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Discrete and Continuous Dynamical Systems - B

July 2022 , Volume 27 , Issue 7

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Global dynamics and bifurcations in a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate
Qin Pan, Jicai Huang and Qihua Huang
2022, 27(7): 3533-3561 doi: 10.3934/dcdsb.2021195 +[Abstract](983) +[HTML](376) +[PDF](1818.72KB)
Abstract:

In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by \begin{document}$ I $\end{document}) exceeds a certain level, the incidence rate is a decreasing function with respect to \begin{document}$ I $\end{document}. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with \begin{document}$ I $\end{document} until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken.Our analysis indicates that there exists a critical value \begin{document}$ \widetilde{I_0} $\end{document} \begin{document}$ ( = \frac{b}{d}) $\end{document} for the infective level \begin{document}$ I_0 $\end{document} at which the health care system reaches its capacity such that:(i) When \begin{document}$ I_0 \geq \widetilde{I_0} $\end{document}, the transmission dynamics of the model is determined by the basic reproduction number \begin{document}$ R_0 $\end{document}: \begin{document}$ R_0 = 1 $\end{document} separates disease persistence from disease eradication. (ii) When \begin{document}$ I_0 < \widetilde{I_0} $\end{document}, the model exhibits very rich dynamics and bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.

Existence and continuity of global attractors for ternary mixtures of solids
Mirelson M. Freitas, Anderson J. A. Ramos, Baowei Feng, Mauro L. Santos and Helen C. M. Rodrigues
2022, 27(7): 3563-3583 doi: 10.3934/dcdsb.2021196 +[Abstract](742) +[HTML](465) +[PDF](418.16KB)
Abstract:

In this paper, we study the long-time dynamics of a system modelinga mixture of three interacting continua with nonlinear damping, sources terms and subjected to small perturbations of autonomousexternal forces with a parameter \begin{document}$ \epsilon $\end{document}, inspired by the modelstudied by Dell' Oro and Rivera [12]. We establish astabilizability estimate for the associated gradient dynamicalsystem, which as a consequence, implies the existence of a compactglobal attractor with finite fractal dimension andexponential attractors. This estimate is establishedindependent of the parameter \begin{document}$ \epsilon\in[0,1] $\end{document}. We also prove thesmoothness of global attractors independent of the parameter\begin{document}$ \epsilon\in[0,1] $\end{document}. Moreover, we show that the family of globalattractors is continuous with respect to the parameter \begin{document}$ \epsilon $\end{document} ona residual dense set \begin{document}$ I_*\subset[0,1] $\end{document} in the same sense proposed inHoang et al. [15].

Practical partial stability of time-varying systems
Abdelfettah Hamzaoui, Nizar Hadj Taieb and Mohamed Ali Hammami
2022, 27(7): 3585-3603 doi: 10.3934/dcdsb.2021197 +[Abstract](870) +[HTML](343) +[PDF](364.93KB)
Abstract:

In this paper we investigate the practical asymptotic and exponential partial stability of time-varying nonlinear systems. We derive some sufficient conditions that guarantee practical partial stability of perturbed systems using Lyapunov's theory where a converse theorem is presented. Therefore, we generalize some works which are already made in the literature. Furthermore, we present some illustrative examples to verify the effectiveness of the proposed methods.

Positive solutions of iterative functional differential equations and application to mixed-type functional differential equations
Jun Zhou and Jun Shen
2022, 27(7): 3605-3624 doi: 10.3934/dcdsb.2021198 +[Abstract](799) +[HTML](340) +[PDF](362.29KB)
Abstract:

In this paper we consider the existence, uniqueness, boundedness and continuous dependence on initial data of positive solutions for the general iterative functional differential equation \begin{document}$ \dot{x}(t) = f(t,x(t),x^{[2]}(t),...,x^{[n]}(t)). $\end{document} As \begin{document}$ n = 2 $\end{document}, this equation can be regarded as a mixed-type functional differential equation with state-dependence \begin{document}$ \dot{x}(t) = f(t,x(t),x(T(t,x(t)))) $\end{document} of a special form but, being a nonlinear operator, \begin{document}$ n $\end{document}-th order iteration makes more difficulties in estimation than usual state-dependence. Then we apply our results to the existence, uniqueness, boundedness, asymptotics and continuous dependence of solutions for the mixed-type functional differential equation. Finally, we present two concrete examples to show the boundedness and asymptotics of solutions to these two types of equations respectively.

