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Discrete & Continuous Dynamical Systems - B

October 2021 , Volume 26 , Issue 10

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The threshold dynamics of a discrete-time echinococcosis transmission model
Cuicui Li, Lin Zhou, Zhidong Teng and Buyu Wen
2021, 26(10): 5197-5216 doi: 10.3934/dcdsb.2020339 +[Abstract](1080) +[HTML](393) +[PDF](2022.64KB)
Abstract:

In this paper, based on the transmission mechanism of echinococcosis in China, we propose a discrete-time dynamical model for the transmission of echinococcosis. The research results indicate that transmission dynamics of this discrete-time model are determined by basic reproduction number \begin{document}$ R_{0} $\end{document}. It is shown that when \begin{document}$ R_{0}\leq1 $\end{document} then the disease-free equilibrium is globally asymptotically stable and when \begin{document}$ R_{0}>1 $\end{document} then the model is permanent while the disease-free equilibrium is unstable. Finally, on the basis of the theoretical results established in this paper, we come up with some specific measures to control the transmission of echinococcosis.

A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition
Shenglan Xie, Maoan Han and Peng Zhu
2021, 26(10): 5217-5226 doi: 10.3934/dcdsb.2020340 +[Abstract](1020) +[HTML](347) +[PDF](1118.56KB)
Abstract:

A reliable and efficient a posteriori error estimator is presented for a weak Galerkin finite element method without stabilizer for the second order elliptic equation with mixed boundary conditions. The upper bound of the estimator is proved by Helmholtz decomposition technique and lower bound is hold naturally. The performance of the estimator is illustrated by numerical experiments.

Qualitative analysis of a simple tumor-immune system with time delay of tumor action
Jianquan Li, Xin Xie, Dian Zhang, Jia Li and Xiaolin Lin
2021, 26(10): 5227-5249 doi: 10.3934/dcdsb.2020341 +[Abstract](901) +[HTML](341) +[PDF](788.89KB)
Abstract:

In this paper, we propose a simple tumor-immune system model with time delay of tumor action, where two kinds of effects of the tumor cells (i.e. stimulation and neutralization) on the effector cells are considered. The local stability of the model is obtained by analyzing the characteristic equations of the model at the corresponding equilibria, the sufficient conditions on the global stability are found by applying the Fluctuation Lemma and constructing the different convergent sequences. The obtained results show that, compared to the results for the model without time delay, the time delay of tumor action can affect the stability of tumor equilibrium of the model as the stimulation effect of the tumor cells is strong enough, while the delay is harmless for the stability of tumor equilibrium under the neutralization of tumor cells. For the appropriate neutralization of tumor cells on effector cells, the bistability of the tumor free equilibrium and the stronger tumor equilibrium can appear. In the case of stimulation of tumor cells, the sufficiently large time delay can lead to the appearance of a stable periodic solution by Hopf bifurcation, and the numerical simulation illustrates that the amplitude of the periodic orbit increases with time delay. We also discuss the dependence of the tumor equilibrium and the time delay threshold, determining the stability of the tumor equilibrium, on tumor action. The related conditions determining dynamics of the model are expressed by certain formulae with biological meanings.

Complex oscillatory patterns near singular Hopf bifurcation in a two-timescale ecosystem
Susmita Sadhu
2021, 26(10): 5251-5279 doi: 10.3934/dcdsb.2020342 +[Abstract](1040) +[HTML](326) +[PDF](3279.12KB)
Abstract:

We consider an ecological model consisting of two species of predators competing for their common prey with explicit interference competition. With a proper rescaling, the model is portrayed as a singularly perturbed system with one fast (prey dynamics) and two slow variables (dynamics of the predators). The model exhibits a variety of rich and interesting dynamics, including, but not limited to mixed-mode oscillations (MMOs), featuring concatenation of small and large amplitude oscillations, relaxation oscillations and bistability between a semi-trivial equilibrium state and a coexistent oscillatory state. More interestingly, in a neighborhood of singular Hopf bifurcation, long lasting transient dynamics in the form of chaotic MMOs or relaxation oscillations are observed as the system approaches the periodic attractor born out of supercritical Hopf bifurcation or a semi-trivial equilibrium state respectively. The transient dynamics could persist for hundreds or thousands of generations before the ecosystem experiences a regime shift. The time series of population cycles with different types of irregular oscillations arising in this model stem from a biological realistic feature, namely, by the variation in the intraspecific competition amongst the predators. To explain these oscillations, we use bifurcation analysis and methods from geometric singular perturbation theory. The numerical continuation study reveals the rich bifurcation structure in the system, including the existence of codimension-two bifurcations such as fold-Hopf and generalized Hopf bifurcations.

