Introduction to Population Modeling by James C. Frauenthal (auth.)

By James C. Frauenthal (auth.)

The textual content of this monograph represents the author's lecture notes from a path taught within the division of utilized arithmetic and facts on the country college of latest York at Stony Brook within the Spring of 1977. because of its beginning as lecture notes, a few sections of the textual content are telegraphic standard whereas different parts are overly specified. This stylistic foible has no longer been converted because it doesn't seem to detract heavily from the clarity and it does aid to point which issues have been under pressure. The viewers for the direction at Stony Brook used to be composed virtually solely of fourth 12 months undergraduates majoring within the mathematical sciences. All of those scholars had studied at the least 4 semesters of calculus and one in every of chance; few had any earlier event with both differential equations or ecology. it kind of feels prudent to show that the author's historical past is in engineering and utilized arithmetic and never within the organic sciences. it truly is was hoping that this isn't painfully visible. -vii- the point of interest of the monograph is at the formula and resolution of mathematical versions; it makes no pretense of being a textual content in ecology. the belief of a inhabitants is hired more often than not as a pedagogic software, offering cohesion and intuitive attract the numerous mathematical rules brought. If the organic surroundings is stripped away, what continues to be might be interpreted as subject matters at the qualitative habit of differential and distinction equations.

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Extra info for Introduction to Population Modeling

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Assume that the intrinsic growth rate r(N) includes a resource recovery time of one time interval, and is logistic in nature; thus let -45- a. b. c. Determine the stability of the equilibrium at N = K. For convenience, let 6t = T. Make a plot of the root(s) of the Characteristic Equation in the complex plane, clearly illustrating the regions of stability and instability. Compare the results of the above model with those found earlier for the discrete time logistic equation without explicit time delays.

Note that this method can be used with the Pure Birth Process too. (t). 1 Note that the sum runs over all possible population sizes. Although we will assume the population initially numbers j at t = 0, so that E(N) = j at t = 0, since deaths can occur as well as births, the possibility of a population size smaller than j at times t > 0 does not vanish. b. simplify notation: PN(t) PN) dEd(tN) = L {-(A+~)i2p. 2 I.. 2 . -lPi+1Pi-I-1Pi_1 ) y i=l and by redefining the indices in the summations, 00 dE(N) = A I Pk{- k 2 + (k+I)2 - (k+l)) dt k=O 00 - ~ I Pk{k 2 - (k-l) 2 - (k-l)} k=O 00 = A I k=O 00 kPk - ~ L k=O kPk = (A-~)E(N).

6119. 092. It is possible to interpret our approximate result quite generally. Q,n R > 0, thus Prob{extinction} O. -+ In other words, the probability that the population becomes eventually extinct or grows without bound depends upon whether the true mean of the logarithm of the random variable R is negative or positive. On the other hand, the expected population size eventually becomes extinct or grows without bound according to whether the true mean of the random variable R itself, is less than or greater than unity.

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