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Discrete & Continuous Dynamical Systems - B

2009 , Volume 12 , Issue 4

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Existence of travelling-wave solutions and local well-posedness of the Fowler equation
Borys Alvarez-Samaniego and Pascal Azerad
2009, 12(4): 671-692 doi: 10.3934/dcdsb.2009.12.671 +[Abstract](557) +[PDF](281.6KB)
Abstract:
We study the existence of travelling-waves and local well-posedness in a subspace of $C_b^1(\R)$ for a nonlinear evolution equation recently proposed by Andrew C. Fowler to describe the dynamics of dunes. The travelling-waves we obtained however, were more bore-like than solitary-wave-like.
Explosive behavior in spatially discrete reaction-diffusion systems
Ana Carpio and Gema Duro
2009, 12(4): 693-711 doi: 10.3934/dcdsb.2009.12.693 +[Abstract](536) +[PDF](862.1KB)
Abstract:
Explosive instabilities in spatially discrete reaction-diffusion systems are studied. We identify classes of initial data developing singularities in finite time and obtain predictions of the blow-up times, whose accuracy is checked by comparison with numerical solutions. We present averaged and local blow-up estimates. Local blow-up results show that it is possible to have blow-up after blow-up. Conditions excluding or implying blow-up at space infinity are discussed.
Homogenization in domains randomly perforated along the boundary
Gregory A. Chechkin, Tatiana P. Chechkina, Ciro D’Apice and Umberto De Maio
2009, 12(4): 713-730 doi: 10.3934/dcdsb.2009.12.713 +[Abstract](580) +[PDF](490.5KB)
Abstract:
We study the asymptotic behavior of the solution of the Laplace equation in a domain perforated along the boundary. Assuming that the boundary microstructure is random, we construct the limit problem and prove the homogenization theorem. Moreover we apply those results to some spectral problems.
A model describing the growth and the size distribution of multiple metastatic tumors
Anne Devys, Thierry Goudon and Pauline Lafitte
2009, 12(4): 731-767 doi: 10.3934/dcdsb.2009.12.731 +[Abstract](536) +[PDF](513.1KB)
Abstract:
Cancer is one of the greatest killers in the world, particularly in western countries. A lot of the effort of the medical research is devoted to cancer and mathematical modeling must be considered as an additional tool for the physicians and biologists to understand cancer mechanisms and to determine the adapted treatments. Metastases make all the seriousness of cancer. In 2000, Iwata et al. [9] proposed a model which describes the evolution of an untreated metastatic tumors population. We provide here a mathematical analysis of this model which brings us to the determination of a Malthusian rate characterizing the exponential growth of the population. We provide as well a numerical analysis of the PDE given by the model.
Automated computation of robust normal forms of planar analytic vector fields
Tomas Johnson and Warwick Tucker
2009, 12(4): 769-782 doi: 10.3934/dcdsb.2009.12.769 +[Abstract](562) +[PDF](221.1KB)
Abstract:
We construct an auto-validated algorithm that calculates a close to identity change of variables which brings a general saddle point into a normal form. The transformation is robust in the underlying vector field, and is analytic on a computable neighbourhood of the saddle point. The normal form is suitable for computations aimed at enclosing the flow close to the saddle, and the time it takes a trajectory to pass it. Several examples illustrate the usefulness of this method.
Mixed entropy estimates for the porous-medium equation with convection
Ansgar Jüngel and Ingrid Violet
2009, 12(4): 783-796 doi: 10.3934/dcdsb.2009.12.783 +[Abstract](599) +[PDF](262.1KB)
Abstract:
In this paper, we answer the question under which conditions the porous-medium equation with convection and with periodic boundary conditions possesses gradient-type Lyapunov functionals (first-order entropies). It is shown that the weighted sum of first-order and zeroth-order entropies are Lyapunov functionals if the weight for the zeroth-order entropy is sufficiently large, depending on the strength of the convection. This provides new a priori estimates for the convective porous-medium equation. The proof is based on an extension of the algorithmic entropy construction method which is based on systematic integration by parts, formulated as a polynomial decision problem.
Dependence of propagation speed on invader species: The effect of the predatory commensalism in two-prey, one-predator system with diffusion
Sungrim Seirin Lee
2009, 12(4): 797-825 doi: 10.3934/dcdsb.2009.12.797 +[Abstract](482) +[PDF](486.5KB)
Abstract:
