Non-Gaussian dynamics of a tumor growth system with immunization
Mengli Hao Ting Gao Jinqiao Duan Wei Xu
Inverse Problems & Imaging 2013, 7(3): 697-716 doi: 10.3934/ipi.2013.7.697
This paper is devoted to exploring the effects of non-Gaussian fluctuations on dynamical evolution of a tumor growth model with immunization, subject to non-Gaussian $\alpha$-stable type Lévy noise. The corresponding deterministic model has two meaningful states which represent the state of tumor extinction and the state of stable tumor, respectively. To characterize the time for different initial densities of tumor cells staying in the domain between these two states and the likelihood of crossing this domain, the mean exit time and the escape probability are quantified by numerically solving differential-integral equations with appropriate exterior boundary conditions. The relationships between the dynamical properties and the noise parameters are examined. It is found that in the different stages of tumor, the noise parameters have different influences on the time and the likelihood inducing tumor extinction. These results are relevant for determining efficient therapeutic regimes to induce the extinction of tumor cells.
keywords: stochastic dynamics of tumor growth Tumor growth with immunization escape probability. non-Gaussian Lévy process mean exit time Lévy jump measure
Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation
Xiangjun Wang Jianghui Wen Jianping Li Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2012, 17(5): 1575-1584 doi: 10.3934/dcdsb.2012.17.1575
The Thermohaline Circulation, which plays a crucial role in the global climate, is a cycle of deep ocean due to the change in salinity and temperature (i.e., density). The effects of non-Gaussian noise on the Stommel box model for the Thermohaline Circulation are considered. The noise is represented by a non-Gaussian $\alpha$-stable Lévy motion with $0<\alpha < 2$. The $\alpha$ value may be regarded as the index of non-Gaussianity. When $\alpha=2$, the $\alpha$-stable Lévy motion becomes the usual (Gaussian) Brownian motion.
    Dynamical features of this stochastic model is examined by computing the mean exit time for various $\alpha$ values. The mean exit time is simulated by numerically solving a deterministic differential equation with nonlocal interactions. It has been observed that some salinity difference levels remain in certain ranges for longer times than other salinity difference levels, for different $\alpha$ values. This indicates a lower variability for these salinity difference levels. Realizing that it is the salinity differences that drive the thermohaline circulation, this lower variability could mean a stable circulation, which may have further implications for the global climate dynamics.
keywords: colored noise. mean exit time Stommel box model thermohaline circulation $\alpha$-stable Lévy motion
Stochastic averaging principle for dynamical systems with fractional Brownian motion
Yong Xu Rong Guo Di Liu Huiqing Zhang Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2014, 19(4): 1197-1212 doi: 10.3934/dcdsb.2014.19.1197
Stochastic averaging for a class of stochastic differential equations (SDEs) with fractional Brownian motion, of the Hurst parameter $H$ in the interval $(\frac{1}{2},1)$, is investigated. An averaged SDE for the original SDE is proposed, and their solutions are quantitatively compared. It is shown that the solution of the averaged SDE converges to that of the original SDE in the sense of mean square and also in probability. It is further demonstrated that a similar averaging principle holds for SDEs under stochastic integral of pathwise backward and forward types. Two examples are presented and numerical simulations are carried out to illustrate the averaging principle.
keywords: stochastic differential equations Correlated noise averaging principle stochastic calculus fractional Brownian motion.
Dynamics of the thermohaline circulation under wind forcing
Hongjun Gao Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2002, 2(2): 205-219 doi: 10.3934/dcdsb.2002.2.205
The ocean thermohaline circulation, also called meridional overturning circulation, is caused by water density contrasts. This circulation has large capacity of carrying heat around the globe and it thus affects the energy budget and further affects the climate. We consider a thermohaline circulation model in the meridional plane under external wind forcing. We show that, when there is no wind forcing, the stream function and the density fluctuation (under appropriate metrics) tend to zero exponentially fast as time goes to infinity. With rapidly oscillating wind forcing, we obtain an averaging principle for the thermohaline circulation model. This averaging principle provides convergence results and comparison estimates between the original thermohaline circulation and the averaged thermohaline circulation, where the wind forcing is replaced by its time average. This establishes the validity for using the averaged thermohaline circulation model for numerical simulations at long time scales.
