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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 9

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A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver
Andreas C. Aristotelous, Ohannes Karakashian and Steven M. Wise
2013, 18(9): 2211-2238 doi: 10.3934/dcdsb.2013.18.2211 +[Abstract](95) +[PDF](2655.3KB)
In this paper we devise and analyze a mixed discontinuous Galerkin finite element method for a modified Cahn-Hilliard equation that models phase separation in diblock copolymer melts. The time discretization is based on a convex splitting of the energy of the equation. We prove that our scheme is unconditionally energy stable with respect to a spatially discrete analogue of the continuous free energy of the system, unconditionally uniquely solvable, and convergent in the natural energy norm with optimal rates. We describe an efficient nonlinear multigrid solver for advancing our semi-implicit scheme in time and conclude the paper with numerical tests confirming the predicted convergence rates and suggesting the near-optimal complexity of the solver.
Epidemic models with age of infection, indirect transmission and incomplete treatment
Liming Cai, Maia Martcheva and Xue-Zhi Li
2013, 18(9): 2239-2265 doi: 10.3934/dcdsb.2013.18.2239 +[Abstract](95) +[PDF](446.8KB)
An infection-age-structured epidemic model with environmental bacterial infection is investigated in this paper. It is assumed that the infective population is structured according to age of infection, and the infectivity of the treated individuals is reduced but varies with the infection-age. An explicit formula for the reproductive number $ \Re_0$ of the model is obtained. By constructing a suitable Lyapunov function, the global stability of the infection-free equilibrium in the system is obtained for $\Re_0<1$. It is also shown that if the reproduction number $\Re_0>1$, then the system has a unique endemic equilibrium which is locally asymptotically stable. Furthermore, if the reproduction number $\Re_0>1$, the system is permanent. When the treatment rate and the transmission rate are both independent of infection age, the system of partial differential equations (PDEs) reduces to a system of ordinary differential equations (ODEs). In this special case, it is shown that the global dynamics of the system can be determined by the basic reproductive number.
Evolutionary branching patterns in predator-prey structured populations
Marcello Delitala and Tommaso Lorenzi
2013, 18(9): 2267-2282 doi: 10.3934/dcdsb.2013.18.2267 +[Abstract](76) +[PDF](866.8KB)
Predator-prey ecosystems represent, among others, a natural context where evolutionary branching patterns may arise. Moving from this observation, the paper deals with a class of integro-differential equations modeling the dynamics of two populations structured by a continuous phenotypic trait and related by predation. Predators and preys proliferate through asexual reproduction, compete for resources and undergo phenotypic changes. A positive parameter $\varepsilon$ is introduced to model the average size of such changes. The asymptotic behavior of the solution of the mathematical problem linked to the model is studied in the limit $\varepsilon \rightarrow 0$ (i.e., in the limit of small phenotypic changes). Analytical results are illustrated and extended by means of numerical simulations with the aim of showing how the present class of equations can mimic the formation of evolutionary branching patterns. All simulations highlight a chase-escape dynamics, where the preys try to evade predation while predators mimic, with a certain delay, the phenotypic profile of the preys.
Dynamics of a ratio-dependent predator-prey system with a strong Allee effect
Yujing Gao and Bingtuan Li
2013, 18(9): 2283-2313 doi: 10.3934/dcdsb.2013.18.2283 +[Abstract](143) +[PDF](1061.2KB)
A ratio-dependent predator-prey model with a strong Allee effect in prey is studied. We show that the model has a Bogdanov-Takens bifurcation that is associated with a catastrophic crash of the predator population. Our analysis indicates that an unstable limit cycle bifurcates from a Hopf bifurcation, and it disappears due to a homoclinic bifurcation which can lead to different patterns of global population dynamics in the model. We study the heteroclinic orbits and determine all possible phase portraits when the Bogdanov-Takens bifurcation occurs. We also provide the conditions for nonexistence of limit cycle under which the global dynamics of the model can be determined.
Blow-up results for semilinear wave equations in the superconformal case
Mohamed-Ali Hamza and Hatem Zaag
2013, 18(9): 2315-2329 doi: 10.3934/dcdsb.2013.18.2315 +[Abstract](75) +[PDF](445.1KB)
We consider the semilinear wave equation in higher dimensions with power nonlinearity in the superconformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Vişan [22]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [16] for perturbations of the pure power conformal case with lower order terms.
On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response
Sze-Bi Hsu, Shigui Ruan and Ting-Hui Yang
2013, 18(9): 2331-2353 doi: 10.3934/dcdsb.2013.18.2331 +[Abstract](62) +[PDF](591.1KB)
In this paper we study a two-consumers-one-resource competing system with Beddington-DeAngelis functional response. The two consumers competing for a renewable resource have intraspecific competition among their own populations. Firstly we investigate the extinction and uniform persistence of the predators, local and global stability of the equilibria, and existence of Hopf bifurcation at the positive equilibrium. Then we compare the dynamic behavior of the system with and without interference effects. Analytically we study the competition of two identically species with different interference effects. We also study the relaxation oscillation in the case of interference effects. Finally we present extensive numerical simulations to understand the interference effects on the competition outcomes.
The infected frontier in an SEIR epidemic model with infinite delay
Zhigui Lin, Yinan Zhao and Peng Zhou
2013, 18(9): 2355-2376 doi: 10.3934/dcdsb.2013.18.2355 +[Abstract](68) +[PDF](574.9KB)
An SEIR epidemic model with infinite delay and the Neumann boundary condition is investigated, as well as the corresponding free boundary problem, in which the free boundary exactly describes the spreading frontier of the disease. For the problem in a fixed domain with null Neumann boundary condition, the transmission dynamics of the disease is determined by the basic reproduction number $R_0$. More specifically, whether the disease will die out or not depends on $R_0<1$ or $R_0>1$; while for the free boundary problem, we show that under certain conditions the disease will die out even $R_0>1$. Our results indicate that besides the basic reproduction number, the initial size of the infected domain and the diffusivity of the disease in a new region also produce a non-negligible influence to the disease transmission, and it seems more reasonable and acceptable.
