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Aggregated dispersion of chinch bugs (an agricultural pest of cereal crops) in space and time; light green = low density, blue = high density

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Chinch bug example illustrates typical case of population variation in time n Population size changes as function of birth, death processes (i.e., demography), and by immigration & emigration n Immigration and emigration create complications in terms of spatial-temporal dynamics… (Metapopulation considerations) Emi- gration Birth Death Immi- gration Population Size

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Life table provides simple way to track how population grows and changes n Look at pop structure also n originally invented by insurance industry, calculate odds of peoples living to particular ages. n Two kinds of life tables in nature: u Best type generally is cohort life table = dynamic life table; (cohort = all individuals within an area born/germinated in population in same time interval) u BUT What about long-lived, or highly mobile animals (like sequoia tree, whale, grizzly bear) that cant 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 populations age-distribution

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Diagramatic illustration of dynamic versus static life table Time since origin of cohort Age t0t0 t1t1 t2t2 t3t3 Birth Static life table from one time period Cohort life table based on one cohort Each individual represented by a diagonal line segment

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Some life-table assumptions: n For all life tables, the sample of individuals must be representative of populations age-structure n In cohort life table, cohort needs to be representative (or multiple cohorts studied); cohorts often differ, e.g., some fish cohorts survive much longer than others n Ideally in static life-table, population is neither increasing nor decreasing; population that is increasing or decreasing will have skewed age- distributions n Life tables represent local ecological conditions, which may or may not be representative (e.g., population density)

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Cohort effect, i.e., dominance of population dynamics by particular cohorts- -in this case, resulting from excellent Lake Erie whitefish spawning in 1944

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Typical life table parameters n x = Age interval n n x = the number alive at start of age interval x (in cemetery lab l x used for n x ) n l x = 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, n 0, or standardized to cohort of Thus l x /1000 = n x /n 0 ; and then l x = 1000*(n x /n 0 ). n d x = n x - n x+1 (d = dead) n q x = d x / n x ; 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 q x = (d x / n x )*1000 (=no. die x /1000)

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Life table parameters, cont. n s x = 1 - q x ; s x = survival rate, the proportion of remaining individuals that will still be alive at the end of age interval x n b x = fecundity, the average number of offspring produced by a female during age interval x (sometimes represented as m x ! Think of b x as births) n L x = (n x + n x+1 )/2 = average number alive in age x T x = x--> L x ; This is the sum of all individuals living in age interval x plus all future ages--an intermediate step in calculating e x. n e x = T x / n x = life expectancy at time interval x, i.e., the average number of additional age classes an individual can expect to reach at each age

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Whew, thats a heck of a lot of math! Whats it good for? n These parameters allow one to do lots of cool stuff u Compare survivorship patterns directly (n 0 = 1000) using survivorship curve: log(l x ) as function of age u Age-specific mortality risk, fecundity, life expectancy, etc. u We can calculate population growth rates (next lecture) n Some factors that influence life tables u Age affects demography dramatically in many species (in plants stage often more impt. than age) u Life tables can also vary by year (e.g., climate & cohort effect), sex, and certainly by species u Life tables are partially heritable, under natural selection

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Example: Dall mountain sheep studied on Mt. Denali, n 0 = 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)

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Survivorship curve for Dall Mountain Sheep

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Three different types of idealized survivorship curve may be recognized: Age Log 10 number of survivors Type I; e.g., many mammals Type II; e.g., many birds, small mammals, lizards, turtles Type III; e.g., many invertebrates, anadromous fish

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Type II survivorship curve, 1978 cohort of Daphne Island (Galapagos) cactus finch, showing annual variation

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Conclusions: n Spatial structure of populations can tell us much about the ecological mechanisms involved n Temporal 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) n 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

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Population growth: lecture content

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What is population growth? n Simplest kind of population growth conceptually: u Each reproducing individual independent of others, and produces some average number of offspring per lifetime. u New population given by old population multiplied by each individuals contribution to population next generation (e.g., this contribution = 1 if individual just replaces itself on average) n Thus, population growth is multiplicative process, not additive & not incremental u Thus, population change over time depends just on population size at particular time, and on each individuals contribution to population size in its lifetime

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Organisms with discrete, seasonal reproduction, described best by Geometric model of population growth

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Geometric model of population growth n Assumptions of model (difference equation): u Constant environment-->constant growth rate u Unlimited resources-->continual population growth u Reproduction in discrete time periods (seasonal) n Model: N(t+1) = N(t)* = rate of pop. change, per capita; t = time N(1) = N(0) N(0) = initial population size N(2) = N(1) N(0) N(0) 2 And in general, by inductive reasoning, N(t) = N(0) t The beauty of this model (and other population models) is that it allows us to project population into future!

