Presentation on theme: "Aggregated dispersion of chinch bugs (an agricultural pest of cereal crops) in space and time; light green = low density, blue = high density."— Presentation transcript:
1 Aggregated dispersion of chinch bugs (an agricultural pest of cereal crops) in space and time; light green = low density, blue = high density
2 Chinch bug example illustrates typical case of population variation in time BirthPopulation size changes as function of birth, death processes (i.e., demography), and by immigration & emigrationImmigration and emigration create complications in terms of spatial-temporal dynamics… (Metapopulation considerations)++-Population SizeImmi-grationEmi-gration-Death
3 Life table provides simple way to track how population grows and changes Look at pop structure alsooriginally invented by insurance industry, calculate odds of people’s living to particular ages.Two kinds of life tables in nature:Best type generally is cohort life table = dynamic life table; (cohort = all individuals within an area born/germinated in population in same time interval)BUT What about long-lived, or highly mobile animals (like sequoia tree, whale, grizzly bear) that can’t be tracked easily from cradle to grave? Ecologists use a second approach, static life table = time-specific life table; this gives a snapshot in time, based on population’s age-distribution
4 Diagramatic illustration of dynamic versus static life table Each individual represented by a diagonal line segmentStatic life table from one time periodCohort life table based on one cohortAgeBirtht0t1t2t3Time since origin of cohort
5 Some life-table assumptions: For all life tables, the sample of individuals must be representative of population’s age-structureIn cohort life table, cohort needs to be representative (or multiple cohorts studied); cohorts often differ, e.g., some fish cohorts survive much longer than othersIdeally in static life-table, population is neither increasing nor decreasing; population that is increasing or decreasing will have skewed age-distributionsLife tables represent local ecological conditions, which may or may not be representative (e.g., population density)
6 “Cohort effect”, i.e., dominance of population dynamics by particular cohorts--in this case, resulting from excellent Lake Erie whitefish spawning in 1944
7 Typical life table parameters x = Age intervalnx = the number alive at start of age interval x (in cemetery lab l’x used for nx)lx = survivorship, the number still living at the start of age interval x, expressed either as a proportion of the number at the start of first interval, n0, or standardized to cohort of Thus lx /1000 = nx/n0; and then lx = 1000*(nx /n0).dx = nx - nx+1 (d = dead)qx = dx/ nx; This is age-specific mortality rate, the probability of dying during age interval x, i.e., the proportion of those living that die during interval x. Note: lab defines qx = (dx/ nx)*1000 (=no. diex/1000)
8 Life table parameters, cont. sx = 1 - qx ; sx = survival rate, the proportion of remaining individuals that will still be alive at the end of age interval xbx = fecundity, the average number of offspring produced by a female during age interval x (sometimes represented as mx! Think of bx as births)Lx = (nx + nx+1)/2 = average number alive in age xTx = Sx-->∞ Lx ; This is the sum of all individuals living in age interval x plus all future ages--an intermediate step in calculating ex.ex = Tx/ nx = life expectancy at time interval x, i.e., the average number of additional age classes an individual can expect to reach at each age
9 Whew, that’s a heck of a lot of math! What’s it good for? These parameters allow one to do lots of cool stuffCompare survivorship patterns directly (n0 = 1000) using survivorship curve: log(lx) as function of ageAge-specific mortality risk, fecundity, life expectancy, etc.We can calculate population growth rates (next lecture)Some factors that influence life tablesAge affects demography dramatically in many species (in plants stage often more impt. than age)Life tables can also vary by year (e.g., climate & cohort effect), sex, and certainly by speciesLife tables are partially heritable, under natural selection
10 Example: Dall mountain sheep studied on Mt Example: Dall mountain sheep studied on Mt. Denali, n0 = 608 sheep dying before 1937; age at death based on horn size used to estimate age-structure of population (static life table) See overheads (StilingText, Table 6.1; Fig. 6.2)
12 Three different types of idealized survivorship curve may be recognized: 1000Type I; e.g., many mammalsType II; e.g., many birds, small mammals, lizards, turtles100Log10 number of survivors10Type III; e.g., many invertebrates, anadromous fish10.1Age
13 Type II survivorship curve, 1978 cohort of Daphne Island (Galapagos) cactus finch, showing annual variation
14 Conclusions:Spatial structure of populations can tell us much about the ecological mechanisms involvedTemporal structure of populations can be summarized and compared using a variety of parameters taken from life tables, i.e., from patterns of birth and death (demography)Idealized patterns of mortality can be recognized, but most real populations are more complicated, because of cohort effects, variable ecological conditions, and complicated changes in age- (and sex-specific) ecological patterns
16 What is population growth? Simplest kind of population growth conceptually:Each reproducing individual independent of others, and produces some average number of offspring per lifetime.New population given by old population multiplied by each individual’s contribution to population next generation (e.g., this contribution = 1 if individual just replaces itself on average)Thus, population growth is multiplicative process, not additive & not incrementalThus, population change over time depends just on population size at particular time, and on each individual’s contribution to population size in its lifetime
17 4/1/2017Organisms with discrete, seasonal reproduction, described best by Geometric model of population growth
18 Geometric model of population growth Assumptions of model (difference equation):Constant environment-->constant growth rateUnlimited resources-->continual population growthReproduction in discrete time periods (seasonal)Model:N(t+1) = N(t)*l; l = rate of pop. change, per capita; t = timeN(1) = N(0)*l; N(0) = initial population sizeN(2) = N(1)*l = N(0)*l*l = N(0)*l2And in general, by inductive reasoning, N(t) = N(0)*ltThe beauty of this model (and other population models) is that it allows us to project population into future!
