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In this paper, a tumor-immune system with impulse comprehensive therapy and stochastic perturbation is investigated. The combination of pulsed chemotherapy and pulsed immunotherapy and the effect of environmental random disturbance are reflected in this model. The existence and uniqueness of global positive solution of the system are proved. It is determined that the expectation of the solution is always less than a constant by utilizing the comparison theorems of impulsive differential equations and that the tumor cells will become weakly persistent in the mean or extinct under some sufficient conditions. Our results and numerical simulations show that random disturbance can inhibit the growth of tumor cells, and the combination of chemotherapy and immunotherapy can reduce the damage of therapy to the healthy cells.

In this paper, we consider the dynamics of a stochastic Gilpin–Ayala model with regime switching and impulsive perturbations. The Gilpin–Ayala parameter is also allowed to switch. Sufficient conditions for extinction, nonpersistence in the mean, weak persistence, and stochastic permanence are provided. The critical number among the extinction, nonpersistence in the mean, and weak persistence is obtained. Our results demonstrate that the dynamics of the model have close relations with the impulses and the Markov switching.

A nonlinear age-structured tumor cell population model with quiescence and bidirectional transition is presented and studied. The division rate of the proliferating cells is assumed to be nonlinear due to the limitation of the nutrient and space. The model includes a proportion of newborn cells that enter directly the quiescent phase with age zero. This proportion can reflect the effect of treatment by drugs such as erlotinib. The bidirectional transition between proliferating cells and quiescent cells is considered. The local and global stabilities of the trivial steady state are investigated. The existence and local stability of the positive steady state are also analyzed. Numerical simulations are performed to verify the results and to examine the impacts of parameters on the nonlinear dynamics of the model.

We propose a differential evolution algorithm based on adaptive fractional gradient descent (DE-FGD) to address the defects of existing bio-inspired algorithms, such as slow convergence speed and local optimum. The crossover and selection processes of the differential evolution algorithm are discarded and the adaptive fractional gradients are adopted to enhance the global searching capability. For the benchmark functions, our proposed algorithm Specifically, our method has higher searching accuracy than several state of the art bio-inspired algorithms. Furthermore, we apply our method to specific tasks - parameters estimation of system response functions and approximate value functions. Experiment results validate that our proposed algorithm produces accurate estimations and improves searching efficiency

In this paper, we present and analyze an age-structured tumor immune model. Based on the fact that tumor cells of different ages tend to exhibit different physiological behaviors, we consider the age structure of tumor cells, age-based proliferation function and age-dependent death function in the model. The threshold R0 for the existence of tumor-free steady state is derived. It is found that if R0 > 1, the tumor-free steady state is not only locally stable but also globally stable. Moreover, numerical simulation shows that the threshold R0 may be regarded as an index to reflect the ability of “tumor immune surveillance”, that is, the smaller the R0, the better the ability of tumor immune surveillance. If R0 > 1, it is proved that the tumor steady state is existent and uniformly persistent. The local stability of the tumor steady state is investigated under some further conditions besides R0 > 1. In the end, we estimate the system parameters, verify the theoretical results and analyze some system parameters’ sensitivities.

Intermittent androgen deprivation therapy is often used to treat prostate cancer, but there are few mathematical modelling studies of it. To explore the mechanisms of such therapy, we describe intermittent therapy with impulsive differential equations, then we propose a novel mathematical model of intermittent androgen deprivation therapy with white noise. We first studied the model’s basic properties including the existence and uniqueness of the solution. By using the theory of stochastic differential equations, we investigated the thresholds for the extinction and persistence of prostate cancer cells, which are markedly affected by the antigenicity of tumours and noise parameters. Moreover, sufficient conditions for the stationary distribution and ergodicity of the system are provided. The results show that reducing the period of pulsed interventions or increasing the dosages (or frequencies) of the therapy will be helpful for curing prostate cancer.

A nonlinear age-structured tumor cell population model is presented and analyzed. The population is divided into proliferating and quiescent cell compartments. The nonlinearity of the birth rate is designed to halt population’s exponential growth and force the system to converge to stable equilibria or stable cycles. The existence and uniqueness of solutions are studied. The local and global stabilities of the trivial steady state are investigated. Moreover, the existence and local stability of the positive steady state are analyzed. Numerical examples are performed to verify the validity of the results and to discuss the impacts of parameters on the nonlinear dynamics of the model.

