AICME II abstracts
Theory and observations of invasion processes
Modelling Dispersal and Coexistence of Two-Species with Integro-Difference Equations Claudia Carrillo1 . Dispersal of propagules is very often the only mobile stage in the life history of plants. Therefore, it is fundamental in determining the range of a plant population. This in turn is of the highest importance when population distributions and biological invasions are being studied. As far as mathematical models are concerned, the reaction-diffusion framework has traditionally been favored when modelling the spread of genes/populations and biological invasions — Fisher (1937) and Skellam (1951) are classical references; Shigesada and Kawasaki (1997) make a thorough revision of reaction-diffusion models for biological invasions — even though the implicit pattern in it is a Gaussian distribution around the centre of spread. However, dispersal of propagules is more accurately described by distributions known as leptokurtic: with less propagules around the entre of the distribution and more in the tails, whence they are also called fat-tailed distributions (Crawford, 1984; Okubo and Levin, 1989; Nathan and Mueller-Landau, 2000). These distributions are more accurate in that they reflect the few but non-negligible propagules that disperse far from the centre and establish spread outposts away from the parent plant. When a Gaussian distribution is used, this kind of longrange dispersal is neglected, which results in underestimated rates of spread. In this work, two ways of implementing long-range dispersal are investigated. The first is to add an advection term to the traditional reaction-diffusion approach. This term simulates a flow, of wind or water, for example, that may take some propagules further away than the mean of their siblings. The second way is to change the mathematical framework from a system of PDEs in the reactiondiffusion setting to a system of integro-difference equations. These equations are particularly suitable for the life history of plants, where growth and dispersal are clearly differentiated stages (Kot et al., 1996). They have the advantage that they admit different kinds of dispersal kernels. In this work, both a Gaussian and a leptokurtic kernel have been used in a system of two integro-difference equations which model the growth followed by dispersal of a two-species population. 1 Department
of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom (e-mail: claudia
[email protected]).
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Theory and observations of invasion processes
AICME II abstracts
The results and limitations of each of these approaches are discussed. It is seen, in particular, that the use of an advection term only modifies the results of the classical reaction-diffusion model as much as the speed of the flow, whereas the modifications in the integro-difference approach are more dramatic and include coexistence of the two-species among the possible results of the model. Finally, a brief discussion of an integrated approach, stratified diffusion, is given. Far from invalidating the reaction-diffusion approach, the conclusion drawn is that it has to be generalized to appropriately reflect long-range dispersal. Indeed, most propagules disperse in a random-walk fashion around the parent plant. This is very well reflected by a reaction-diffusion model. It is important to add those few propagules that disperse further from the mean and succeed in establishing a centre of spread. The total range and rate of spread of the whole population is a combination of how these subpopulations spread and whether they coalesce or not with the ’parent’ population, which is in its turn spreading.
References [1] Crawford, T. J. 1984. What is a population? In: Evolutionary Ecology. (Shorrocks, B. ed.), pp. 135-173. Oxford: Blackwell Scientific Publications. [2] Fisher, R.A. 1937. The wave of advance of advantageous genes. Annal of Eugenics 7: 225-369. [3] Kot, M., Lewis, M.A. and van den Driessche, P. 1996. Dispersal data and the spread of invading organisms. Ecology 77: 2027-2042. [4] Nathan, R., and Mueller-Landau, H.C. 2000. Spatial patterns of seed dispersal, their determinants and consequences. Trend in Ecology and Evolution 15(7): 278-285. [5] Okubo, A. and Levin, S. A. 1989. A theoretical framework for data analysis of wind dispersal of seeds and pollen. Ecology 70: 329-338. [6] Shigesada, N. and Kawasaki, K. 1997. Biological Invasions: Theory and Practice (Oxford Series in Ecology and Evolution). Oxford: Oxford University Press. [7] Skellam, J.G. 1951. Random dispersal in theoretical populations. Biometrika 38: 196-218.
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