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1531-3492

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## Discrete & Continuous Dynamical Systems - B

May 2016 , Volume 21 , Issue 3

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2016, 21(3): 737-761
doi: 10.3934/dcdsb.2016.21.737

*+*[Abstract](2183)*+*[PDF](1345.4KB)**Abstract:**

In this paper we propose a prey-predator model in multiple patches through the stage structured maturation time delay with migrations among patches. Focus on the case with two patches, we discuss the existence of equilibrium points and the uniform persistence. In particular, when the maturation times are the same in the patches, we study the local and global attractivity of boundary equilibrium point with general migration function and the local stability of the positive equilibrium with constant migration rate. Numerical simulations are provided to demonstrate the theoretical results, to illustrate the effect of the maturation time, the migration rate on the dynamical behavior of the system.

2016, 21(3): 763-779
doi: 10.3934/dcdsb.2016.21.763

*+*[Abstract](1862)*+*[PDF](424.8KB)**Abstract:**

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these equations defined in the upper half-plane. Under suitable growth conditions on the flux function, we are able to establish the global well-posedness in a Sobolev class. When the initial- and boundary-data become more regular, the corresponding solutions are shown to be classical. In addition, the continuous dependence on the data is also obtained.

2016, 21(3): 781-801
doi: 10.3934/dcdsb.2016.21.781

*+*[Abstract](2056)*+*[PDF](442.5KB)**Abstract:**

In this paper, we consider the initial boundary value problem for a fourth order pseudo-parabolic equation with memory and source terms. Under suitable assumptions on the function $g$, the initial data and the parameters in the equation, we not only prove the existence of global weak solutions by the combination of the Galerkin method and potential well theory, but also establish an explicit decay rate estimate of the energy adopting the ideas of Marcelo M. Cavalcanti et al. (J. Differ. Equations 203 (2004) 119-158) and Patrick Martinez (ESAIN: Control Optim. Calc. Var. 4 (1999) 419-444). Furthermore, the finite time blow up results for the solutions with both positive and negative initial energy are obtained under certain conditions.

2016, 21(3): 803-813
doi: 10.3934/dcdsb.2016.21.803

*+*[Abstract](2010)*+*[PDF](615.6KB)**Abstract:**

Cooperative behaviour is often accompanied by the incentives to defect, i.e., to reap the benefits of others' efforts without own contribution. We provide evidence that cooperation and defection can coexist under very broad conditions in the framework of evolutionary games on graphs under deterministic imitation dynamics. Namely, we show that for all graphs there exist coexistence equilibria for certain game-theoretical parameters. Similarly, for all relevant game-theoretical parameters there exists a graph yielding coexistence equilibria. Our proofs are constructive and robust with respect to various utility functions which can be considered. Finally, we briefly discuss bounds for the number of coexistence equilibria.

2016, 21(3): 815-836
doi: 10.3934/dcdsb.2016.21.815

*+*[Abstract](1978)*+*[PDF](1101.3KB)**Abstract:**

In this article, we give an existence-comparison theorem for wavefront solutions in a general class of reaction-diffusion systems. With mixed quasi-monotonicity and Lipschitz condition on the set bounded by coupled upper-lower solutions, the existence of wavefront solution is proven by applying the Schauder Fixed Point Theorem on a compact invariant set. Our main result is then applied to well-known examples: a ratio-dependent predator-prey model, a three-species food chain model of Lotka-Volterra type and a three-species competition model of Lotka-Volterra type. For each model, we establish conditions on the ecological parameters for the presence of wavefront solutions flowing towards the coexistent states through suitably constructed upper and lower solutions. Numerical simulations on those models are also demonstrated to illustrate our theoretical results.

