# American Institute of Mathematical Sciences

ISSN:
1531-3492

eISSN:
1553-524X

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## Discrete & Continuous Dynamical Systems - B

2014 , Volume 19 , Issue 1

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2014, 19(1): 1-26 doi: 10.3934/dcdsb.2014.19.1 +[Abstract](217) +[PDF](771.1KB)
Abstract:
We investigate a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model describes both the evolution of primitive hematopoietic stem cells and the maturation of these cells as they differentiate to form the three kinds of progenitors and mature blood cells (red blood cells, white cells and platelets). The three types of progenitors and mature cells are coupled to each other via their common origin in primitive hematopoietic stem cells compartment. The resulting system is composed by eleven age-structured partial differential equations. To analyze this model, we don't take into account cell age-dependence of coefficients, that prevents a usual reduction of the structured system to an unstructured delay differential system. We study the existence of stationary solutions: trivial, axial and positive steady states. Then we give conditions for the local asymptotic stability of the trivial steady state and by using a Lyapunov function, we obtain a sufficient condition for its global asymptotic stability. In some particular cases, we analyze the local asymptotic stability of the positive steady state by using the characteristic equation. Finally, by numerical simulations, we illustrate our results and we show that a change in the duration of cell cycle can cause oscillations.
2014, 19(1): 27-53 doi: 10.3934/dcdsb.2014.19.27 +[Abstract](276) +[PDF](2714.0KB)
Abstract:
In this article, we are interested in determining spectral minimal $k$-partitions for angular sectors. We first deal with the nodal cases for which we can determine explicitly the minimal partitions. Then, in the case where the minimal partitions are not nodal domains of eigenfunctions of the Dirichlet Laplacian, we analyze the possible topologies of these minimal partitions. We first exhibit symmetric minimal partitions by using a mixed Dirichlet-Neumann Laplacian and then use a double covering approach to catch non symmetric candidates. In this way, we improve the known estimates of the energy associated with the minimal partitions.
2014, 19(1): 55-72 doi: 10.3934/dcdsb.2014.19.55 +[Abstract](349) +[PDF](1890.0KB)
Abstract:
Activation of CD$8^+$ cytotoxic T lymphocytes (CTLs) is naturally regarded as a major antitumor mechanism of the immune system. In contrast, CD$4^+$ T cells are commonly classified as helper T cells (HTCs) on the basis of their roles in providing help to the generation and maintenance of effective CD$8^+$ cytotoxic and memory T cells. In order to get a better insight on the role of HTCs in a tumor immune system, we incorporate the third population of HTCs into a previous two dimensional ordinary differential equations (ODEs) model. Further we introduce the adoptive cellular immunotherapy (ACI) as the treatment to boost the immune system to fight against tumors. Compared tumor cells (TCs) and effector cells (ECs), the recruitment of HTCs changes the dynamics of the system substantially, by the effects through particular parameters, i.e., the activation rate of ECs by HTCs, $p$ (scaled as $\rho$), and the HTCs stimulation rate by the presence of identified tumor antigens, $k_2$ (scaled as $\omega_2$). We describe the stability regions of the interior equilibria $E^*$ (no treatment case) and $E^+$ (treatment case) in the scaled $(\rho,\omega_2)$ parameter space respectively. Both $\rho$ and $\omega_2$ can destabilize $E^*$ and $E^+$ and cause Hopf bifurcations. Our results show that HTCs might play a crucial role in the long term periodic oscillation behaviors of tumor immune system interactions. They also show that TCs may be eradicated from the patient's body under the ACI treatment.
2014, 19(1): 73-88 doi: 10.3934/dcdsb.2014.19.73 +[Abstract](253) +[PDF](739.6KB)
Abstract:
The paper provides a scheme for phase separation and transition by accounting for diffusion, dynamic equations and consistency with thermodynamics. The constituents are compressible fluids thus improving the model of a previous approach. Moreover a possible saturation effect for the concentration of a constituent is made explicit. The mass densities of the constituents are independent of temperature. The evolution of concentration is described by the standard equation for mixtures but the balance of energy and entropy of the mixture are stated as for a single constituent. However, due to the non-simple character of the mixture, an extra-energy flux is allowed to occur. Also motion and diffusion effects are considered by letting the stress in the mixture have additive viscous terms and, remarkably, the chemical potential contains a quadratic term in the stretching tensor. As a result a whole set of evolution equations is set up for the concentration, the velocity, and the temperature. Shear-induced mixing and demixing are examined. A maximum theorem is proved which implies that the concentration of the mixture has values from 0 to 1 as is required from the physical standpoint.
