Probability and Dispersal in One Space Dimension Dispersal via Random Walks and the Diffusion Equation Many of the basic probability kernels associated with population dispersal find their basis in partial differential equations. For example, the normal distribution is associated withthe diffusion equation. Now, suppose a population of organisms is dispersed over space with an initial density of.

One can think of the number of individuals located in a small interval of size at a location as. These indidviduals disperse randomly according to the distribution ; thus, their probabilistic location at a later time will be. Convolutions and Fourier Transforms The Fourier Transform of a function defined on an interval from to is. In wave space, evaluate the product of and. Invert the transform of to obtain. The only turd in this particular punchbowl is that FTs are not all that easy to calculate or invert.

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Fortunately, there is a very simple computational routine, called the Fast Fourier Transform FFT which will perform these operations numerically. The extra factor of allows for a change in the length of the interval from 1 to. As a note of interest, the fraction is equal to the discrete step size in space,. Invert the FFT to arrive at. An added wrinkle is that, although it is theoretically natural to center the interval at , in discrete terms this is unnatural. In general a shift has to be performed on the discrete version of so that the value of at is associated with the first element of a vector, as in the following diagram: Matlab Implementation We can use FFTs in M ATLAB to calculate convolutions rapidly.

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Let's begin by illustrating the procedure for a population initially localized uniformly between -1 and 1, dispersing over time into a much larger domain. The FFT assumes periodicity, so if we define a space interval between and 10 with nodes remember the FFT is much more efficient for computations involving elements , we need to take into account that the node corresponding to 10 must be left off the FFT assumes that whatever data might be specified there is identical with the data at. First off, the command fftshift basically chops a vector in half and interchanges the first and second halves as blocks.

In this context it is necessary because the M ATLAB FFT is built on the assumption that functions are defined on the interval from 0 to 1, as opposed to centered around 0. Secondly, the factor of dx.

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Since we have implemented a product of FFTs, we need to divide out a factor of np. Now let's see what we have done: plot x,p0,'r',x,p1,'y' You should also get a warning about imaginary numbers being ignored.

This occurs because the FFT requires complex numbers, and numerical rounding creates very small imaginary components, even for perfectly real data fields. The best way to get around this is to take the real part after convolution, which we will do in the future. Now it is possible to keep iterating the random dispersal of the population. Does it look reasonable? Use the up-arrow and edit the previous command, plot, and see the population after dispersal over three time units.

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Dispersal and Settling with a Localized Source Another interesting dispersal model involves assuming random motion for propagules seeds, spores, pollen, motile larvae while they are in the medium and then settling onto a substrate at some rate. If propagules are initially released at the point the PDE modelling the density of propagules in the medium is. E XERCISE 7 : For a uniform distribution of sources between -1 and 1 investigate the dispersal of propagules according to the probability distribution given above. On the same set of axes plot the effect of changing on the dispersal of the population.

By studying the heat equation by considering the individual random particles, however, one gains further intuition into the problem.

While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equation and the closely related notion of harmonic functions from a probabilistic perspective.

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The theme of the first two chapters of the book is the relationship between random walks and the heat equation. The first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case.

Random walk and Brownian motion are introduced and developed from first principles.

## Takagi : A generalized simple random walk in one dimension related to the Gaussian polynomials

The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.

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