BIOL 4120: Principles of Ecology
Lecture 10: Population
Growth
Dafeng Hui
Office: Harned Hall 320
Phone: 963-5777
Email: dhui@tnstate.edu
World population
Population growth
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Definition: how the number of individuals
in a population increases or decreases
with time
Growth is controlled by rates of birth,
immigration and death and emigration.
Open or closed population: no immigration
and emigration, or immig.=emig.
In closed population, growth is determined
by birth rate and death rate.
10.1 Population growth reflects the
difference between rates of birth and death
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Model development
A population of freshwater hydra growing in an aquarium in
the laboratory.
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Population size N(t) when time is t.
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This is a closed population.
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Population size change is related to birth rate (b) and death
rate (d)
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dN/dt=(b-d)N=rN
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The difference between birth rate and death rate is the
intrinsic growth rate (r) (instantaneous per capita rate of
growth).
r=b-d
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In a closed population, population size
change is related to birth rate (b) and
death rate (d)
The difference between birth rate and
death rate is the intrinsic growth rate (r)
(instantaneous per capita rate of growth).
r=b-d
Population growth is related to this
intrinsic growth rate (r).
Exponential population growth
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Equations:
• 1) dN/dt=rN (differential equation form)
• 2) N(t)=N(0) exp(rt) (exponential growth
model)
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Conditions:
• Initial population is small
• No food or resource limitation
An example
Started in 1910
with only 4
males and 22
females
In 1940, there
were nearly 2000
Reindeer, St. Paul, Alaska.
Whooping crane, an
endangered species
recovered from near
extinction in 1941
How to calculate r?
Software, Excel (trendline)
Aransas National Wildlife Refuge
Properties of exponential growth
Properties of exponential growth
Widely used
in biology
r determines the shape of the growth.
r=0, no change in population size
r<0, decrease in population size; r>0, increase in population size.
Prediction of population growth
N(t)=N(0)Exp(rt)
Give a time t, we can predict the population size.
An Example:
Deer population: N(0)=300, r=0.5, after 5 years,
what’s the population size?
N(5)=N(0)Exp(rt)=300*exp(0.5*5)=3655
(495, 815, 1344, 2216, 3655) t=10, ?
44524
10.2 Life table
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Life table is an
age-specific
account of
mortality.
Purpose of life
table: to provide a
clear and
systematic picture
of mortality and
survival within a
population
How to construct a life table?
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1. start with a
cohort: a group of
individuals born in
the same period of
time;
2. Add a column of
lx as the probability
at birth of
surviving to any
given age;
How to construct a life table
(cont.)?
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3. calculate dx, a
measure of agespecific mortality
4. Calculate agespecific mortality
rate, qx
10.3 Different types of life table
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Two types
• Cohort or dynamic life table
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as the above gray squirrel
• Time-specific life table
Elf opine
10.4 Life tables provide data for
mortality and survivorship curves
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Table is better than words, but a
graph is worth one thousand words.
Mortality curve and survivorship
curve.
Mortality curves
Survivorship curves
Log scale for Y axis
Three basic types of survivorship
Type I (convex)
Humans and other
mammals and some
plants (k-selection)
Type II (survival rates do
not vary with age)
Adult birds, rodent, and
reptiles, perennial plants
Type III. Concave
Mortality rate high in the
beginning (r-selection)
Oysters, Fish,
many plant species (most
trees)
10.5 Birthrate is age-specific
Crude birthrate
(demographers): # of birth
over a period of time divided
by population size at the
beginning of the period*1000
Age-specific birthrates, bx
Mean # of females birth to a female in
each age group.
(Only females give birth; birth rates
vary with ages)
Gross reproduction rate: sum
of the bx values across all
age classes, provides an
estimate of average offspring
born to a female over her
lifetime.
10.6 Birth rate and survivorship
determine net reproductive rate
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Fecundity table:
take survivorship
column, lx, from life
table and add agespecific birthrate,
bx .
Net reproduction
rate, R0: number of female R0: depends on
survivorship and
offspring a female at birth can
produce (or average # of
fecundity
females that will be produced
(left) during a lifetime by a
 R0=1, >1 or <1
newborn females.)
10.7 Project population growth
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Given a population with age structure and
some other information (age-specific
mortality rates and birthrates), we can
project future changes of the population
size.
For example, a population of squirrel with
10 adults (1-yr) and 20 juveniles
females, what would happen in the next
10 years?
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What do we need to project future
population size change?
Calculate age-specific survivor rate: sx=1-qx
bx is age-specific birthrate
How to construct a population
projection table?
How to construct a population
projection table?
Population size (N) increases every year.
Lambda (finite multiplication rate): =N(t+1)/N(t).
Age distribution
Stable age distribution: by year 7, the proportion of each age group
remain the same year after year.
Population is still growing.
Geometric growth vs exponential
growth
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N(t)=N(0) t
N(t)=N(0)exp(rt)
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=exp(r)
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or r=ln()
These models are used to describe
dynamics of populations. Geometric
growth is used for population generations
not overlap (discrete time interval),
exponential growth model is for
continuous population.
Fig. 11.3
10.8 Stochastic processes can
influence population dynamics
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What’s stochastic process?
Deterministic process: Given a set of
initial conditions (N(0), r), the
exponential growth will predict only one
exact outcome.
But the age-specific mortality rates, birth
rates represent probability and averages
derived from the cohort or population
under study (bx=2? 0,1,2,3).
Stochasticity
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Demographic stochasticity: stochastic (or
random) variations in birth and death
rates that occur in populations from year
to year. (Cause change in r).
Environmental stochasticity: Random
variation in the environment, such as
annul variation in climate and natural
disasters can have a direct influence on
average birth and death rates within the
population.
10.9 Population extinction
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If r becomes negative
(birth rate < death
rate), population
declines and will go
extinction.
Factors: Extreme
environmental events
(droughts, floods, cold
or heat etc), loss of
Overgraze, only 8 in 1950
habitat (human).
Allee effect, genetic drift,
Small populations are
inbreeding (mating
susceptible to extinction between relatives)
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Hackney and McGraw (West Virginia
University) examined the reproductive
limitations by small population size on
American ginseng (Panax
quinquefolius)
• Fruit production per plant declined with
decreasing population size due to
reduced visitation by pollination
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Small population size may result in
the breakdown of social structures
that are integral to successful
cooperative behaviors (mating,
foraging, defense)
The Allee effect is the decline in
reproduction or survival under
conditions of low population density
There is less genetic variation in a
small population and this may affect
the population’s ability to adapt to
environmental change
END
Geometric Growth
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When generations do not overlap,
growth can be modeled geometrically.
Nt = Not
• Nt = Number of individuals at time t.
• No = Initial number of individuals.
•  = Geometric rate of increase.
• t = Number of time intervals or
generations.
Exponential Growth
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Continuous population growth in an
unlimited environment can be modeled
exponentially.
Appropriate for populations with
overlapping generations.
• As population size increases, rate of
population increase gets larger.