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Discrete & Continuous Dynamical Systems - B

2013 , Volume 18 , Issue 8

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Global analysis of age-structured within-host virus model
Cameron J. Browne and Sergei S. Pilyugin
2013, 18(8): 1999-2017 doi: 10.3934/dcdsb.2013.18.1999 +[Abstract](129) +[PDF](405.6KB)
A mathematical model of a within-host viral infection with explicit age-since-infection structure for infected cells is presented. A global analysis of the model is conducted. It is shown that when the basic reproductive number falls below unity, the infection dies out. On the contrary, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium that attracts all positive solutions of the model. The global stability analysis combines the existence of a compact global attractor and a Lyapunov function.
Adaptive full state hybrid function projective synchronization of financial hyperchaotic systems with uncertain parameters
Guoliang Cai, Lan Yao, Pei Hu and Xiulei Fang
2013, 18(8): 2019-2028 doi: 10.3934/dcdsb.2013.18.2019 +[Abstract](86) +[PDF](605.1KB)
This paper further investigates a new type synchronization called full state hybrid function projective synchronization (FSHFPS). Based on the Lyapunov stability theory, the adaptive control law and the parameter update laws are derived to make FSHFPS between two financial hyperchaotic systems. And FSHFPS of financial hyperchaotic systems is first studied in this paper. The method is successfully applied to the synchronization between two identical financial hyperchaotic systems and two different financial hyperchaotic systems when the parameters unknown. Numerical simulations are presented to demonstrate the effectiveness of the proposed controllers.
Blow-up dynamics of self-attracting diffusive particles driven by competing convexities
Vincent Calvez and Lucilla Corrias
2013, 18(8): 2029-2050 doi: 10.3934/dcdsb.2013.18.2029 +[Abstract](67) +[PDF](1169.0KB)
In this paper, we analyze the dynamics of an $N$ particles system evolving according the gradient flow of an energy functional. The particle system is an approximation of the Lagrangian formulation of a one parameter family of non-local drift-diffusion equations in one spatial dimension. We shall prove the global in time existence of the trajectories of the particles (under a sufficient condition on the initial distribution) and give two blow-up criteria. All these results are consequences of the competition between the discrete entropy and the discrete interaction energy. They are also consistent with the continuous setting, that in turn is a one dimension reformulation of the parabolic-elliptic Keller-Segel system in high dimensions.
Mean-square convergence of numerical approximations for a class of backward stochastic differential equations
Chuchu Chen and Jialin Hong
2013, 18(8): 2051-2067 doi: 10.3934/dcdsb.2013.18.2051 +[Abstract](165) +[PDF](397.6KB)
This paper is devoted to the fundamental convergence theorem on the mean-square order of numerical approximations for a class of backward stochastic differential equations with terminal condition $\chi=\varphi(W_{T}+x)$. Our theorem shows that the mean-square order of convergence of a numerical method depends on the order of the one-step approximation for the mean-square deviation only. And some numerical schemes as examples are presented to verify the theorem.
Triple collisions of invariant bundles
Jordi-Lluís Figueras and Àlex Haro
2013, 18(8): 2069-2082 doi: 10.3934/dcdsb.2013.18.2069 +[Abstract](59) +[PDF](549.7KB)
We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with $3$ different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.
Convergence, non-negativity and stability of a new Milstein scheme with applications to finance
Desmond J. Higham, Xuerong Mao and Lukasz Szpruch
2013, 18(8): 2083-2100 doi: 10.3934/dcdsb.2013.18.2083 +[Abstract](73) +[PDF](445.1KB)
We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multi-level Monte Carlo simulations for mean-reverting financial models with polynomial growth in the diffusion term. We introduce a double implicit Milstein scheme and show that it possesses desirable properties. It converges strongly and preserves non-negativity for a rich family of financial models and can reproduce linear and nonlinear stability behaviour of the underlying SDE without severe restriction on the time step. Although the scheme is implicit, we point out examples of financial models where an explicit formula for the solution to the scheme can be found.
Bifurcation analysis in a predator-prey model with constant-yield predator harvesting
Jicai Huang, Yijun Gong and Shigui Ruan
2013, 18(8): 2101-2121 doi: 10.3934/dcdsb.2013.18.2101 +[Abstract](79) +[PDF](3028.2KB)
