All Issues

Volume 27, 2022

Volume 26, 2021

Volume 25, 2020

Volume 24, 2019

Volume 23, 2018

Volume 22, 2017

Volume 21, 2016

Volume 20, 2015

Volume 19, 2014

Volume 18, 2013

Volume 17, 2012

Volume 16, 2011

Volume 15, 2011

Volume 14, 2010

Volume 13, 2010

Volume 12, 2009

Volume 11, 2009

Volume 10, 2008

Volume 9, 2008

Volume 8, 2007

Volume 7, 2007

Volume 6, 2006

Volume 5, 2005

Volume 4, 2004

Volume 3, 2003

Volume 2, 2002

Volume 1, 2001

Discrete and Continuous Dynamical Systems - B

July 2019 , Volume 24 , Issue 7

Select all articles


$ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity
Mengyao Ding and Sining Zheng
2019, 24(7): 2971-2988 doi: 10.3934/dcdsb.2018295 +[Abstract](3103) +[HTML](582) +[PDF](451.26KB)

This paper studies the parabolic-elliptic Keller-Segel system with supercritical sensitivity: \begin{document} $u_{t} = \nabla·(D(u)\nabla u)-\nabla ·(S(u)\nabla v)$ \end{document}, \begin{document} $0 = Δ v -v+u$ \end{document} in \begin{document} $Ω× (0,T)$ \end{document}, where the bounded domain \begin{document} $Ω\subset\mathbb{R}^n$ \end{document}, \begin{document} $n≥2$ \end{document}, subject to the non-flux boundary conditions, \begin{document} $D(u)≥ a_0(u+1)^{-q}$ \end{document}, \begin{document} $0≤ S(u)≤ b_0u(u+1)^{α-q-1}$ \end{document} with \begin{document} $q \in \mathbb{R}$ \end{document}, \begin{document} $α>\frac{2}{n}$ \end{document}, and \begin{document} $a_0, b_0>0$ \end{document}. It is proved that the problem possesses a unique globally bounded solution for \begin{document} $α>\frac{2}{n}$ \end{document} whenever \begin{document} $\|u_0\|_{L^{\frac{nα}{2}}}$ \end{document} is sufficiently small. In addition, we establish the large-time behavior of solutions when \begin{document} $q = 0$ \end{document}.

Comparison theorem and correlation for stochastic heat equations driven by Lévy space-time white noises
Min Niu and Bin Xie
2019, 24(7): 2989-3009 doi: 10.3934/dcdsb.2018296 +[Abstract](3079) +[HTML](582) +[PDF](465.07KB)

Two properties of stochastic heat equations driven by impulsive noises, which are also called Lévy space-time white noises, are mainly investigated in this paper. We first study the comparison theorem for two stochastic heat equations driven by same noises under some sufficient condition, which is proved via the application of Itô's formula. In particular, we obtain the non-negativity of solutions with non-negative initial data. Then, we investigate the positive correlation of the solutions as the application of the comparison theorem. We prove that the total masses of two solutions relative to two different stochastic heat equations with same noise become nearly uncorrelated after a long time.

Analysis and computation of some tumor growth models with nutrient: From cell density models to free boundary dynamics
Jian-Guo Liu, Min Tang, Li Wang and Zhennan Zhou
2019, 24(7): 3011-3035 doi: 10.3934/dcdsb.2018297 +[Abstract](4415) +[HTML](650) +[PDF](12146.87KB)

In this paper, we study a tumor growth equation along with various models for the nutrient component, including a in vitro model and a in vivo model. At the cell density level, the spatial availability of the tumor density \begin{document}$ n$\end{document} is governed by the Darcy law via the pressure \begin{document}$ p(n) = n^{γ}$\end{document}. For finite \begin{document}$ γ$\end{document}, we prove some a priori estimates of the tumor growth model, such as boundedness of the nutrient density, and non-negativity and growth estimate of the tumor density. As \begin{document}$ γ \to ∞$\end{document}, the cell density models formally converge to Hele-Shaw flow models, which determine the free boundary dynamics of the tumor tissue in the incompressible limit. We derive several analytical solutions to the Hele-Shaw flow models, which serve as benchmark solutions to the geometric motion of tumor front propagation. Finally, we apply a conservative and positivity preserving numerical scheme to the cell density models, with numerical results verifying the link between cell density models and the free boundary dynamical models.

