Population dynamics oscillations

Keywords Porous media Hydrogen Reactive transport Bacteria Methanogenic microorganisms Population dynamics Oscillations Chemotaxis... 

In order to relate the dressed state population dynamics to the more intuitive semiclassical picture of a laser-driven charge oscillation, we analyze the induced dipole moment n) t) and the interaction energy V)(0 of the dipole in the external field. To this end, we insert the solution of the TDSE (6.27) into the expansion of the wavefunction Eq. (6.24) and determine the time evolution of the charge density distribution p r, t) = -e r, f)P in space. Erom the density we calculate the expectation value of the dipole operator... 

Figure 6.10 Ultrafast efficient switching in the five-state system via SPODS based on multipulse sequences from sinusoidal phase modulation (PL). The shaped laser pulse shown in (a) results from complete forward design of the control field. Frame (b) shows die induced bare state population dynamics. After preparation of the resonant subsystem in a state of maximum electronic coherence by the pre-pulse, the optical phase jump of = —7r/2 shifts die main pulse in-phase with the induced charge oscillation. Therefore, the interaction energy is minimized, resulting in the selective population of the lower dressed state /), as seen in the dressed state population dynamics in (d) around t = —50 fs. Due to the efficient energy splitting of the dressed states, induced in the resonant subsystem by the main pulse, the lower dressed state is shifted into resonance widi die lower target state 3) (see frame (c) around t = 0). As a result, 100% of the population is transferred nonadiabatically to this particular target state, which is selectively populated by the end of the pulse.

Figure 6.19 Quantum dynamics simulations for the two distinct situations of selective population of the (a) upper and (b) the lower target state. The frame (iv) shows the population dynamics induced by the shaped laser field pictured in (iii). The remaining panels depict (ii) the oscillations of the laser field together with the induced dipole moment and (1) the induced energetic splitting in the X-A-subsystem along with the accessibility of the target states. Gray backgrounds highlight the relevant time windows that are discussed in the text.

Figures 6a-c show the population dynamics encountered in a three-level system (see Fig. 4) interacting resonantly with two Fourier-transform-limited laser pulses with three different delay times between the two pulses. The calculation was done assuming that the chosen Rabi frequencies fulfill the relation > 1/pulse duration) in all three cases. This relation ensures that the typical time for a Rabi oscillation of the population in an isolated two-level system is shorter than the pulse duration. Ionization from level 2 was introduced as a fast laser intensity-dependent decay of level 2 [6, 60], and resonant laser frequencies were assumed.

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Fig. 1 Population dynamics of a damped harmonic oscillator. The populations of the ground state (n = 0) up to the third excited state (n = 3) are shown while initially all population is in the third excited state. The parameters are o-v = tao/2, / = 0.544ivo, and (3 = l/ivo. The results for the TNL theory are shown by the solid curve, those for the TL approach by the dotted curve and those of the Markovian limit by the dashed curve. (Reproduced from Ref. [29]. Copyright 2004, American Institute of Physics.)...

The recursive state observed here is not necessarily a fixed point with regard to the population dynamics of the chemical concentrations. In some case, the chemical concentrations oscillate in time, but the nature of the oscillation is not altered by the process of cell division. 

Oscillations can also arise from the nonlinear interactions present in population dynamics (e.g. predator-prey systems). Mixing in this context is relevant for oceanic plankton populations. Phytoplankton-zooplankton (PZ) and other more complicated plankton population models often exhibit oscillatory solutions (see e.g. Edwards and Yool (2000)). Huisman and Weissing (1999) have shown that oscillations and chaotic fluctuations generated by the plankton population dynamics can provide a mechanism for the coexistence of the huge number of plankton species competing for only a few key resources (the plankton paradox ). In this chapter we review theoretical, numerical and experimental work on unsteady (mainly oscillatory) systems in the presence of mixing and stirring. 

To summarize, the field exhibits slow and fast oscillations that correspond to the timescales of vibrational and electronic motion, respectively. It is interesting to change the objective and aim at a selective population of the second excited state 2). This is a more demanding problem because the ground state is not directly coupled to the target state and thus transitions have to proceed via the intermediate state. The LCT field and the population dynamics for the selective... 

A fragmented liquid core is simulated by injecting large drops which break up into smaller and smaller product droplets, until the latter reach a stable condition. The primary breakup, that is, the first drop breakup after injection, is modeled by delaying the initial drop breakup in accordance with experimental correlations. The drop distortion and the breakup criterion are obtained from Taylor s drop oscillator. The properties of the product droplets are derived from principles of population dynamics and are modeled after experimentally observed droplet breakup mechanisms. 

Population dynamics involves autocatalysis similar to that discussed in chapter on chemical oscillations. Lotka-Volterra model was proposed involving this concept which involves following steps ... 

The examples to be presented illustrate the diversity of fields of applications, but they are mentioned in outline form only. Many biological phenomena used to be modelled by real or formal kinetic models. A biochemical control theory that is partially based on non-mass-action-type enzyme kinetics seems to be under elaboration, and certain aspects will be illustrated. A few specific models of fluctuation and oscillation phenomena in neurochemical systems will be presented. The formal structure of population dynamics is quite similar to that of chemical kinetics, and models referring to different hierarchical levels from elementary genetics to ecology are well-known examples. Polymerisation, cluster formation and recombination kinetics from the physical literature will be mentioned briefly. Another question to be discussed is how electric-circuit-like elements can be constructed in terms of chemical kinetics. Finally, kinetic theories of selection will be mentioned. 

Figure 6 shows the results for the more challenging model. Model IVb, comprising three strongly coupled vibrational modes. Overall, the MFT method is seen to give only a qualitatively correct picture of the electronic dynamics. While the oscillations of the adiabatic population are reproduced quite well for short time, the MFT method predicts an incorrect long-time limit for both electronic populations and fails to reproduce the pronounced recurrence in the diabatic population. In contrast to the results for the electronic dynamics, the MFT is capable of describing the almost undamped coherent vibrational motion of the vibrational modes. 

A further important property of a MQC description is the ability to correctly describe the time evolution of the electronic coefficients. A proper description of the electronic phase coherence is expected to be particularly important in the case of multiple curve-crossings that are frequently encountered in bound-state relaxation dynamics [163]. Within the limits of the classical-path approximation, the MPT method naturally accounts for the coherent time evolution of the electronic coefficients (see Fig. 5). This conclusion is also supported by the numerical results for the transient oscillations of the electronic population, which were reproduced quite well by the MFT method. Similarly, it has been shown that the MFT method in general does a good job in reproducing coherent nuclear motion on coupled potential-energy surfaces. 

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