{"title":"An optimization method for treating solid tumors with combined therapy using the Great Deluge algorithm","authors":"A. Glick, A. Mastroberardino","doi":"10.3934/dcdsb.2023143","DOIUrl":"https://doi.org/10.3934/dcdsb.2023143","url":null,"abstract":"","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"1 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88253966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a birth-death process with generator $ mathcal{L} $ and reversible invariant probability measure $ pi. $ We identify explicitly the Lipschitzian norm of the solution of the Poisson equation $ -mathcal{L} G = g-pi(g) $ for $ |g|levarphi $. This leads to some transportation-information inequalities, concentration inequalities and Cheeger-type isoperimetric inequalities. Several examples are provided to illustrate the results.
{"title":"Lipschitzian norms and functional inequalities for birth-death processes","authors":"Wei Liu","doi":"10.3934/dcdsb.2023177","DOIUrl":"https://doi.org/10.3934/dcdsb.2023177","url":null,"abstract":"In this paper, we consider a birth-death process with generator $ mathcal{L} $ and reversible invariant probability measure $ pi. $ We identify explicitly the Lipschitzian norm of the solution of the Poisson equation $ -mathcal{L} G = g-pi(g) $ for $ |g|levarphi $. This leads to some transportation-information inequalities, concentration inequalities and Cheeger-type isoperimetric inequalities. Several examples are provided to illustrate the results.","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135157752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper investigates the synchronization problem of a kind of discrete-time Kuramoto oscillators based on delayed position states. For this kind of Kuramoto model, the oscillators can only obtain delayed phase and frequency states relative to each other at a series of discrete time instants. Some simple sufficient conditions on the time step, the time delay, the natural frequencies, the coupling strength, and the historical and initial values of the phases and frequencies are derived analytically to ensure the phase synchronization of homogeneous oscillators and the frequency synchronization of heterogeneous oscillators, respectively. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.
{"title":"Synchronization of discrete time Kuramoto oscillators with delayed states","authors":"Hua Zhang, Sisi Xiao","doi":"10.3934/dcdsb.2023183","DOIUrl":"https://doi.org/10.3934/dcdsb.2023183","url":null,"abstract":"This paper investigates the synchronization problem of a kind of discrete-time Kuramoto oscillators based on delayed position states. For this kind of Kuramoto model, the oscillators can only obtain delayed phase and frequency states relative to each other at a series of discrete time instants. Some simple sufficient conditions on the time step, the time delay, the natural frequencies, the coupling strength, and the historical and initial values of the phases and frequencies are derived analytically to ensure the phase synchronization of homogeneous oscillators and the frequency synchronization of heterogeneous oscillators, respectively. Finally, some numerical simulations are performed to verify the correctness of the theoretical results.","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135319478","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Some efficient temporal first-, second- and higher-order numerical schemes are constructed for the coupled nonlinear Schrödinger-Boussinesq (CNSB) equations based on discretizing the Hamiltonian formula by an exponential method in a compact representation, discrete gradient method and composition method. The schemes are fully decoupling of the three solution components, which is distinct from the partially decoupled scheme in the literature, and their compact representation reduces the storage requirement and operation count. Rigorous analyses are carried out to show the exact preservation of the discrete energy for the proposed schemes. Numerical experiments verify the theoretical results and confirm the satisfactory solution accuracy and excellent efficiency of the present schemes.
{"title":"Fully-decoupled conservative exponential approaches for the coupled nonlinear Schrödinger-Boussinesq equations","authors":"Jiaxiang Cai, Juan Chen","doi":"10.3934/dcdsb.2023186","DOIUrl":"https://doi.org/10.3934/dcdsb.2023186","url":null,"abstract":"Some efficient temporal first-, second- and higher-order numerical schemes are constructed for the coupled nonlinear Schrödinger-Boussinesq (CNSB) equations based on discretizing the Hamiltonian formula by an exponential method in a compact representation, discrete gradient method and composition method. The schemes are fully decoupling of the three solution components, which is distinct from the partially decoupled scheme in the literature, and their compact representation reduces the storage requirement and operation count. Rigorous analyses are carried out to show the exact preservation of the discrete energy for the proposed schemes. Numerical experiments verify the theoretical results and confirm the satisfactory solution accuracy and excellent efficiency of the present schemes.","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135505510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose and study a predator-prey model with a Smith growth function and the addition predation term described by a Holling Type Ⅱ functional response in prey. This additive predation term can lead to Allee effects in prey population dynamics that can generate complicated dynamics in the corresponding predator-prey model. We provide a through analysis of the global dynamics of the proposed model, including the equilibrium stability, Hopf bifurcation and its directions, existence of a heteroclinic orbit loop and limit cycles. We show that when the predator-prey model exhibits Allee effects, Hopf bifurcation is either backward and supercritical or forward and subcritical. In the strong Allee effect case, the model has a heteroclinic orbit loop connecting two boundary saddle points. Our results show that the coexistence can be achieved by controlling the attack rate of other potential predators so that the model exhibits weak Allee effects or no Allee effect. Both the small additional predation rate and the large replacement rate of mass can improve the coexistence probability of two species. The main difference of dynamics between the model exhibiting weak Allee effect and no Allee effect lies in the pattern of coexistence: If no Allee effect, the coexistence can be a steady state while in the weak Allee case, the coexistence may be periodic.
