Exam # 1 Study Guide Lectures: 1 – 5Lecture 1 (August 26th)Historical Origins of Population TheoryT.R. Malthus’ PrincipleThe Origin of Species (Darwin)Hierarchal Organization of Organisms (Nested)IndividualsPopulationsSpeciesGuildCommunitiesEcosystemsBiomesEvolutionary & PhylogeneticGenesIndividualsPopulationsSpeciesGeneraFamiliesOrdersClassPhylaKingdomPopulationEcologically – interacting individuals of the same speciesEvolutionarily – conspecific individuals that form an interbreeding groupPopulation sampling methodEasiest with plantsTelemetryCamera traps to census cryptic populationDNA analysis of fur/skinNetsMark recapture experimentsM2/n2 =n1/NDispersal pattern affects samplingSurrogate measures can affect precision (ex. Scat count)Statistical measuresLecture 2 (August 28th)Density Independent Growth ModelsExponentialGeometricImagine a chemostat with 2 algal species – they grow at different rates – why?CompetitionBirth/death ratesA real life factor would be immigration/emigration∆N = (B+I) – (D+E) = (B-D) + (I-E)(change in population size is equal to the number of births and immigrations minus the number of deaths and emigrations)Assumptionsindividuals have same probability of death and reproductionvital rates do not change overtimeno structure – genetic, age, sizecontinuous growth (no lag time)emigration/immigration are in balance ( or zero)… ∆N =B-DB=bNb = instantaneous birth rate/individual (per capita birth rate)D=dNd = instantaneous death rate/individual (per capita death rate)∆N= B-D =dN/dt =rNr = intrinsic rate of increase or instantaneous growth rater = (b-d)r can be less than, greater than, or just zerotake the integral to predict the population time in the futurelnNt-lnNt0= rt-rt0­if t0=0 then: Nt=Nt0ertDescribes exponential population growth – this equation allows you to predict population sizeGeometric growth rate of discrete cohortsCohort – a group of individuals in a population, all born at the same time, that are followed until death.Seasonal reproductionN(t) = population size at time trd = discrete growth factorNt+1 = Nt+ rd Ntλ = 1 +rdλ – finite rate of increase, which is a ratio of consecutive population sizes in an exponentially growing populationif the population growth rate remains constant, thenNt =λt N0Comparing continuous and discrete models:after algebraic manipulation: λ = erwhen:r > 0, λ > 1r = 0, λ = 1r < 0, λ < 1Lecture 3 (Sept 2nd)Stochasticity- randomness is included in the model; uses average values of parametersStochastic model has some parameters that vary unpredictably with time. It reflects a random or chance event in nature, or complex, changing phenomena that are too complicated to model directly. The population track will reflect an element of uncertainty. Consequently, if the model is run twice with the same starting conditions, it will generate somewhat different answers each time. Although each run of a stochastic model is unique, if the model is run many times, there is usually an expected mean and variance for the predicted population size.Two types:Environmental stochasticity – uncertainty due to variation in environmental conditions. The environment can be modeled as a series of unpredictably good and bad years for population growth. In an exponential growth model, this uncertainty is expressed in the mean and variance in r, the instantaneous growth rate of increase. In an exponential growth model with environmental stochasticity, a population is at risk of extinction if the variation in r is too large relative to the mean of r. in contrast, a population is never at risk of extinction in the deterministic model of exponential growth as long as r is greater than 0.Demographic stochasticity – uncertainty due to variation in the sequence of births and deaths in a population. Even in a constant environment (no variation in r), discrete births and deaths can cause population numbers to vary unpredictably. Demographic stochasticity is analogous to genetic drift, in which allele frequencies in a population vary by chance. Demographic stochasticity is not important in large populations because this source of random variation tends to average itself out over the long run. But in small populations, demographic stochasticity can generate a substantial risk of extinction, even in a population growth model where birth rates exceed death rates. In contrast, small populations are never at risk of extinction in a deterministic model of exponential growth, as long as r is greater than 0.Our model of density independent population growth was deterministicDeterministic – uses single point estimates of parameters in model; nothing is left to chance. If the starting conditions are not altered, a deterministic model will always produce the same result.Continuous growth Nt=N0ertDiscrete growth Nt=λtN0Environmental stochasticity reflects good and bad yrs for population growthNt=N0ert model for future population size under environmental stochasticityGrizzly Bear population size can be modeled with environmental stochasticityListed as threatened in 1975 – delisted in 2007Now, once again threatened in Yellowstone if the current grizzly bear population is 700 and r = 0.053, how long will it take the population to grow to an average size of 1000 bears?6.7 yearsModel for future population size under environmental stochasticityDemographic stochasticity reflects randomness in the sequence of births and deathsEven if r is positive, demographic stochasticity can cause population decline.Model for future population size under demographic stochasticity:Nt=N0ertProbability of extinction:P(extinction) = (d/b)N(0)Density Dependent Population GrowthExponential population growth is inherently unstableModify “b” and “d” to reflect density dependence:Continuous exponential model: r = (b-d)We now need to consider the per capita birth rate and the per capita death rates as varying with population sizeAs breeding pairs increase, productivity decreasesFactors related to reduced productivity:Distance to nearest conspecific breeding pairProximity to small supplemental feeding pointsDensity forces stabilize population size at a ‘sweet spot’Result: logistic growth of populationLogistic growth model – a model of population growth that incorporates the concept of resource limitation and density dependence in the instantaneous birth rate and/or the instantaneous death rate. It generates a characteristic S-shaped curve in