Combined therapy for treating solid tumors with chemotherapy and angiogenic inhibitors
Adam Glick and Antonio Mastroberardino
2021, 26(10): 5281-5304 doi: 10.3934/dcdsb.2020343 +[Abstract](945) +[HTML](407) +[PDF](1046.86KB)
Abstract:

Anti-angiogenesis therapy has been an emerging cancer treatment which may be further combined with chemotherapy to enhance overall survival of cancer patients. In this paper, we investigate a system of nonlinear ordinary differential equations describing a microenvironment consisting of host cells, tumor cells, immune cells and endothelial cells while incorporating treatment combination with chemotherapy and anti-angiogenesis therapy. We perform a dynamical systems analysis demonstrating that our model is able to capture the three phases of cancer immunoediting: elimination, equilibrium, and escape. In addition, we present transcritical bifurcations for relevant parameter values that correspond to the progression from the elimination phase to the equilibrium phase. A range of medically useful tumor doubling times were simulated to determine how combined therapy affects the tumor microenvironment over the course of a 250 day treatment. This analysis found two additional bifurcation parameters that move the system of equations from the equilibrium phase to the elimination phase. We determine that the most important aspect of an effective therapy is the activation of the anti-tumor immune response.

Codimension one and two bifurcations in Cattaneo-Christov heat flux model
Zhouchao Wei, Wei Zhang, Irene Moroz and Nikolay V. Kuznetsov
2021, 26(10): 5305-5319 doi: 10.3934/dcdsb.2020344 +[Abstract](882) +[HTML](414) +[PDF](819.18KB)
Abstract:

Layek and Pati (Phys. Lett. A, 2017) studied a nonlinear system of five coupled equations, which describe thermal relaxation in Rayleigh-Benard convection of a Boussinesq fluid layer, heated from below. Here we return to that paper and use techniques from dynamical systems theory to analyse the codimension-one Hopf bifurcation and codimension-two double-zero Bogdanov-Takens bifurcation. We determine the stability of the bifurcating limit cycle, and produce an unfolding of the normal form for codimension-two bifurcation for the Layek and Pati's model.

Singular support of the global attractor for a damped BBM equation
Jianhua Huang, Yanbin Tang and Ming Wang
2021, 26(10): 5321-5335 doi: 10.3934/dcdsb.2020345 +[Abstract](896) +[HTML](354) +[PDF](371.32KB)
Abstract:

The singular support of the global attractor is introduced. It is shown that the singular support of the global attractor for a damped BBM equation equals to the singular support of the force term. This gives a delicate description of the local regularity, which roughly says that the attractor is smooth exactly where the force is smooth.

Qualitative analysis of a generalized Nosé-Hoover oscillator
Qianqian Han and Xiao-Song Yang
2021, 26(10): 5337-5354 doi: 10.3934/dcdsb.2020346 +[Abstract](670) +[HTML](313) +[PDF](366.69KB)
Abstract:

In this paper, we analyze the qualitative dynamics of a generalized Nosé-Hoover oscillator with two parameters varying in certain scope. We show that if a solution of this oscillator will not tend to the invariant manifold \begin{document}$ \{(x,y,z)\in \mathbb R^3|x = 0,y = 0\} $\end{document}, it must pass through the plane \begin{document}$ z = 0 $\end{document} infinite times. Especially, every invariant set of this oscillator must have intersection with the plane \begin{document}$ z = 0 $\end{document}. In addition, we show that if a solution is quasiperiodic, it must pass through at least five quadrants of \begin{document}$ \mathbb R^3 $\end{document}.

