POPULATION DYNAMICSReadingChanges in population sizeDetermining population sizeMark-recapture basicsMark-recapture exampleCensus HistoriesCensus history examplesMinimum Doubling TimePopulation Size at a particular timePopulation Change using birth & deaths and migrationThe Whooping CraneCensus history of Whooping CraneGeometric Growth ModelExponential Growth ModelExponential Growth ModelRelationships of  and rModifications of growth modelThe “logistic” model of growthParameciumThe Logistic EquationMetapopulationsMeta-populationsPopulation Abundance CyclesHare & Lynx populationsOutbreaksCensus History with OutbreakVocabularyExam 3 lecture #4 UIC BioS 101 Nyberg 1POPULATION DYNAMICSToday we focus on population censuses and models that count all individuals equally (using the variable N only, i.e. without age or sex) and that do not measure resource availability.Exam 3 #4 UIC BioS 101 Nyberg 2Reading Chapter 52.2. Box 52.2 (p1043-44) -read carefully Box 52.3 (p1048) on Mark-Recapture Review of the x1 07 lecture may be useful to understand population growth. Chapter 52.1 introduces demographic models using age of individuals.Exam 3 #4 UIC BioS 101 Nyberg 3Changes in population size In a sustainable world, we expect population sizes of animals and plant species to stay about the same, N constant thru time. Reproduction gives organisms the potential to grow exponentially. Population growth eventually exhausts the resources and maintenance of population is dependent on renewal of resources.Exam 3 #4 UIC BioS 101 Nyberg 4Determining population size Census = Count all the individuals Generally tough to do Sampling Create subpopulations (often based on area), count individuals in subpopulations, extrapolate to entire population/area Mark-Recapture Studies Sample, mark, release, resampleExam 3 #5 UIC BioS 101 Nyberg 5Mark-recapture basics Box 52.3 (p1048) Perhaps simplest to understand in small lake Capture individuals, mark (tag) them, release n1marked individuals, now n1/ N is fraction marked. Assume the marked animals disperse and mix with unmarked animals in lake Capture n2individuals, if m2is # marked, the fraction marked is m2/ n2& should be equal to n1/ N . then N = total # in population = n1•n2/ m2Mark-recapture example You catch 14 butterflies and mark the thorax with white paint and then release the butterflies. Two days later you go out and net 18 butterflies and find 4 of them marked. The Estimate of the total population = 14 x 18/4 = 63 butterflies.Exam 3 #5 UIC BioS 101 Nyberg 6Exam 3 #5 UIC BioS 101 Nyberg 7Census Histories The Census history is the record of the numbers of individuals through time Some populations show a pattern of constant doubling for a period of time. Many populations are stable in size. Some species have a pattern of steady decrease. Many insects have population “outbreaks”with large fluctuations from year to yearCensus history examplesExam 3 #5 UIC BioS 101 Nyberg 8Exam 3 #5 UIC BioS 101 Nyberg 9Minimum Doubling Time The time it takes for a species to double the number of individuals even when resources are abundant is called the doubling time. Both geometric and exponential growth imply a constant doubling time, doubling time simply related to r, growth rate, is a parameter of the species.Exam 3 #5 UIC BioS 101 Nyberg 10Population Size at a particular time N is the symbol for the variable population size N t= means the population size at time t, as a subscript it implies the geometric or discrete time model. N t + 1= population size one generation after t The exponential or continuous time model would be written as N(t), verbally ‘N as a function of time’.Exam 3 #5 UIC BioS 101 Nyberg 11Population Change using birth & deaths and migration Plus Births, B is number of births Minus Deaths, D is number that died Plus Immigrants, I = # that moved in Minus Emigrants, E = # that leftN t+1 = N t + B – D + I - EIf population is closed, N t+1 = N t + B – DΔN = N t+1 -N t = B – D = change in sizeExam 3 #5 UIC BioS 101 Nyberg 12The Whooping Crane The species is ENDANGEREDaccording to US law. The population was once at least 10,000 birds (always rare). The population was reduced to only 20 birds in the 1940s. The population has been growing exponentially for about 60 years.Exam 3 #5 UIC BioS 101 Nyberg 13Census history of Whooping CraneExam 3 #5 UIC BioS 101 Nyberg 14Geometric Growth ModelNt+1= λ•Ntλ, lambda,is the multiplier from one generation to the next.If λ = 1, the population size stays the same = is constant.Exam 3 #5 UIC BioS 101 Nyberg 15Exponential Growth ModelN(t) = N0•er•t The parameter that measures growth is r, which measure the instantaneous per capita growth rate per unit time. If r = 0.04 yr-1the population grows 4% per year, as e0.04= 1.04 approximately. If r = 0, the population size does not changeExam 3 #5 UIC BioS 101 Nyberg 16Exponential Growth Model Can also be written in “differential” form:dN/dt = r•N where dN/dt is the change in abundance per unit time change and r is the per capita growth rate.Note that if r =0 the population is not changing in size, i.e. dN/dt =0, and if r is negative the population is decreasing.Exam 3 #5 UIC BioS 101 Nyberg 17Relationships of λ and r λ = ertor erif time is one unit long. If λ < 1, then r will be negative, i.e. the population is declining (exponential decay). If λ > 1, then r will be greater than zero and the population will increase geometrically.Exam 3 #5 UIC BioS 101 Nyberg 18Modifications of growth model Populations don’t grow indefinitely, but rather reach a maximum density. The logistic model is a simple modification of exponential growth that leads to curve (sometimes referred to a ‘s’ shaped) that conforms to observations of batch cultures.Exam 3 #5 UIC BioS 101 Nyberg 19The “logistic” model of growth We take our exponential model (per capita growth constant) and add a new parameter, a, that reduces the growth rate in proportion to the population size dN/dt = r•N – a•N2 per capita growth rate dN/N•dt = r – a•NdN/ N•dtNParameciumExam 3 #5 UIC BioS 101 Nyberg 20Exam 3 #5 UIC BioS 101 Nyberg 21The Logistic EquationNTimeSee Fig. 52.7aExam 3 #5 UIC BioS 101 Nyberg 22Metapopulations Species are usually made up of patches of populations with few individuals found in