In this letter, a new scheme based on Born iterative method is proposed to solve the inverse scattering problems with noise disturbance. By controlling the balance factor in the new scheme, the convergence speed of the inversion algorithm is improved while ensuring that the inversion has a stable solution. This means that a new degree of freedom in deciding the performance of the inversion algorithm is proposed and can be controlled by the researcher. This letter discusses the significance of the balance factor and its impact on the performance of the algorithm. Besides, the method for choosing an appropriate balance factor is introduced. Finally, the effectiveness of the new scheme is verified by numerical analysis.

In this letter, a subspace-based variational Born iterative method (S-VBIM) is proposed to solve the inverse scattering problem. The proposed method retrieves the deterministic subspace of the variational induced current at the beginning of the iterative computation, and thus, the process of the total electric field approximation is more accurate than the Born approximation does in VBIM. Therefore, S-VBIM achieves faster convergence speed and higher solution accuracy than VBIM. In addition, since the singular value decomposition, which is the main process of retrieving the variational induced current for radiation operator matrix, only needs to be performed once, the computational cost is reduced significantly. Finally, the numerical results validate the effectiveness of S-VBIM and comparisons with VBIM are made.

In this paper, an impulsive quaternion-valued neural networks (QVNNs) model with leakage, discrete, and distributed delays is considered. Based on the homeomorphic mapping method, Lyapunov stability theorem, and linear matrix inequality (LMI) approach, sufficient conditions for the existence, uniqueness, and global robust stability of the equilibrium point of the impulsive QVNNs are provided. A numerical example is provided to confirm the obtained results. A conclusion is presented in the end.

In this paper, we construct an infection-age model to study the interaction between viruses and the immune system within the host. In the model, the mortality rate of infected cells, the rate that cytotoxic T lymphocytes (CTL) kill infected cells, the rate that infected cells produce new virus, and the CTL proliferate rate may depend on the infection-age. The basic reproduction number and the threshold for the existence of steady states are obtained. Local stability of both the infection-free and infection steady states is studied by analyzing the linearized systems. Global stability of the infection-free steady state is obtained by investigating a renewal integral equation and global stability of the infection steady state is obtained by constructing a Liapunov functional. Numerical simulations are presented to verify the theoretical results.

In this paper, a one-prey-n-predator impulsive reaction-diffusion periodic predator-prey system with ratio-dependent functional response is investigated. On the basis of the upper and lower solution method and comparison theory of differential equation, sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Examples and numerical simulations are presented to verify the feasibility of our results. A discussion is conducted at the end.

A periodic mathematical model of cancer treatment by radiotherapy followed by chemotherapy is presented and studied in this paper. Sufficient conditions on the coexistence of the healthy and cancer cells are obtained. Conditions on the existence and globally asymptotic stability of the positive periodic solution, the cancer eradication periodic solution and the cancer win periodic solution are derived. Numerical examples are shown to verify the validity of the results. A discussion is presented for further study.

A methodology of the the Equivalent-Input-Disturbance control system based on H-infinity $(H_{\infty})$ is proposed in this paper. This design method aims to solve the problem that the parameters of the control system lack unified standard to set. It converts the controller design problem into a convex optimization problem with LMI constraints. The parameters of the control system are obtained by using the LMI toolbox in Matlab. Firstly, the model of the Equivalent-Input-Disturbance system containing state feedback controller is built; secondly, the controller design problem is transformed into a standard $H_{\infty}$ state-feedback design problem and a mathematical model of the transformed $H_{\infty}$ standard control model is established; finally, under the $H_{\infty}$ formulation, the state feedback gain is obtained by solving the LMI constraint inequations and the stability is considered based on the small gain theorem. Simulation results show that the proposed method is effective and easy to implement.

In this paper, an equivalent distorted Born iteration method (DBIM) is proposed to solve inverse scattering problem based on volume integral equation (VIE) method. For its forward problem, the object region is regarded as the scatterer and free space is considered to be the background, which relates to a free space Green's function. The equivalent DBIM inversion model is constructed base on the forward model which has the same convergent rate with the conventional DBIM. Differently, the integration kernel, the free space Green's function, remains unchanged in the equivalent DBIM while that of the conventional DBIM need to be computed and updated in every iterations. In this way, the computational complexity of the DBIM has been reduced remarkably, and it is easy to be employed in solving three-dimensional (3-D) inverse problem.

