Ch11 Population Growth and Regulation.docx

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Spencer Barrett

chapter 11: POPULATION GROWTH AND REGULATION  human pop. growth during the last 10 000 years has been one of the most significant ecological developments in earth’s history  food supply no longer limited survival and reproduction after the advent of agriculture  the Industrial Revolution brought improvements w/ public health and medicine and increasing material wealth  pop. doubled from 3 billion to 6 billion in only 40 years (1.7% growth per year)  further growth will stress the biosphere and lead to the continued degradation of many natural envmts  predicting future is difficult b/c many unknown possibilities  changes in technology, emergence of epidemic diseases of humans or their crops and livestock, and changes in material wealth, education, culture  concern for human pop. growth has led to  the develpmt of mathematical techniques to predict the growth of pop. (one aspect of demography [the study of populations]  and to intensive study of natural and laboratory pop. to learn about mechanism of pop. regulation  demography began in 17 century, London cloth merchant John GraunJohn Graunt  developed a variety of pop. statistics (probability of death at diff. ages – led to life insurance industry, and the rate of growth of the human pop.)  18 century, British economist ThomThomas Malthus  calculated that the human pop. would soon outstrip its food supply (this inspired Charles Darwin to develop his theory of evolution by natural selection) Populations Grow by Multiplication  pop. increases in proportion to its size  pop. growth depends on reproduction and deaths of individuals (rate of growth is described on a per-individual [peper capitabasis)  to compare, pop. growth has to be expressed as a rate (ex. # of indiv. per unit of time)  time flows continuously and change can occur at every instant  thus, biologists can use a ccontinuous-timeapproach to model the way in which pop. change instantaneously (but more convenient to work w/ time intervals)  discrete-time approach is the method used when working with time intervals that match the natural cycles of activities in pop. and the ways in which ecologists sample pop.  currently, hupop. is growing at ~2.5 indiv./s  this is very unusual in natural pop.  reproduction is typ. restricted to the time of year when resources are most abundant  many pop. grow during reproductive season, then decline b/w one reproductive season and the next  long-term pop. growth rate must be measured by comparing the same month each year  geometric growth is the increase or decrease in a measured over discrete indiv. in which the increment is proportional to the # of individuals at the beginning of the interval  indiv. are counted at the same time each year so that all counts are separated by the same cycle of birth and death processes  *populations with discrete reproductive seasons increase by geometric growth* Calculating Population Growth Rates  the rate of geometric growth is most conveniently expressed as a ratio of a pop.’s size in one year to its size in the preceding year (or other time interval)  this ratio is represented as λ, expresses the factor by which a pop. changes from one time interval to the next  if N(t) is the size of a pop. at time t, then N(t+1) = N(t)λ is its size one time interval later  never a –ve # of indiv., thus λ is always +ve  N(t) = N(0)λ t  ex. when a pop. increases by 50% over one time interval, the # of indiv. increases by a factor of 1.5 (λ = 1.50)  initial pop. of 100 indiv. would grow to: o N(0)λ = 100 × 1.50 = 150 at the end of year 1 2 o N(0)λ 10225 at the end of 2 years o N(0)λ = 5767 at the end of 10 years  geometric growth has the equation as exponeexponential growth= N(0)e rt  variable r is the exponential growth rate  constant e is the base of natural logarithms (~2.72)  geometric and exponential growth are related by λ = e and log λ = r e Geometric growth (λ) Exponential growth (r) Decreasing population 0 < λ < 1 r < 0 Constant population size λ = 1 r = 0 Increasing population λ > 1 r > 0 Calculating Changes in Population Size  when a pop. undergoes geometric growth at a constant rate, the # of indiv. added to or removed from the pop. varies w/ the size of the pop.  change in pop. size from one time period to the next is ∆N  change over one interval is the diff. b/w the pop. size at the beginning and at the end of the interval  ∆N = N(t+1) – N(t)  ∆N = N(t)λ – N(t)  ∆N = (λ-1)N(t)  when λ > 1, +∆N, pop. grows  when λ < 1, -∆N, pop. declines  change in pop. size is more conveniently expressed for exponential growth  in this case, change is instantaneous, and the rate at which indiv. are added to or removed from a pop. is the derivative of the exponential eqn. N(t) = N(0)e rt  the exponential growth rate (r) expresses pop. increase (or decrease) on a per capita basis  the rate of increase of the pop. as a whole (dN/dt) varies in direct proportion to the size of the pop. (N) [ ] [ ] [ ]  there is another reason to express pop. growth as