Comparison Theorems For Differential Equations Research Articles

Stability criterion of a nonautonomous 3-species ratio-dependent diffusive predator-prey model

The global stability of a nonautonomous 3-species ratio-dependent diffusive predator-prey model is studied in this paper. Firstly, some easily verifiable sufficient conditions which guarantee the existence of the strictly positive space homogenous periodic solution (SHPS) of the ratio- dependent predator-prey model (RDPPM) with diffusive and variable coefficient are achieved by using a comparison theorem of differential equation and fixed point theorem. At the same time, some new analysis techniques are developed as a byproduct. Secondly, some sufficient conditions ensuring the globally asymptotically stability of the strictly positive SHPS of the diffusive nonautonomous predator-prey model are given by using the method of upper and lower solutions (UALS) for the parabolic partial differential equations and Lyapunov stability theory. In addition, two numerical examples are given to validate the theoretical results obtained in this paper.

Complex dynamics of a nonlinear impulsive control predator-prey model with Leslie-Gower and B-D functional response

<abstract><p>A good pulse control strategy should depend on the numbers of pests and natural enemies as determined via an integrated pest control strategy. Taking this into consideration, here, a nonlinear impulsive predator-prey model with improved Leslie-Gower and Beddington-DeAngelis functional response terms is qualitatively analyzed. The existence of a periodic solution for pest eradication has been obtained and the critical condition of global asymptotic stability has been established by using the impulsive differential equation Floquet theory. Furthermore, the conditions for the lasting survival of the system has been proved by applying a comparison theorem for differential equations. Additionally, a stable positive periodic solution has been obtained by applying bifurcation theory. To understand how nonlinear pulses affect the dynamic behavior of a system, MATLAB was used to conduct numerical simulations to show that the model has very complex dynamical behavior.</p></abstract>

Global Stability of a Commensal Symbiosis Model With Holling Ii Functional Response and Feedback Controls

A commensal symbiosis model with Holling II functional response and feedback controls is proposed and studied in this paper. The system admits four equilibria, and three boundary equilibria are unstable, only positive equilibrium is locally asymptotically stable. By applying the comparison theorem of differential equation, we show that the unique positive equilibrium is globally attractive. Numeric simulations show the feasibility of the main result.

Dynamic behaviors of a nonautonomous predator\u2013prey system with Holling type II schemes and a prey refuge

In this paper, we consider a nonautonomous predator–prey model with Holling type II schemes and a prey refuge. By applying the comparison theorem of differential equations and constructing a suitable Lyapunov function, sufficient conditions that guarantee the permanence and global stability of the system are obtained. By applying the oscillation theory and the comparison theorem of differential equations, a set of sufficient conditions that guarantee the extinction of the predator of the system is obtained.

The Extinction of a Non-Autonomous Allelopathic Phytoplankton Model with Nonlinear Inter-Inhibition Terms and Feedback Controls

A non-autonomous allelopathic phytoplankton model with nonlinear inter-inhibition terms and feedback controls is studied in this paper. Based on the comparison theorem of differential equation, some sufficient conditions for the permanence of the system are obtained. We study the extinction of one of the species by using some suitable Lyapunov type extinction function. Our analyses extend those of Xie et al. (Extinction of a two species competitive system with nonlinear inter-inhibition terms and one toxin producing phytoplankton. Advances in Difference Equations, 2016, 2016, 258) and show that the feedback controls and toxic substances have no effect on the permanence of the system but play a crucial role on the extinction of the system. Some known results are extended.

Global asympototic stability of predator-prey systems with stage structure and nonlinear birth rate

A predator-prey model with nonlinear birth rate and stage structure on prey species is revisited in this paper. By using the comparison theorem of differential equation and constructing some suitable Lyapunov function, we are able to show that the conditions which ensure the locally asymptotically stable of the equilibria is enough to ensure its globally asymptotically stable. Finally, the obtained results have substantially improved and extended the corresponding results of predecessors.

一类具有脉冲免疫治疗的HIV-1感染模型的动力学分析

Based on a class of HIV-1 infection immunotherapy models, an HIV-1 infection model with pulsed immunotherapy was addressed. The non-negativity and uniform boundedness of the solutions to the pulsed immunotherapy model were studied with the pulsed differential equation theory. According to the Floquet multiplier theory and the comparison theorem for differential equations, the threshold conditions for the local and global asymptotic stability of the infection-free periodic solution as well as uniform persistence of HIV-1 were obtained. Through numerical simulation, the therapeutic effects of 3 different treatment regimens were compared, and the effectiveness of the pulsed immunotherapy was verified. The numerical results show that, the virus can be effectively controlled or eliminated theoretically when the drug input is large enough or the dosing interval is short properly.

Global asymptotical stability of almost periodic solutions for a non-autonomous competing model with time-varying delays and feedback controls

In this paper, we are concerned with a non-autonomous competing model with time delays and feedback controls. Applying the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. An example with its numerical simulations is given to illustrate the feasibility of our results.

Stability analysis of uncertain complex-variable delayed nonlinear systems via intermittent control with multiple switched periods

In this paper, an intermittent control approach with multiple switched periods is proposed for the robust exponential stabilization of uncertain complex-variable delayed nonlinear systems with parameters perturbation, in which the considered complex systems have bounded parametric uncertainties. Based on the Lyapunov stability theory and comparison theorem of differential equations, some stability criteria are established for a class of uncertain complex delayed nonlinear systems with parameters perturbation. Finally, some numerical simulations are given to show the effectiveness and the benefits of the theoretical results.

