Population oscillations

Fig. 13. The anisotropy parameters for the OH(X, = 0,1V) products. The two oscillation sites correspond exactly to the population oscillations exhibited in Fig. 11.

Traub, R. D., Miles, R. and Wong, R. K. Model of the origin of rhythmic population oscillations in the hippocampal slice. Science 243 1319-1325,1989. 

One of the interesting features of difference equations such as these is that they are much more unstable than the differential equations we have used throughout this text. If the reproduction rate exceeds a certain value, the system is predicted to exhibit population oscillations with high and low populations in alternate years. At higher birth rates, the system exhibits period doubling, and at even higher birth rates, the system exhibits chaos, in which the population in any year cannot be predicted from the population in a previous year. Similar phenomena are observed in real ecological systems, and even these simple models can capture this behavior. 

Figure 21 Anti-Stokes transients for acetonitrile (ACN) with C-H stretch pumping (left) and with combination band (C=N + C-C stretch) pumping (right). With combination band pumping, population builds up instantaneously in the C=N stretch and C-C stretch, the decay of the C=N stretch is about 10 times faster, and a population oscillation is seen in the C-C stretch. (From Ref. 47.)...

This is an up-conversion process (57). The C=N stretch and C-C stretch excitations are in fast equilibrium with C-H stretch excitations, but the C-H stretch decays with a 5 ps time constant. That explains the C=N stretch decay in 5 ps. The population oscillation in the C-C stretch data occurs as follows. First we see C-C stretch excitation pumped by the laser. This is actually a sign of combination band excitation (C=N + C-C stretch). Then the C-C stretch starts to be annihilated by the 5 ps up-conversion and C-H stretch decay process. C-H stretch decay produces C-H bending excitations, which subsequently repopulate the C-C stretch, causing the second rise in the C-C stretch transient (47). 

Even though population oscillations - which can take place in either a regular (Lotka 1910) or chaotic (May 1976) manner - may still occur, there will be some kind of labile equilibrium among biochemical autocatalysis, increased use of resources and the various... 

The times ta and r/> spent at each static detuning value A a and A/> arc proportional to the periods Ta and Tb of the static population-oscillations, respectively. The coefficients linking 754 and tb to Ta and Tb depend on the difference A4 — A/> but not on the initial value A a or A b- The dynamic asymptotic value of the excited-state population is mainly controlled by the initial value of the detuning whereas the oscillations amplitude is controlled by... 

Figure 3. Dressed state basis for atomic collisions. A - The square of the transfer matrix between the excitation Fock state and the dressed state bases for N = M = 100. Darker areas correspond to larger probability. B - Damping spectrum between the N = M = 5000 manifold and the N = 4999, M = 5000 manifold. Dashed line k = 3.2, dotted line k = 1.6 and solid line k = 0.7, q = k/ /2. Inset energy-conserving surfaces for the two center frequencies of the solid line and for elastic damping from mode k (dashed line). The splitting in the spectrum is due to the nonlinear population oscillations due to three-wave mixing of the modes in the time domain. This behavior is analogous to that of a strongly driven two level atom (Mollow splitting).

The population oscillates with the Rabi frequency of the g) —. v) transition and at certain times Ps(t) = 1, indicating that all the population is in the symmetric state. This happens at times... 

The solutions of (3.68)-(3.69) for positive parameters and generic initial conditions are oscillations, of amplitude fixed by the initial conditions, around the fixed point Z — a /a2, P = b2/b. The equation of this family of closed trajectories is ai In Z + b2 In P — biP — a2Z = constant. The oscillations are suggestive of the population oscillations observed in some real predator-prey systems, but they suffer from an important drawback the existence of the continuous family of oscillating trajectories is structurally unstable systems similar to (3.68)-(3.69) but with small additional terms either lack the oscillations, or a single limit cycle is selected out of the continuum. Thus, the model (3.68)-(3.69) can not be considered a robust model of biological interactions, which are never known with enough... 

First, assume for tlie moment that the mean field is zero. Tlien all the elements iii the population oscillate independently, and theii contributions to the mean field nearly cancel each other. Even if the frequencies of these ascillations are identical, bnt their pliases are independent, the average of the outputs of all elements of the ensemble is small if compared witli the amplitude of a single oscillator. (According to the law of large mmibers, it tends to zero when tlie uuniber of interacting oscillators tends to infinity the fluctuations of the mean field are of the order Thus, the... 

FIGURE 6.20.3 One-hundred-year record of population cycles of the snowshoe hare Lepus americanus) and the Canada lynx (Lynx canadensis), based on pelt records of the Hudson s Bay Company in Canada. Lack of anticipation in predator-prey systems lead to unstable population oscillations. (From Gotelli, N.J., A Primer of Ecology, Sinauer Associates, Sunderland, MA, 1998. With permission.)... 

The time evolution of the atomic system can be calculated by the Liouville equation. It is found that population oscillations, represented by the diagonal elements of the density matrix, show up between states with identical values of mj = m + mg. Their oscillation frequencies are exactly given by the fine structure splitting. In our case, due to the excitation only coherence between the substates (mqlms) = (ll- ) and (0 ) ig ex-... 

Other coherent interactions include optical nutation and free induction decay, in which the population oscillates be-... 

This is the usual magnetic resonance lineshape for transitions in a two-level system without damping. At resonance the population oscillates sinusoidally between the two states (this is known as Rabi oscillation). A n-pulse is an on-resonance pulse with 2bx = Tt, which transfers all the population from state (0) to state (c). In Section 15.4.3 we will discuss how this can be used in a molecular beam to map out the fields along the beamline. An on-resonance 7r/2-pulse 2bz = n/2) drives the transition only half-way, creating an equal superposition of states (0) and (c) with a definite relative phase. The density matrix element describing this coherence at the end of... 

The general Rabi flopping frequency 12 gives the frequency of population oscillation in a two-level system in an electromagnetic field with amplitude Eq. 

This last equation shows that, in a two-level system interacting with a monochromatic continuous laser field, the population oscillates between the two states. The amplitude of the oscillations depends on the detuning. In particular, the populations oscillate between 0 and 1 when the laser field is exactly resonant. The period of the oscillations depends both on the amplitude and on the detuning. This phenomenon is called oscillations. 

However, in the case of incoherent interaction, the picture looks simpler, for there are no level population oscillations at all. In that case, to describe the evolution of level populations, use can be made as before of rate equations, such as eqn (2.61), but the quantities ni and ri2 should now be understood to be the total populations of each level. The difference in level populations given by eqn (2.48) should then be modified for naturally polarized light as follows ... 

---