Interacting populations

A.D. Bazikin, Mathematical Biophysics of Interacting Populations (Nauka, Moscow, 1985). 

Despite these obvious difficulties we decided to develop a model of the effects of DBS using a systems biology approach. Our aim is to propose a large scale and multiscale computational model of DBS effects in PD based on physiology of individual neurons, population of neurons, and interacting populations of neurons. 

To illustrate how to proceed using the cumulant generating functions, the well-known two-compartment model and the enzymatic reaction will be presented as examples of linear and nonlinear systems, respectively. In these examples, there are two interacting populations (m = 2) and the cumulant generating function is... 

As we have seen in the text, the equations governing interacting populations in a chemostat-like environment eventually take the form... 

Any effects on populations may ultimately be manifested as effects on communities because, by definition, communities are collections of interacting populations of several species (e.g., an aquatic community may consist of populations of fish, worms, plants, insects). Individual populations within a community may interact by competing for resources (food, habitat, etc.) or by predator/prey relationships. Environmental contaminants can affect the structure of communities as well as the interactions of species within them. For example, it is well known that exposure to chemicals may cause a reduction in community diversity (e.g., relative number of species), and changes in community composition. In addition, the trophic structure of fish and invertebrate communities may also be affected by exposure to anthropogenic chemicals. Changes in community structure and diversity may be determined by field sampling or manipulative studies. Alternatively, computer simulations using food web or linked population models may be used to assess community-level effects. 

Figure 1 Molecular orbital diagram for an octahedral complex with o--only Interactions. Population of the molecular orbitals by 18-electrons results In occupation of six bonding and three nonbonding orbitals, with no occupation of antibonding orbitals. As such, it provides a simple rationalization for the concept of the 18-electron rule a 20-electron complex with occupied antibonding orbitals would be expected to be unstable and dissociate a ligand to transform to an 18-electron complex, while a 16-electron complex would be able to bind an additional ligand and transform to an 18-electron complex. However, the nature of the molecular orbital diagram depends critically on the molecular structure (for example, see Figure 2) and so this explanation is necessarily over simplistic.

Linear diffusion satisfactorily describes the transport mechanism for a single population. For interacting populations, linear diffusion terms imply that the populations are able to mix completely, with the movement of one cell type unaffected by the presence of cells of the other type. The reality is exactly the opposite. Cell movement is typically halted by contact with another cell. This phenomenon is known as contact inhibition and is very well documented for many types of cells. Sherratt introduced a phenomenological model to account for contact inhibition [402]. Consider the interaction between normal and tumor cells with concentrations pm(xj) and Pt(x, t), respectively. The overall cell flux of both populations is given by x(Pn + Pt)- a fraction Pn/(Pn + Pt) of this flux corresponds to normal cells, so that the flux of normal cells is - [pn/(Pn + Pr)] x(Pn + Pr)> a similar expression for the flux of tumor cells. These expressions indicate that the movement of one population is inhibited by the presence of the other. The system of dimensionless reaction-diffusion equations reads [402]... 

The Nyquist plot in Figure 9 is a classical Debye response. Debye response is the dielectric response of an ideal, non-interacting population of dipoles to an alternating external electric field. It involves only a single relaxation time because the dipoles do not interact with each other or induce other dipoles surrounding it. They aligned tliemselves with the direction of external field within the same time. This relaxation model was introduced by Peter Debye in 1913. 

The ecosystem is the basic concept of Ecodynamics. It can be defined as a set of interacting populations of different species, the population of a species being defined as a set of objects, or elements of some kind, which are similar enough to be... 

Weidlich, W. Haag, G. (1983). Concepts and models of a quantitative sociology The dynamics of interacting populations), (Springer Series in Synergetics, Vol. 14). Springer Verlag, Berlin. 

Goel, N.S., Maitra, S.C., Montroll, E.W. On the Volterra and other Nonlinear Models of Interacting Populations, New York, London Academic Press 1971. 

Dynamic behavior and oscillations are also found in nature, such as predator-prey interactions. A classical example of interacting populations is shown in Figure 4.10.30 for the snowshoe hare and the Canadian lynx, a specialist predator. The lynx-hare... 

L Peters, D. C. Webster, J. A Chlouber, C. S. Proc. Aphid-Plant Interactions Populations to Molecules. 1990. 

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