Dynamic behaviors of an obligate Gilpin-Ayala system

In this paper, a nonautonomous obligate Gilpin-Ayala system is proposed and studied. The persistence and extinction of the system are discussed by using the comparison theorem of differential equations. The results show that, depending on the cooperation intensity between the species, the first species will be driven extinct or be permanent. After that, by using the Lyapunov function method, series of sufficient conditions are obtained which ensure the global attractivity of the system. Finally, two examples are given to illustrate the feasibility of the main results.

Synchronization analysis of time delay complex-variable chaotic systems with discontinuous coupling

In this paper, the synchronization problem between two complex-variable delayed chaotic systems with discontinuous coupling is analyzed, where the interactions between two systems are considered as on–off coupling. Based on the stability theory and comparison theorem of differential equations, some sufficient conditions are given to ensure synchronization in the complex-variable delayed chaotic systems with on–off coupling. The theoretical results show that two complex-variable delayed chaotic systems can achieve synchronization even if the two complex-variable delayed chaotic systems are switched off sometimes and the synchronization speed is closely related to the coupling strength and the on–off rate. Finally, numerical examples are examined to illustrate the feasibility and effectiveness of the analytical results.

Almost periodic solutions for a competition and cooperation model of two enterprises with time-varying delays and feedback controls

In this paper, we propose and deal with a competition and cooperation model of two enterprises with time-varying delays and feedback controls, which can effectively describe the competition and cooperation of two enterprises in real economic environments. With the help of the comparison theorem of differential equations and by constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are established. We present an example to explain the economical significance of mathematical results. The paper ends with brief conclusions.

Influence of single feedback control variable on an autonomous Holling-II type cooperative system

An autonomous two-species Holling-II type cooperative system with single feedback control is considered in this paper. Firstly, by applying the comparison theorem of differential equation, sufficient conditions which ensure the permanence of the system are obtained. Secondly, by constructing some suitable Lyapunov function and using the average Inequality, some sufficient conditions for the globally asymptotical stability and extinction of the system are obtained. Our results show that by choosing suitable values of feedback control variables, stable species could become extinct, or still keep the property of globally stable.

Synchronization analysis of complex‐Variable chaotic systems with discontinuous unidirectional coupling

This article is concerned with the problem of synchronization between two uncertain complex‐variable chaotic systems with parameters perturbation and discontinuous unidirectional coupling. Based on the stability theory and comparison theorem of differential equations, some sufficient conditions for the complete synchronization and generalized synchronization are obtained. The theoretical results show that the two uncertain complex‐variable chaotic systems with discontinuous unidirectional coupling can achieve synchronization if the time‐average coupling strength is large enough. Finally, numerical examples are examined to illustrate the feasibility and effectiveness of the analytical results. © 2015 Wiley Periodicals, Inc. Complexity 21: 343–355, 2016

Synchronization of two different chaotic systems with discontinuous coupling

This paper is concerned with the problem of synchronization between two different chaotic sys- tems with discontinuous coupling. Based on the stabil- ity theory and the comparison theorem of differential equations, we derive less restrictive synchronization conditions than those resulting from the Lyapunov the- ory. The theoretical results show that generalized syn- chronization between two different chaotic systems can be achieved if the time-average coupling strength is large enough. Finally, the corresponding numerical simulations are presented to demonstrate the effective- ness of proposed schemes.

Synchronization of chaotic systems with on–off periodic coupling

In this paper, we study the complete synchronization between two chaotic systems with on–off periodic coupling. Using the stability theory and the comparison theorem of differential equations, we derive less restrictive synchronization conditions than those resulting from the Lyapunov theory. The theoretical results show that two chaotic systems can achieve complete synchronization if the time-average coupling strength is large enough. Finally, numerical simulations fully verify the effectiveness of the analytical results.

Permanence for a two-species Gause-type ratio-dependent predator–prey system with time delay in a two-patch environment

This paper is devoted to a two-species periodic Gause-type ratio-dependent predator–prey system with time delay in a two-patch environment. By using the comparison theorem of differential equations, we establish sufficient conditions for the permanence of the system. As corollaries, some applications are listed. In particular, our results generalize and improve some known ones.

Global asymptotical stability of a unique almost periodic solution for enterprise clusters based on ecology theory with time-varying delays and feedback controls

This paper is concerned with some dynamic behavior of enterprises cluster constituted by n satellite enterprises and a dominant enterprise. We present a model involving time-varying delays and feedback controls based on ecology theory, which effectively describe the competition and cooperation of enterprises cluster in real economic environments. Applying the comparison theorem of differential equations and constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotically stable nonnegative almost periodic solution of the system are obtained. Finally, we present an example to explain the economical significance of mathematical results.

Estimates in deterministic fate modelling of environmental chemicals

Ecological modelling is frequently confronted with the problem of unavailable parameters, or input data of poor quality. This makes the interpretation of the results difficult, and predictions questionable, unless the quality of the parameters is taken into account. In this paper, we propose a mathematical approach to address the problem of large parameter variability in deterministic models, and especially in linear compartmental models. The method is based on a classical comparison theorem for differential equations, and yields upper and lower estimates for solutions when upper and lower estimates for the model input functions are given. Such estimates may be checked against experimental data, and thus play a crucial role in verification or falsification of model predictions. Applications are presented with regard to fate modelling of environmental chemicals in a pond sediment.

The depth of a steep evaporating grain boundary groove: Application of comparison theorems

The nonlinear Mullins diffusion equation for the development of a surface groove by evaporation-condensation is reconsidered. Comparison theorems for differential equations are employed to obtain upper and lower bounds on the profile of the groove itself and on the growth rate of the groove at the grain boundary. It is shown that as the dihedral angle at the grain boundary approaches zero, the groove growth rate diverges logarithmically. This phenomenon is in disparity with the finite growth rate found in the development of grooves by surface diffusion.