19 Example: Geometric population growth of Ringed neck pheasants introduced on Protection Island, Washington State, an environment free of predators (from G.E. Hutchinson, 1978, An Introduction to Population Ecology, Yale University Press.)
20 Exponential population model What about species like humans that grow and reproduce continually???Assumptions of model (differential equation, based on calculus)Constant environment-->constant growth rateUnlimited resources, so that population not limitedGrowth, Reproduction continuousdN/dt = r*N = (b-d)*Nr = per capita rate of population growth: dN/dt*(1/N) = rr = b-d; thus, dN/dt = (b-d)*NSolution of differential equation (I won’t do derivation here): (I.e. to get the population SIZE at any particular time t)N(t) = N(0)*ert ; e = base of natural logarithms
23 Exponential growth of coffee consumption (TP, 9/13/97)
24 Populations growing exponentially (or geometrically) double in fixed time To calculate doubling time, start with N(t) = N(0)*ertDivide both sides by N(0), we get N(t)/N(0) = ertPopulation that is doubling fits relationship N(t)/N(0) = 2…why?Thus, we can write 2 = ert, and solve for t as function of rTaking log of both sides, loge(2) = loge(er*t) = r*tSolving for t, we get t = loge(2)/r = /rHuman population is doubling now roughly every 40 years! Given that we reached 6 billion on October 12th, 1999, this means we will reach 12 billion in 2039, assuming growth rate remains constant!Also seen in invasives, colonizers, recently conserved orgs.
25 Geometric growth closely related to exponential growth Let N(t) = N(t) in both models to see exactly how the two models are related (because each model calculates N(t) as a function of other entities including N(0) = initial population size)Then models are identical if and only if N(0)*lt = N(0)*ertBy algebraic simplification (divide both sides by N(0)), l = er; Alternatively, r = loge(l)Both models give “J-shaped” curves
26 Geometric (left) and exponential (right) models are equivalent (superimposable) for particular values of l, r; here l = 1.6, r = 0.47
27 Population grows when l >1, r > 0; declines when l <1, r < 0
28 On log scale, Geometric (& exponential) models are linear If population size in geometric model is expressed on log scale, we get straight line relationship, with slope = population growth rateProof of this relationshipTake geometric equation, N(t) = N(0)*ltTake logs of both sides-->equation for a lineLeft side of equation gives loge(N(t)) = YRight side gives loge(N(0) lt) = loge(N(0)) + t* loge(l); this simplifies to Y-intercept + r*t; recall that loge(l) = r
29 Rate of population growth (r, l) can be calculated from life-table! l estimated by R0R0 = Sx lx*mx = net reproductive rateIn words, net reproductive rate is the number of offspring an individual produces over its lifetimeNote: mx same as bx--i.e.,age-specific fecundity. Also note that lx here is a PROPORTION, i.e., l0 = 1.0rm is the intrinsic rate of increase, or Malthusian parameter, after Thomas Malthusrm approximately equal to ln(R0)/TcTc = generation time of population = S x*lx*mx/(S lx*mx)Generation time, in words, is the age-weighted reproductive effort, divided by the total reproductive effort, which gives the average age of reproduction!rm thus larger with big reproductive effort, and/or with short generation time
30 Example: calculation of net reproductive rate, generation time from hypothetical life-table; ra is approximate estimate
31 Euler’s equation provides more precise way to calculate r, l: 1 = S e-rxlxbxor (discrete form): 1 = S l-xlxbxNote: These different forms of the Euler equation are solved by trial-and-error, recursively, i.e., by plugging in values for r, or l (depending on model); & checking fit of right hand side of the equation to the value on the left (“1”)
32 Population growth rate depends on ecological conditions--e. g Population growth rate depends on ecological conditions--e.g., two grain beetle species (imp later, competition!)
33 Conclusions:Simplest conceptual models of population growth (geometric, exponential) are multiplicative processes, depending just on population size, population growth rateThese two models are closely related, but apply to different kinds of populations (in seasonal, versus non-seasonal environments, respectively)These conceptual models can be related back explicitly to nature, via life-tablesModeling is a powerful process in ecology (and science in general):process of abstracting essential aspects of systemAllows projection of system into future, exploration
34 Acknowledgements: Most illustrations for this lecture from R. E Acknowledgements: Most illustrations for this lecture from R.E. Ricklefs The Economy of Nature, 5th Edition. W.H. Freeman and Company, New York.
35 Acknowledgements: Illustrations for this lecture from R. E. Ricklefs Acknowledgements: Illustrations for this lecture from R.E. Ricklefs The Economy of Nature, 5th Edition. W.H. Freeman and Company, New York.