The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence
Wuyuan Jiang
2019, 24(7): 3037-3050 doi: 10.3934/dcdsb.2018298 +[Abstract](3258) +[HTML](544) +[PDF](417.92KB)

We consider a compound Poisson risk process perturbed by a Brownian motion through using a potential measure where the claim sizes depend on inter-claim times via the Farlie-Gumbel-Morgenstern copula. We derive an integro-differential equation with certain boundary conditions for the distribution of the maximum surplus before ruin. This distribution can be calculated through the probability that the surplus process attains a given level from the initial surplus without first falling below zero. The explicit expressions for this distribution are derived when the claim amounts are exponentially distributed.

Inverse parameter-dependent Preisach operator in thermo-piezoelectricity modeling
Pavel Krejčí and Giselle A. Monteiro
2019, 24(7): 3051-3066 doi: 10.3934/dcdsb.2018299 +[Abstract](3114) +[HTML](535) +[PDF](397.3KB)

Hysteresis is an important issue in modeling piezoelectric materials, for example, in applications to energy harvesting, where hysteresis losses may influence the efficiency of the process.The main problem in numerical simulations is the inversion of the underlying hysteresis operator.Moreover, hysteresis dissipation is accompanied with heat production, which in turn increases thetemperature of the device and may change its physical characteristics. More accurate models thereforehave to take the temperature dependence into account for a correct energy balance.We prove here that the classical Preisach operator with a fairly general parameter-dependenceadmits a Lipschitz continuous inverse in the space of right-continuous regulated functions, propose a time-discrete and memory-discrete inversion algorithm, and show that higher regularity of the inputs leads to a higher regularity of the output of the inverse.

Uniqueness of traveling front solutions for the Lotka-Volterra system in the weak competition case
Yang Wang and Xiong Li
2019, 24(7): 3067-3075 doi: 10.3934/dcdsb.2018300 +[Abstract](3294) +[HTML](682) +[PDF](350.21KB)

In this paper, we will prove the uniqueness of traveling front solutions with critical and noncritical speeds, connecting the origin and the positive equilibrium, for the classical competitive Lotka-Volterra system with diffusion in the weak competition, which partially answers the open problem presented by Tang and Fife in [17]. In fact, once these traveling front solutions have the same wave speed and the same asymptotic behavior at $ξ = ±∞$, they are unique up to translation.

On the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system
Shixing Li and Dongming Yan
2019, 24(7): 3077-3088 doi: 10.3934/dcdsb.2018301 +[Abstract](3241) +[HTML](552) +[PDF](347.86KB)

In this paper, the steady state bifurcation of the Cahn-Hilliard/Allen-Cahn system is investigated. By using the Lyapunov-Schmidt method, combining with the implicit function theorem, we prove that this system bifurcates from the trivial solution to the nontrivial solution branch as parameter crosses certain critical value. The expression of bifurcated solution is also obtained.

Attractivity of saturated equilibria for Lotka-Volterra systems with infinite delays and feedback controls
Yoshiaki Muroya and Teresa Faria
2019, 24(7): 3089-3114 doi: 10.3934/dcdsb.2018302 +[Abstract](3420) +[HTML](543) +[PDF](548.12KB)

In this paper, we apply a Lyapunov functional approach to Lotka-Volterra systems with infinite delays and feedback controls and establish that the feedback controls have no influence on the attractivity properties of a saturated equilibrium. This improves previous results by the authors and others, where, while feedback controls were used mostly to change the position of a unique saturated equilibrium, additional conditions involving the controls had to be assumed in order to preserve its global attractivity. The situation of partial extinction is further analysed, for which the original system is reduced to a lower dimensional one which maintains its global dynamics features.

