Here’s an adapted extract from part of my original thesis. I removed this in the end, since it wasn’t directly relevant to the spread of invasive species and the thesis was already too long. This text does however link several different fields that are interested in the spread of something, and I find connections across scientific fields interesting because these days they are rife with fertile research directions.

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The seminal works of R. H. Fisher (1937) described the propagation and diffusion of advantageous alleles in a population, and a lot of current theory on spread and dispersal has it’s root in population genetics.

Recent work on the spread of humans, with heredity between individuals, have indicated that certain mutations are either maintained with low frequencies at their origin or are propagated along wave fronts. If these mutations are tracked then it is possible to establish their origin (Ibrahim, 2004).

In the discipline of sociology Hägerstrand (1952) was the first to develop a predictive math model showing how phenomena diffuse in space and time, specifically innovation diffusion, or the speed at which the adoption of new technologies spreads.

In mathematics and physics, diffusion specifically concerns change in the density of something due to an random movement of a large number of things. Brownian motion is a frequently assumed category of random particle or individual movement, and results in a Gaussian/Normal distribution for the entire population if individuals or particles are all released from a central point. Diffusion is often equated with dispersal (Pielou 1969, 1977, 1979) particularly in the humanities, such as in economics (e.g. Brown 1981) but also in sciences such as human epidemiology (Cliff 1981). However this is not always true, and in other fields can indicate an intentional or directed movement.

Diffusion is often described by the power law:

msd(*t*) ≈ 6*Dt*^{α}

where *D* is the diffusion coefficient and *t* is the elapsed time. Typically, in a diffusive process, the mean squared displacement (msd) of a particle is a linear function of time (α = 1). The term **anomalous diffusion** is used to describe a diffusive process with a non-linear dependence on time. Additionally, if &alpha > 1, the phenomenon is called **super-diffusion**. In cellular biology, super-diffusion can be the result of active cellular transport processes (Caspi 2002). If &alpha < 1, dispersing particles undergo **sub-diffusion**. Sub-diffusion has been proposed as a measure of macromolecular crowding in the cytoplasm (Weiss 2004).

The spread of cancer occurs as cancerous cells replicate in an uncontrolled manner, resulting in a growing tumour. Metastasis, where cancerous cells break off from the primary tumour and establish elsewhere in the body, is akin to long-distance dispersal of invasive species, accelerating the rate of spread and often becoming the reason for cancer mortality (MacDonald 2002).

One subject in which spread models have become particularly advanced is in the prediction of wild fire behaviour. The models take into account wind direction, underlying vegetation, and also include long distance dispersal events which are so influential in invasive species spread. In wild fire spread, these long distance dispersal events are referred to as “spotting” (Xu 1994).

Another aspect of human health that involves spread is epidemiology – the study of heath and illness of populations. For example, *SARS* or *Severe Acute Respiratory Syndrome* spreads by common local transmission but also air passenger travel provided incidental large distance transmission (Bell 2003) again akin long-distance dispersal. Another example related to the impact of increasing travel and trade, shows that the increase in human travel had resulted in population size substituting distance as the most important factor for the spread of cholera in North America (Cliff 1981).

These varying subjects approach spread of objects or organisms in different ways, but as discussed seem to often demonstrate dispersal processes occurring at different scales, such as local growth coupled with diffusion and a stochastic long distance dispersal process.

In other subjects, Hengeveld (1989) notes that the area encompassed by the dispersal kernel is analogous to the:

**neighbourhood area**– in genetics when dealing with the transfer of genetic information in a population;**contact area**– in epidemiology, describing the chance of pathogen transfer and spread;**information field**– in human sociology when concerned with the spread of innovation and ideas. Hagerstand (1968) called this the “information probability field”, Brown (1981) called it the “mean information

field”;

Is it possible to combine these in to a general synthesis of spread? I don’t know, but I have attempted to make a generic iterative spread model that can share aspects of each between individual models. The only bias here however is that it’s constrained to geographic spread, so trying to model the spread of a 3D process, although possible through 3D raster maps and voxels, is probably a lot more trouble than it’s worth (when there are more specific modelling tools out there).

**References**

- Fisher, R.A. (1937) The wave of advance of advantageous genes.
*Ann. Eugenics*. - Ibrahim, K. M. (2004) Simulations of human colonization history.
*Heredity*93:124-125 - Hägerstrand, T. (1952) The propagation of innovation waves Lund, Sweden: Gleerup;
*Lund Studies in Geography*. - Pielou, E. C. (1969) An Introduction to Mathematical Ecology. Wiley, New York.
- Pielou, E. C. (1977) Mathematical Ecology. John WIley & Sons, New York, New York, USA.
- Pielou, E. C. (1979) Biogeography. John Wiley & Sons, New York.
- Cliff, A. D., P. Haggett, J. D. Ord & G. R. Versey. (1981) Spatial Diffusion Cambridge University Press, 1981
- Brown, L. A. Innovation Diffusion Methuan, London.
- Caspi, A.; Granek, R. & Elbaum, M. (2002) Diffusion and directed motion in cellular transport.
*Physical Review E*66 - Weiss, M.; Elsner, M.; Kartberg, F. & Nilsson, T. (2004) Anomalous Subdiffusion Is a Measure for Cytoplasmic Crowding in Living Cells.
*Biophysical Journal*87:3518-3524 - MacDonald, I. C.; Groom, A. C. & Chambers, A. F. (2002) Cancer spread and micrometastasis development: quantitative approaches for in vivo models.
*BioEssays*. 24:885-893 - Xu, J. (1994) Simulating the spread of wildfires using a geographic information system and remote sensing. PhD thesis. Rutgers, The State University of New Jersey.
- Bell, D. M. (2003) Public health interventions and SARS spread.
*Emerging Infectious Diseases*. - Hengeveld, R. (1989) Dynamics of biological invasions. Chapman & Hall, London.
- Hägerstand, T. (1968) Innovation Diffusion as a Spatial Process University of Chicago Press, Chicago.

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