Optimal control strategies for an online game addiction model with low and high risk exposure
Youming Guo and Tingting Li
2021, 26(10): 5355-5382 doi: 10.3934/dcdsb.2020347 +[Abstract](1015) +[HTML](339) +[PDF](745.33KB)
Abstract:

In this paper, we establish a new online game addiction model with low and high risk exposure. With the help of the next generation matrix, the basic reproduction number \begin{document}$ R_{0} $\end{document} is obtained. By constructing a suitable Lyapunov function, the equilibria of the model are Globally Asymptotically Stable. We use the optimal control theory to study the optimal solution problem with three kinds of control measures (isolation, education and treatment) and get the expression of optimal control. In the simulation, we first verify the Globally Asymptotical Stability of Disease-Free Equilibrium and Endemic Equilibrium, and obtain that the different trajectories with different initial values converges to the equilibria. Then the simulations of nine control strategies are obtained by forward-backward sweep method, and they are compared with the situation of without control respectively. The results show that we should implement the three kinds of control measures according to the optimal control strategy at the same time, which can effectively reduce the situation of game addiction.

Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions
Zhilei Liang and Jiangyu Shuai
2021, 26(10): 5383-5405 doi: 10.3934/dcdsb.2020348 +[Abstract](780) +[HTML](302) +[PDF](467.21KB)
Abstract:

We study the Cauchy problem for the equations describing a viscous compressible and heat-conductive fluid in two dimensions. By imposing a weight function to initial density to deal with Sobolev embedding in critical space, and constructing an ad-hoc truncation to control the quadratic nonlinearity appeared in energy equation, we establish the local in time existence of unique strong solution with large initial data. The vacuum state at infinity or the compactly supported density is permitted. Moreover, we provide a different approach and slightly improve the weighted \begin{document}$ L^{p} $\end{document} estimates in [19,Theorem B.1].

Effective reduction of a three-dimensional circadian oscillator model
Shuang Chen, Jinqiao Duan and Ji Li
2021, 26(10): 5407-5419 doi: 10.3934/dcdsb.2020349 +[Abstract](861) +[HTML](299) +[PDF](745.43KB)
Abstract:

We investigate the dynamics of a three-dimensional system modeling a molecular mechanism for the circadian rhythm in Drosophila. We first prove the existence of a compact attractor in the region with biological meaning. Under the assumption that the dimerization reactions are fast, in this attractor we reduce the three-dimensional system to a simpler two-dimensional system on the persistent normally hyperbolic slow manifold.

Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise
Leanne Dong
2021, 26(10): 5421-5448 doi: 10.3934/dcdsb.2020352 +[Abstract](955) +[HTML](318) +[PDF](484.9KB)
Abstract:

In this paper we prove that the stochastic Navier-Stokes equations with stable Lévy noise generate a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Lévy noise (finite dimensional). We also deduce the existence of a Feller Markov Invariant Measure.

Norm inflation for the Boussinesq system
Zongyuan Li and Weinan Wang
2021, 26(10): 5449-5463 doi: 10.3934/dcdsb.2020353 +[Abstract](817) +[HTML](293) +[PDF](340.36KB)
Abstract:

We prove the norm inflation phenomena for the Boussinesq system on \begin{document}$ \mathbb T^3 $\end{document}. For arbitrarily small initial data \begin{document}$ (u_0,\rho_0) $\end{document} in the negative-order Besov spaces \begin{document}$ \dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty} $\end{document}, the solution can become arbitrarily large in a short time. Such largeness can be detected in \begin{document}$ \rho $\end{document} in Besov spaces of any negative order: \begin{document}$ \dot{B}^{-s}_{\infty, \infty} $\end{document} for any \begin{document}$ s>0 $\end{document}.

On an initial and final value problem for fractional nonclassical diffusion equations of Kirchhoff type
Nguyen Huy Tuan
2021, 26(10): 5465-5494 doi: 10.3934/dcdsb.2020354 +[Abstract](976) +[HTML](302) +[PDF](718.84KB)
Abstract:

We study for nonlinear Kirchhoff's model of pseudo parabolic type by considering its two different problems.

\begin{document}$ \bullet $\end{document} For initial value problem, we obtain the results on the existence and regularity of solutions. Moreover, we also prove that the solutions \begin{document}$ u $\end{document} corresponding with \begin{document}$ \beta < 1 $\end{document} of the problem convergence to \begin{document}$ u $\end{document} for \begin{document}$ \beta = 1 $\end{document}.

\begin{document}$ \bullet $\end{document} For final value problem, we show that the ill-posed property in the sense of Hadamard is occurring. Using the Fourier truncation method to regularize the problem. We establish some stability estimates in the \begin{document}$ H^1 $\end{document} and \begin{document}$ L^p $\end{document} norms under some a-priori conditions on the sought solution.