Positive solutions of a diffusive two competitive species model with saturation
Aung Zaw Myint
2022, 27(7): 3625-3641 doi: 10.3934/dcdsb.2021199 +[Abstract](848) +[HTML](339) +[PDF](495.5KB)
Abstract:

In this paper, the positive solutions of a diffusive two competitive species model with Bazykin functional response are investigated. We give the a priori estimates and compute the fixed point indices of trivial and semi-trivial solutions. And obtain the existence of solution and demonstrate the bifurcation of a coexistence state emanating from semi-trivial solutions. Finally, multiplicity and stability results are presented.

Effects of fear and anti-predator response in a discrete system with delay
Ritwick Banerjee, Pritha Das and Debasis Mukherjee
2022, 27(7): 3643-3661 doi: 10.3934/dcdsb.2021200 +[Abstract](841) +[HTML](404) +[PDF](906.58KB)
Abstract:

In this paper a discrete-time two prey one predator model is considered with delay and Holling Type-Ⅲ functional response. The cost of fear of predation and the effect of anti-predator behavior of the prey is incorporated in the model, coupled with inter-specific competition among the prey species and intra-specific competition within the predator. The conditions for existence of the equilibrium points are obtained. We further derive the sufficient conditions for permanence and global stability of the co-existence equilibrium point. It is observed that the effect of fear induces stability in the system by eliminating the periodic solutions. On the other hand the effect of anti-predator behavior plays a major role in de-stabilizing the system by giving rise to predator-prey oscillations. Finally, several numerical simulations are performed which support our analytical findings.

The truncated Milstein method for super-linear stochastic differential equations with Markovian switching
Weijun Zhan, Qian Guo and Yuhao Cong
2022, 27(7): 3663-3682 doi: 10.3934/dcdsb.2021201 +[Abstract](774) +[HTML](363) +[PDF](379.96KB)
Abstract:

In this paper, to approximate the super-linear stochastic differential equations modulated by a Markov chain, we investigate a truncated Milstein method with convergence order 1 in the mean-square sense. Under Khasminskii-type conditions, we establish the convergence result by employing a relationship between local and global errors. Finally, we confirm the convergence rate by a numerical example.

Spatiotemporal dynamics in a diffusive Holling-Tanner model near codimension-two bifurcations
Daifeng Duan, Ben Niu and Junjie Wei
2022, 27(7): 3683-3706 doi: 10.3934/dcdsb.2021202 +[Abstract](728) +[HTML](357) +[PDF](3492.59KB)
Abstract:

We investigate spatiotemporal patterns near the Turing-Hopf and double Hopf bifurcations in a diffusive Holling-Tanner model on a one- dimensional spatial domain. Local and global stability of the positive constant steady state for the non-delayed system is studied. Introducing the generation time delay in prey growth, we discuss the existence of Turing-Hopf and double Hopf bifurcations and give the explicit dynamical classification near these bifurcation points. Finally, we obtain the complicated dynamics, including periodic oscillations, quasi-periodic oscillations on a three-dimensional torus, the coexistence of two stable nonconstant steady states, the coexistence of two spatially inhomogeneous periodic solutions, and strange attractors.

Counterexamples to local Lipschitz and local Hölder continuity with respect to the initial values for additive noise driven stochastic differential equations with smooth drift coefficient functions with at most polynomially growing derivatives
Arnulf Jentzen, Benno Kuckuck, Thomas Müller-Gronbach and Larisa Yaroslavtseva
2022, 27(7): 3707-3724 doi: 10.3934/dcdsb.2021203 +[Abstract](776) +[HTML](524) +[PDF](343.62KB)
Abstract:

In the recent article [A. Jentzen, B. Kuckuck, T. Müller-Gronbach, and L. Yaroslavtseva, J. Math. Anal. Appl. 502, 2 (2021)] it has been proved that the solutions to every additive noise driven stochastic differential equation (SDE) which has a drift coefficient function with at most polynomially growing first order partial derivatives and which admits a Lyapunov-type condition (ensuring the existence of a unique solution to the SDE) depend in the strong sense in a logarithmically Hölder continuous way on their initial values. One might then wonder whether this result can be sharpened and whether in fact, SDEs from this class necessarily have solutions which depend in the strong sense locally Lipschitz continuously on their initial value. The key contribution of this article is to establish that this is not the case. More precisely, we supply a family of examples of additive noise driven SDEs, which have smooth drift coefficient functions with at most polynomially growing derivatives and whose solutions do not depend in the strong sense on their initial value in a locally Lipschitz continuous, nor even in a locally Hölder continuous way.

Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise
Tian Zhang and Chuanhou Gao
2022, 27(7): 3725-3747 doi: 10.3934/dcdsb.2021204 +[Abstract](1123) +[HTML](428) +[PDF](914.67KB)
Abstract:

This paper focuses on the \begin{document}$ p $\end{document}th moment and almost sure stability with general decay rate (including exponential decay, polynomial decay, and logarithmic decay) of highly nonlinear hybrid neutral stochastic pantograph differential equations driven by L\begin{document}$ \acute{e} $\end{document}vy noise (NSPDEs-LN). The crucial techniques used are the Lyapunov functions and the nonnegative semi-martingale convergence theorem. Simultaneously, the diffusion operators are permitted to be controlled by several additional functions with time-varying coefficients, which can be applied to a broad class of the non-autonomous hybrid NSPDEs-LN with highly nonlinear coefficients. Besides, \begin{document}$ H_\infty $\end{document} stability and the almost sure asymptotic stability are also concerned. Finally, two examples are offered to illustrate the validity of the obtained theory.

Well-posedness of the initial-boundary value problem for the fourth-order nonlinear Schrödinger equation
Boling Guo and Jun Wu
2022, 27(7): 3749-3778 doi: 10.3934/dcdsb.2021205 +[Abstract](900) +[HTML](411) +[PDF](463.81KB)
Abstract:

The main purpose of this paper is to study local regularity properties of the fourth-order nonlinear Schrödinger equations on the half line. We prove the local existence, uniqueness, and continuous dependence on initial data in low regularity Sobolev spaces. We also obtain the nonlinear smoothing property: the nonlinear part of the solution on the half line is smoother than the initial data.

Predator-prey interactions under fear effect and multiple foraging strategies
Susmita Halder, Joydeb Bhattacharyya and Samares Pal
2022, 27(7): 3779-3810 doi: 10.3934/dcdsb.2021206 +[Abstract](891) +[HTML](421) +[PDF](3443.02KB)
Abstract:

We propose and analyze the effects of a generalist predator-driven fear effect on a prey population by considering a modified Leslie-Gower predator-prey model. We assume that the prey population suffers from reduced fecundity due to the fear of predators. We investigate the predator-prey dynamics by incorporating linear, Holling type Ⅱ and Holling type Ⅲ foraging strategies of the generalist predator. As a control strategy, we have considered density-dependent harvesting of the organisms in the system. We show that the systems with linear and Holling type Ⅲ foraging exhibit transcritical bifurcation, whereas the system with Holling type Ⅱ foraging has a much more complex dynamics with transcritical, saddle-node, and Hopf bifurcations. It is observed that the prey population in the system with Holling type Ⅲ foraging of the predator gets severely affected by the predation-driven fear effect in comparison with the same with linear and Holling type Ⅱ foraging rates of the predator. Our model simulation results show that an increase in the harvesting rate of the predator is a viable strategy in recovering the prey population.

Stabilization for hybrid stochastic differential equations driven by Lévy noise via periodically intermittent control
Yong Ren and Qi Zhang
2022, 27(7): 3811-3829 doi: 10.3934/dcdsb.2021207 +[Abstract](709) +[HTML](401) +[PDF](524.0KB)
Abstract:

In this work, the issue of stabilization for a class of continuous-time hybrid stochastic systems with Lévy noise (HLSDEs, in short) is explored by using periodic intermittent control. As for the unstable HLSDEs, we design a periodic intermittent controller. The main idea is to compare the controlled system with a stabilized one with a periodic intermittent drift coefficient, which enables us to use the existing stability results on the HLSDEs. An illustrative example is proposed to show the feasibility of the obtained result.

An analysis approach to permanence of a delay differential equations model of microorganism flocculation
Songbai Guo, Jing-An Cui and Wanbiao Ma
2022, 27(7): 3831-3844 doi: 10.3934/dcdsb.2021208 +[Abstract](933) +[HTML](372) +[PDF](306.4KB)
Abstract:

In this paper, we develop a delay differential equations model of microorganism flocculation with general monotonic functional responses, and then study the permanence of this model, which can ensure the sustainability of the collection of microorganisms. For a general differential system, the existence of a positive equilibrium can be obtained with the help of the persistence theory, whereas we give the existence conditions of a positive equilibrium by using the implicit function theorem. Then to obtain an explicit formula for the ultimate lower bound of microorganism concentration, we propose a general analysis method, which is different from the traditional approaches in persistence theory and also extends the analysis techniques of existing related works.