2014, 19(1): 89-130 doi: 10.3934/dcdsb.2014.19.89 +[Abstract](354) +[PDF](8850.3KB)
Abstract:
The focus here is on the study disease dynamics under the assumption that a critical mass of susceptible individuals is required to guarantee the population's survival. Specifically, the emphasis is on the study of the role of an Allee effect on a Susceptible-Infectious (SI) model where the possibility that susceptible and infected individuals reproduce, with the S-class being the best fit. It is further assumed that infected individuals loose some of their ability to compete for resources, the cost imposed by the disease. These features are set in motion as simple model as possible. They turn out to lead to a rich set of dynamical outcomes. This toy model supports the possibility of multi-stability (hysteresis), saddle node and Hopf bifurcations, and catastrophic events (disease-induced extinction). The analyses provide a full picture of the system under disease-free dynamics including disease-induced extinction and proceed to identify required conditions for disease persistence. We conclude that increases in (i) the maximum birth rate of a species, or (ii) in the relative reproductive ability of infected individuals, or (iii) in the competitive ability of a infected individuals at low density levels, or in (iv) the per-capita death rate (including disease-induced) of infected individuals, can stabilize the system (resulting in disease persistence). We further conclude that increases in (a) the Allee effect threshold, or (b) in disease transmission rates, or in (c) the competitive ability of infected individuals at high density levels, can destabilize the system, possibly leading to the eventual collapse of the population. The results obtained from the analyses of this toy model highlight the significant role that factors like an Allee effect may play on the survival and persistence of animal populations. Scientists involved in biological conservation and pest management or interested in finding sustainability solutions, may find these results of this study compelling enough to suggest additional focused research on the role of disease in the regulation and persistence of animal populations. The risk faced by endangered species may turn out to be a lot higher than initially thought.
2014, 19(1): 131-152 doi: 10.3934/dcdsb.2014.19.131 +[Abstract](218) +[PDF](2026.4KB)
Abstract:
We first conduct a comparative numerical study of two recently proposed two-species chemotaxis models. We show that different scenarios are possible: depending on the initial masses, either one or both cell densities may blow up, or a global solution may exist. In particular, our numerical results indicate answers on some open questions of possible blow up stated in [4,7]. We then introduce two regularizations of the studied models and demonstrate that their solutions are capable of developing spiky structure without blowing up.
2014, 19(1): 153-172 doi: 10.3934/dcdsb.2014.19.153 +[Abstract](290) +[PDF](529.8KB)
Abstract:
This paper develops and analyzes an age-structured model of HIV infection with various compartments, including target cells, infected cells, viral loads and immune effector cells, to provide a better understanding of the interaction between HIV and the immune system. We show that the proposed model has one uninfected steady state and several infected steady states. We conduct a local stability analysis of these steady states by using a generalized Jacobian matrix method in conjunction with the Laplace transform. In addition, we consider various techniques and ideas from optimal control theory to derive optimal therapy protocols by using two types of dynamic treatment methods representing reverse transcriptase inhibitors and protease inhibitors. We derive the necessary conditions (an optimality system) for optimal control functions by considering the first variations of the Lagrangian. Further, we obtain optimal therapy protocols by solving a large optimality system of equations through the use of a difference scheme based on the Runge-Kutta method. The results of numerical simulations indicate that the optimal therapy protocols can facilitate long-term control of HIV through a strong immune response after the discontinuation of the therapy.
2014, 19(1): 173-187 doi: 10.3934/dcdsb.2014.19.173 +[Abstract](295) +[PDF](412.6KB)
Abstract:
In this paper, we consider a predator-prey system with stage structure and mutual interference. By analyzing the characteristic equations, we study the local stability of the interior equilibrium of the system. Using an iterative method, we investigate the global stability of this equilibrium.
2014, 19(1): 189-215 doi: 10.3934/dcdsb.2014.19.189 +[Abstract](212) +[PDF](3015.6KB)
Abstract:
