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Abstract Competitive behavior, like predation, widely exists in the world which has influences on the dynamics of ecosystems. Though a good knowledge for the patten formation and selection of predator–prey models with diffusion is known in the literature, little is known for that of a diffused competition model. As a result, a competition model with saturation effect and self- and cross-diffusion is proposed and analyzed in this paper. Firstly, by making use of nullcline analysis, we derive the condition of existence and stability of positive equilibrium and prove global stability. Then using the linear stability analysis, the Turing bifurcation critical value and the condition of the occurrence of Turing pattern are obtained when control parameter is selected. Finally, we deduce the amplitude equations around the Turing bifurcation point by using the standard multiple scale analysis. Meanwhile, a series of numerical simulations are given to expand our theoretical analysis. This work shows that cross-diffusion plays a key role in the formation of spatial patterns for competitive model with saturation effect. [ABSTRACT FROM AUTHOR]

An SVEIR epidemic model with imperfect vaccination and nonlinear incidence, and a general latent distribution is formulated. By constructing Lyapunov functionals, it is shown that the disease will die out if the vaccination reproduction number R v a c ≤ 1 and the disease becomes endemic if R v a c > 1 . Furthermore, vaccination effects are analyzed. Two special forms the probability of remaining in latent class are discussed. When the probability is negatively exponentially distributed, we present an efficient approach of proving global stability of the endemic equilibrium of the SVEIR system of ordinary differential equations (ODEs), which may improve some known approaches. When the probability is a step-function, the delay differential equation (DDE) system derived is used to study the impacts of vaccination and saturated incidence on the mumps transmission. [ABSTRACT FROM AUTHOR]

This paper presents a nonautonomous impulsive stochastic differential model of competition between species which contains the classic two-species competitive Lotka–Volterra model as a special case. With the theory of impulsive stochastic differential equations, a good understanding of stochastic permanence and extinction of system is obtained. Two specific numerical examples are presented to support the analytical results. Conclusions are also given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, an almost periodic system of impulsive differential equations used in modeling two-species in competition is considered. By the comparison between the solutions of impulsive system and the corresponding non-impulsive system and constructing Lyapunov function, sufficient conditions ensuring the uniformly asymptotic stability of a unique positive almost periodic solution of the system are derived. Finally, an example and its corresponding numerical simulations are presented to explain our theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper, we propose a periodic delayed two-species system modeling facultative mutualism. By using the method of coincidence degree and Lyapunov functional, easily verifiable sufficient conditions for the existence and globally asymptotic stability of positive periodic solutions of the above system. [ABSTRACT FROM AUTHOR]

Abstract: Based on two types of well known periodic single-species population growth models with time delay, we propose two corresponding periodic competitive systems with multiple delays. By using some new analysis techniques, we derive the same criteria for the existence and globally asymptotic stability of positive periodic solutions of the above two competitive systems. The easily verifiable criteria show that the existence and globally asymptotic stability of positive periodic solutions of two delayed nonautonomous competitive systems correspond to the existence and global stability of positive equilibrium of corresponding undelayed autonomous system. As an application, some special cases of the above systems are examined and some earlier results are extended and improved. Biological interpretations on the main results are also given. [Copyright &y& Elsevier]

Abstract: Inspired by a delayed single-species population growth model with so-called hereditary effect, we propose a delayed two-species system modelling “facultative mutualism” in this paper. Easily verifiable sufficient conditions are derived for the existence and globally asymptotic stability of positive periodic solutions of system by employing some analysis techniques and Lyapunov functional. Later, we extend the above results to a more general two-species facultative mutualism system involving multiple delays. Some previous results are extended and improved. Biological interpretations on the main results are also given. [Copyright &y& Elsevier]

Abstract: In this paper, we establish a periodic delayed predator–prey system according to two types of single-species growth population models with delay. By using some analysis methods, easily verifiable sufficient conditions for the existence and global asymptotic stability of positive periodic solution of the above system are derived. Some previous results are generalized and improved. Biological interpretations on the main results are also given. [Copyright &y& Elsevier]

• A novel nonlinear SVEIS epidemic model with temporary immunity is formulated. • The incidence rate of the model is a generalization of numerous nonlinear ones. • Global dynamics of the model proposed arempletely addressed by a new geometric criterion. • Global stability for several existing nonlinear SVEIS models is fully solved. • Model parameters are estimated to study the effects of multiple measures on the H1N1 pandemic. This work applies a novel geometric criterion for global stability of nonlinear autonomous differential equations generalized by Lu and Lu (2017) to establish global threshold dynamics for several SVEIS epidemic models with temporary immunity, incorporating saturated incidence and nonmonotone incidence with psychological effect, and an SVEIS model with saturated incidence and partial temporary immunity. Incidentally, global stability for the SVEIS models with saturated incidence in Cai and Li (2009), Sahu and Dhar (2012) is completely solved. Furthermore, employing the DEDiscover simulation tool, the parameters in Sahu and Dhar'model are estimated with the 2009–2010 pandemic H1N1 case data in Hong Kong China, and it is validated that the vaccination programme indeed avoided subsequent potential outbreak waves of the pandemic. Finally, global sensitivity analysis reveals that multiple control measures should be utilized jointly to cut down the peak of the waves dramatically and delay the arrival of the second wave, thereinto timely vaccination is particularly effective. [ABSTRACT FROM AUTHOR]

This work studies the problem of robust finite-time sampled-data control of linear systems subject to random occurring delays and its application to Four-Tank system. The input time delays are described by Bernoulli distributed white sequences. By employing a newly augmented Lyapunov–Krasovskii functional with Jensen’s inequality, free-weighting matrix method and Wirtinger-based double integral inequality approach, some less conservative delay-dependent conditions for finite-time sampled-data control law are obtained. Finally, to reveal the proposed technique, an application to a Four-Tank system is addressed to illustrate the effectiveness of developed techniques. [ABSTRACT FROM AUTHOR]

In this contribution, the dynamic behaviors of a turbidostat model with discrete delay are investigated. With the delay of digestion chosen to be a bifurcation parameter, we find that Hopf bifurcations occur when the delay crosses some critical values. Also, by employing the normal form theory and the center manifold theorem, we determine the type and stability of the bifurcating periodic solutions. Finally, numerical simulations are offered to support our results. [ABSTRACT FROM AUTHOR]