On oscillations to a 2D age-dependent predation equations characterizing Beddington-DeAngelis type schemes
Peng Yang and Yuanshi Wang
2022, 27(7): 3845-3895 doi: 10.3934/dcdsb.2021209 +[Abstract](650) +[HTML](299) +[PDF](955.89KB)
Abstract:

In this study, a 2D age-dependent predation equations characterizing Beddington\begin{document}$ - $\end{document}DeAngelis type schemes are established to investigate the evolutionary dynamics of population, in which the predator is selected to be depicted with an age structure and its fertility function is assumed to be a step function. The dynamic behaviors of the equations are derived from the integrated semigroup method, the Hopf bifurcation theorem, the center manifold reduction and normal form theory of semilinear equations with non-dense domain. It turns out that the equations appear the oscillation phenomenon via Hopf bifurcation (positive equilibrium age distribution lose its stability and give rise to periodic solutions), as the bifurcation parameter moves across certain threshold values. Additionally, the explicit expressions are offered to determine the properties of Hopf bifurcation (the direction the Hopf bifurcation and the stability of the bifurcating periodic solutions). This technique can also be employed to other epidemic and ecological equations. Eventually, some numerical simulations and conclusions are executed to validating the major results of this work.

On an upper bound for the spreading speed
Mohammed Mesk and Ali Moussaoui
2022, 27(7): 3897-3912 doi: 10.3934/dcdsb.2021210 +[Abstract](1138) +[HTML](325) +[PDF](486.29KB)
Abstract:

In this paper, we use the exponential transform to give a unified formal upper bound for the asymptotic rate of spread of a population propagating in a one dimensional habitat. We show through examples how this upper bound can be obtained directly for discrete and continuous time models. This upper bound has the form \begin{document}$ \min_{s>0} \ln (\rho(s))/s $\end{document} and coincides with the speeds of several models found in the literature.

Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms
Willian Cintra, Carlos Alberto dos Santos and Jiazheng Zhou
2022, 27(7): 3913-3931 doi: 10.3934/dcdsb.2021211 +[Abstract](773) +[HTML](355) +[PDF](374.64KB)
Abstract:

In this paper, we present results about existence and non-existence of coexistence states for a reaction-diffusion predator-prey model with the two species living in a bounded region with inhospitable boundary and Holling type II functional response. The predator is a specialist and presents self-diffusion and cross-diffusion behavior. We show the existence of coexistence states by combining global bifurcation theory with the method of sub- and supersolutions.

Uniform attractors for nonautonomous reaction-diffusion equations with the nonlinearity in a larger symbol space
Xiangming Zhu and Chengkui Zhong
2022, 27(7): 3933-3945 doi: 10.3934/dcdsb.2021212 +[Abstract](701) +[HTML](307) +[PDF](342.41KB)
Abstract:

Existence and structure of the uniform attractors for reaction-diffusion equations with the nonlinearity in a weaker topology space are considered. Firstly, a weaker symbol space is defined and an example is given as well, showing that the compactness can be easier obtained in this space. Then the existence of solutions with new symbols is presented. Finally, the existence and structure of the uniform attractor are obtained by proving the \begin{document}$ (L^{2}\times \Sigma, L^{2}) $\end{document}-continuity of the processes generated by solutions.

Optimization of a control law to synchronize first-order dynamical systems on Riemannian manifolds by a transverse component
Adolfo Damiano Cafaro and Simone Fiori
2022, 27(7): 3947-3969 doi: 10.3934/dcdsb.2021213 +[Abstract](706) +[HTML](333) +[PDF](1746.47KB)
Abstract:

The present paper builds on the previous contribution by the second author, S. Fiori, Synchronization of first-order autonomous oscillators on Riemannian manifolds, Discrete and Continuous Dynamical Systems – Series B, Vol. 24, No. 4, pp. 1725 – 1741, April 2019. The aim of the present paper is to optimize a previously-developed control law to achieve synchronization of first-order non-linear oscillators whose state evolves on a Riemannian manifold. The optimization of such control law has been achieved by introducing a transverse control field, which guarantees reduced control effort without affecting the synchronization speed of the oscillators. The developed non-linear control theory has been analyzed from a theoretical point of view as well as through a comprehensive series of numerical experiments.

