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Discrete & Continuous Dynamical Systems - B

2014 , Volume 19 , Issue 2

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Local stability implies global stability for the planar Ricker competition model
E. Cabral Balreira , Saber Elaydi and  Rafael Luís
2014, 19(2): 323-351 doi: 10.3934/dcdsb.2014.19.323 +[Abstract](29) +[PDF](599.4KB)
Abstract:
Under certain analytic and geometric assumptions we show that local stability of the coexistence (positive) fixed point of the planar Ricker competition model implies global stability with respect to the interior of the positive quadrant. This result is a confluence of ideas from Dynamical Systems, Geometry, and Topology that provides a framework to the study of global stability for other planar competition models.
Isotropic realizability of electric fields around critical points
Marc Briane
2014, 19(2): 353-372 doi: 10.3934/dcdsb.2014.19.353 +[Abstract](25) +[PDF](529.1KB)
Abstract:
In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $u$ is solution to the equation div$\left(\sigma\nabla u\right)=0$ for some isotropic conductivity $\sigma>0$. The case of a smooth function $u$ without critical point was investigated in [7] thanks to a dynamical system approach which yields a global isotropic realizability result in $\mathbb{R}^d$. The presence of a critical point $x^*$ needs a specific treatment according to the behavior of the gradient flow in the neighborhood of $x^*$. The case where the hessian matrix $\nabla^2 u(x^*)$ is invertible with both positive and negative eigenvalues is the most favorable: the anisotropic realizability is a consequence of Morse's lemma, while the Hadamard-Perron theorem leads us to a characterization of the isotropic realizability around $x^*$ through some boundedness condition involving the laplacian of $u$ along the gradient flow. When the matrix $\nabla^2 u(x^*)$ has $d$ positive eigenvalues or $d$ negative eigenvalues, we get a strong maximum principle under the same boundedness condition. However, when the matrix $\nabla^2 u(x^*)$ is not invertible, the derivation of the isotropic realizability is much more intricate: the Hartman-Wintner theorem gives necessary conditions for the isotropic realizability in dimension two, while the dynamical system approach provides a criterion of non realizability in any dimension. The two methods are illustrated by a two-dimensional and a three-dimensional example.
Nonlocal convection-diffusion volume-constrained problems and jump processes
Qiang Du , Zhan Huang and  Richard B. Lehoucq
2014, 19(2): 373-389 doi: 10.3934/dcdsb.2014.19.373 +[Abstract](37) +[PDF](729.8KB)
Abstract:
We introduce the Cauchy and time-dependent volume-constrained problems associated with a linear nonlocal convection-diffusion equation. These problems are shown to be well-posed and correspond to conventional convection-diffusion equations as the region of nonlocality vanishes. The problems also share a number of features such as the maximum principle, conservation and dispersion relations, all of which are consistent with their corresponding local counterparts. Moreover, these problems are the master equations for a class of finite activity Lévy-type processes with nonsymmetric Lévy measure. Monte Carlo simulations and finite difference schemes are applied to these nonlocal problems, to show the effects of time, kernel, nonlocality and different volume-constraints.
Stability analysis for a size-structured juvenile-adult population model
Xianlong Fu and  Dongmei Zhu
2014, 19(2): 391-417 doi: 10.3934/dcdsb.2014.19.391 +[Abstract](35) +[PDF](591.3KB)
Abstract:
In this paper, we discuss the asymptotic behavior of a size-structured juvenile-adult population equation with resource-dependent and delayed birth process. The linearization about stationary solutions is analyzed by using semigroup and spectral methods. The juvenile-adult interaction, resource-dependent and delayed boundary condition are considered deliberately for the system to investigate their influences on the asymptotic behavior of solutions. We obtain the stability and instability of the stationary solutions by given some biologically meaningful conditions in two important cases. Finally, two examples are presented and simulated to illustrate the obtained results.
Hyperbolic quenching problem with damping in the micro-electro mechanical system device
Jong-Shenq Guo and  Bo-Chih Huang
2014, 19(2): 419-434 doi: 10.3934/dcdsb.2014.19.419 +[Abstract](26) +[PDF](389.4KB)
Abstract:
We study the initial boundary value problem for the damped hyperbolic equation arising in the micro-electro mechanical system device with local or nonlocal singular nonlinearity. For both cases, we provide some criteria for quenching and global existence of the solution. We also derive the existence of the quenching curve for the corresponding Cauchy problem with local source.
