Diffusion population dynamics

D. G. Aronson and H.F. Weinberger. Multidimensional nonlinear diffusion arising in population dynamics. Adv. Math., 30 33, 1978. 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

In the second chapter of this book, we shall represent and discuss a few examples of physical or chemical models for biological phenomena like transport across membranes, membrane excitation, control of metabolism, and population dynamic interaction between different species. All these models will be of the type of a reaction kinetic model, i.e., the model processes are chemical reactions and diffusion of molecules or may at least be interpreted like that. Thus, the physical background of the various models is irreversible thermodynamics of reactions and diffusion. 

To address some of these issues, Cheng et al. [35,44,120] followed a hybrid approach that uses a discrete, stochastic model based on CA to describe the population dynamics of migrating, interacting, and proliferating cells. The diffusion and consumption of a key nutrient or GF can be modeled by a... 

The diffusion-reaction problem in a scaffold that progressively fills with cells is mathematically more challenging than the classical isothermal diffusion-reaction problem that has been extensively studied in the engineering literature [131]. Similar to the active sites in a catalyst particle, cells act as sinks for the nutrient that is transported into the scaffold. In the case of a cellularized scaffold, however, these sinks move constantly, multiply and may even die by apoptosis or necrosis. At first, this problem may seem intractable because of the complex interplay between mass transport and cell population dynamics induced by the temporal and spatial variations of the cell distribution function y. 

Molecules can be classified both structurally and dynamically in terms of their intramolecular reactivity. The three structural classes are (1) those with a reactive geometry as the most stable or only conformation, (2) those with a reactive geometry that interconverts with other more stable but unreactive conformations, and (3) those with no reactive conformations populated. Dynamically, the three classes of excited states are (1) those in which interconversion among conformers is much faster than decay reactions of the individual conformers, such that conformational equilibrium is established before reaction (2) those in which conformational interconversions are all slower than decay, such that ground-state conformational preferences control photoreactivity and (3) those in which unreactive conformers undergo irreversible rotation into a reactive conformer, which reacts faster than it can rotate back into the unreactive geometry. In this third case, reaction of the excited state is subject to rotational control, the intramolecular equivalent of intermolecular diffusion control. 

It is conceivable that diffusion of kinks, or overdamped solitons, along the DNA could act to relax the FPA with a time dependence similar to that predicted for torsional deformation/31 32) High levels of intercalated dyes would be expected to alter both the equilibrium population of kinks and their mobility along the DNA. Hence, this question is addressed by examining the effect of intercalating dyes on the torsional dynamics. 

The q-space imaging method, which deals with signals only after long diffusion times, discards all information relevant to dynamic aspects of water diffusion and transport, especially the restriction of water transport by membrane and cell wall permeability barriers in cellular tissues. This information is contained in the functional dependence of the pulsed gradient spin echo amplitude S(q,A,x) on the three independent variables q, A, and x (x is the 90-180 degree pulse spacing) [53]. As the tool to explore the q and A dependence of S(q,A,x), generalized diffusion times and their associated fractional populations are introduced and a multiple exponential time series expansion is used to analyze the dependence [53]. 

The decay of C(t) in the midpoint state (either a or ft) is characterized by an even wider dynamical range. The short time scales in the cases of freely diffusing and repulsive surface-immobilized polypeptides are attributed to dynamics within the folded and unfolded sub-populations mentioned above. An... 

Interfacial hole transfer dynamics from titanium dioxide (Degussa P 25) to SCN has been investigated by Colombo and Bowman using femtosecond time-resolved diffuse reflectance spectroscopy [6c]. A dramatic increase in the population of trapped electrons was observed within the first few picoseconds, demonstrating that interfacial charge transfer of an electron from the SCN" to a hole on the photoexcited titanium dioxide effectively competes with electron-hole recombination (reactions (7.12) - (7.15)) on an ultrafast time scale [6c]. 

Van Nimwegen and Crutchfield (1999a) have constructed a theory for the optimization of evolutionary searches involving epochal dynamics. They showed that the destabilization of the epochs due to fluctuations in the finite population occurs near the optimal mutation rate and population size. Under these conditions, the epoch time is only constrained by the diffusion of the population to a neutral network boundary. Often the optimal parameters are very close to the region in which destabilization is an important effect. This emphasizes that, to utilize neutral evolution, it is important to tune the evolutionary parameters (such as mutation rate and population size) so that the time spent in an epoch is minimized without destabilizing the search. 

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