Difference equations population dynamics

The derivation of regular patterns of cellular behavior that can be assigned to certain defined subpopulations (or single cells) from measured data necessarily involves modeling approaches that allow assignment of mechanistic (e.g., metabohc) models to subpopulations with individual parameters or other variations. Further, such models must allow the tracking of individual cell s dynamic behavior even when they fluctuate between different populations. Two principal approaches can be distinguished systems of partial different equations (population balance systems, PBE) and stochastic cell ensemble models (CEMs)... 

It has been demonstrated that the whole photoexcitation dynamics in m-LPPP can be described considering the role of ASE in the population depletion process [33], Due to the collective stimulated emission associated with the propagation of spontaneous PL through the excited material, the exciton population decays faster than the natural lifetime, while the electronic structure of the photoexcited material remains unchanged. Based on the observation that time-integrated PL indicates the presence of ASE while SE decay corresponds to population dynamics, a numerical simulation was used to obtain a correlation of SE and PL at different excitation densities and to support the ASE model [33]. The excited state population N(R.i) at position R and time / within the photoexcited material is worked out based on the following equation ... 

The simulation of a continuous, evaporative, crystallizer is described. Four methods to solve the nonlinear partial differential equation which describes the population dynamics, are compared with respect to their applicability, accuracy, efficiency and robustness. The method of lines transforms the partial differential equation into a set of ordinary differential equations. The Lax-Wendroff technique uses a finite difference approximation, to estimate both the derivative with respect to time and size. The remaining two are based on the method of characteristics. It can be concluded that the method of characteristics with a fixed time grid, the Lax-Wendroff technique and the transformation method, give satisfactory results in most of the applications. However, each of the methods has its o%m particular draw-back. The relevance of the major problems encountered are dicussed and it is concluded that the best method to be used depends very much on the application. 

Equation (33) assumes that IV// is large compared to 2J (i.e., no electronic and vibrational recurrences). In addition, Eq. (33) deals only with population dynamics Interferences between different Franck-Condon factors are neglected. These interferences do influence the rate, and the interplay between electronic and vibrational dynamics can be quite complex [25], Finally, as discussed by Jean et al. [22], Eq. (33) does not separate the influence of pure dephasing (T-T) and population relaxation (Ti). These two processes (defined as the site representation [22]) can have significantly different effects on the overall rate. For example, when (T () becomes small compared to Eq. (33) substantially overestimates the rate compared to... 

Figure 6. Coherent population dynamics calculated using the density matrix equation (3) for different delays (a-c) of the laser pulses. Upper part Time evolution of the Rabi frequencies of both laser pulses. Lower part Calculated time evolution of the level populations for three different delays.

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. 

This difference equation is called a logistic map, and represents a simple deterministic system, where given a yi one can calculate the consequent point y% and so on. We are interested in solutions yi > 0 with 6 > 0. This model describes the dynamics of a single species population [32]. For this map, the fixed points y on the first iteration are solutions of... 

This is a difference equation widely used as a model in ecology and population dynamics (May (1974, 1987), Gleick (1987), Devaney (1992), Ott (1993)). Let Xn be the (normalized) number of individuals of some biological species present in year n. Then, the prescription (1.2.1) predicts the number of individuals in the following year n -I-1. The logistic map... 

A great deal of interest has been sparked by the realization of simple models for the description of population dynamics of organisms with nonoverlapping generations. May (1974), and May and Oster (1976) demonstrated that the use of difference equations such as that for population growth can yield a... 

This is an ordinary differential equation because it contains a function and its first derivative ordinary means that the only independent variable is time, t. The parameter r is the population s per capita growth rate and includes the difference of the per capita birth and death rates. The purpose of the equation is to determine the solution, Nil), that is, the population dynamics. In this case, the solution is simple to find because we know that only the exponential function is equal to its derivative ... 

Matrix models are sets of mostly linear difference equations. Each equation describes the dynamics of 1 class of individuals. Matrix models are based on the fundamental observation that demographic rates, that is, fecundity and mortality, are not constant throughout an organism s life cycle but depend on age, developmental stage, or size. Ecological interactions, natural disturbances, or pesticide applications usually will affect different classes of individuals in a different way, which can have important implications for population dynamics and risk. In the following, I will only consider age-structured models, but the rationale of the other types of matrix models is the same. For an example of this approach applied to pesticide risk assessment, see Stark (Chapter 5). 

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... 

The governing equations to model the particle formation dynamics are identical for the gas and Uquid phase, whereby of course, the reaction mechanisms and the kinetic parameters (e.g., for the mass transfer) differ. The population balance equation (PBE) can be considered as the master equation for formation of particles by both top-dovm and bottom-up methods (Eq. 5) ... 

This equation describes the evolution over time of the number of neurons per state (a density) depending on a flux term. To illustrate this, let us imagine a fluid inside a basin the higher the fluid is, the higher the molecular density is at this position. The evolution over time of this molecular density will depend of currents inside the fluid which represent the flux term. Our population density of neurons is very similar to this fluid the difference is that the current inside the fluid is driven by individual neuronal dynamics and neuronal interactions. 

The essence of optically controlled enantioselectivity in this scenario lies in Eq. (8.22) and the effect of these relationships on the dynamical equations for the level populations [Eq. (8.23)]. Note specifically that the equation for hD(t) is differ- ent than the equation for bL(t), due to the sign difference in the last term in Eq. 

In the previous chapter it was shown that the simple chemostat produces competitive exclusion. It could be argued that the result was due to the two-dimensional nature of the limiting problem (and the applicability of the Poincare-Bendixson theorem) or that this was a result of the particular type of dynamics produced by the Michaelis-Menten hypothesis on the functional response. This last point was the focus of some controversy at one time, inducing the proposal of alternative responses. In this chapter it will be shown that neither additional populations nor the replacement of the Michaelis-Menten hypothesis by a monotone (or even nonmonotone) uptake function is sufficient to produce coexistence of the competitors in a chemostat. This illustrates the robustness of the results of Chapter 1. It will also be shown that the introduction of differing death rates (replacing the parameter D by D, in the equations) does not change the competitive exclusion result. 

A more interesting question from the ecological point of view is that of how many different populations can coexist in an -vessel gradostat. [JST] shows that this number cannot exceed n. Some numerical simulations and conjectures appear in [BWu CB], but very little is known about this question. New techniques must be developed to handle this problem, since the resulting equations do not generate a monotone dynamical system when the number of competitors exceeds two (see [JST]). 

The population difference results in macroscopic magnetization that is measurable. It is not possible to measure the magnetization of an individual nuclear spin in the experiment. Hence, we must treat the dynamics either of the macroscopic magnetization, which is described by Bloch equations, or by an ensemble of nuclear spins, which require a Master equation [5]. 

The importance of initial conditions. Although the equations governing the populations are identical, a slight 1 /10000 difference in the initial conditions results in very different dynamics. 

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