keywords: wind forcing. exponential decay Geophysical flows averaging principle
Approximation for random stable manifolds under multiplicative correlated noises
Tao Jiang Xianming Liu Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2016, 21(9): 3163-3174 doi: 10.3934/dcdsb.2016091
The usual Wong-Zakai approximation is about simulating individual solutions of stochastic differential equations(SDEs). From the perspective of dynamical systems, it is also interesting to approximate random invariant manifolds which are geometric objects useful for understanding how complex dynamics evolve under stochastic influences. We study a Wong-Zakai type of approximation for the random stable manifold of a stochastic evolutionary equation with multiplicative correlated noise. Based on the convergence of solutions on the invariant manifold, we approximate the random stable manifold by the invariant manifolds of a family of perturbed stochastic systems with smooth correlated noise (i.e., an integrated Ornstein-Uhlenbeck process). The convergence of this approximation is established.
keywords: Wong-Zakai approximation integrated Ornstein-Uhlenbeck processes. correlated noise random invariant manifolds stochastic evolutionary equations
Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor
Chunyou Sun Daomin Cao Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2008, 9(3&4, May): 743-761 doi: 10.3934/dcdsb.2008.9.743
The dynamical behavior of non-autonomous strongly damped wave-type evolutionary equations with linear memory, critical nonlinearity, and time-dependent external forcing is investigated. The time-dependent external forcing is assumed to be only translation-bounded, instead of translation-compact. First, the asymptotic regularity of solutions is proved, and then the existence of the compact uniform attractor together with its structure and regularity is obtained.
keywords: Non-autonomous systems with memory critical exponent uniform attractor. asymptotic regularity wave equations
Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation
Yanfeng Guo Jinqiao Duan Donglong Li
Discrete & Continuous Dynamical Systems - S 2016, 9(6): 1701-1715 doi: 10.3934/dcdss.2016071
Random invariant manifolds are considered for a stochastic Swift-Hohenberg equation with multiplicative noise in the Stratonovich sense. Using a stochastic transformation and a technique of cut-off function, existence of random invariant manifolds and attracting property of the corresponding random dynamical system are obtained by Lyaponov-Perron method. Then in the sense of large probability, an approximation of invariant manifolds has been investigated and this is further used to describe the geometric shape of the invariant manifolds.
keywords: stochastic transformation Stochastic partial differential equations approximation of invariant manifolds Lyaponov-Perron method. random invariant manifolds
Dispersion in flows with obstacles and uncertainty
Jinqiao Duan Vincent J. Ervin Daniel Schertzer
Conference Publications 2001, 2001(Special): 131-136 doi: 10.3934/proc.2001.2001.131
Please refer to Full Text.
Long-time behavior of a class of nonlocal partial differential equations
Chang Zhang Fang Li Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2018, 23(2): 749-763 doi: 10.3934/dcdsb.2018041
This work is devoted to investigate the well-posedness and long-time behavior of solutions for the following nonlocal nonlinear partial differential equations in a bounded domain
$\begin{align*}u_t+(-Δ)^{σ/2}u +f(u) = g.\end{align*}$
Firstly, due to the lack of an upper growth restriction of the nonlinearity $f$, we have to utilize a weak compactness approach in an Orlicz space to obtain the well-posedness of weak solutions for the equations. We then establish the existence of
$(L^2_0(Ω), L^2_0(Ω))$
-absorbing sets and
$(L^2_0(Ω), H^{σ/2}_0(Ω))$
-absorbing sets for the solution semigroup
$\{S(t)\}_{t≥q 0}$
. Finally, we prove the existence of
$(L^2_0(Ω), L^2_0(Ω))$
-global attractor and
$(L^2_0(Ω), H^{σ/2}_0(Ω))$
-global attractor by a asymptotic compactness method.
keywords: Fractional Laplacian long-time behavior weak solution nonlocal partial differential equation bounded domain
Recurrent motions in the nonautonomous Navier-Stokes system
Vena Pearl Bongolan-walsh David Cheban Jinqiao Duan
Discrete & Continuous Dynamical Systems - B 2003, 3(2): 255-262 doi: 10.3934/dcdsb.2003.3.255
We prove the existence of recurrent or Poisson stable motions in the Navier-Stokes fluid system under recurrent or Poisson stable forcing, respectively. We use an approach based on nonautonomous dynamical systems ideas.
keywords: Navier-Stokes equations Nonautonomous dynamical system skew-product flow Poisson stable motion. recurrent motion

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