2016, 21(3): 837-847
doi: 10.3934/dcdsb.2016.21.837

*+*[Abstract](1905)*+*[PDF](368.1KB)**Abstract:**

This short paper revisits a free boundary problem which is used to describe the spreading of a new or invasive species. Our main goal is to understand how the underlying long-time dynamical behaviors response to the initial data. To this end, we parameterize the initial function as $u_0=\sigma\phi^*$, where $\sigma$ is regarded as a variable parameter and $\phi^*$ is a given function. Our main result suggests that when the diffusion rate is small, the species can persist in the long run (called spreading) for any $\sigma>0$; while if the diffusion rate is large, the species will go to extinction finally (called vanishing) for small $\sigma>0$. Maybe of more interest is that for some intermediate diffusion rates, there appears a sharp threshold value $\sigma^*\in(0, \infty)$ such that vanishing happens provided $0<\sigma\leq\sigma^*$ and spreading happens provided $\sigma>\sigma^*$. This result can be seen as an improvement of Theorem 1.2 in [8].

2016, 21(3): 849-861
doi: 10.3934/dcdsb.2016.21.849

*+*[Abstract](1802)*+*[PDF](3203.2KB)**Abstract:**

Uniqueness of nonzero positive solutions of a Laplacian elliptic equation arising in combustion theory is of great interest in combustion theory since it can be applied to determine where the extinction phenomenon occurs. We study the uniqueness whenever the orders of the reaction rates are in $(-\infty,1]$. Previous results on uniqueness treated the case when the orders belong to $[0,1)$. When the orders are negative or 1, it is physically meaningful and the bimolecular reaction rate corresponds to the order 1, but there is little study on uniqueness. Our results on the uniqueness are completely new when the orders are negative or 1, and also improve some known results when the orders belong to $(0,1)$. Our results provide exact intervals of the Frank-Kamenetskii parameters on which the extinction phenomenon never occurs. The novelty of our methodology is to combine and utilize the results from Laplacian elliptic

*inequalities*and equations to derive new results on uniqueness of nonzero positive solutions for general Laplacian elliptic equations.

2016, 21(3): 863-881
doi: 10.3934/dcdsb.2016.21.863

*+*[Abstract](1932)*+*[PDF](882.1KB)**Abstract:**

Human T-cell Lymphotropic virus type 1(HTLV-I) causes HAM/T SP and other illnesses. HTLV-I mainly infects $CD4^+$ T cells and activates HTLV-I-specific immune response. In this paper, we formulate a mathematical model of HTLV-I to investigate the role of selective mitotic transmission, Tax expression, and CTL response in vivo. We define two parameters ($R_0$ and $R_1$) to study the model dynamics. The unique infection-free equilibrium $P_0$ is globally asymptomatic stable if $R_0<1$. There exists the chronic-infection equilibrium $P_1$ if $R_1 < 1 < R_0$. There exists a unique chronic-infection equilibrium $P_2$ if $R_1 > 1$. There is a backward bifurcation of chronic-infection equilibria with CTL response if $R_1 < 1 < R_0$. The numerical simulations shown that the existence of backward bifurcation may lead to the existence of periodic solutions.

2016, 21(3): 883-908
doi: 10.3934/dcdsb.2016.21.883

*+*[Abstract](2061)*+*[PDF](479.7KB)**Abstract:**

This paper deals with a $p$-Kirchhoff type problem involving sign-changing weight functions. It is shown that under certain conditions, by means of variational methods, the existence of multiple nontrivial nonnegative solutions for the problem with the subcritical exponent are obtained. Moreover, in the case of critical exponent, we establish the existence of the solutions and prove that the elliptic equation possesses at least one nontrivial nonnegative solution.

2016, 21(3): 909-918
doi: 10.3934/dcdsb.2016.21.909

*+*[Abstract](2133)*+*[PDF](611.0KB)**Abstract:**

A system of four coupled ordinary differential equations is considered, which are coupled through migration of both prey and predator model with logistic type growth. Combined effect of quiescence provides a more realistic way of modeling the complex dynamical behavior. The global stability and Hopf bifurcation solutions are investigated.

2016, 21(3): 919-941
doi: 10.3934/dcdsb.2016.21.919

*+*[Abstract](1951)*+*[PDF](451.3KB)**Abstract:**

In this paper, we prove the uniqueness of weak solutions to the two dimensional full Ericksen-Leslie system with the Leslie stress and general Ericksen stress under the physical constrains on the Leslie coefficients. This question remains unknown even in the case when the Leslie stress is vanishing. The main technique used in the proof is Littlewood-Paley analysis performed in a very delicate way. Different from the earlier result in [28], we introduce a new metric and explore the algebraic structure of the molecular field.

