Global stability of fuzzy cognitive maps

Complex systems can be effectively modelled by fuzzy cognitive maps. Fuzzy cognitive maps (FCMs) are network-based models, where the connections in the network represent causal relations. The conclusion about the system is based on the limit of the iteratively applied updating process. This iteration may or may not reach an equilibrium state (fixed point). Moreover, if the model is globally asymptotically stable, then this fixed point is unique and the iteration converges to this point from every initial state. There are some FCM models, where global stability is the required property, but in many FCM applications, the preferred scenario is not global stability, but multiple fixed points. Global stability bounds are useful in both cases: they may give a hint about which parameter set should be preferred or avoided. In this article, we present novel conditions for the global asymptotical stability of FCMs, i.e. conditions under which the iteration leads to the same point from every initial vector. Furthermore, we show that the results presented here outperform the results known from the current literature.

Dynamics of Stage-Structured Predator–Prey Model with Beddington–DeAngelis Functional Response and Harvesting

In this paper, we investigate the stability of equilibrium in the stage-structured and density-dependent predator–prey system with Beddington–DeAngelis functional response. First, by checking the sign of the real part for eigenvalue, local stability of origin equilibrium and boundary equilibrium are studied. Second, we explore the local stability of the positive equilibrium for τ=0 and τ≠0 (time delay τ is the time taken from immaturity to maturity predator), which shows that local stability of the positive equilibrium is dependent on parameter τ. Third, we qualitatively analyze global asymptotical stability of the positive equilibrium. Based on stability theory of periodic solutions, global asymptotical stability of the positive equilibrium is obtained when τ=0; by constructing Lyapunov functions, we conclude that the positive equilibrium is also globally asymptotically stable when τ≠0. Finally, examples with numerical simulations are given to illustrate the obtained results.

Dynamics of a diffusive predator–prey system with ratio-dependent functional response and time delay

In this paper, we investigate the qualitative behaviors of a predator–prey system with ratio-dependent function. The system accommodates the diffusion effect to model the migration of individuals and the time delay induced by reproduction. We start with some basic properties of the system. Then the sufficient condition independent of time delay and diffusion effect for global asymptotical stability of the boundary equilibrium is obtained by using the comparison principle. Afterwards, based on the LaSalle’s invariance principle and Lyapunov functional, we investigate the global attractiveness of the positive equilibrium, arriving at its global asymptotical stability. Further, Hopf bifurcation induced by time delay around the positive equilibrium is explored. Finally, numerical examples are listed to verify the corresponding analytical results.

Global Asymptotical Stability of Neutral-Type Neural Networks with D-Operator and Mixed Delays

In this manuscript, we investigate the stability problems of neutral-type neural networks with D-operator and mixed delays. Some sufficient conditions are obtained for guaranteeing the existence, uniqueness, and global asymptotical stability of periodic solutions to the considered neural networks. Finally, a numerical example is performed to illustrate the theoretical results.

Dynamics of an N-Species Gilpin–Ayala Impulsive Competition System

In this paper, we consider an N-species Gilpin–Ayala impulsive competition system. By using comparison theorem, Lyapunov functional, and almost periodic functional hull theory of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence, global asymptotical stability, and almost periodic solution of the model. Our results extend some previously known results. The method used in this paper provides a possible method to study the permanence, global asymptotical stability, and almost periodic solution of the models with impulsive perturbations in biological populations.

Dynamics of two-species delayed competitive stage-structured model described by differential-difference equations

Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.

Global Asymptotical Stability Analysis for Fractional Neural Networks with Time-Varying Delays

In this paper, the global asymptotical stability of Riemann-Liouville fractional-order neural networks with time-varying delays is studied. By combining the Lyapunov functional function and LMI approach, some sufficient criteria that guarantee the global asymptotical stability of such fractional-order neural networks with both discrete time-varying delay and distributed time-varying delay are derived. The stability criteria is suitable for application and easy to be verified by software. Lastly, some numerical examples are presented to check the validity of the obtained results.