Nonlinear dynamics of a mathematical model on action potential duration and calcium transient in paced cardiac cells
Jiying Ma and Dongmei Xiao
2013, 18(9): 2377-2396 doi: 10.3934/dcdsb.2013.18.2377 +[Abstract](120) +[PDF](1088.8KB)
In the paper we focus on the dynamics of a two-dimensional discrete-time mathematical model, which describes the interaction between the action potential duration (APD) and calcium transient in paced cardiac cells. By qualitative and bifurcation analysis, we prove that this model can undergo period-doubling bifurcation and Neimark-Sacker bifurcation as parameters vary, respectively. These results provide theoretical support on some experimental observations, such as the alternans of APD and calcium transient, and quasi-periodic oscillations between APD and calcium transient in paced cardiac cells. The rich and complicated bifurcation phenomena indicate that the dynamics of this model are very sensitive to some parameters, which might have important implications for the control of cardiovascular disease.
Strong and weak Allee effects and chaotic dynamics in Richards' growths
J. Leonel Rocha, Danièle Fournier-Prunaret and Abdel-Kaddous Taha
2013, 18(9): 2397-2425 doi: 10.3934/dcdsb.2013.18.2397 +[Abstract](152) +[PDF](1144.7KB)
In this paper we define and investigate generalized Richards' growth models with strong and weak Allee effects and no Allee effect. We prove the transition from strong Allee effect to no Allee effect, passing through the weak Allee effect, depending on the implicit conditions, which involve the several parameters considered in the models. New classes of functions describing the existence or not of Allee effect are introduced, a new dynamical approach to Richards' populational growth equation is established. These families of generalized Richards' functions are proportional to the right hand side of the generalized Richards' growth models proposed. Subclasses of strong and weak Allee functions and functions with no Allee effect are characterized. The study of their bifurcation structure is presented in detail, this analysis is done based on the configurations of bifurcation curves and symbolic dynamics techniques. Generically, the dynamics of these functions are classified in the following types: extinction, semi-stability, stability, period doubling, chaos, chaotic semistability and essential extinction. We obtain conditions on the parameter plane for the existence of a weak Allee effect region related to the appearance of cusp points. To support our results, we present fold and flip bifurcations curves and numerical simulations of several bifurcation diagrams.
Advantages for controls imposed in a proper subset
Gengsheng Wang and Yashan Xu
2013, 18(9): 2427-2439 doi: 10.3934/dcdsb.2013.18.2427 +[Abstract](64) +[PDF](392.9KB)
In this paper, we study some time optimal control problems for heat equations on $\Omega\times \mathbb{R}^+$. Two properties under consideration are the existence and the bang-bang properties of time optimal controls. It is proved that those two properties hold when controls are imposed on most proper subsets of $\Omega$; while they do not stand when controls are active on the whole $\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ with Dirichlet boundary condition is revealed.
Spreading speed and traveling waves for a two-species weak competition system with free boundary
Chang-Hong Wu
2013, 18(9): 2441-2455 doi: 10.3934/dcdsb.2013.18.2441 +[Abstract](118) +[PDF](385.3KB)
In this paper, we will focus on the spreading speed for a Lotka-Volterra type weak competition model with free boundary in one-dimensional habitat. Based on the comparison principle for free boundary problems, we provide some estimates of the spreading speed. Also, we deal with traveling wave solutions for the same model and show that there exists a traveling wave solution with monotone profile using a shooting method and the Schauder's fixed point theorem.
A study on the positive nonconstant steady states of nonlocal chemotaxis systems
Tian Xiang
2013, 18(9): 2457-2485 doi: 10.3934/dcdsb.2013.18.2457 +[Abstract](80) +[PDF](532.8KB)
In one spatial dimension, we perform global bifurcation analysis on a general nonlocal Keller-Segel chemotaxis model, showing that positive nonconstant steady states exist, if the chemotactic coefficient $\chi$ is larger than a bifurcation value $\overline{\chi}_1$, which is expressible in terms of the parameters and the nonlocal sampling radius in the model. We then show that the positive solutions of the nonlocal model converge at least in $C^2([0,l])\times C^2([0,l])$ to that of the corresponding ``local" model as the nonlocal sampling radius $\rho\rightarrow 0+$. Finally, we use Helly's compactness theorem to establish the profiles of these steady states, when the ratio of the chemotactic coefficient and the cell diffusion rate is large and the nonlocal sampling radius is small, exhibiting whether they are either spiky, of transition layer structure or just flat everywhere. Our results supply understandings on how the biological parameters affect pattern formation for the nonlocal model. In the limit of $\rho\rightarrow 0+$, our results agree with those of local models studied in Wang and Xu [29].
On positive solutions and the Omega limit set for a class of delay differential equations
Changjing Zhuge, Xiaojuan Sun and Jinzhi Lei
2013, 18(9): 2487-2503 doi: 10.3934/dcdsb.2013.18.2487 +[Abstract](76) +[PDF](3401.6KB)
This paper studies positive solutions of a class of delay differential equations of two delays that are originated from a mathematical model of hematopoietic dynamics. We give an optimal condition on initial conditions for $t\leq 0$ such that the solutions are positive for $t>0$. Long time behaviors of these positive solutions are also discussed through a dynamical system defined at a space of continuous functions. Characteristic description of the $\omega$ limit set of this dynamical system is obtained. This $\omega$ limit set provides informations for the long time behaviors of positive solutions of the delay differential equation.

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