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Abstract: In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of . The function describing stability of HB is periodic with respect to . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data. [Copyright &y& Elsevier]

Abstract: A bounded rationality duopoly game with delay is formulated. Its dynamical evolution is analyzed. The existence of an economic equilibrium of the game is derived. The local stability analysis has been carried out. The analysis showed that firms using delayed bounded rationality have a higher chance of reaching a Nash equilibrium point. Numerical simulations were used to show bifurcation diagrams and phase portraits. [Copyright &y& Elsevier]

Abstract: Recently, there were several differential delay models of the glucose–insulin system analyzed in the literature. One of the mostly used models is generally known as the which was first published in 1979 and modified in 1986. This minimal model has been challenged by De Gaetano and Arino in 2000 from both physiological and modeling aspects. Instead, they proposed a new and mathematically more reasonable model, called . Following dynamic model, Li, Kuang and Li proposed a more popular, general and realistic model. Panunzi, Palumbo and De Gaetano believed that it requires a reasonably simple model, with as few parameters to be estimated as practicable, and with a qualitative behavior consistent with physiology, and they proposed a discrete single delay model (SDM). These models were proved to have the reasonable qualitative behavior, such as uniform persistence and global stability of equilibrium. In this paper we investigate a more general model, which includes SDM and one of the models in Li et al. (2001) as its special cases. The model is shown to admit globally stable equilibrium under certain conditions of the parameters. For the two special models (SDM and one in ), it is shown that these conditions are slightly better. It is also shown that the model admits oscillating behavior due to the existence of Hopf-bifurcation. [Copyright &y& Elsevier]

Abstract: In this paper, a reaction-diffusion predator–prey system with nonlocal delay due to the gestation of the predator and homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each boundary steady state is established. The existence of Hopf bifurcations at the positive steady state is also discussed. Sufficient conditions are derived for the global stability of the positive steady state and the semi-trivial steady state of the proposed problem by using the method of upper–lower solutions and its associated monotone iteration scheme, respectively. Numerical simulations are carried out to illustrate the main results. [Copyright &y& Elsevier]

Abstract: Mathematical models are used to determine if infection wave fronts could occur by traveling geographically in a loop around a region or continent. These infection wave fronts arise by Hopf bifurcation for some spatial models for infectious disease transmission with distributed-contacts. Periodic traveling waves are shown to exist for the spatial analog of the SIRS endemic model, in which the temporary immunity is described by a delay, but they do not exist in a similar spatial SIRS endemic model without a delay. Specifically, we found that the ratio of the delay in the recovered class and the average infectious period must be sufficiently large for Hopf bifurcation to occur. [Copyright &y& Elsevier]

This investigation is aimed at describing the reduction of a class of nonlinear delay differential equations having single and multiple time delays. We use the theory of delay dynamical systems, whose underlying mathematics is summarized. The delay versions of Duffing, van der Pol and Tobias equations are considered as illustrative examples. [Copyright &y& Elsevier]

In this paper, we set up the dynamical models for an industrial system by means of ordinary or delay differential equations. By the qualitative analyses to the equilibrium points of the differential equations, we indicate how the parameters influence the equilibrium states of the system. [Copyright &y& Elsevier]