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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.)

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Exponential population model n What about species like humans that grow and reproduce continually??? n Assumptions of model (differential equation, based on calculus) u Constant environment-->constant growth rate u Unlimited resources, so that population not limited u Growth, Reproduction continuous n dN/dt = r*N = (b-d)*N u r = per capita rate of population growth: dN/dt*(1/N) = r u r = b-d; thus, dN/dt = (b-d)*N u Solution of differential equation (I wont do derivation here): (I.e. to get the population SIZE at any particular time t) n N(t) = N(0)*e rt ; e = base of natural logarithms

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Yeast begin growing exponentially...

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Is human population growing exponentially?

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Exponential growth of coffee consumption (TP, 9/13/97)

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Populations growing exponentially (or geometrically) double in fixed time n To calculate doubling time, start with N(t) = N(0)*e rt n Divide both sides by N(0), we get N(t)/N(0) = e rt n Population that is doubling fits relationship N(t)/N(0) = 2…why? n Thus, we can write 2 = e rt, and solve for t as function of r n Taking log of both sides, log e (2) = log e (e r*t ) = r*t n Solving for t, we get t = log e (2)/r = /r n Human 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! n Also seen in invasives, colonizers, recently conserved orgs.

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Geometric growth closely related to exponential growth n 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) t = N(0)*e rt By algebraic simplification (divide both sides by N(0)), e r ; Alternatively, r = log e ( ) n Both models give J-shaped curves

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Geometric (left) and exponential (right) models are equivalent (superimposable) for particular values of r; here = 1.6, r = 0.47

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Population grows when >1, r > 0; declines when <1, r < 0

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On log scale, Geometric (& exponential) models are linear n If population size in geometric model is expressed on log scale, we get straight line relationship, with slope = population growth rate n Proof of this relationship Take geometric equation, N(t) = N(0)* t u Take logs of both sides-->equation for a line u Left side of equation gives log e (N(t)) = Y Right side gives log e (N(0) t ) = log e (N(0)) + t* log e ( ); this simplifies to Y-intercept + r*t; recall that log e ( ) = r

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Rate of population growth (r, ) can be calculated from life-table! estimated by R 0 R 0 = x l x *m x = net reproductive rate u In words, net reproductive rate is the number of offspring an individual produces over its lifetime u Note: m x same as b x --i.e.,age-specific fecundity. Also note that l x here is a PROPORTION, i.e., l 0 = 1.0 n r m is the intrinsic rate of increase, or Malthusian parameter, after Thomas Malthus u r m approximately equal to ln(R 0 )/T c T c = generation time of population = x l x *m x /( l x *m x ) u Generation time, in words, is the age-weighted reproductive effort, divided by the total reproductive effort, which gives the average age of reproduction! u r m thus larger with big reproductive effort, and/or with short generation time

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Example: calculation of net reproductive rate, generation time from hypothetical life-table; r a is approximate estimate

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Eulers equation provides more precise way to calculate r, 1 = e -rx l x b x or (discrete form): 1 = -x l x b x Note: These different forms of the Euler equation are solved by trial-and-error, recursively, i.e., by plugging in values for r, or (depending on model); & checking fit of right hand side of the equation to the value on the left (1)

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Population growth rate depends on ecological conditions--e.g., two grain beetle species (imp later, competition!)

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Conclusions: n Simplest conceptual models of population growth (geometric, exponential) are multiplicative processes, depending just on population size, population growth rate n These two models are closely related, but apply to different kinds of populations (in seasonal, versus non-seasonal environments, respectively) n These conceptual models can be related back explicitly to nature, via life-tables n Modeling is a powerful process in ecology (and science in general): u process of abstracting essential aspects of system u Allows projection of system into future, exploration

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Acknowledgements: Most illustrations for this lecture from R.E. Ricklefs The Economy of Nature, 5th Edition. W.H. Freeman and Company, New York.

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Acknowledgements: Illustrations for this lecture from R.E. Ricklefs The Economy of Nature, 5th Edition. W.H. Freeman and Company, New York.

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