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Biology

BIO120H1

Spencer Barrett

Fall

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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 pop.as 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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