High-order numerical method for two-dimensional Riesz space fractional advection-dispersion equation
Abdollah Borhanifar, Maria Alessandra Ragusa and Sohrab Valizadeh
2021, 26(10): 5495-5508 doi: 10.3934/dcdsb.2020355 +[Abstract](842) +[HTML](292) +[PDF](382.47KB)
Abstract:

In this paper, by combining of fractional centered difference approach with alternating direction implicit method, we introduce a mixed difference method for solving two-dimensional Riesz space fractional advection-dispersion equation. The proposed method is a fourth order centered difference operator in spatial directions and second order Crank-Nicolson method in temporal direction. By reviewing the consistency and stability of the method, the convergence of the proposed method is achieved. Several numerical examples are considered aiming to demonstrate the validity and applicability of the proposed technique.

A diffusive weak Allee effect model with U-shaped emigration and matrix hostility
Nalin Fonseka, Jerome Goddard II, Ratnasingham Shivaji and Byungjae Son
2021, 26(10): 5509-5517 doi: 10.3934/dcdsb.2020356 +[Abstract](755) +[HTML](322) +[PDF](833.91KB)
Abstract:

We study positive solutions to steady state reaction diffusion equations of the form:

where \begin{document}$ u $\end{document} is the population density, \begin{document}$ f(u) = \frac{1}{a}u(u+a)(1-u) $\end{document} represents a weak Allee effect type growth of the population with \begin{document}$ a\in (0,1) $\end{document}, \begin{document}$ \alpha(u) $\end{document} is the probability of the population staying in the habitat \begin{document}$ \Omega $\end{document} when it reaches the boundary, and positive parameters \begin{document}$ \lambda $\end{document} and \begin{document}$ \gamma $\end{document} represent the domain scaling and effective exterior matrix hostility, respectively. In particular, we analyze the case when \begin{document}$ \alpha(s) = \frac{1}{[1+(A - s)^2 + \epsilon]} $\end{document} for all \begin{document}$ s \in [0,1] $\end{document}, where \begin{document}$ A\in (0,1) $\end{document} and \begin{document}$ \epsilon\geq 0 $\end{document}. In this case \begin{document}$ 1-\alpha(s) $\end{document} represents a U-shaped relationship between density and emigration. Existence, nonexistence, and multiplicity results for this model are established via the method of sub-super solutions.

The spatially heterogeneous diffusive rabies model and its shadow system
Yuxin Zhang
2021, 26(10): 5519-5549 doi: 10.3934/dcdsb.2020357 +[Abstract](894) +[HTML](308) +[PDF](429.11KB)
Abstract:

In this paper, we consider a class of spatially heterogeneous reaction diffusion rabies model which was used to describe population dynamics of the rabies epidemic disease observed in Europe. The dynamics of both the original non-degenerate reaction-diffusion system and its corresponding shadow system are investigated in great details. Firstly, we prove that under certain conditions, the in-time solutions of both the original non-degenerate reaction-diffusion system and its shadow system exist globally and remain uniformly bounded. Secondly, we are capable of showing that the shadow system is the nice approximations for the original non-degenerate reaction-diffusion system when the diffusion rate \begin{document}$ d_R $\end{document} of the infectious rabid individuals (R) is sufficiently large. This implies that the dynamics of the shadow system can say as much as possible about the dynamics of the original system when \begin{document}$ d_R $\end{document} is sufficiently large. Finally, we characterize the basic reproduction number for the shadow system, and study the stability/instability of the disease-free steady state.