This paper tests the inflation hedging ability of four categories of important financial assets in China: Commodity futures, real estate, gold and industry stock and select the assets that have significant inflation hedging effect. Then the authors construct the mean-variance model under the inflation factor, using the selected assets to construct the inflation hedging portfolio, solving the model and obtain the optimal investment strategy with inflation protection function. The result shows that the portfolio constructed by the model have more stable real returns and its inflation hedging ability can be even better if the short selling restriction of stocks is eliminated.

The present study considers the robust stability for impulsive complex-valued neural networks (CVNNs) with discrete time delays. By applying the homeomorphic mapping theorem and some inequalities in a complex domain, some sufficient conditions are obtained to prove the existence and uniqueness of the equilibrium for the CVNNs. By constructing appropriate Lyapunov-Krasovskii functionals and employing the complex-valued matrix inequality skills, the study finds the conditions to guarantee its global robust stability. A numerical simulation illustrates the correctness of the proposed theoretical results.

In this paper, a mathematical model is established to describe the dynamics of the dengue fever with quarantine measures. Firstly, we discuss the existence and the local stability of the equilibria of the model. Secondly, by constructing Lyapunov function, we gain the sufficient conditions for global stability of the equilibria.

A new augmented combined field integral equation formulation to address the low frequency instability issue is presented. The new integral equation formulation is achieved by combing the magnetic field integral operator and the augmented electric field integral equation (A-EFIE). In addition to that, a constraint preconditioning is also introduced to fast up the convergence speed of the system matrix. Numerical examples are given to show the validity and efficiency of the proposed formulation.

This paper aims at studying the model proposed by Kuznetsov and Taylor in 1994. Inspired by Mayer et al., time delay is introduced in the general model. The dynamic behaviors of this model are studied, which include the existence and stability of the equilibria and Hopf bifurcation of the model with discrete delays. The properties of the bifurcated periodic solutions are studied by using the normal form on the center manifold. Numerical examples and simulations are given to illustrate the bifurcation analysis and the obtained results.

An impulsive reaction–diffusion periodic food-chain system with ratio-dependent functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the food-chain system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.

An n+1-dimensional impulsive reaction-diffusion periodic predator-prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions on the ultimate boundedness and permanence of the predator-prey system are established based on the comparison theory of differential equation and upper and lower solution method. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.

An impulsive two species periodic predator-prey Lotka–Volterra type dispersal system with mixed functional responses is presented and studied in this paper. Conditions for the permanence and extinction of the predator-prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.

This paper studies an impulsive nonautonomous two species predator-prey Lotka-Volterra type dispersal system in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. On the basis of inequality estimation techniques and the comparison theory of impulsive differential equations, sufficient conditions of integrable form for the permanence and extinction are obtained. By constructing appropriate Lyapunov function, conditions for the global attractivity of the system are established.

Non-negative matrix factorization (NMF) is a new approach to deal with the multivariate nonnegative data. Although the classic multiplicative update algorithm can solve the NMF problems, it fails to find sparse and localized object parts. Then a Gibbs random field (GRF) modeling based NMF algorithm, called the GRF-NMF algorithm, try to directly model the prior object structure of the components into the NMF problem. In this paper, the convergence of the GRF-NMF algorithm and its advantages are investigated. Based on a classic model, the equilibrium points are obtained. Some invariant sets are constructed to prepare for the analysis of the convergence of the GRF-NMF algorithm. Then using stability theory of the equilibrium point, the convergence of the algorithm is proved and the convergence conditions of the algorithm are obtained. We theoretically present the advantages of the GRF-NMF algorithm in the end.

We study a single-species periodic logistic type dispersal system in a patchy environment with impulses. On the basis of inequality estimation technique, sufficient conditions of integrable form for the permanence and extinction of the system are obtained. By constructing an appropriate Lyapunov function, conditions for the existence of a unique globally attractively positive periodic solution are also established. Numerical examples are shown to verify the validity of our results and to further discuss the model.

This paper studies an impulsive two species periodic predator–prey Lotka–Volterra type dispersal system with Holling type III functional response in a patchy environment, in which the prey species can disperse among n different patches, but the predator species is confined to one patch and cannot disperse. Conditions for the permanence and extinction of the predator–prey system, and for the existence of a unique globally stable periodic solution are established. Numerical examples are shown to verify the validity of our results.

This paper deals with the problem of the stochastic admissibility for a class of singular Markovian jump systems with mode-dependent time delays. The singular matrices E r ( t ) in the considered systems are mode-dependent. Based on the Lyapunov functional method and the free-weighting matrix method, sufficient conditions are derived in terms of strict linear matrix inequalities (LMIs) to guarantee the considered systems to be stochastically admissible. Numerical examples are provided to show the effectiveness of the proposed approaches.

