Random Dispersal in Random Dispersal in Theoretical PopulationsTheoretical Populations

By: J.G. SkellamBy: J.G. Skellam

J.G. SkellamJ.G. Skellam

““Traditional biology course lay far too Traditional biology course lay far too much emphasis on the direct much emphasis on the direct acquisition of information. acquisition of information. Insufficient attention is given to the Insufficient attention is given to the interpretation of facts or to the interpretation of facts or to the drawing of conclusions from drawing of conclusions from observations and experience. The observations and experience. The student is given little opportunity to student is given little opportunity to apply scientific principles to new apply scientific principles to new situations.” situations.”

Random?Random?

From the perspective of Skellam the From the perspective of Skellam the best way to understand the random best way to understand the random dispersal amongst populations was dispersal amongst populations was by first understanding the principle by first understanding the principle of random walks.of random walks.

So as a reminder of what a random So as a reminder of what a random walk is: A random process consisting walk is: A random process consisting of a sequence of discrete steps of of a sequence of discrete steps of fixed length. fixed length.

SOmE MoRe Randomness!SOmE MoRe Randomness!

Random walks Random walks have interesting have interesting mathematical mathematical properties that properties that vary greatly vary greatly depending on the depending on the dimension in which dimension in which the walk occurs the walk occurs and whether it is and whether it is confined to a confined to a lattice. lattice.

Skellam’s PerspectiveSkellam’s Perspective

With regards to random walks, With regards to random walks, Skellam proposed the following:Skellam proposed the following:

Consider a plane using the Euclidean Consider a plane using the Euclidean coordinate system.coordinate system.

In the immediate neighborhood of In the immediate neighborhood of the origin let there be a particle that the origin let there be a particle that tends to leave the origin to gradually tends to leave the origin to gradually form a circular representation of the form a circular representation of the previous graph.previous graph.

uuuggguuuggg Now, this might seem Now, this might seem

to resemble the to resemble the concept of Brownian concept of Brownian Motion of a particle in Motion of a particle in a viscous substance a viscous substance but here in lies the but here in lies the difference:difference:

““The distribution of The distribution of the position of a the position of a particle of the nth particle of the nth generation with be generation with be henceforth”henceforth”

Even more uuugggEven more uuuggg

Skellam’s polar Skellam’s polar transformation of transformation of this particle this particle positioning of n-positioning of n-generations turned generations turned out to be the out to be the following:following:

Are we getting anywhere with this?Are we getting anywhere with this?

Alas!! Integrating over Alas!! Integrating over θθ gives us the radical gives us the radical probability density:probability density:

a^2 = the mean-square dispersion per a^2 = the mean-square dispersion per generation analogous with the mean-square generation analogous with the mean-square velocity in Maxwell’s distribution.velocity in Maxwell’s distribution.

Soooooo?Soooooo?

So from what we have gathered thus So from what we have gathered thus far is that an organism or particle far is that an organism or particle with tend to move away from its with tend to move away from its origin in a semicircular pattern.origin in a semicircular pattern.

From the previous equations we are From the previous equations we are then able to calculate its probable then able to calculate its probable whereabouts with regards to random whereabouts with regards to random distributing.distributing.

Interesting..Interesting..

““of the population spread out after n-of the population spread out after n-generations that proportion lying outside generations that proportion lying outside a circle of radius R is:”a circle of radius R is:”

Awww wook at the fuzzy wuzziesAwww wook at the fuzzy wuzzies

The results of the particle motion can be The results of the particle motion can be made applicable to the dispersal of small made applicable to the dispersal of small animals such as worms and snails.animals such as worms and snails.

A bug example:A bug example: If the random mean square dispersion If the random mean square dispersion

(RMSD) per minute of a wingless beetle (RMSD) per minute of a wingless beetle wandering at random is 1 yard ^2 the wandering at random is 1 yard ^2 the after a season of 6 months RMSD of the after a season of 6 months RMSD of the resulting probability distributions is only resulting probability distributions is only 500 yards.500 yards.

The bug example continued…The bug example continued… The probability that after 6 months the The probability that after 6 months the

beetle wanders more than a mile from the beetle wanders more than a mile from the starting point is less than 8 in a million starting point is less than 8 in a million (wow, wonder how he figured that out?).(wow, wonder how he figured that out?).