The effect of surface pattern property on the advancing motion of three-dimensional droplets
Hua Zhong, Xiaolin Fan and Shuyu Sun
2021, 26(10): 5551-5566 doi: 10.3934/dcdsb.2020366 +[Abstract](778) +[HTML](327) +[PDF](3788.07KB)
Abstract:

We investigate numerically the advancing motion of 3D droplets spreading on physically flat chemically heterogeneous surfaces with periodic structures. We use the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions to model the motion of droplets. Based on a convex splitting scheme, we have done numerical simulations and compared the results between different surface patterns quantitatively. We study the effect of pattern property on the advancing motion of three phase contact lines, the critical volume at the contact line jump and the effective advancing angles. By increasing the volume of droplet slowly on heterogeneous surfaces with different pattern property, we find that the advancing contact line approaches an equiangular octagon for the patterned surface with periodic squares separated by channels and approaches a regular hexagon for the patterned surface with periodic circles in regular hexagonal arrays. The shape of three-phase contact line is much more determined by the macro structure of the pattern than the micro structure of the pattern in each period.

Strong convergence rates for markovian representations of fractional processes
Philipp Harms
2021, 26(10): 5567-5579 doi: 10.3934/dcdsb.2020367 +[Abstract](747) +[HTML](319) +[PDF](4161.45KB)
Abstract:

Many fractional processes can be represented as an integral over a family of Ornstein–Uhlenbeck processes. This representation naturally lends itself to numerical discretizations, which are shown in this paper to have strong convergence rates of arbitrarily high polynomial order. This explains the potential, but also some limitations of such representations as the basis of Monte Carlo schemes for fractional volatility models such as the rough Bergomi model.

Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve
Huanhuan Tian and Maoan Han
2021, 26(10): 5581-5599 doi: 10.3934/dcdsb.2020368 +[Abstract](834) +[HTML](289) +[PDF](1105.89KB)
Abstract:

This paper deals with the number of limit cycles for planar piecewise smooth near-Hamiltonian or near-integrable systems with a switching curve. The main task is to establish a so-called first order Melnikov function which plays a crucial role in the study of the number of limit cycles bifurcated from a periodic annulus. We use the function to study Hopf bifurcation when the periodic annulus has an elementary center as its boundary. As applications, using the first order Melnikov function, we consider the number of limit cycles bifurcated from the periodic annulus of a linear center under piecewise linear polynomial perturbations with three kinds of quadratic switching curves. And we obtain three limit cycles for each case.

Mathematical analysis of a three-tiered food-web in the chemostat
Sarra Nouaoura, Radhouane Fekih-Salem, Nahla Abdellatif and Tewfik Sari
2021, 26(10): 5601-5625 doi: 10.3934/dcdsb.2020369 +[Abstract](829) +[HTML](309) +[PDF](1153.48KB)
Abstract:

A mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web is investigated. The model is a six-dimensional system of ordinary differential equations. In our study, the phenol and the hydrogen inflowing concentrations are taken into account as well as the maintenance terms. The case of a large class of growth kinetics is considered, instead of specific kinetics. We show that the system can have up to eight types of steady states and we analytically determine the necessary and sufficient conditions for their existence according to the operating parameters. In the particular case without maintenance, the local stability conditions of all steady states are determined. The bifurcation diagram shows the behavior of the process by varying the concentration of influent chlorophenol as the bifurcating parameter. It shows that the system exhibits a bi-stability where the positive steady state can lose stability undergoing a supercritical Hopf bifurcation with the emergence of a stable limit cycle.

Stable transition layers in an unbalanced bistable equation
Maicon Sônego
2021, 26(10): 5627-5640 doi: 10.3934/dcdsb.2020370 +[Abstract](857) +[HTML](279) +[PDF](384.44KB)
Abstract:

In this paper we are concerned with the existence of stable stationary solutions for the problem \begin{document}$ u_t = \epsilon^2(k_1^2(x) u_x)_x+k_2^2(x)g(u,x) $\end{document}, \begin{document}$ (t,x)\in\mathbb{R}^+\times (0,1) $\end{document} subject to Neumann boundary condition. We suppose that \begin{document}$ k_1,k_2\in C^1(0,1) $\end{document} are positive functions and \begin{document}$ g $\end{document} is an unbalanced bistable function. We prove the existence of a family of stable stationary solutions developing internal transition layers in a specific sub-interval of \begin{document}$ (0,1) $\end{document}. For this, we provide a general variational method inspired by the \begin{document}$ \Gamma $\end{document}-convergence theory.

