessary admissible everywhere (i.e. the singular control can take values outside the admissible control set). Thanks to Pontrygin's Principle, we provide an optimal synthesis of the problem that exhibits a frame point at the intersection of the singular arc and a switching curve. Numerical simulations allow to compute this curve and the frame point.
We study a minimal time control problem under the presence of a saturation point on the singular locus. The system describes a fed-batch reactor with one species and one substrate. Our aim is to find an optimal feedback control steering the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc which is non-necessary admissible everywhere (i.e. the singular control can take values outside the admissible control set). Thanks to Pontrygin's Principle, we provide an optimal synthesis of the problem that exhibits a frame point at the intersection of the singular arc and a switching curve. Numerical simulations allow to compute this curve and the frame point.
2015, 20(1): 59-75
doi: 10.3934/dcdsb.2015.20.59
+[Abstract](993)
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Abstract:
We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator.
It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.
We consider a Large Eddy Simulation model for a homogeneous incompressible Newtonian fluid in a box space domain with periodic boundary conditions on the lateral boundaries and homogeneous Dirichlet conditions on the top and bottom boundaries, thus simulating a horizontal channel. The model is obtained through the application of an anisotropic horizontal filter, which is known to be less memory consuming from a numerical point of view, but provides less regularity with respect to the standard isotropic one defined as the inverse of the Helmholtz operator.
It is known that there exists a unique regular weak solution to this model that depends weakly continuously on the initial datum. We show the existence of the global attractor for the semiflow given by the time-shift in the space of paths. We prove the continuity of the horizontal components of the flow under periodicity in all directions and discuss the possibility to introduce a solution semiflow.
2015, 20(1): 77-92
doi: 10.3934/dcdsb.2015.20.77
+[Abstract](1006)
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Abstract:
In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time.
In this paper, we consider a drift-diffusion system describing the corrosion of an iron based alloy in a nuclear waste repository. In comparison with the classical drift-diffusion system arising in the modeling of semiconductor devices, the originality of the corrosion model lies in the boundary conditions which are of Robin type and induce an additional coupling between the equations. We prove the existence of a weak solution by passing to the limit on a sequence of approximate solutions given by a semi-discretization in time.
2015, 20(1): 93-105
doi: 10.3934/dcdsb.2015.20.93
+[Abstract](993)
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In this paper, the chaotic behaviour of a forced discretized version of the Mackey-Glass delay differential equation is considered for different levels of noise intensity. The existence and stability of the equilibria of the skeleton are studied. The modified straight-line stabilization method is used to control chaos. The autocorrelation structure is discussed. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents, bifurcations, time series diagrams and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation.
In this paper, the chaotic behaviour of a forced discretized version of the Mackey-Glass delay differential equation is considered for different levels of noise intensity. The existence and stability of the equilibria of the skeleton are studied. The modified straight-line stabilization method is used to control chaos. The autocorrelation structure is discussed. Numerical simulations are employed to show the model's complex dynamics by means of the largest Lyapunov exponents, bifurcations, time series diagrams and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation.
2015, 20(1): 107-127
doi: 10.3934/dcdsb.2015.20.107
+[Abstract](1276)
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Abstract:
We introduce the optimal inflow control problem for buffer restricted production systems involving a conservation law with discontinuous flux. Based on an appropriate numerical method inspired by the wave front tracking algorithm, we present two techniques to solve the optimal control problem efficiently. A numerical study compares the different optimization procedures and comments on their benefits and drawbacks.
We introduce the optimal inflow control problem for buffer restricted production systems involving a conservation law with discontinuous flux. Based on an appropriate numerical method inspired by the wave front tracking algorithm, we present two techniques to solve the optimal control problem efficiently. A numerical study compares the different optimization procedures and comments on their benefits and drawbacks.