{"title":"Global dynamics of a predator-prey model with a Smith growth function and the additive predation in prey","authors":"Dingyong Bai, Jiale Zheng, Yun Kang","doi":"10.3934/dcdsb.2023161","DOIUrl":"https://doi.org/10.3934/dcdsb.2023161","url":null,"abstract":"We propose and study a predator-prey model with a Smith growth function and the addition predation term described by a Holling Type Ⅱ functional response in prey. This additive predation term can lead to Allee effects in prey population dynamics that can generate complicated dynamics in the corresponding predator-prey model. We provide a through analysis of the global dynamics of the proposed model, including the equilibrium stability, Hopf bifurcation and its directions, existence of a heteroclinic orbit loop and limit cycles. We show that when the predator-prey model exhibits Allee effects, Hopf bifurcation is either backward and supercritical or forward and subcritical. In the strong Allee effect case, the model has a heteroclinic orbit loop connecting two boundary saddle points. Our results show that the coexistence can be achieved by controlling the attack rate of other potential predators so that the model exhibits weak Allee effects or no Allee effect. Both the small additional predation rate and the large replacement rate of mass can improve the coexistence probability of two species. The main difference of dynamics between the model exhibiting weak Allee effect and no Allee effect lies in the pattern of coexistence: If no Allee effect, the coexistence can be a steady state while in the weak Allee case, the coexistence may be periodic.","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"63 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135595513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We develop a new technique for the path approximation of one-dimensional stochastic processes. Our results apply to the Brownian motion and to some families of stochastic differential equations whose distributions could be represented as a function of a time-changed Brownian motion (usually known as $ L $ and $ G $-classes). We are interested in the $ varepsilon $-strong approximation. We propose an explicit and easy-to-implement procedure that jointly constructs, the sequences of exit times and corresponding exit positions of some well-chosen domains. In our main results, we prove the convergence of our scheme and how to control the number of steps, which depends on the covering of a fixed time interval by intervals of random sizes. The underlying idea of our analysis is to combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also developed in order to complete the theoretical results.
提出了一种新的一维随机过程路径逼近方法。我们的结果适用于布朗运动和一些随机微分方程族，其分布可以表示为随时间变化的布朗运动的函数(通常称为$ L $和$ G $-类)。我们感兴趣的是$ varepsilon $ strong近似。我们提出了一个明确且易于实现的过程，该过程可以联合构建一些选定的域的退出时间序列和相应的退出位置。在我们的主要结果中，我们证明了我们的方案的收敛性以及如何控制步数，这取决于随机大小的区间覆盖固定的时间区间。我们分析的基本思想是将布朗退出时间从随时间域(一维热球)和经典更新理论的结果结合起来。为了完善理论结果，还开发了数值实例和问题。
{"title":"Strong approximation of some particular one-dimensional diffusions","authors":"Madalina Deaconu, Samuel Herrmann","doi":"10.3934/dcdsb.2023164","DOIUrl":"https://doi.org/10.3934/dcdsb.2023164","url":null,"abstract":"We develop a new technique for the path approximation of one-dimensional stochastic processes. Our results apply to the Brownian motion and to some families of stochastic differential equations whose distributions could be represented as a function of a time-changed Brownian motion (usually known as $ L $ and $ G $-classes). We are interested in the $ varepsilon $-strong approximation. We propose an explicit and easy-to-implement procedure that jointly constructs, the sequences of exit times and corresponding exit positions of some well-chosen domains. In our main results, we prove the convergence of our scheme and how to control the number of steps, which depends on the covering of a fixed time interval by intervals of random sizes. The underlying idea of our analysis is to combine results on Brownian exit times from time-depending domains (one-dimensional heat balls) and classical renewal theory. Numerical examples and issues are also developed in order to complete the theoretical results.","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"134 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135700138","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the predator-prey system with density-dependent motilities and pursuit-evasion interaction under homogeneous Neumann boundary conditions. The main obstacle of analysis comes from the term produced by pursuit-evasion interaction. With the $ L^p $-estimate techniques and Moser iteration, we show that the system possesses a global bounded classical solution. Furthermore, with the aid of Lyapunov functional, we establish the asymptotic behavior of solutions to this system under appropriate parameter conditions.
{"title":"Boundedness and asymptotic stability in a predator-prey system with density-dependent motilities","authors":"Yunxi Li, Chunlai Mu, Xu Pan","doi":"10.3934/dcdsb.2023173","DOIUrl":"https://doi.org/10.3934/dcdsb.2023173","url":null,"abstract":"In this paper, we consider the predator-prey system with density-dependent motilities and pursuit-evasion interaction under homogeneous Neumann boundary conditions. The main obstacle of analysis comes from the term produced by pursuit-evasion interaction. With the $ L^p $-estimate techniques and Moser iteration, we show that the system possesses a global bounded classical solution. Furthermore, with the aid of Lyapunov functional, we establish the asymptotic behavior of solutions to this system under appropriate parameter conditions.","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136367265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local discontinuous Galerkin schemes for an ultrasonic propagation equation with fractional attenuation","authors":"Can Li, Min-Min Li, Z. Fellah","doi":"10.3934/dcdsb.2023063","DOIUrl":"https://doi.org/10.3934/dcdsb.2023063","url":null,"abstract":"","PeriodicalId":51015,"journal":{"name":"Discrete and Continuous Dynamical Systems-Series B","volume":"19 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72687035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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