Volatility in options formulae for general stochastic dynamics
Kais Hamza , Fima C. Klebaner and  Olivia Mah
2014, 19(2): 435-446 doi: 10.3934/dcdsb.2014.19.435 +[Abstract](31) +[PDF](320.7KB)
Abstract:
It is well-known that the Black-Scholes formula has been derived under the assumption of constant volatility in stocks. In spite of evidence that this parameter is not constant, this formula is widely used by financial markets. This paper addresses the question of whether an alternative model for stock price exists for which the Black-Scholes or similar formulae hold. The results obtained in this paper are very general as no assumptions are made on the dynamics of the model, whether it be the underlying price process, the volatility process or how they relate to each other. We show that if the formula holds for a continuum of strikes and three terminal times then the volatility must be constant. However, when it only holds for finitely many strikes, and three or more maturity times, we obtain a universal bound on the variation of the volatility. This bound yields that the implied volatility is constant when the sequence of strikes increases to cover the entire half-line. This recovers the result for a continuum of strikes by a different approach.
Cost-effectiveness evaluation of gender-based vaccination programs against sexually transmitted infections
Jane M. Heffernan , Yijun Lou , Marc Steben and  Jianhong Wu
2014, 19(2): 447-466 doi: 10.3934/dcdsb.2014.19.447 +[Abstract](39) +[PDF](653.1KB)
Abstract:
The ultimate goal of a vaccination program is to interrupt pathogen transmission so as to eradicate the disease from the population in the future, and/or to decrease morbidity and mortality due to the disease in the short term. For sexually transmitted infections (STI) the determination of an optimal vaccination program is not straightforward since (1) the transmission probabilities between two different sexes are normally unequal (weighted to a greater probability from males to females than vice versa), (2) demographic parameters between the two sexes are unequal, (3) the prevalence of disease in one sex may have a greater impact on the morbidity and mortality of the next generation (transmission to the neonate) and, (4) the existence of pathogens closely related to the STI in question (i.e. herpes - HSV-1 vs. HSV-2, different strains of Chlamydia trachomatis, different strains of Neisseria which cause Gonorrhea, and others) may induce immunity in individuals that render a vaccine ineffective.
    We have developed two models of sexually transmitted infections (with and without age structure) to evaluate the cost-efficacy of gender-based vaccination programs in the context of STI control. The first model ignores age structure for qualitative analysis of points (1-3), while the second refined one incorporates the age structure, reflecting the effects of immunity gained from infection of closely related strains (point 4), which is important for HSV-2 vaccination strategies. For both models, we find that the stability of the system and ultimate eradication of the disease depends explicitly on the corresponding reproduction number. We also find that vaccinating females is more cost-effective, providing a greater reduction in disease prevalence in the population and number of infected females of childbearing age. This result is counter-intuitive since vaccinating super-transmitters (males) over sub-transmitters (females) usually has the greatest impact on disease prevalence. Sensitivity analysis is implemented to investigate how the parameters affect the control reproduction numbers and infectious population sizes.
Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold
Wan-Tong Li , Guo Lin , Cong Ma and  Fei-Ying Yang
2014, 19(2): 467-484 doi: 10.3934/dcdsb.2014.19.467 +[Abstract](86) +[PDF](414.7KB)
Abstract:
This paper is concerned with the traveling wave solutions of a diffusive SIR system with nonlocal delay. We obtain the existence and nonexistence of traveling wave solutions, which formulate the propagation of disease without outbreak threshold. Moreover, it is proved that at any fixed moment, the faster the disease spreads, the more the infected individuals, and the larger the recovery/remove ratio is, the less the infected individuals.
Identification of focus and center in a 3-dimensional system
Lingling Liu , Bo Gao , Dongmei Xiao and  Weinian Zhang
2014, 19(2): 485-522 doi: 10.3934/dcdsb.2014.19.485 +[Abstract](41) +[PDF](802.5KB)
Abstract:
In this paper we identify focus and center for a generalized Lorenz system, a 3-dimensional quadratic polynomial differential system with four parameters $a$, $b$, $c$, $\sigma$. The known work computes the first order Lyapunov quantity on a center manifold and shows the appearance of a limit cycle for $a\neq b$, but the order of weak foci was not determined yet. Moreover, the case that $a=b$ was not discussed. In this paper, for $a\neq b$ we use resultants to decompose the algebraic varieties of Lyapunov quantities so as to prove that the order is at most 3. Further, we apply Sturm's theorem to determine real zeros of the first order Lyapunov quantity over an extension field so that we give branches of parameter curves for each order of weak foci. For $a=b$ we prove its Darboux integrability by finding an invariant surface, showing that the equilibrium of center-focus type is actually a rough center on a center manifold.