2015, 20(1): 129-152
doi: 10.3934/dcdsb.2015.20.129
+[Abstract](1091)
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Abstract:
We study coupled maps where a map representing an `active phase' is coupled to the identity which represents a `quiescent phase'. The resulting system in double dimension is a natural analogue of differential equations with quiescent phases that have been thoroughly studied. In the continuous time case quiescent phases with equal rates for all components stabilize against the onset of Hopf bifurcations (but not against eigenvalues passing through zero) while unequal rates may induce Hopf bifurcations unless the Jacobian matrix has a `strong stability' property. Here we show that similar effects occur in the discrete time case. In the case of equal rates we determine the exact stability boundary as an algebraic curve of fourth order. It is shown that large quiescence rates may completely inhibit period doubling bifurcations. If the rates are unequal, quiescent phases may destabilize a stationary point. In this case we find (for two components) a notion of `strong stability' for the Jacobian matrix such that the stationary point cannot be excited. Discrete time predator prey models serve as examples for the damping and excitation phenomena.
We study coupled maps where a map representing an `active phase' is coupled to the identity which represents a `quiescent phase'. The resulting system in double dimension is a natural analogue of differential equations with quiescent phases that have been thoroughly studied. In the continuous time case quiescent phases with equal rates for all components stabilize against the onset of Hopf bifurcations (but not against eigenvalues passing through zero) while unequal rates may induce Hopf bifurcations unless the Jacobian matrix has a `strong stability' property. Here we show that similar effects occur in the discrete time case. In the case of equal rates we determine the exact stability boundary as an algebraic curve of fourth order. It is shown that large quiescence rates may completely inhibit period doubling bifurcations. If the rates are unequal, quiescent phases may destabilize a stationary point. In this case we find (for two components) a notion of `strong stability' for the Jacobian matrix such that the stationary point cannot be excited. Discrete time predator prey models serve as examples for the damping and excitation phenomena.
2015, 20(1): 153-160
doi: 10.3934/dcdsb.2015.20.153
+[Abstract](1236)
+[PDF](434.5KB)
Abstract:
In this note we work on the construction of positive preserving numerical schemes for a class of multidimensional stochastic differential equations. We use the semi discrete idea that we have proposed before proposing now a numerical scheme that preserves positivity on some multidimensional stochastic differential equations converging strongly in the mean square sense to the true solution.
In this note we work on the construction of positive preserving numerical schemes for a class of multidimensional stochastic differential equations. We use the semi discrete idea that we have proposed before proposing now a numerical scheme that preserves positivity on some multidimensional stochastic differential equations converging strongly in the mean square sense to the true solution.
2015, 20(1): 161-171
doi: 10.3934/dcdsb.2015.20.161
+[Abstract](1325)
+[PDF](357.2KB)
Abstract:
This paper investigates the existence of traveling waves and their propagation speeds for the Lotka-Volterra predator-prey reaction-diffusion models with no predator diffusion. We prove the existence of traveling waves with any positive speed. Our mathematical tool is the shooting argument in the phase space based on the Wazewski theorem.
This paper investigates the existence of traveling waves and their propagation speeds for the Lotka-Volterra predator-prey reaction-diffusion models with no predator diffusion. We prove the existence of traveling waves with any positive speed. Our mathematical tool is the shooting argument in the phase space based on the Wazewski theorem.
2015, 20(1): 173-187
doi: 10.3934/dcdsb.2015.20.173
+[Abstract](1191)
+[PDF](536.3KB)
Abstract:
Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton. In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations. In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters. The resulting systems give a traveling pulse and a self-replication pattern with appropriate initial data and parameters. The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern. A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern and a self-replication pattern are also constructed.
Ultradiscretization is a limiting procedure transforming a given difference equation into a cellular automaton. In addition the cellular automaton constructed by this procedure preserves the essential properties of the original equation, such as the structure of exact solutions for integrable equations. In this article, we propose a discretization and an ultradiscretization of Gray-Scott model which is not an integrable system and which gives various spatial patterns with appropriate initial data and parameters. The resulting systems give a traveling pulse and a self-replication pattern with appropriate initial data and parameters. The ultradiscrete system is directly related to the elementary cellular automaton Rule 90 which gives a Sierpinski gasket pattern. A $(2+1)$D ultradiscrete Gray-Scott model that gives a ring pattern and a self-replication pattern are also constructed.