Positive solutions for critical quasilinear Schrödinger equations with potentials vanishing at infinity
Guofa Li and Yisheng Huang
2022, 27(7): 3971-3989 doi: 10.3934/dcdsb.2021214 +[Abstract](889) +[HTML](325) +[PDF](384.57KB)
Abstract:

In this paper, we study the existence of positive solutions for the following quasilinear Schrödinger equations

where \begin{document}$ \kappa>0 $\end{document}, \begin{document}$ \lambda>0, \mu>0, h\in C(\mathbb{R}, \mathbb{R}) $\end{document} is superlinear at infinity, the potentials \begin{document}$ V(x) $\end{document} and \begin{document}$ K(x) $\end{document} are vanishing at infinity. In order to discuss the existence of solutions we apply minimax techniques together with careful \begin{document}$ L^{\infty} $\end{document}-estimates. For the subcritical case (\begin{document}$ \mu = 0 $\end{document}) we can deal with large \begin{document}$ \kappa>0 $\end{document}. For the critical case we treat that \begin{document}$ \kappa>0 $\end{document} is small.

Isochronicity of bi-centers for symmetric quartic differential systems
Wilker Fernandes, Viviane Pardini Valério and Patricia Tempesta
2022, 27(7): 3991-4006 doi: 10.3934/dcdsb.2021215 +[Abstract](666) +[HTML](310) +[PDF](369.66KB)
Abstract:

In this paper we investigate the simultaneous existence of isochronous centers for a family of quartic polynomial differential systems under four different types of symmetry. Firstly, we find the normal forms for the system under each type of symmetry. Next, the conditions for the existence of isochronous bi-centers are presented. Finally, we study the global phase portraits of the systems possessing isochronous bi-centers. The study shows the existence of seven non topological equivalent global phase portraits, where three of them are exclusive for quartic systems under such conditions.
 
Addendum: "W. Fernandes and P. Tempesta are partially supported by FAPESP Grand number 2019/07316-0." is added under Fund Project. We apologize for any inconvenience this may cause.

Blow-up prevention by quadratic degradation in a higher-dimensional chemotaxis-growth model with indirect attractant production
Jianing Xie
2022, 27(7): 4007-4022 doi: 10.3934/dcdsb.2021216 +[Abstract](614) +[HTML](361) +[PDF](385.36KB)
Abstract:

This paper deals with a boundary-value problem in three-dimensional smoothly bounded domains for a coupled chemotaxis-growth system generalizing the prototype

in a smoothly bounded domain \begin{document}$ \Omega\subset\mathbb{R}^N(N\geq1) $\end{document} under zero-flux boundary conditions, which describe the spread and aggregative behavior of the Mountain Pine Beetle in forest habitat, where the parameters \begin{document}$ \mu $\end{document} as well as \begin{document}$ \delta $\end{document} and \begin{document}$ \tau $\end{document} are positive. Based on an new energy-type argument combined with maximal Sobolev regularity theory, it is proved that global classical solutions exist whenever

and the initial data \begin{document}$ (u_0,v_0,w_0) $\end{document} are sufficiently regular. Here \begin{document}$ \lambda_0 $\end{document} is a positive constant which is corresponding to the maximal Sobolev regularity. This extends some recent results by several authors.

Chaotic switching in driven-dissipative Bose-Hubbard dimers: When a flip bifurcation meets a T-point in $ \mathbb{R}^4 $
Andrus Giraldo, Neil G. R. Broderick and Bernd Krauskopf
2022, 27(7): 4023-4075 doi: 10.3934/dcdsb.2021217 +[Abstract](880) +[HTML](334) +[PDF](14603.45KB)
Abstract:

The Bose-Hubbard dimer model is a celebrated fundamental quantum mechanical model that accounts for the dynamics of bosons at two interacting sites. It has been realized experimentally by two coupled, driven and lossy photonic crystal nanocavities, which are optical devices that operate with only a few hundred photons due to their extremely small size. Our work focuses on characterizing the different dynamics that arise in the semiclassical approximation of such driven-dissipative photonic Bose-Hubbard dimers. Mathematically, this system is a four-dimensional autonomous vector field that describes two specific coupled oscillators, where both the amplitude and the phase are important. We perform a bifurcation analysis of this system to identify regions of different behavior as the pump power \begin{document}$ f $\end{document} and the detuning \begin{document}$ \delta $\end{document} of the driving signal are varied, for the case of fixed positive coupling. The bifurcation diagram in the \begin{document}$ (f, \delta) $\end{document}-plane is organized by two points of codimension-two bifurcations——a \begin{document}$ \mathbb{Z}_2 $\end{document}-equivariant homoclinic flip bifurcation and a Bykov T-point——and provides a roadmap for the observable dynamics, including different types of chaotic behavior. To illustrate the overall structure and different accumulation processes of bifurcation curves and associated regions, our bifurcation analysis is complemented by the computation of kneading invariants and of maximum Lyapunov exponents in the \begin{document}$ (f, \delta) $\end{document}-plane. The bifurcation diagram displays a menagerie of dynamical behavior and offers insights into the theory of global bifurcations in a four-dimensional phase space, including novel bifurcation phenomena such as degenerate singular heteroclinic cycles.

Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model
Yu Ma, Chunlai Mu and Shuyan Qiu
2022, 27(7): 4077-4095 doi: 10.3934/dcdsb.2021218 +[Abstract](645) +[HTML](521) +[PDF](375.04KB)
Abstract:

This work deals with a Neumann initial-boundary value problem for a two-species predator-prey chemotaxis system

in a bounded domain \begin{document}$ \Omega\subset \mathbb{R}^n \,\,(n = 2,3) $\end{document} with smooth boundary \begin{document}$ \partial\Omega $\end{document}, where the parameters \begin{document}$ d_1, d_2, d_3,\chi, \xi,\lambda,\mu,\alpha,\beta_1,\beta_2, a, b $\end{document} are positive. It is shown that for any appropriate regular initial date \begin{document}$ u_0 $\end{document}, \begin{document}$ v_0 $\end{document}, the corresponding system possesses a global bounded classical solution in \begin{document}$ n = 2 $\end{document}, and also in \begin{document}$ n = 3 $\end{document} for \begin{document}$ \chi $\end{document} being sufficiently small. Moreover, by constructing some suitable functionals, it is proved that if \begin{document}$ b\lambda<\mu $\end{document} and the parameters \begin{document}$ \chi $\end{document} and \begin{document}$ \xi $\end{document} are sufficiently small, then the solution \begin{document}$ (u,v,w) $\end{document} of this system converges to \begin{document}$ (\frac{\lambda+a\mu}{1+ab}, \frac{\mu-b\lambda}{1+ab}, \frac{\beta_1(\lambda+a\mu)+\beta_2(\mu-b\lambda)}{\alpha(1+ab)}) $\end{document} exponentially as \begin{document}$ t\rightarrow \infty $\end{document}; if \begin{document}$ b\lambda\geq \mu $\end{document} and \begin{document}$ \chi $\end{document} is sufficiently small and \begin{document}$ \xi $\end{document} is arbitrary, then the solution \begin{document}$ (u,v,w) $\end{document} converges to \begin{document}$ (\lambda,0,\frac{\beta_1\lambda}{\alpha}) $\end{document} with exponential decay when \begin{document}$ b\lambda> \mu $\end{document}, and with algebraic decay when \begin{document}$ b\lambda = \mu $\end{document}.

The impact of time delay and angiogenesis in a tumor model
Zejia Wang, Haihua Zhou and Huijuan Song
2022, 27(7): 4097-4119 doi: 10.3934/dcdsb.2021219 +[Abstract](660) +[HTML](297) +[PDF](404.77KB)
Abstract:

We consider a free boundary tumor model under the presence of angiogenesis and time delays in the process of proliferation, in which the cell location is incorporated. It is assumed that the tumor attracts blood vessels at a rate proportional to \begin{document}$ \alpha $\end{document}, and a parameter \begin{document}$ \mu $\end{document} is proportional to the 'aggressiveness' of the tumor. In this paper, we first prove that there exists a unique radially symmetric stationary solution \begin{document}$ \left(\sigma_{*}, p_{*}, R_{*}\right) $\end{document} for all positive \begin{document}$ \alpha $\end{document}, \begin{document}$ \mu $\end{document}. Then a threshold value \begin{document}$ \mu_\ast $\end{document} is found such that the radially symmetric stationary solution is linearly stable if \begin{document}$ \mu<\mu_\ast $\end{document} and linearly unstable if \begin{document}$ \mu>\mu_\ast $\end{document}. Our results indicate that the increase of the angiogenesis parameter \begin{document}$ \alpha $\end{document} would result in the reduction of the threshold value \begin{document}$ \mu_\ast $\end{document}, adding the time delay would not alter the threshold value \begin{document}$ \mu_\ast $\end{document}, but would result in a larger stationary tumor, and the larger the tumor aggressiveness parameter \begin{document}$ \mu $\end{document} is, the greater impact of time delay would have on the size of the stationary tumor.

2021 Impact Factor: 1.497
5 Year Impact Factor: 1.527
2021 CiteScore: 2.3

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