1	POPULATION DYNAMICS Today 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.
2	Reading Chapter 52, particularly 52.2. Box 52.3 on Mark-Recapture Review of the x1 07 lecture may be useful to understand population growth. Chapter 52 starts with the more complicated models using age.
3	Changes in population size In an unchanging world, we expect population sizes of animals and plant species to stay about the same because births equal deaths. Reproduction gives organisms the potential to grow exponentially. Exponential growth eventually exhausts the resources and maintenance of population is dependent on renewal of resources.
4	Determining population size 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, resample
5	Mark-recapture basics Box 52.3 Perhaps simplest to understand in small lake Capture individuals, mark(tag) them, release n₁ marked individuals Assume the marked animals disperse and mix with unmarked animals in lake Capture n₂ individuals, if m₂ is # marked, then N = total # in population = n₁ • n₂/ m₂
6	Census 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 year
7	Minimum 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.
8	Population 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’.
9	Population 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 left N _t+1 = N _t + B – D + I - E If population is closed, N _t+1 = N _t + B – D ΔN = N _t+1 - N _t = B – D = change in size
10	The Whooping Crane The species is ENDANGERED according 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.
11	Census history of Whooping Crane
12	Geometric Growth Model N_t+1 = l• N_t l, lambda,is the multiplier from one generation to the next. If l = 1, the population size stays the same = is constant.
13	Exponential Growth Model N(t) = N₀•e^r•t The parameter that measures growth is r, which measure the instantaneous per capita growth rate per unit time. If r = 0.04 yr^-1 the population grows 4% per year, as e^0.04 = 1.04 approximately. If r = 0, the population size does not change
14	Exponential 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.
15	Relationships of l and r l = e^rtor e^rif time is one unit long. If l < 1, then r will be negative, i.e. the population is declining (exponential decay). If l > 1, then r will be greater than zero and the population will increase geometrically.
16	Modifications 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.
17	The “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•N
18	The Logistic Equation
19	Metapopulations Species are usually made up of patches of populations with few individuals found in the area between the patches. The population is said to have a metapopulation structure if population extinction and colonization of empty suitable patches is frequent.
20	Population Abundance Cycles Some populations show regular fluctuations of population size. Evenly repeated highs and lows are known as population cycles. The Lynx (a cat that eats hares) and hare (rabbit) populations cycle with an 11 year period. The Lynx hi and low trails the hi and low of the hare.
21	Outbreaks It is not unusual for populations of insects to vary greatly from year to year Very great increases in abundance are called “outbreaks” Examples in 2003 include the Painted Lady butterfly & the Asian Ladybug
22	Census History with Outbreak
23	Vocabulary Constant Doubling time Geometric growth Exponential growth metapopulations parameter Emigrants, immigrants Census history Outbreak N₀•e^r•t Logistic growth Cycle l, lambda