The Study Of Stochastic Resonance In Tumor-immune System With Correlatively Bounded Noises And Anomalous Transport Of Inertial Brownian Particles

The stochastic resonance (SR) of a tumor-immune system and the diffusive transport of underdamped Brownian particles are studied in this thesis. Firstly, the research status of the SR and the transport of underdamped Brownian par-ticles driven by time-periodic forces in a spatially symmetric periodic potential is summarized. Secondly, the SR in a mathematical model of tumor-immune interaction under a periodic immunotherapy treatment is investigated. Thirdly, effects of delay on transport in an active Brownian particle (ABP) are reported. Moreover, the anomalous diffusion and enhancement of diffusion in a vibrational motor are studied. Finally, uphill anomalous transport in a deterministic ABP is investigated. The main results are as follows:In the first part of the work, a mathematical model of the tumor-immune interaction, subject to a periodic immunotherapy treatment (imitated by a peri-odic signal), correlative and bounded stochastic fluctuations and time delays, is investigated by numerical simulations for its signal power amplification (SPA). The immunotherapy is one of the most recent approaches in cancer therapy. Within the tailored parameter regime, the synchronous response of tumor growth to the immunotherapy, stochastic resonance (SR), versus both the noises and delays is obtained. The details are as follows (i) the peak values of SPA versus the noise intensity (A) in the proliferation term of tumor cells decrease as the frequency of periodic signal increases, i.e. an increase of the frequency restrains the SR;(ii) an increase of the amplitude of periodic signal restrains the SR versus A, but boosts up the SR versus the noise intensity B in the immune term;(iii) there is an optimum cross-correlated degrees between the two bounded noises, at which the system exhibits the strongest SR versus the delay time Tα (the reaction time of tumor cell population to its surrounding environment constraints);(iv) upon increasing the delay time Tα, double SR versus the delay time Tβ (the time taken by both the tumor antigen identification and tumor-stimulated proliferation of ef-fectors) emerges. These results may be helpful for an immunotherapy treatment for the sufferer.In the second part of the work, the transport properties of the active Brownian particle with a time-delayed feedback and an external bias are investigated theoret-ically. By virtue of the perturbation theory for small delay, analytical expressions for the mean velocity and effective diffusion coefficient are derived. There exists a critical absolute value of the bias, below and above which the delay, respectively, enhances and weakens the diffusion, for a fixed noise intensity. The effects of delay above are more pronounced for weaker noise. These results are further verified by the direct numerical simulations.In the third part of the work, the diffusion properties of a vibrational motor, in which an additional time-dependent driving brings the system out of equilib-rium and the other time-periodic driving fills the role usually played by noise, are investigated. Within the tailored parameter regime, the diffusion coefficient evolving after considerably long time develops a sharp peak closely related to the inflection points of mean velocity, upon an increase of the damping constant. The diffusion peak here depends on the superdiffusion motion of the particles. Also, the negative velocity (for a positive bias) depending the subdiffusive motion, i.e. dispersionless transport in the asymptotic long time limit, is obtained. Moreover, the enhancement of diffusion phenomenon can be observed. These results may be helpful for separating particles under some conditions by adjusting the external forces applied.In the final of work, we investigate the transport of a deterministic Brownian particle theoretically, which moves in simple one-dimensional, symmetric peri- odic potentials under the influence of both a time periodic and a static biasing forces. The physical system employed contains a friction coefficient that is speed-dependent. Within the tailored parameter regime, the absolute negative mobility, in which a particle can travel in the direction opposite to a constant applied force, is observed. This behavior is robust and can be maximized at two regimes upon variation of the characteristic factor of friction coefficient. Further analysis reveals that this uphill motion is subdiffusive, dispersionless transport in the asymptotic long time limit. Whereas, the most of the downhill motion evolves chaotically with the normal diffusion.