an exponential function  whereas the change in absolute # of indiv. depends on the pop. size, the change in the logarithm of pop. size is linear over time  if we plot the natural logarithm (ln) of N as a function of time, we obtain a straight line whose slope is the value of r  we can see this by taking the logarithm of the expression for the exponential growth, N(t) = N(0)e , which is ln[N(t)] = ln[N(0)] + rt o this is the eqn for the straight line w/ an intercept at t = 0 of ln[N(0)] and a slope of r  the growth rates of diff pop., or of a single pop. over time, can be compared readily by plotting the logarithm of pop. size over time Calculating Population Growth Rates from Birth and Death Rates  exponential growth: the indiv. contribution to pop. growth (r) is the diff. b/w birth rate (b) and death rate (d) calculated on a per capita basis, thus r = b – d  geometric growth: the per capita growth rate per unit of time (R) is the diff. b/w the per capita rates of birth (B) and death (D) per unit of time, thus R = B – D & λ = 1 + R  in the case of individuals moving b/w subpopulations: r = b – d + i – e  immigration (i), emigration (e)  birth and death rates pertain to populations not to indiv.  ex. an indiv. groundhog dies only once, it can’t have a personal death rate  when birth and deaths are averaged over a pop., they take on meaning as rates of demographic events  rates represent the probability for an indiv. to be born or to die  exponential growth r = b – d, and dN/dt  simpler and widely adopted in the development of pop. theory  geometric growth: ∆N = B – D, so λ = 1 + B – D Age Structure Influences Population Growth Rate  future pop. size can be estimated from total pop. size (N) when birth and death rates have the same values for all members of a pop.  when birth/death rates vary w/ ages, younger and older pop. growth must be calculated separately  the pop.’s aage structureis made up of the proportions of individuals in each age class  if 2 pop. have identical birth/death rates, but diff. age structures, they will grow at diff. rates  ex. a pop. composed wholly of prereproductive adolescents and post reproductive oldsters can’t increase until the young reach reproductive age (fecundity)  projecting population in the future  b/c we are projecting growth over discrete intervals, we should calculate the geometric rate of pop. growth, which is the ratio of the pop. size after one year to that at the beginning of the year  sometimes, the pop. first grows erratically with fluctuating λ, but provided that the age-specific rates of survival and fecundity remain unchanged, the pop. eventually assumes a ststable age distribution  each age class in a pop. grows or declines at the same rate, and therefore so does the total size of the pop.  stable age distribution and constant growth rate achieved by a particular pop. depend on the age-specific survival and fecundity rates of its indiv. o any change in those rates alters the stable age distribution and results in a new rate of pop. growth  # of indiv. at time t, n (x), x is age  pop. was projected by multiplying the # of indiv. in each age class by the survival to obtain the # in the next older age class in the next time period: n (t) = n (x-1)s x-1 x  # of indiv. in each age class was multiplied by its fecundity to obtain the # of new borns: n (0) = Σn (x)b x  age structures of human pop. represented by pop. pyramids A Life Table Summarizes Age-Specific Schedules of Survival and Fecundity  life tables can be used to model the addition and removal of indiv. in a pop. (in the absence of immigration and emigration)  life tables are usually based on females since it is hard to ascertain paternity in many species  life table: Age (x), Survival (s ),xFecundity (b ), Nxmber of Individuals (n ) x  x = max. age, s = x  x = 0, b x 0  when reproduction occurs during a brief reproductive season each year, each age class is composed of a discrete group of indiv. born at approx. the same time  when reproduction is continuous (ex. humans), each age class can be designated arbitrarily as comprising indiv. b/w ages x - ½  the ffecundityof females is often expressed in terms of female offspring produced per reproductive season or age interval and is designated by b (b for “births) x  mortality portrayal in life tables  probability of survival (s )xb/w ages x and x + 1  probabilities of survival over many age intervals are summarized by survivorshipsurvivorship age x, designated by l (l xor “living”), which is the prob. that a newborn will be alive at age x  it is possible to compile a life table from age 0 only for pop. in which the fates of all indiv. can be followed by birth  in these cases, practical realities dictate that the life table begin at the age of first reproduction, after which mvmnts are limited  survival rate (s )xiw # of indiv. alive at age x + 1 divided by the # alive at age x, thus s = n /n x x+1 x  survivorship (l )x  ex. all indiv. are alive at age 1, l = 1  the prop. of these indiv. alive at age 2 is the prob.
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