FW662 Lecture 4 – Stochastic Models 1Lecture 4. Population Models Incorporating Stochasticity.Reading: Renshaw (1991) Chapters 2 Simple birth-death processes, and 3 General birth-deathprocesses, Pages 15-86.Optional:Lebreton, J.-D. 1990. Modeling density dependence, environmental variability, anddemographic stochasticity from population counts: an example using WythamWood Great Tits. Pages 89-102 in Population Biology of Passerine Birds, J.Blondel (ed.). Springer-Verlag Berlin Heidelberg.Paraphrase of Renshaw (1991:2):Why would we not accept the weight of one Scotchman as representative of theentire population of Scotland, but will accept the time trace of a populationas representative of the population's dynamics?How important is stochasticity? Consider a simple example patterned from an examplein Morris and Doak (2002:26) where 8t can take on 2 values with equalprobability: 0.83 and 1.17. Thus, the arithmetic mean of the possible values of 8is exactly 1.0, so the population should remain stable. Right? WRONG! Here’swhy. Change in population size is expressed as the product of the 8t values, . (Tt'18tHalf the time, 8t is equal to 0.83, and half the time it is 1.17. The product of these2 values is 0.9711, and is thus the expected value of the change in the populationover a 2-year interval is the geometric mean of 0.9711, not the arithmetic mean of1.0. Therefore, the expected value of the above product is 0.9711T/2, and is thus<1, so that the population will decline. The key point of this example is that thegeometric mean is what is important, and the geometric mean of a series is alwaysless than the arithmetic mean. Variation in values of 8 is important because thegeometric mean of these values can be <1 even when the arithmetic mean is >1.Types of stochasticity. Variation in population sizes can be classified into 2 generalmechanisms. The true population may vary through time and space, even thougha deterministic model predicts a constant population. Variation in the truepopulation size is termed process variation, because of stochasticity in thepopulation growth process. Several mechanisms can cause process variation.Bernoulli or penny-flip variation (Demographic).Individual heterogeneity, including phenotypic and genotypic.Spatial variation (Environmental).Temporal variation (Environmental).Small annual variation, such as typical weather patterns mightgenerate.Catastrophes, such as severe storms might generate (impact of aFW662 Lecture 4 – Stochastic Models 20 20 40 60 80 100 N(t)0 20 40 60 80 100 Time (t)hurricane or tornado on a habitat).In contrast to process variation, sampling variation is not variation in thepopulation size, but only our inability to measure the population withouterror. Thus, we observe only estimates of population size, , instead ofˆNttrue population size .NtEnvironmental stochasticity can be incorporated into a model by making the parametersrandom variables. As an example, consider the logistic growth model. By setting equal to 100 and making aNt%1' Nt[1 % R0(1 & Nt/K)] K R0random variable that is normally distributed with mean 0.15 and standarddeviation 0.075, population growth might look like the following. Variation inpopulation growth is predominantly at small population sizes, with little variationat populations around .KFW662 Lecture 4 – Stochastic Models 30 20 40 60 80 100 120 N(t)0 20 40 60 80 100 Time (t)0 20 40 60 80 100 120 N(t)0 20 40 60 80 100 Time (t)Making just a random variable with mean 100 and standard deviation 20 andKsetting to 0.15 results in the following graph. Note that the variation inR0population growth is now predominantly at higher levels.By making both and random variables with the same normal distributionsK R0as used above, the following graph results. Now, variation in population growthis evident at all population sizes.We can envision how such variation occurs in the natural world: could easilyKvary from year to year because of weather effects. Good rains result in lushgrowth for herbivores, whereas drought results in little growth and poorconditions, with the population above carrying capacity because of the smallervalue of . Note that the average population size is not the mean of theKdistribution of because the population will drop faster to reach a decreased K Kthan it will growth to reach an increased (Gotelli 1998:38). Likewise, lushKFW662 Lecture 4 – Stochastic Models 4f(t) ' 8e&8t,FT(t) ' Pr(T # t) ' 1 & e&8t.Pr(T $ t) ' 1 & F(T) ' e&8t.growth should result in larger values of , whereas drought conditions wouldR0result in smaller, even negative, values of this parameter.Incorporation of demographic stochasticity into a population model can be in the form ofa differential equation with continuous time, or in a difference equation withdiscrete time. In the continuous time models, the time to the next event is therandom variable modeled. This is the approach generally presented in Renshaw(1991). If time to the next event is modeled as an exponential random variable,then analytical solutions can generally be derived, and a huge mathematicalliterature exists on this type of stochastic process (known as a Poisson process). For the exponential distribution, the probability density function isgiving a mean of and variance of . The cumulative density function is1/81/82Thus, the probability of surviving at least T or longer is given by If the proportion of the population undergoing the process (giving birth or dying)over a finite time interval is taken as the random variable, then a binomial processresults. Just as difference equations become identical to differential equations asthe time interval becomes infinitely small, so do binomial process models becomethe same as Poisson process models. For most of the populations we areinterested in this course, a binomial process is much more reasonable. Births are apulse in time, and mortality is not a constant rate across time, but more severe atcertain months of the year. The lack of a constant instantaneous mortality rate(hazard rate) makes the simple Poisson process models unrealistic for mostfishery and wildlife populations.Pure Birth Process -- Binomial modelEach individual undergoes a Bernoulli trial to determine if it gives birth (splits) inthis finite interval.Only binomial variation is present.FW662 Lecture 4 – Stochastic Models 5Pr(B) 'NBpB(1 & p)(N & B)For a population of N