Bounds on the Hausdorff dimension of random attractors for infinite-dimensional random dynamical systems on fractals
Markus Böhm and Björn Schmalfuss
2019, 24(7): 3115-3138 doi: 10.3934/dcdsb.2018303 +[Abstract](3465) +[HTML](590) +[PDF](537.96KB)

We consider a stochastic nonlinear evolution equation where the domain is given by a fractal set. The linear part of the equation is given by a Laplacian defined on the fractal. This equation generates a random dynamical system. The long time behavior is given by an attractor which has a finite Hausdorff dimension. We would like to reveal the connections between upper and lower estimates of this Hausdorff dimension and the geometry of the fractal. In particular, the parameter which determines these bounds is the spectral exponent of the fractal. Especially for the lower estimate we construct a local unstable random Lipschitz manifold.

Interlocked multi-node positive and negative feedback loops facilitate oscillations
Qingqing Li and Tianshou Zhou
2019, 24(7): 3139-3155 doi: 10.3934/dcdsb.2018304 +[Abstract](3825) +[HTML](676) +[PDF](1325.15KB)

Positive and negative feedback loops in biological regulatory networks appear often in a multi-node manner since regulatory processes are in general multi-step. Although it is well known that interlocked positive and negative feedback loops (iPNFLs) can generate sustained oscillations, how the number of nodes in each loop affects the oscillations remains elusive. By analyzing a model of iPNFLs with multiple nodes, we find that the node number of the negative loop mainly plays a role of amplifying oscillation amplitudes whereas that of the positive loop mainly plays a role of reducing oscillatory regions, both depending on the (competitive or noncompetitive) way of interaction between the two loops. We also find that given an iPNFL network of the same structure, the noncompetitive model is more likely to produce large-amplitude oscillations than the competitive model. These results not only indicate that multi-node iPNFLs are an effective mechanism of promoting oscillations but also are helpful for the design of synthetic oscillators.

On the backward uniqueness of the stochastic primitive equations with additive noise
Boling Guo and Guoli Zhou
2019, 24(7): 3157-3174 doi: 10.3934/dcdsb.2018305 +[Abstract](3327) +[HTML](683) +[PDF](463.3KB)

The previous works focus on the uniqueness for the initial-value problems of stochastic primitive equations. Uniqueness for the initial-value problems means that if the two initial conditions are the same, then the two solutions coincide with each other. However there is no work to answer what will happen to the solutions if the two initial conditions are different. This problem for the stochastic three dimensional primitive equations is addressed by the backward uniqueness established in this article. The backward uniqueness means that if two solutions intersect at time $t>0, $ then they are equal everywhere on the interval $(0, t).$ In other words, given two different initial-value conditions, the corresponding two solutions will never cross in the future. Hence this article can be viewed as a further study of the dependence of the solutions on the initial data.

Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates
Dong Deng and Ruikuan Liu
2019, 24(7): 3175-3193 doi: 10.3934/dcdsb.2018306 +[Abstract](3525) +[HTML](625) +[PDF](492.58KB)

The main aim of this paper is to study the bifurcation solutions associated with the spinor Bose-Einstein condensates. Based on the Principle of Hamilton Dynamics and the Principle of Lagrangian Dynamics, a general pattern formation equation for the spinor Bose-Einstein condensates is established. Moreover, three kinds of critical conditions for eigenvalues are obtained under spectrum analysis and the different external confining potentials. With the change of different external potentials, the different topological structures of bifurcation solutions for the spinor Bose-Einstein condensates system are derived from steady state bifurcation theory.

Regularity of solutions to time fractional diffusion equations
Binjie Li and Xiaoping Xie
2019, 24(7): 3195-3210 doi: 10.3934/dcdsb.2018340 +[Abstract](3457) +[HTML](412) +[PDF](401.46KB)

We derive some regularity estimates of the solution to a time fractional diffusion equation by using the Galerkin method. The regularity estimates partially unravel the singularity structure of the solution with respect to the time variable. We show that the regularity of the weak solution can be improved by subtracting some particular forms of singular functions.