Blow-up in second and fourth order semi-linear parabolic partial differential equations (PDEs) is considered in bounded regions of one, two and three spatial dimensions with uniform initial data. A phenomenon whereby singularities form at multiple points simultaneously is exhibited and explained by means of a singular perturbation theory. In the second order case we predict that points furthest from the boundary are selected by the dynamics of the PDE for singularity. In the fourth order case, singularities can form simultaneously at multiple locations, even in one spatial dimension. In two spatial dimensions, the singular perturbation theory reveals that the set of possible singularity points depends subtly on the geometry of the domain and the equation parameters. In three spatial dimensions, preliminary numerical simulations indicate that the multiplicity of singularities can be even more complex. For the aforementioned scenarios, the analysis highlights the dichotomy of behaviors exhibited between the second and fourth order cases.
2014, 19(1): 217-230 doi: 10.3934/dcdsb.2014.19.217 +[Abstract](248) +[PDF](384.7KB)
Abstract:
In this paper, we are concerned with some properties of the global attractor of weakly damped wave equations. We get the existence of multiple stationary solutions for wave equations with weakly damping. Furthermore, we provide some approaches to verify the small neighborhood of the origin is an attracting domain which is important to obtain the multiple equilibrium points in global attractor.
2014, 19(1): 231-256 doi: 10.3934/dcdsb.2014.19.231 +[Abstract](241) +[PDF](496.4KB)
Abstract:
We study a one-dimensional parabolic PDE with degenerate diffusion and non-Lipschitz nonlinearity involving the derivative. This evolution equation arises when searching radially symmetric solutions of a chemotaxis model of Patlak-Keller-Segel type. We prove its local in time wellposedness in some appropriate space, a blow-up alternative, regularity results and give an idea of the shape of solutions. A transformed and an approximate problem naturally appear in the way of the proof and are also crucial in [22] in order to study the global behaviour of solutions of the equation for a critical parameter, more precisely to show the existence of a critical mass.
2014, 19(1): 257-279 doi: 10.3934/dcdsb.2014.19.257 +[Abstract](238) +[PDF](814.8KB)
Abstract:
Energy Balance Models (EBM) are conceptual models which have proved useful in the study of planetary climate. The focus of EBM is placed on large scale climate components such as incoming solar radiation, albedo, outgoing longwave radiation and heat transport, and their interactions. Until recently, their study has centered on equilibrium solutions of an associated model equation, with no consideration of the dynamical nature of these solutions. In this paper we continue and expand upon recent efforts aimed at placing EBM in a more mathematical, dynamical systems context. In particular, the dynamical behavior of several variants of the Budyko-Sellers model, all but one of which involve the movement of glaciers, is shown to reduce to the study of the system on an attracting one-dimensional invariant manifold in an appropriately defined state space.
2014, 19(1): 281-298 doi: 10.3934/dcdsb.2014.19.281 +[Abstract](258) +[PDF](496.1KB)
Abstract:
In this paper, inspired by the work by A. Iserles and G. Söderlind [Global bounds on numerical error for ordinary differential equations, J. Complexity, 9 (1993), pp. 97-112], we present comprehensive discussion on Dirichlet series for dynamical systems of first-order ordinary differential equations (ODEs). We first derive the scheme of Dirichlet approximation for scalar dynamical systems and present the bounds on the terms of Dirichlet series. The global error and the right choice of a term in Dirichlet series are analysed and two numerical experiments are carried out to demonstrate the efficiency of Dirichlet approximation. Then we consider applying Dirichlet series to multivariate dynamical systems and present a new scheme of Dirichlet approximation for such systems. Some discussion and a numerical experiment are accordingly carried out for the new Dirichlet approximation. Compared with routine time-stepping algorithms, Dirichlet series does not need time stepping and yields a continuous solution that is equally valid along an interval, which is significant for obtaining long-time numerical solution. As a result of the special nature of Dirichlet series, the Dirichlet approximation delivers considerable information on dynamical systems of first-order ODEs and provides a novel and effective approach to numerical solutions of these dynamical systems.
2014, 19(1): 299-322 doi: 10.3934/dcdsb.2014.19.299 +[Abstract](265) +[PDF](1136.6KB)
Abstract:
We propose an efficient Legendre-Gauss collocation algorithm for second-order nonlinear ordinary differential equations (ODEs). We also design a Legendre-Gauss-type collocation algorithm for time-dependent second-order nonlinear partial differential equations (PDEs), which can be implemented in a synchronous parallel fashion. Numerical results indicate the high accuracy and effectiveness of the suggested algorithms.

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