2016, 21(3): 943-957
doi: 10.3934/dcdsb.2016.21.943

*+*[Abstract](1480)*+*[PDF](366.3KB)**Abstract:**

The black hole core of a galaxy attracts a large amounts of gases around it, forming an active galactic nucleus (AGN). An AGN emits huge quantities of energy, leading to AGN jets. In 16, Ma and Wang established a model governing the AGN, in which they obtain the driving force of AGN jets. In this paper, we generalize their model to couple magnetic fields describing the AGN plasma, and derive the huge explosive electromagnetic energy as proposed in (1.13) of 16.

2016, 21(3): 959-975
doi: 10.3934/dcdsb.2016.21.959

*+*[Abstract](1500)*+*[PDF](473.0KB)**Abstract:**

We consider the general class of two-component reaction-diffusion systems on a finite domain that admit interface solutions in one of the components, and we study the dynamics of $n$ interfaces in one dimension. In the limit where the second component has large diffusion, we fully characterize the possible behaviour of $n$ interfaces. We show that after the transients die out, the motion of $n$ interfaces is described by the motion of a

*single*interface on the domain that is $1/n$ the size of the original domain. Depending on parameter regime and initial conditions, one of the following three outcomes results: (1) some interfaces collide; (2) all $n$ interfaces reach a symmetric steady state; (3) all $n$ interfaces oscillate indefinitely. In the latter case, the oscillations are described by a simple harmonic motion with even-numbered interfaces oscillating in phase while odd-numbered interfaces are oscillating in anti-phase. This extends a recent work by [McKay, Kolokolnikov, Muir, DCDS B(17), 2012] from two to any number of interfaces.

2016, 21(3): 977-996
doi: 10.3934/dcdsb.2016.21.977

*+*[Abstract](1924)*+*[PDF](444.6KB)**Abstract:**

A multi-group epidemic model with general nonlinear incidence and vaccination age structure has been formulated and studied. Mathematical analysis shows that the global stability of disease-free equilibrium and endemic equilibrium of the model are determined by the basic reproduction number $\mathcal{R}_0$: the disease-free equilibrium is globally asymptotically stable if $\mathcal{R}_0<1$, the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. The Lyapunov functionals for the global dynamics of the multi-group model are constructed by applying the theory of non-negative matrices and a novel grouping technique in estimating the derivative.

2016, 21(3): 997-1008
doi: 10.3934/dcdsb.2016.21.997

*+*[Abstract](2168)*+*[PDF](373.7KB)**Abstract:**

In this paper we study a free boundary problem for the growth of avascular tumors. The establishment of the model is based on the diffusion of nutrient and mass conservation for the two process proliferation and apoptosis(cell death due to aging). It is assumed the supply of external nutrients is periodic. We mainly study the long time behavior of the solution, and prove that in the case $c$ is sufficiently small, the volume of the tumor cannot expand unlimitedly. It will either disappear or evolve to a positive periodic state.

2016, 21(3): 1009-1022
doi: 10.3934/dcdsb.2016.21.1009

*+*[Abstract](2783)*+*[PDF](425.1KB)**Abstract:**

We formulate a mathematical model to explore the impact of vaccination and treatment on the transmission dynamics of tuberculosis (TB). We develop a technique to prove that the basic reproduction number is the threshold of global stability of the disease-free and endemic equilibria. We then incorporate a control term and evaluate the cost of control strategies, and then perform an optimal control analysis by Pontryagin's maximum principle. Our numerical simulations suggest that the maximum vaccination strategy should be enforced regardless of its efficacy.

2016, 21(3): 1023-1026
doi: 10.3934/dcdsb.2016.21.1023

*+*[Abstract](1593)*+*[PDF](707.3KB)**Abstract:**

N/A

2018 Impact Factor: 1.008

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