This paper deals with the H"~ filtering problem for a class of singular Markovian jump systems with time-varying delay. The singular matrices E"r"("t") in the considered system are mode-dependent. We aim to design a filter ensuring that the filtering error system is mean-square exponentially admissible and satisfies a prescribed H"~ performance level. Using the Lyapunov functional method and the free-weighting matrix method, a delay-dependent sufficient condition for the existence of a desired filter is presented in the form of linear matrix inequalities (LMIs). Finally, some numerical examples are provided to show the effectiveness of the proposed approach.

An n species stochastic impulsive migration Lotka–Volterra model with Markovian switching in N different patches is presented and studied in this paper. By constructing appropriate Lyapunov functions, some sufficient conditions on the global positivity, the ultimate boundedness in mean and the extinction in mean are established. Real world examples are provided to illustrate the validity of our results. A discussion is given in the end.

An impulsive reaction–diffusion periodic predator–prey system with Holling type III functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator–prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing an appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given at the end.

A periodic mathematical model of cell populations affected by periodic radiation is presented and studied in this paper. We obtain some sufficient conditions on the permanence and extinction of the system. Furthermore, criteria on the existence and global asymptotic stability of unique positive periodic solutions are established. Some numerical examples are shown to verify our results. A discussion is presented for further study.

In this paper, we study a periodic predator–prey system with prey impulsively unilateral diffusion in two patches. Firstly, based on the results in [41] , sufficient conditions on the existence, uniqueness and globally attractiveness of periodic solution for predator-free and prey-free systems are presented. Secondly, by using comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the permanence and extinction of prey species x with predator have other food source are established. Finally, the theoretical results both for non-autonomous system and corresponding autonomous system are confirmed by numerical simulations, from which we can see some interesting phenomena happen.

An impulsive reaction-diffusion periodic predator-prey system with ratio-dependent functional response is investigated in the present paper. Sufficient conditions for the ultimate boundedness and permanence of the predator-prey system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are presented to verify our results. A discussion is given in the end of the paper.

In this paper, we study a periodic predator–prey system with prey impulsive diffusion in two patches. On the basis of comparison theorem of impulsive differential equation and other analysis methods, sufficient and necessary conditions on the predator–prey system where predator have not other food source are established. Two examples and numerical simulations are presented to illustrate the feasibility of our results. A conclusion is given in the end.

In this paper, the effects of time delay on Open image in new window control of Lur’e systems are considered. Using a convex representation of the nonlinearity, a novel delay-dependent Open image in new window stability criterion is derived for the Lur’e systems via proportional-derivative (PD) feedback law. The Lyapunov–Krasovskii functional based on the delay discretization approach is used for the purpose of obtaining the Open image in new window stability condition by utilizing Projection Lemma. The criterion is utilized to not only guarantee stability of systems but also reduce the effect of external disturbance to an Open image in new window norm constraint. Finally, two numerical examples show the effectiveness of the proposed method.

An impulsive reaction-diffusion periodic food-chain system with Holling type III functional response is presented and studied in this paper. Sufficient conditions for the ultimate boundedness and permanence of the food-chain system are established based on the upper and lower solution method and comparison theory of differential equation. By constructing appropriate auxiliary function, the conditions for the existence of a unique globally stable positive periodic solution are also obtained. Some numerical examples are shown to illustrate our results. A discussion is given in the end of the paper.

In this paper, it is studied that two species predator–prey Lotka–Volterra type dispersal system with delay and Holling type II response function, in which the prey species can disperse among n patches, while the density-independent predator species is confined to one of the patches and cannot disperse. Sufficient conditions of integrable form for the boundedness, permanence, extinction and the existence of positive periodic solution are established, respectively.

In this paper, we study a periodic single-species logistic system with impulsive diffusion in two patches. By using the iterative method, sufficient and necessary conditions on the existence, uniqueness and global attractivity of positive periodic solution and the extinction of species for this system are established. Two examples and numerical simulations are presented to illustrate the feasibility of our results.

Erratum to ‘‘n Species impulsive migration model with Markovian switching’’ [J. Theor. Biol. 307 (2012) 62–69] Zijian Liu , Shouming Zhong , Zhidong Teng d a Department of Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, PR China b School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China c Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, PR China d College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, PR China