Biological invasion theory is one of important subjects in a biological control, an environmental preservation problem, a propagation of infectious diseases. I propose an propagation speed of traveling waves induced by an invasion of alien species for two-prey, one-predator modesl in which the commensalism induced by a predator between two prey species is considered. I investigate a spreading phenomenon and a minimal propagation speed for two cases that invader species is one species or more than one species. By numerical simulations and mathematical analysis, I conclude that the minimal speed is contingent only on the mobility of invasive species, furthermore, on that of one invader species even if two invader species invade at the same time. It is also shown that the commensalism via predator species affects spreading phenomena and a propagation speed, which is contingent on the type and the number of invasive species.
A mathematical analysis of malaria and tuberculosis co-dynamics
Expeditho Mtisi, Herieth Rwezaura and Jean Michel Tchuenche
2009, 12(4): 827-864 doi: 10.3934/dcdsb.2009.12.827 +[Abstract](787) +[PDF](761.0KB)
Abstract:
We formulate and analyze a deterministic mathematical model which incorporates some basic epidemiological features of the co-dynamics of malaria and tuberculosis. Two sub-models, namely: malaria-only and TB-only sub-models are considered first of all. Sufficient conditions for the local stability of the steady states are presented. Global stability of the disease-free steady state does not hold because the two sub-models exhibit backward bifurcation. The dynamics of the dual malaria-TB only sub-model is also analyzed. It has different dynamics to that of malaria-only and TB-only sub-models: the dual malaria-TB only model has no positive endemic equilibrium whenever $R_{MT}^d<1$, - its disease free equilibrium is globally asymptotically stable whenever the reproduction number for dual malaria-TB co-infection only $R_{MT}^d<1$ - it does not exhibit the phenomenon of backward bifurcation. Graphical representations of this phenomenon is shown, while numerical simulations of the full model are carried out in order to determine whether the two diseases will co-exist whenever their partial reproductive numbers exceed unity. Finally, we perform sensitivity analysis on the key parameters that drive the disease dynamics in order to determine their relative importance to disease transmission.
Distributed susceptibility: A challenge to persistence theory in infectious disease models
Horst R. Thieme
2009, 12(4): 865-882 doi: 10.3934/dcdsb.2009.12.865 +[Abstract](508) +[PDF](202.3KB)
Abstract:
We consider an S-I(-R) type infectious disease model where the susceptibles differ by their susceptibility to infection. This model presents several challenges. Even existence and uniqueness of solutions is non-trivial. Further it is difficult to linearize about the disease-free equilibrium in a rigorous way. This makes disease persistence a necessary alternative to linearized instability in the superthreshold case. Application of dynamical systems persistence theory faces the difficulty of finding a compact attracting set. One can work around this obstacle by using integral equations and limit equations making it the special case of a persistence theory where the state space is just a set.
Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip
Peixuan Weng
2009, 12(4): 883-904 doi: 10.3934/dcdsb.2009.12.883 +[Abstract](546) +[PDF](287.8KB)
Abstract:
We derive an age-structured population model for the growth of a single species on a 2-dimensional (2D) lattice strip with Neumann boundary conditions. We show that the dynamics of the mature population is governed by a lattice reaction-diffusion system with delayed global interaction. Using theory of asymptotic speed of spread and monotone traveling waves for monotone semiflows, we obtain the asymptotic speed of spread $c^$*, the nonexistence of traveling wavefronts with wave speed $0 < c < c^$*, and the existence of traveling wavefront connecting the two equilibria $w\equiv 0$ and $w\equiv w^+$ for $c\geq c^$*.
Error estimates of the $\theta$-scheme for backward stochastic differential equations
Weidong Zhao, Jinlei Wang and Shige Peng
2009, 12(4): 905-924 doi: 10.3934/dcdsb.2009.12.905 +[Abstract](765) +[PDF](288.6KB)
Abstract:
In this paper, we study the error estimate of the $\theta$-scheme for the backward stochastic differential equation $y_t=\varphi(W_T)+\int_t^Tf(s,y_s)ds-\int_t^Tz_sdW_s$. We show that this scheme is of first-order convergence in $y$ for general $\theta$. In particular, for the case of $\theta=\frac{1}{2}$ (i.e., the Crank-Nicolson scheme), we prove that this scheme is of second-order convergence in $y$ and first-order in $z$. Some numerical examples are also given to validate our theoretical results.

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