2015, 20(1): 189-213
doi: 10.3934/dcdsb.2015.20.189
+[Abstract](1172)
+[PDF](622.1KB)
Abstract:
Cancer cell migration is an essential feature in the process of tumor spread and establishing of metastasis. It characterizes the invasion observed on the level of the cell population, but it is also tightly connected to the events taking place on the subcellular level. These are conditioning the motile and proliferative behavior of the cells, but are also influenced by it. In this work we propose a multiscale model linking these two levels and aiming to assess their interdependence. On the subcellular, microscopic scale it accounts for integrin binding to soluble and insoluble components present in the peritumoral environment, which is seen as the onset of biochemical events leading to changes in the cell's ability to contract and modify its shape. On the macroscale of the cell population this leads to modifications in the diffusion and haptotaxis performed by the tumor cells and implicitly to changes in the tumor environment. We prove the (local) well posedness of our model and perform numerical simulations in order to illustrate the model predictions.
Cancer cell migration is an essential feature in the process of tumor spread and establishing of metastasis. It characterizes the invasion observed on the level of the cell population, but it is also tightly connected to the events taking place on the subcellular level. These are conditioning the motile and proliferative behavior of the cells, but are also influenced by it. In this work we propose a multiscale model linking these two levels and aiming to assess their interdependence. On the subcellular, microscopic scale it accounts for integrin binding to soluble and insoluble components present in the peritumoral environment, which is seen as the onset of biochemical events leading to changes in the cell's ability to contract and modify its shape. On the macroscale of the cell population this leads to modifications in the diffusion and haptotaxis performed by the tumor cells and implicitly to changes in the tumor environment. We prove the (local) well posedness of our model and perform numerical simulations in order to illustrate the model predictions.
2015, 20(1): 215-230
doi: 10.3934/dcdsb.2015.20.215
+[Abstract](1247)
+[PDF](646.4KB)
Abstract:
The 2002-2003 SARS outbreaks exhibited some distinct features such as rapid spatial spread, massive media reports, and fast self-control. These features were shared by the 2009 pandemic influenza and will be experienced by other emerging infectious diseases. We focus on the dynamic interaction of media reports, epidemic outbreak and behavior change in the population and formulate a compartmental model, that tracks the evolution of the human population. Such population is characterized by the disease progression (susceptible, infected, hospitalized, and recovered) and by the extent to which the media has impacted, so individuals have modified their behaviors to reduce their transmissibility and infectivity. The model also describes the dynamics of media reports by considering how media is influenced by the disease statistics (numbers of infected and hospitalized individuals, for example). We then conduct linear stability analysis and numerical simulations to study how interaction of media reports and disease progress affects the disease transmission dynamics, so as to shed light on what type of media will be the most effective for the control of an epidemic.
The 2002-2003 SARS outbreaks exhibited some distinct features such as rapid spatial spread, massive media reports, and fast self-control. These features were shared by the 2009 pandemic influenza and will be experienced by other emerging infectious diseases. We focus on the dynamic interaction of media reports, epidemic outbreak and behavior change in the population and formulate a compartmental model, that tracks the evolution of the human population. Such population is characterized by the disease progression (susceptible, infected, hospitalized, and recovered) and by the extent to which the media has impacted, so individuals have modified their behaviors to reduce their transmissibility and infectivity. The model also describes the dynamics of media reports by considering how media is influenced by the disease statistics (numbers of infected and hospitalized individuals, for example). We then conduct linear stability analysis and numerical simulations to study how interaction of media reports and disease progress affects the disease transmission dynamics, so as to shed light on what type of media will be the most effective for the control of an epidemic.
2015, 20(1): 231-248
doi: 10.3934/dcdsb.2015.20.231
+[Abstract](1083)
+[PDF](454.1KB)
Abstract:
This paper concerns with the stability of bifurcating steady states obtained in [13] of several chemotaxis systems. By spectral analysis and the principle of the linearized stability, we prove that the bifurcating steady states are stable when the parameters satisfy some certain conditions.