Expanding Baker Maps as models for the dynamics emerging from 3D-homoclinic bifurcations
Antonio Pumariño , José Ángel Rodríguez , Joan Carles Tatjer and  Enrique Vigil
2014, 19(2): 523-541 doi: 10.3934/dcdsb.2014.19.523 +[Abstract](29) +[PDF](1056.2KB)
Abstract:
For certain 3D-homoclinic tangencies where the unstable manifold of the saddle point involved in the homoclinic tangency has dimension two, many different strange attractors have been numerically observed for the corresponding family of limit return maps. Moreover, for some special value of the parameter, the respective limit return map is conjugate to what was called bidimensional tent map. This piecewise affine map is an example of what we call now Expanding Baker Map, and the main objective of this paper is to show how many of the different attractors exhibited for the limit return maps resemble the ones observed for Expanding Baker Maps.
Stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders
Quan Wang
2014, 19(2): 543-563 doi: 10.3934/dcdsb.2014.19.543 +[Abstract](35) +[PDF](430.9KB)
Abstract:
In this paper, our objective is to apply the attractor bifurcation theory to study the stability and bifurcation of a viscous incompressible plasma fluid contained between two concentric rotating cylinders. We get a dimensionless parameter $T$ which can describe the stability and bifurcation of the plasma fluid through calculation. When $T$ is smaller than a critical number $T_0$, the plasma fluid is stable. When $T$ crosses the critical number $T_0$, the plasma fluid becomes unstable and will generate a new magnetic field which has an interesting structure.
Transverse instability for a system of nonlinear Schrödinger equations
Yohei Yamazaki
2014, 19(2): 565-588 doi: 10.3934/dcdsb.2014.19.565 +[Abstract](30) +[PDF](514.3KB)
Abstract:
In this paper, we consider the transverse instability for a system of nonlinear Schrödinger equations on $\mathbb{R} \times \mathbb{T}_L $. Here, $\mathbb{T}_L$ means the torus with a $2\pi L$ period. It was shown by Colin-Ohta [11] that this system on $\mathbb{R}$ has a stable standing wave. In this paper, we regard this standing wave as the standing wave of this system on $\mathbb{R} \times \mathbb{T}_L$. Then, we show that there exists the critical period $L_{\omega}$ which is the boundary between the stability and the instability of the standing wave on $\mathbb{R} \times \mathbb{T}_L$.
Periodic unimodal Allee maps, the semigroup property and the $\lambda$-Ricker map with Allee effect
Yi Yang and  Robert J. Sacker
2014, 19(2): 589-606 doi: 10.3934/dcdsb.2014.19.589 +[Abstract](33) +[PDF](434.1KB)
Abstract:
The $\lambda$-Ricker equation has, for certain values of the parameters, an unstable fixed point giving rise to the Allee effect, and an attracting fixed point, the carrying capacity. The $k$-periodic $\lambda$-Ricker equation is studied and parameter intervals are determined for which there exist a $k$-periodic Allee state and a $k$-periodic attracting state.
Dynamic transitions for Landau-Brazovskii model
Hao Zhang , Kai Jiang and  Pingwen Zhang
2014, 19(2): 607-627 doi: 10.3934/dcdsb.2014.19.607 +[Abstract](30) +[PDF](237.1KB)
Abstract:
We study the disordered and ordered phase transitions modeled by the Landau-Brazovskii (LB) equation using the dynamic transition theory. It is shown that the linear instability of the disordered phase always leads to phase transitions to ordered phases, and type of transitions is dictated by the sign of a nondimensional parameter, measuring the strengths of the quadratic and cubic nonlinearity of the model. By the center manifold reduced theory, we analysis the lower dimension system. The analysis shows that the second-order disordered-ordered phase transition occurs in two ways: one is due to the combined affect of the surface energy and the rectangular geometry of the spatial domain, and another other is due to mixed transition. For the mixed transition, there exist two regions of fluctuations where the first and second-order transitions occur respectively. We also present the general results of the phase transition for the rectangle domain as if the first eigenvalue is simple, which is the most likely to happen. It is shown that for case of second order transition, the spatial period of the ordered phases is essentially independent of the size of the domain, and for the first order transition case, the disordered phase undergoes a drastic change, leading to more richer ordered patterns. In addition, we show the stability of some classical ordered phases near the homogeneous phase, like HPC, BCC and Doulbe Gyroid phase.

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