Matrix models linear difference equations

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). 

This equation is the equivalent of Eq. (9-12) for the induced dipole model but has one important difference. Equation (9-13), the derivative of Eq. (9-12), is linear and standard matrix methods can be used to solve for the p. because Eq. (9-12) is a quadratic function of p , while Eq. (9-54) is not a quadratic function of d and thus matrix methods are usually not used to find the Drude particle displacements that minimize the energy. 

The models discrete in space and continuous in time as well as those continuous in space and time, led many times to non-linear differential equations for which an analytical solution is extremely difficult or impossible. In order to solve the equations, simplifications, e.g. linearization of expressions and assumptions must be carried out. However, if this is not sufficient, one must apply numerical solutions. This led the author to a major conclusion that there are many advantages of using Markov chains which are discrete in time and space. The major reason is that physical models can be presented in a unified description via state vector and a one-step transition probability matrix. Additional reasons are detailed in Chapter 1. It will be shown later that this presentation coincides also with the fact that it yields the finite difference equations of the process under consideration on the basis of which the differential equations have been derived. 

The correlation functions of the partly quenched system satisfy a set of replica Ornstein-Zernike equations (21)-(23). Each of them is a 2 x 2 matrix equation for the model in question. As in previous studies of ionic systems (see, e.g.. Refs. 69, 70), we denote the long-range terms of the pair correlation functions in ROZ equations by qij. Here we apply a linearized theory and assume that the long-range terms of the direct correlation functions are equal to the Coulomb potentials which are given by Eqs. (53)-(55). This assumption represents the mean spherical approximation for the model in question. Most importantly, (r) = 0 as mentioned before, the particles from different replicas do not interact. However, q]f r) 7 0 these functions describe screening effects of the ion-ion interactions between ions from different replicas mediated by the presence of charged obstacles, i.e., via the matrix. The functions q j (r) need to be obtained to apply them for proper renormalization of the ROZ equations for systems made of nonpoint ions. 

The above model was solved numerically by writing finite difference approximations for each term. The equations were decoupled by writing the reaction terms on the previous time steps where the concentrations are known. Similarly the equations were linearized by writing the diffusivities on the previous time step also. The model was solved numerically using a linear matrix inversion routine, updating the solution matrix between iterations to include the proper concentration dependent diffusivities and reactions. 

The matrix approach to the solution of a set of simultaneous linear equations is entirely general. Requirements for a solution are that there be a number of equations exactly equal to the number of parameters to be calculated and that the determinant D of the X matrix be nonzero. This latter requirement can be seen from Equations 5.14 and 5.15. Elements a and c of the X matrix associated with the present model are both equal to unity (see Equations 5.10 and 5.7) thus, with this model, the condition for a nonzero determinant (see Equation 5.12) is that element b (xu) not equal element d (x12). When the experimental design consists of two experiments carried out at different levels of the factor xt (xn = x12 see Figure 5.1), the condition is satisfied. 

Linear prediction and state-space methods are grouped together here because, although the philosophy behind the two methods is different, the mathematics involved is very similar. In both cases, the data-fitting problem is made linear by constructing a matrix from the observed data points, and the model equation is then solved by linear means. The nonlinear model parameters are... 

A powerful tool for EM modeling and inversion is the integral equation (IE) method and the corresponding linear and nonlinear approximations, introduced in the previous chapter. One important advantage which the IE method has over the finite difference (FD) and finite element (FE) methods is its greater suitability for inversion. Integral equation formulation readily contains a sensitivity matrix, which can be recomputed at each inversion iteration at little expense. With finite differences, however, this matrix has to be established anew on each iteration at a cost at least equal to the cost of the full forward simulation. 

A lumped mass, finite element or finite difference scheme may be used to model the lines. The line is decomposed into a number of straight elements (bars) with linear shape function. The distributed mass plus hydrodynamic added mass is lumped at the element nodes. The hydrodynamic damping is included for the relative motion between the line and the fluid. Damping levels vary significantly depending on water depth, line makeup, offsets, and top-end excitation. Quite often a modified Morison equation is used to represent the environmental effect. At each time step, a standard set of matrix equation is developed composed of the inertia, damping, and stiffness matrix. 

The hydrodynamic model solves Laplace s equation in 3D to obtain the velocity potential function. The moving body is fixed, and an incoming flow is imposed with the same velocity in the opposite direction at the far upstream boundary. A matrix system is assembled with Laplace s equation for each tetrahedron and the different boundary conditions. As the problem is linear, the potential function is solved by a matrix inversion represented by... 

Based on the assumptions made during the derivation of equation (6.15), it is imperative to choose MFI(T, 0) > MFI(T, (p). In the case of filled systems, this condition is naturally satisfied when the polymeric matrix is taken as the reference medium. Equation (6.15) predicts that a plot of 1/log mh vs. l/4> should be linear, and the propriety of this model has been examined quantitatively in the light of the reported experimental data. Existing viscosity data in the literature available for all filled systems are in the form of viscosity vs. shear rate or shear stress vs. shear rate curve. In each case the data are transformed into specific MFI values using the method discussed in Shenoy and Saini [99]. Figures 6.11-6.16 show plots of l/log< MH vs. l/ for different filled... 

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