Population dynamics growth rates

The point is that the same population dynamics are applicable to remediation systems. The principal difference is that in a soil/water system, one has essentially the growth rate of the bacteria as a limiting condition. This is also akin to another type of system known as the Sequencing Batch Reactor or SBR. 

The first method, the method of moments, is restricted to MSMPR crystallizers with size-independent growth or very simple size-dependent growth rate kinetics. Depending on the control demand, several process outputs can be chosen for the control algorithm. When population densities, or the number or mass of crystals in a size range are to be controlled, the method of moments can not be used because it reveals information on the dynamics of the moments of the crystal size distribution only. 

Using such microbial population dynamics factors, we are now in a position to estimate biodegradation rates for compounds supporting growth like p-cresol. In the situation depicted in Fig. 17.14, we have for the early part of the experiment ... 

Since the proposal of the hypercycle, population dynamics of molecules for such catalytic networks have been developed. However, the hypercycle itself turned out to be weak against parasitic molecules—that is, those which replicate, catalyzed by a molecule in the cycle, but do not catalyze those in the cycle. In contrast to the previous mutant, the growth rate of the population of these molecules is again the product of the populations of two species, and such parasitic molecules can invade. 

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 ... 

Neufeld et al. (2002a) have shown that this behavior can be explained by the interplay between excitable plankton population dynamics and chaotic flow, similarly to the excitable behavior described in the previous section. In a chaotic flow a steady bloom filament profile can be generated, that does not decay until it invades the whole computational domain as an advectively propagating bloom. The condition for the existence of the steady bloom filament solution in the corresponding one-dimensional filament model is that the rate of convergence, quantified by the Lyapunov exponent, should be slower than the phytoplankton growth rate, but faster than the zooplankton reproduction rate. In this case the phytoplankton does not became diluted by the flow and the zooplankton is thus kept at low concentration unable to graze down the bloom filament. 

The underlying processes behind these dynamics are not well understood. One descriptive model that has been proposed for bark beetles and other eruptive herbivores is known as dual equilibria theory (Fig. 4.5). According to this view, population growth rates follow the standard discretized nonlinear curve... 

Quite clearly, the growth of a predator population is in some way dependent upon the abundance of its prey. The most frequently cited model of predator-prey dynamics is the set of linked, non-linear differential equations known as the Lotka-Volterra equations (1 ). This model assumes that in the absence of predator, the prey grows exponentially, while in the absence of prey the predator dies exponentially, and that the predator growth rate is directly proportional to the product of the prey... 

One method is to solve the population balance equation (Equation 64.6) and to take into account the empirical expression for the nucleation rate (Equation 64.10), which is modified in such a way that the expression includes the impeller tip speed raised to an experimental power. In addition, the experimental value, pertinent to each ch ical, is required for the power of the crystal growth rate in the nncleation rate. Besides, the effect of snspension density on the nucleation rate needs to be known. Fnrthermore, an indnstrial suspension crystallizer does not operate in the fully mixed state, so a simplified model, such as Equation 64.6, reqnires still another experimental coefficient that modifies the CSD and depends on the mixing conditions and the eqnipment type. If the necessary experimental data are available, the method enables the prediction of CSD and the prodnction rate as dependent on the dimensions of the tank and on the operating conditions. One such method is that developed by Toyokura [23] and discussed and modified by Palosaari et al. [24]. However, this method deals with the CTystaUization tank in average and does not distinguish what happens at various locations in the tank. The more fundamental and potentially far more accurate simulation of the process can be obtained by the application of the computational fluid dynamics (CFD). It will be discussed in the following section. 

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