Permanence, extinction and periodic solution of a stochastic single-species model with Lévy noises
Jiangtao Yang
2021, 26(10): 5641-5660 doi: 10.3934/dcdsb.2020371 +[Abstract](889) +[HTML](296) +[PDF](630.87KB)
Abstract:

This paper considers a stochastic single-species model with Lévy noises and time periodic coefficients. By Lyapunov functions and stochastic estimates, the threshold conditions between the time-average persistence in probability and extinction for the model are derived where Lévy noises play an important role in persistence and extinction of populations. It is shown that the time-average persistence in probability of the model implies the existence and uniqueness of positive periodic solution and the existence and uniqueness of periodic measure of the model. An example and its numerical simulations are given to verify the effectiveness of the theoretical results.

Parameter identification on Abelian integrals to achieve Chebyshev property
Xianbo Sun, Zhanbo Chen and Pei Yu
2021, 26(10): 5661-5679 doi: 10.3934/dcdsb.2020375 +[Abstract](648) +[HTML](278) +[PDF](539.88KB)
Abstract:

Chebyshev criterion is a powerful tool in the study of limit cycle bifurcations in dynamical systems based on Abelian integrals, but it is difficult when the Abelian integrals involve parameters. In this paper, we consider the Abelian integrals on the periodic annuli of a Hamiltonian with one parameter, arising from the generalized Liénard system, and identify the parameter values such that the Abelian integrals have Chebyshev property. In particular, the bounds on the number of zeros of the Abelian integrals are derived for different parameter intervals. The main mathematical tools are transformations and polynomial boundary theory, which overcome the difficulties in symbolic computations and analysis, arising from large parametric-semi-algebraic systems.

Dynamics of non-autonomous fractional reaction-diffusion equations on $ \mathbb{R}^{N} $ driven by multiplicative noise
Kaixuan Zhu, Ji Li, Yongqin Xie and Mingji Zhang
2021, 26(10): 5681-5705 doi: 10.3934/dcdsb.2020376 +[Abstract](849) +[HTML](364) +[PDF](479.67KB)
Abstract:

We examine the asymptotic behavior of the non-autonomous non-local fractional stochastic reaction-diffusion equations on \begin{document}$ \mathbb{R}^{N} $\end{document} with the nonlinearity \begin{document}$ f $\end{document} satisfying the polynomial growth of arbitrary order \begin{document}$ p-1 $\end{document} \begin{document}$ (p\geq2) $\end{document}. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time. Then, we prove that the solution process is continuous from \begin{document}$ L^{2}(\mathbb{R}^{N}) $\end{document} to \begin{document}$ H^{s}(\mathbb{R}^{N}) $\end{document} with respect to initial data for \begin{document}$ s\in(0, 1) $\end{document}. As an application of the higher-order integrability and the continuity, we obtain the pullback \begin{document}$ \mathcal{D} $\end{document}-random attractors in \begin{document}$ L^{p}(\mathbb{R}^{N}) $\end{document} and \begin{document}$ H^{s}(\mathbb{R}^{N}) $\end{document}, respectively. This is a natural and necessary extension of current existing results to further understand the dynamics of the underlying problem.

Survival analysis for tumor growth model with stochastic perturbation
Dongxi Li, Ni Zhang, Ming Yan and Yanya Xing
2021, 26(10): 5707-5722 doi: 10.3934/dcdsb.2021041 +[Abstract](766) +[HTML](289) +[PDF](442.44KB)
Abstract:

In this paper, we investigate the dynamical behavior of extinction and survival in tumor growth model with immunization under stochastic perturbation. Firstly, the model describing the growth of cancer cells monitored by immune cells is established. Then, the steady probability distribution of tumor cells for different noise intensities and immune parameter intensities, and necessary conditions for extinction and different survival of cancer cells are obtained by numerical and theoretical method. Besides, it is found that the extinction and survival of cancer cells rely on the state of immunization and noise. Finally, stochastic simulations are taken to test the theoretical analytical results. The results of our work are beneficial to discover the evolution mechanism and design effective immunotherapy of tumor.

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

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