RandomDispersalinTheoreticalPopulations在隨機種群擴散理論課件

1、Random Dispersal in Theoretical PopulationsBy: J.G. Skellam第1頁，共21頁。J.G. Skellam“Traditional biology course lay far too much emphasis on the direct acquisition of information. Insufficient attention is given to the interpretation of facts or to the drawing of conclusions from observations and experi

2、ence. The student is given little opportunity to apply scientific principles to new situations.” 第2頁，共21頁。Random?From the perspective of Skellam the best way to understand the random dispersal amongst populations was by first understanding the principle of random walks.So as a reminder of what a ran

3、dom walk is: A random process consisting of a sequence of discrete steps of fixed length. 第3頁，共21頁。SOmE MoRe Randomness!Random walks have interesting mathematical properties that vary greatly depending on the dimension in which the walk occurs and whether it is confined to a lattice. 第4頁，共21頁。Skella

4、ms PerspectiveWith regards to random walks, Skellam proposed the following:Consider a plane using the Euclidean coordinate system.In the immediate neighborhood of the origin let there be a particle that tends to leave the origin to gradually form a circular representation of the previous graph.第5頁，共

5、21頁。uuugggNow, this might seem to resemble the concept of Brownian Motion of a particle in a viscous substance but here in lies the difference:“The distribution of the position of a particle of the nth generation with be henceforth”第6頁，共21頁。Even more uuugggSkellams polar transformation of this parti

6、cle positioning of n-generations turned out to be the following:第7頁，共21頁。Are we getting anywhere with this?Alas! Integrating over gives us the radical probability density:a2 = the mean-square dispersion per generation analogous with the mean-square velocity in Maxwells distribution.第8頁，共21頁。Soooooo?

7、So from what we have gathered thus far is that an organism or particle with tend to move away from its origin in a semicircular pattern.From the previous equations we are then able to calculate its probable whereabouts with regards to random distributing.第9頁，共21頁。Interesting.“of the population sprea

8、d out after n-generations that proportion lying outside a circle of radius R is:”第10頁，共21頁。Awww wook at the fuzzy wuzziesThe results of the particle motion can be made applicable to the dispersal of small animals such as worms and snails.A bug example:If the random mean square dispersion (RMSD) per

9、minute of a wingless beetle wandering at random is 1 yard 2 the after a season of 6 months RMSD of the resulting probability distributions is only 500 yards.第11頁，共21頁。The bug example continuedThe probability that after 6 months the beetle wanders more than a mile from the starting point is less than

10、 8 in a million (wow, wonder how he figured that out?).Without external aid a period of time equivalent to 1000000 seasons would be required to raise RMSD.Soooo, basically as RMSD increases a great deal the particle or in this case wingless beetle comes a great deal nearer the origin than the farthe

11、rmost position previously reached.第12頁，共21頁。TIMBER!Skellam makes reference to Reids Problem:“We can clearly establish a rigorous conclusion in the form of an equality provided that we can fix appropriate bounds to various parameters.”It turns out the problem that is being referred to is having to do

12、 with the Oak tree.第13頁，共21頁。Oaky DokyThe oak does not produce accorns until it is sixty or seventy years old and even then it is not mature.It then produces acorns over a period of several hundred years.Obviously not all the acorns grow to produce more Oak trees:Some are eaten by mammels, fail to g

13、erminate, or are simply overshadowed by the larger mature trees.第14頁，共21頁。It seems that only 1% of the seedlings are likely to survive the next three years.It is also safe to assume that the oak population is no more than 9 million.We then have R/a 300 sqrt(log 9,000,000) = 1200.In the original form

14、 of the problem as stated by Reid, R is given as 600 miles第15頁，共21頁。Lastly.It then follows that the rootmean square distance of daughter oaks about their parents is greater than a mile and that agents such as small fuzzy wuzzies (aka mammals and birds) played a major role in the dispersal of this po

15、pulation.第16頁，共21頁。Just kidding, Ive got more!Skellam, goes on to explain that many problems on dispersal cannot be formulated unless some law of population growth (in the absence of dispersal) is assumed.As long as the population is small of shows a natural tendency to decrease, the Malthusian law

16、dN/dt =cN is usually satisfactory.If the population is not small the Pearl-Verhulst logistic law is more appropriate.This law may be written in the form:dN/dt = cN lN2第17頁，共21頁。Almost done, really“In practice there is rarely sufficient information to construct the contours of population density with

17、 accuracy”Buuuuuuut here is a well illustrated spread of the muskrat in central Europe since its introduction in 1905.If we are prepared to accept a boundary as being representative of a theoretical contour, then we must regard the area enclosed by that boundary as an estimate of pi*r2第18頁，共21頁。Well thats about it!So the basic principle that I want to emphasize is that there is random