Rate of attraction for a semilinear thermoelastic system with variable coefficients
Fágner D. Araruna, Flank D. M. Bezerra and Milton L. Oliveira
2019, 24(7): 3211-3226 doi: 10.3934/dcdsb.2018316 +[Abstract](3075) +[HTML](436) +[PDF](444.94KB)

The present paper is concerned with the problem of determining the rate of convergence of global attractors of the family of dissipative semilinear thermoelastic systems with variable coefficients

where \begin{document}$ l>0 $\end{document}, \begin{document}$ a_\varepsilon,\kappa_\varepsilon $\end{document} and \begin{document}$ m $\end{document} are regular enough functions, and the nonlinearity \begin{document}$ f $\end{document} is a continuously differentiable function satisfying suitable growth conditions. We show that rate of convergence, as \begin{document}$ \varepsilon\to0^+ $\end{document}, of the global attractors of these problems is proportional the distance of the coefficients \begin{document}$ \|a_\varepsilon-a_0\|_{L^p(0,l)}+\|\kappa_\varepsilon-\kappa_0\|_{L^p(0,l)} $\end{document} for some \begin{document}$ p\geq 2 $\end{document}.

A new model of groundwater flow within an unconfined aquifer: Application of Caputo-Fabrizio fractional derivative
Pierre Aime Feulefack, Jean Daniel Djida and Atangana Abdon
2019, 24(7): 3227-3247 doi: 10.3934/dcdsb.2018317 +[Abstract](3748) +[HTML](558) +[PDF](410.22KB)

In this paper, the groundwater flow equation within an unconfined aquifer is modified using the concept of new derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. Some properties and applications are given regarding the Caputo-Fabrizio fractional order derivative. The existence and the uniqueness of the solution of the modified groundwater flow equation within an unconfined aquifer is presented, the proof of the existence use the definition of Caputo-Fabrizio integral and the powerful fixed-point Theorem. A detailed analysis on the uniqueness is included. We perform on the numerical analysis on which the Crank-Nicolson scheme is used for discretisation. Then we present in particular the proof of the stability of the method, the proof combine the Fourier and Von Neumann stability analysis. A detailed analysis on the convergence is also achieved.

A blow-up criterion for three-dimensional compressible magnetohydrodynamic equations with variable viscosity
Xin Zhong
2019, 24(7): 3249-3264 doi: 10.3934/dcdsb.2018318 +[Abstract](3254) +[HTML](428) +[PDF](444.33KB)

We are concerned with the breakdown of strong solutions to the three-dimensional compressible magnetohydrodynamic equations with density-dependent viscosity. It is shown that for the initial density away from vacuum, the strong solution exists globally if the gradient of the velocity satisfies \begin{document}$ \|\nabla{\bf{u}}\|_{L^{2}(0,T;L^\infty)}<\infty $\end{document}. Our method relies upon the delicate energy estimates and elliptic estimates.

Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $
Linglong Du and Caixuan Ren
2019, 24(7): 3265-3280 doi: 10.3934/dcdsb.2018319 +[Abstract](3260) +[HTML](445) +[PDF](433.35KB)

In this paper, the asymptotic wave behavior of the solution for the nonlinear damped wave equation in \begin{document}$ \mathbb{R}^n_+ $\end{document} is investigated. We describe the double mechanism of the hyperbolic effect and the parabolic effect using the explicit functions. With the absorbing and radiative boundary condition, we show that the Green's function for the half space linear problem can be described in terms of the fundamental solution for the Cauchy problem and the reflected fundamental solution coupled with a boundary operator. Using the Duhamel's principle, we see that due to the fast decay property of the Green's function and the high nonlinearity, the pointwise decaying rate for the nonlinear solution and extra time decaying rate for its first order derivative are obtained.

On a beam model related to flight structures with nonlocal energy damping
Marcio A. Jorge Silva, Vando Narciso and André Vicente
2019, 24(7): 3281-3298 doi: 10.3934/dcdsb.2018320 +[Abstract](3433) +[HTML](471) +[PDF](490.37KB)

This paper deals with new results on existence, uniqueness and stability for a class of nonlinear beams arising in connection with nonlocal dissipative models for flight structures with energy damping first proposed by Balakrishnan-Taylor [2]. More precisely, the following \begin{document}$ n $\end{document}-dimensional model is addressed

where \begin{document}$ \Omega\subset \mathbb{R}^n $\end{document} is a bounded domain with smooth boundary, the coefficient of extensibility \begin{document}$ \kappa $\end{document} is nonnegative, the damping coefficient \begin{document}$ \gamma $\end{document} is positive and \begin{document}$ q\ge 1 $\end{document}. The nonlinear source \begin{document}$ f(u) $\end{document} can be seen as an external forcing term of lower order. Our main results feature global existence and uniqueness, polynomial stability and a non-exponential decay prospect.

Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients
Chunhong Li and Jiaowan Luo
2019, 24(7): 3299-3318 doi: 10.3934/dcdsb.2018321 +[Abstract](3670) +[HTML](505) +[PDF](450.24KB)

In this paper, by the use of martingale property and spectral decomposition theory, we investigate the stochastic invariance for neutral stochastic functional differential equations (NSFDEs) and provide necessary and sufficient conditions for the invariance of closed sets of \begin{document}$ R^{d} $\end{document} with non-Lipschitz coefficients. A pathwise asymptotic estimate example is given to illustrate the feasibility and effectiveness of obtained result.

On the global convergence of frequency synchronization for Kuramoto and Winfree oscillators
Chun-Hsiung Hsia, Chang-Yeol Jung and Bongsuk Kwon
2019, 24(7): 3319-3334 doi: 10.3934/dcdsb.2018322 +[Abstract](3648) +[HTML](470) +[PDF](697.0KB)

We investigate the collective behavior of synchrony for the Kuramoto and Winfree models. We first prove the global convergence of frequency synchronization for the non-identical Kuramoto system of three oscillators. It is shown that the uniform boundedness of the diameter of the phase functions implies complete frequency synchronization. In light of this, we show, under a suitable condition on the coupling strength and deviation of the intrinsic frequencies, that the diameter function of the phases is uniformly bounded. In a similar spirit, we also prove the global convergence of phase-locked synchronization for the Winfree model of \begin{document}$ N $\end{document} oscillators for \begin{document}$ N\ge2 $\end{document}.

Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation
Congming Peng and Dun Zhao
2019, 24(7): 3335-3356 doi: 10.3934/dcdsb.2018323 +[Abstract](3583) +[HTML](518) +[PDF](504.66KB)

In this article, we study the initial-value problem for inhomogeneous fractional nonlinear Schrödinger equation

where \begin{document}$ \frac{1}{2}<s<1, $\end{document} \begin{document}$ N\geq2 $\end{document} and \begin{document}$ \frac{2s-b}{N}\leq \sigma<\frac{2s-b}{N-2s} $\end{document}. We prove a Gagliardo-Nirenberg-type estimate and use it to establish sufficient conditions for global existence in \begin{document}$ H^{s}(\mathbb{R}^{N}) $\end{document}. In addition, we derive a localized Virial estimate for inhomogeneous fractional nonlinear Schrödinger equation in \begin{document}$ \mathbb{R}^{N} $\end{document}, which uses Balakrishnan's formula for the fractional Laplacian \begin{document}$ (-\Delta)^{s} $\end{document} from semigroup theory. By these estimates, we give the blowup criterion of radial solutions in \begin{document}$ \mathbb{R}^{N} $\end{document} for \begin{document}$ L^{2} $\end{document}-critical, \begin{document}$ L^{2} $\end{document}-supercritical and \begin{document}$ H^{s} $\end{document}-subcritical power.

Global existence and boundedness of solution of a parabolic-parabolic-ODE chemotaxis-haptotaxis model with (generalized) logistic source
Ling Liu and Jiashan Zheng
2019, 24(7): 3357-3377 doi: 10.3934/dcdsb.2018324 +[Abstract](3751) +[HTML](455) +[PDF](511.12KB)

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source

in a smooth bounded domain \begin{document}$ \mathbb{R}^N(N\geq1) $\end{document}, with parameter \begin{document}$ r>1 $\end{document}. the parameters \begin{document}$ a\in \mathbb{R}, \mu>0, \chi>0 $\end{document}. It is shown that when \begin{document}$ r>2 $\end{document}, or

the considered problem possesses a global classical solution which is bounded, where \begin{document}$ C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1} $\end{document} is a positive constant which is corresponding to the maximal sobolev regularity. Here \begin{document}$ C_{\beta} $\end{document} is a positive constant which depends on \begin{document}$ \xi $\end{document}, \begin{document}$ \|u_0\|_{C(\bar{\Omega})}, \ \|v_0\|_{W^{1, \infty}(\Omega)} $\end{document} and \begin{document}$ \|w_0\|_{L^\infty(\Omega)} $\end{document}. This result improves or extends previous results of several authors.

