| Noise and time delay are two elemental factors that should be considered in modeling many real systems, and they are sources of complex behaviors. In this thesis, the phenomena of noise enhanced stability (NES) and stochastic resonance (SR) were studied in time-delayed stochastic systems. The whole thesis consists of two parts. Firstly, the background and the current progress on the study of the effects of noise and time delay in nonlinear systems were briefly summarized. The methods of analytical approximation and stochastic simulation for time-delayed stochastic systems were also introduced. Then, by virtue of small delay approximation and stochastic simulation, in terms of mean first-passage time (MFPT) and signal-to-noise ratio (SNR), we investigated the effects of noise correlation and time delay on the NES and SR phenomena in different systems. In particular, the following results have been presented in this thesis:The NES phenomenon was studied in a cubic potential system with cross-correlated noises and different time delays. The results show that:(1) Both internal and external noises can induce the NES effect. and the increasing of the noise correlation strength plays an opposite role on the NES effect induced by external and internal noises:(2) The double NES effect induced by the additive noise (i.e., MFPT as a function of the additive noise intensity shows a two-peak structure) can be observed when the correlation between noises is positive, and the increasing of the noise correlation strength has different impacts on the NES effect occurring at different additive noise intensities:(3) There is a delay-induced critical behavior in the NES effect caused by additive and multiplicative noises for the time delay only existing in the deterministic force and the one simultaneously appearing in the deterministic and stochastic forces. That is, there is a critical value of the delay time, below and above which the increasing of the delay time plays an opposite role on the NES effect;(4) The time delay only existing in the stochastic force has an opposite effect on the NES phenomenon induced by external and internal noises.The effects of noise correlation, different time-delayed feedbacks and pa-rameters of periodic force on the NES phenomenon were investigated in a peri-odically modulated bistable system. The results indicate that:(1) The additive noise always induces the NES effect, whereas the multiplicative noise can cause the NES effect only for the strong correlation between noises. The noise cor-relation has different roles in the NES effect induced by multiplicative and additive noises:(2) The linear and nonlinear time-delayed feedbacks play an opposite role on the NES effect for the case of the presence of a single additive noise. However, there is a delay-induced critical behavior in the NES effect for these two time-delayed feedbacks, in which the critical value of the delay time is different and the increasing of the delay time has different impacts on the NES effect when the delay time is larger or less than its critical value:(3) For the global time-delayed feedback, the increasing of the delay time just suppresses the NES effect due to the competition between linear and nonlinear time-delayed feedbacks;(4) The NES effect can be weakened by increasing the amplitude of the periodic force and enforced by increasing the frequency.The effects of correlated noises and time delay on the spontaneous ex-tinction process of tumor cells in the presence of immunization were analyzed. The results show that:(1) The spontaneous extinction of tumor cells can be dramatically delayed by increasing the noise intensities due to the NES effect caused by external and internal noises;(2) The increasing of the noise correla-tion strength speeds up the extinction of tumor cells for smaller noise intensities and slows it down for larger noise intensities because of the combined action of the noise correlation and the NES effect;(3) The time delay always speeds up the extinction by weakening the NES effect.The control role of a linear delayed feedback on the SR phenomenon in-duced by a multiplicative signal was investigated in a symmetric bistable system driven by cross-correlated noises. The results indicate that:(1) Adjusting the delay time and the feedback strength of the linear delayed feedback can weaken or enforce the SR phenomenon. Moreover, the impacts of this delayed feed-back on the SR phenomenon not only depends on its characteristics(positive or negative feedback), but also on the initial condition of the system due to the symmetry-breaking effect caused by the correlation between noises;(2) The delay time-and the feedback strength-caused critical effects in the SR phe-nomenon can be observed under a certain condition.As an example of asymmetric bistable systems, we investigated the SR phe-nomenon induced by a multiplicative signal in a gene transcriptional regulatory system with time delay and correlated noises. It is revealed that:(1) SNR as a function of the multiplicative noise intensity shows a transition from two peaks to one peak when the correlation between noises becomes negative from posi-tive, implying the double SR phenomenon induced by the multiplicative noise in the case of positive correlation between noises. However, the increasing of the noise correlation strength always weakens the SR phenomenon induced by the additive noise:(2) The increasing of the delay time can weaken or enforce the SR phenomenon, depending on the noise correlation strength;(3) The time delay also causes the same transition of SNR as that induced by the noise cor-relation, and there is a particular value of the noise correlation strength, at which the delay-induced critical behavior in the SR phenomenon induced by the multiplicative noise occurs:(4) Both noise correlation and time delay can cause a resonance-like behavior of SNR (i.e.. SNR as a function of the noise correlation strength and the delay time shows a non-monotonic behavior with a maximum) for a given condition. Moreover, there is a double critical behavior in SNR with respect to the noise correlation strength and the delay time due to the co-existence of these two resonance-like effects.The effects of noise correlation and time delay in the random response of a tumor cell growth system to the periodic treatment were studied. The results indicate that:(1) The response of tumor cells to the periodic treatment can be enhanced by the SR mechanism under the action of external and internal noises;(2) The increasing of the noise correlation strength can restrain the SR phenomenon while the increasing of the delay time can enhance it.The effects of noise correlation and time delay on the SR phenomenon in-duced by a multiplicative signal were investigated in a reduced one-dimensional FitzHugh-Nagumo model. The results show that:(1) There is a noise correlation-induced critical behavior in the SR phenomenon caused by additive and mul-tiplicative noises:(2) The increasing of the delay time enhances the SR phe-nomenon induced by additive and multiplicative noises;(3) For the appropriate parameters, both noise correlation and time delay can cause a resonance-like behavior of SNR. where SNR versus the noise correlation strength shows a transition from two peaks to one peak as the delay time increases, and there is a noise correlation-caused critical behavior in the delay-induced resonance-like effect.
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