This paper concerns with the stability of bifurcating steady states obtained in [13] of several chemotaxis systems. By spectral analysis and the principle of the linearized stability, we prove that the bifurcating steady states are stable when the parameters satisfy some certain conditions.
2015, 20(1): 249-257
doi: 10.3934/dcdsb.2015.20.249
+[Abstract](1026)
+[PDF](349.2KB)
Abstract:
We prove the absolute continuity of the Hausdorff measure with respect to any conformal measure. These results extend Denker and Urbanski's work on parabolic rational functions.
We prove the absolute continuity of the Hausdorff measure with respect to any conformal measure. These results extend Denker and Urbanski's work on parabolic rational functions.
2015, 20(1): 259-280
doi: 10.3934/dcdsb.2015.20.259
+[Abstract](1853)
+[PDF](480.0KB)
Abstract:
This paper is mainly concerned with the issue of stability for multi-group models with dispersal (MGMD). A system on multi-digraph is used to model the MGMD. The popular single graph-based method has been successfully generalized into multi-digraph-based approach. More precisely, by constructing a Lyapunov function for general MGMD, some simple yet less conservative conditions are derived for the stability of MGMD. Furthermore, the graph-theoretic method on multi-graph is successfully applied on predator-prey model with dispersal and coupled oscillators on two digraphs. Subsequently, numerical results are presented to demonstrate the effectiveness of the proposed new technique.
This paper is mainly concerned with the issue of stability for multi-group models with dispersal (MGMD). A system on multi-digraph is used to model the MGMD. The popular single graph-based method has been successfully generalized into multi-digraph-based approach. More precisely, by constructing a Lyapunov function for general MGMD, some simple yet less conservative conditions are derived for the stability of MGMD. Furthermore, the graph-theoretic method on multi-graph is successfully applied on predator-prey model with dispersal and coupled oscillators on two digraphs. Subsequently, numerical results are presented to demonstrate the effectiveness of the proposed new technique.
2015, 20(1): 281-293
doi: 10.3934/dcdsb.2015.20.281
+[Abstract](1394)
+[PDF](352.9KB)
Abstract:
Peng introduced the notions of $G$-expectation and $G$-Brownian motion as well as $G$-Itô formula in 2006. The $G$-Brownian motion has many rich and new properties comparing to classical Brownian motion. In this paper, we present a method to solve stochastic differential equation driven by $G$-Brownian motion without using $G$-Itô formula. Our method is mainly depending on Frobenius's Theorem. Many classical models in mathematical finance are investigated to illustrate the method. As a by-product, this financial models are extended to the case of $G$-Brownian motion.
Peng introduced the notions of $G$-expectation and $G$-Brownian motion as well as $G$-Itô formula in 2006. The $G$-Brownian motion has many rich and new properties comparing to classical Brownian motion. In this paper, we present a method to solve stochastic differential equation driven by $G$-Brownian motion without using $G$-Itô formula. Our method is mainly depending on Frobenius's Theorem. Many classical models in mathematical finance are investigated to illustrate the method. As a by-product, this financial models are extended to the case of $G$-Brownian motion.
2015, 20(1): 295-321
doi: 10.3934/dcdsb.2015.20.295
+[Abstract](1110)
+[PDF](1289.8KB)
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In this paper, we have proposed and analyzed a perturbed eco-epidemiological model with Holling type II functional response by white noise. By Lyapunov analysis methods, we prove the stochastic stability, its long time behavior around the equilibrium of deterministic eco-epidemiological model and the stochastic asymptotic stability. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.
In this paper, we have proposed and analyzed a perturbed eco-epidemiological model with Holling type II functional response by white noise. By Lyapunov analysis methods, we prove the stochastic stability, its long time behavior around the equilibrium of deterministic eco-epidemiological model and the stochastic asymptotic stability. Numerical simulations for a hypothetical set of parameter values are presented to illustrate the analytical findings.
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