In this paper we study the effect of constant-yield predator harvesting on the dynamics of a Leslie-Gower type predator-prey model. It is shown that the model has a Bogdanov-Takens singularity (cusp case) of codimension 3 or a weak focus of multiplicity two for some parameter values, respectively. Saddle-node bifurcation, repelling and attracting Bogdanov-Takens bifurcations, supercritical and subcritical Hopf bifurcations, and degenerate Hopf bifurcation are shown as the values of parameters vary. Hence, there are different parameter values for which the model has a homoclinic loop or two limit cycles. It is also proven that there exists a critical harvesting value such that the predator specie goes extinct for all admissible initial densities of both species when the harvest rate is greater than the critical value. These results indicate that the dynamical behavior of the model is very sensitive to the constant-yield predator harvesting and the initial densities of both species and it requires careful management in the applied conservation and renewable resource contexts. Numerical simulations, including the repelling and attracting Bogdanov-Takens bifurcation diagrams and corresponding phase portraits, two limit cycles, the coexistence of a stable homoclinic loop and an unstable limit cycle, and a stable limit cycle enclosing an unstable multiple focus with multiplicity one, are presented which not only support the theoretical analysis but also indicate the existence of Bogdanov-Takens bifurcation (cusp case) of codimension 3. These results reveal far richer and much more complex dynamics compared to the model without harvesting or with only constant-yield prey harvesting.
Permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates
Yun Kang
2013, 18(8): 2123-2142 doi: 10.3934/dcdsb.2013.18.2123 +[Abstract](80) +[PDF](465.4KB)
The per-capita growth rate of a species is influenced by density-independent, positive and negative density-dependent factors. These factors can lead to nonlinearity with a consequence that species may process multiple nontrivial equilibria in its single state (e.g., Allee effects). This makes the study of permanence of discrete-time multi-species population models very challenging due to the complex boundary dynamics. In this paper, we explore the permanence of a general discrete-time two-species-interaction model with nonlinear per-capita growth rates for the first time. We find a simple sufficient condition for guaranteeing the permanence of the system by applying and extending the ecological concept of the relative nonlinearity to estimate systems' external Lyapunov exponents. Our method allows us to fully characterize the effects of nonlinearities in the per-capita growth functions and implies that the fluctuated populations may devastate the permanence of systems and lead to multiple attractors. These results are illustrated with specific two species competition and predator-prey models with generic nonlinear per-capita growth functions. Finally, we discuss the potential biological implications of our results.
A note on global asymptotic stability of nonautonomous master equations
Christian Lax and Sebastian Walcher
2013, 18(8): 2143-2149 doi: 10.3934/dcdsb.2013.18.2143 +[Abstract](54) +[PDF](282.7KB)
We present sufficient conditions to either preclude or guarantee global asymptotic stability of linear differential equations for time-dependent $\mathbb W$-matrices. These conditions are concerned with integrability or non-integrability of the matrix entries. The proofs employ differential inequalities.
Qualitative analysis of an age- and sex-structured vaccination model for human papillomavirus
Tufail Malik, Abba Gumel and Elamin H. Elbasha
2013, 18(8): 2151-2174 doi: 10.3934/dcdsb.2013.18.2151 +[Abstract](46) +[PDF](477.4KB)
A new model for the transmission dynamics of human pappilomavirus (HPV) is designed and analysed. The model, which stratifies the total population in terms of age and gender, incorporates an imperfect anti-HPV vaccine with some therapeutic benefits. Rigorous qualitative analysis of the resulting age-structured model, which takes the form of a deterministic system of non-linear partial differential equations with separable transmission coefficients, shows that the disease-free equilibrium of the model is locally-asymptotically stable whenever the effective reproduction number (denoted by $\mathcal{R}_v$) is less than unity. It is shown to be globally-asymptotically stable if certain additional conditions hold. Furthermore, it is shown that the model has at least one endemic equilibrium when $\mathcal{R}_v$ exceeds unity. Hence, the effective control of HPV spread in a community, using a vaccine, is governed by the threshold quantity $\mathcal{R}_v$ (the use of the vaccine will lead to effective disease control or elimination only if it reduces the threshold quantity to a value less than unity; and the use of such vaccine will not lead to effective disease control if it fails to make the threshold quantity to be less than unity).
Fractional diffusion with Neumann boundary conditions: The logistic equation
Eugenio Montefusco, Benedetta Pellacci and Gianmaria Verzini
2013, 18(8): 2175-2202 doi: 10.3934/dcdsb.2013.18.2175 +[Abstract](119) +[PDF](641.9KB)
Motivated by experimental studies on the anomalous diffusion of biological populations, we study the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Such operator arises in the continuous limit for long jumps random walks with reflecting barriers. Existence and uniqueness results for positive solutions are proved in the case of indefinite nonlinearities of logistic type by means of bifurcation theory.
Dead-core rates for the heat equation with a spatially dependent strong absorption
Chin-Chin Wu and Zhengce Zhang
2013, 18(8): 2203-2210 doi: 10.3934/dcdsb.2013.18.2203 +[Abstract](67) +[PDF](297.8KB)
This work is to study the dead-core behavior for a semilinear heat equation with a spatially dependent strong absorption term. We first give a criterion on the initial data such that the dead-core occurs. Then we prove the temporal dead-core rate is non-self-similar. This is based on the standard limiting process with the uniqueness of the self-similar solutions in a certain class.

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