Without external aid a period of time Without external aid a period of time equivalent to 1000000 seasons would be equivalent to 1000000 seasons would be required to raise RMSD.required to raise RMSD.

Soooo, basically as RMSD increases a Soooo, basically as RMSD increases a great deal the particle or in this case great deal the particle or in this case wingless beetle comes a great deal nearer wingless beetle comes a great deal nearer the origin than the farthermost position the origin than the farthermost position previously reached.previously reached.

TIMBER!!!TIMBER!!!

Skellam makes reference to Reid’s Skellam makes reference to Reid’s Problem:Problem:

““We can clearly establish a rigorous We can clearly establish a rigorous conclusion in the form of an equality conclusion in the form of an equality provided that we can fix appropriate provided that we can fix appropriate bounds to various parameters.”bounds to various parameters.”

It turns out the problem that is being It turns out the problem that is being referred to is having to do with the referred to is having to do with the Oak tree.Oak tree.

Oaky DokyOaky Doky

The oak does not produce accorns until it The oak does not produce accorns until it is sixty or seventy years old and even then is sixty or seventy years old and even then it is not mature.it is not mature.

It then produces acorns over a period of It then produces acorns over a period of several hundred years.several hundred years.

Obviously not all the acorns grow to Obviously not all the acorns grow to produce more Oak trees:produce more Oak trees:

Some are – eaten by mammels, fail to Some are – eaten by mammels, fail to germinate, or are simply overshadowed by germinate, or are simply overshadowed by the larger mature trees.the larger mature trees.

It seems that only 1% of the seedlings are It seems that only 1% of the seedlings are likely to survive the next three years.likely to survive the next three years.

It is also safe to assume that the oak It is also safe to assume that the oak population is no more than 9 million.population is no more than 9 million.

We then have R/a < 300 sqrt(log We then have R/a < 300 sqrt(log 9,000,000) = 1200.9,000,000) = 1200.

In the original form of the problem as In the original form of the problem as stated by Reid, R is given as 600 milesstated by Reid, R is given as 600 miles

Lastly..Lastly..

It then follows that the rootmean It then follows that the rootmean square distance of daughter oaks square distance of daughter oaks about their parents is greater than ½ about their parents is greater than ½ a mile and that agents such as small a mile and that agents such as small fuzzy wuzzies (aka mammals and fuzzy wuzzies (aka mammals and birds) played a major role in the birds) played a major role in the dispersal of this population.dispersal of this population.

Just kidding, I’ve got more!!Just kidding, I’ve got more!! Skellam, goes on to explain that many problems Skellam, goes on to explain that many problems

on dispersal cannot be formulated unless some on dispersal cannot be formulated unless some law of population growth (in the absence of law of population growth (in the absence of dispersal) is assumed.dispersal) is assumed.

As long as the population is small of shows a As long as the population is small of shows a natural tendency to decrease, the Malthusian law natural tendency to decrease, the Malthusian law dN/dt =cN is usually satisfactory.dN/dt =cN is usually satisfactory.

If the population is not small the Pearl-Verhulst If the population is not small the Pearl-Verhulst logistic law is more appropriate.logistic law is more appropriate.

This law may be written in the form:This law may be written in the form: dN/dt = cN – lN^2dN/dt = cN – lN^2

Almost done, really…Almost done, really…

““In practice there is rarely sufficient In practice there is rarely sufficient information to construct the contours of information to construct the contours of population density with accuracy…”population density with accuracy…”

Buuuuuuut here is a well illustrated spread Buuuuuuut here is a well illustrated spread of the muskrat in central Europe since its of the muskrat in central Europe since its introduction in 1905.introduction in 1905.

If we are prepared to accept a boundary If we are prepared to accept a boundary as being representative of a theoretical as being representative of a theoretical contour, then we must regard the area contour, then we must regard the area enclosed by that boundary as an estimate enclosed by that boundary as an estimate of pi*r^2of pi*r^2

Well that’s about it!Well that’s about it!

So the basic principle that I want to So the basic principle that I want to emphasize is that there is random emphasize is that there is random dispersal in theoretical populations, dispersal in theoretical populations, although not apparent it is virtually although not apparent it is virtually everywhere so the next time you are everywhere so the next time you are keeping an eye on a particle or tree keeping an eye on a particle or tree or beetle or muskrat, just think of or beetle or muskrat, just think of Skellam and his principles of random Skellam and his principles of random dispersal.dispersal.

Preguntas??Preguntas??