Weighted exponential stability of stochastic coupled systems on networks with delay driven by $ G $-Brownian motion
Yong Ren, Huijin Yang and Wensheng Yin
2019, 24(7): 3379-3393 doi: 10.3934/dcdsb.2018325 +[Abstract](3117) +[HTML](412) +[PDF](443.34KB)

This paper investigates the issue of weighted exponentially input to state stability (EISS, in short) of stochastic coupled systems on networks with time-varying delay driven by \begin{document}$ G $\end{document}-Brownian motion (\begin{document}$ G $\end{document}-SCSND, in short). Combining with inequality technique, \begin{document}$ k $\end{document}th vertex-Lyapunov functions and graph-theory, we establish the weighted EISS for \begin{document}$ G $\end{document}-SCSND. An application to the EISS for a class of stochastic coupled oscillators networks with control inputs driven by \begin{document}$ G $\end{document}-Brownian motion and an example are provided to illustrate the effectiveness of the obtained theory.

Pullback attractors for bi-spatial continuous random dynamical systems and application to stochastic fractional power dissipative equation on an unbounded domain
Wenqiang Zhao
2019, 24(7): 3395-3438 doi: 10.3934/dcdsb.2018326 +[Abstract](3103) +[HTML](495) +[PDF](680.33KB)

In this article, a notion of bi-spatial continuous random dynamical system is introduced between two completely separable metric spaces. It is show that roughly speaking, if such a random dynamical system is asymptotically compact and random absorbing in the initial space, then it admits a bi-spatial pullback attractor which is measurable in two spaces. The measurability of pullback attractor in the regular spaces is completely solved theoretically. As applications, we study the dynamical behaviour of solutions of the non-autonomous stochastic fractional power dissipative equation on \begin{document}$ \mathbb{R}^N $\end{document} with additive white noise and a polynomial-like growth nonlinearity of order \begin{document}$ p, p\geq2 $\end{document}. We prove that this equation generates a bi-spatial \begin{document}$ (L^2(\mathbb{R}^N), H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N)) $\end{document}-continuous random dynamical system, and the random dynamics for this system is captured by a bi-spatial pullback attractor which is compact and attracting in \begin{document}$ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N) $\end{document}, where \begin{document}$ H^s(\mathbb{R}^N) $\end{document} is a fractional Sobolev space with \begin{document}$ s\in(0,1) $\end{document}. Especially, the measurability of pullback attractor is individually derived by proving the the continuity of solutions in \begin{document}$ H^s(\mathbb{R}^N) $\end{document} and \begin{document}$ L^p(\mathbb{R}^N) $\end{document} with respect to the sample. A difference estimates approach, rather than the usual truncation estimate and spectral decomposition technique, is employed to overcome the loss of Sobolev compact embedding in \begin{document}$ H^s(\mathbb{R}^N)\cap L^p(\mathbb{R}^N),s\in(0,1),N\geq1 $\end{document}.

Complex dynamics in a discrete-time size-structured chemostat model with inhibitory kinetics
Dan Zhang, Xiaochun Cai and Lin Wang
2019, 24(7): 3439-3451 doi: 10.3934/dcdsb.2018327 +[Abstract](2884) +[HTML](439) +[PDF](1515.3KB)

An inhibitory uptake function is incorporated into the discrete, size-structured nonlinear chemostat model developed by Arino et al. (Journal of Mathematical Biology, 45(2002)). Different from the model with a monotonically increasing uptake function, we show that the inhibitory kinetics can induce very complex dynamics including stable equilibria, cycles and chaos (via the period-doubling cascade). In particular, when the nutrient concentration in the input feed to the chemostat \begin{document}$ S^0 $\end{document} is larger than the upper break-even concentration value \begin{document}$ \mu $\end{document}, the model exhibits three types of bistability allowing a stable equilibrium to coexist with another stable equilibrium, or a stable cycle or a chaotic attractor.

2020 Impact Factor: 1.327
5 Year Impact Factor: 1.492
2020 CiteScore: 2.2




Special Issues

Email Alert

[Back to Top]