Transcript Population growth

Number of Animals
Population Ecology
72 73 74 75 76 77 78 79 80
Year
Goal of Population
Ecology is to
Describe the
Composition of
Populations Through
Time and
Understand
Population
Fluctuations
Describing Population
Composition
Sex Ratio
Age Ratio
Genetic Composition
Spatial Structuring
Sex Ratio Indicates Important
Processes in Population
Sex Ratio (Males:Females)
in Flock of Pinyon Jays
Population growth
potential--greater male
bias = less growth ability
(sexual species)
Breeding System
Dispersal
(Data from Marzluff and
Balda 1992)
72 73 74 75 76 77 78 79 80
Age Pyramids Summarize Age
Structure
Differ for Increasing, Steady, and Declining
Populations
Indicate Bad Years, Bottlenecks in
Reproduction, etc.
Proportion
in each
age class
Increasing
Population
Stable
Population
Declining
Population
Pinyon Jays Were
Studied for 20 Years
Long-term studies of
marked animals are
needed to get
accurate population
growth and
composition
information.
Age Structure Reflects Relative
Productivity of Cohorts
Young (cohort) from productive years constitute large
proportion of population for many years (1977, 1978)
A poor year of reproduction continues to be echoed in
population as a missing cohort (1976)
300
Number
of Jays
in Flock
50
73
1978
1977
1976
74
75
76
(Marzluff & Balda 1992)
77
78
YEAR
79
80
81
82
Importance of Indirect and Direct Selection
Depends on Genetic Composition of
Population
Number
of
Relatives
in Flock
(Marzluff & Balda 1992)
Age of Focal Individual
Density of Great Tits in 4 Areas
Describing Change in Population
Size
(Lack 1966)
Year
Managers are
usually concerned
with monitoring
population SIZE---So,
How do WE Quantify
CHANGE in
Population Size??
Population size and rates of
growth
Population size:
Nt = population size at time t
Nt+1 = population size at time t+1
Nt+1 = Nt + Births + Immigration – Deaths -Emigration
Growth rates:
r = exponential growth rate
λ (‘lambda’) = intrinsic population growth rate
Population growth
Reproduction, births, natality (B)
Immigration (I)
Population
Mortality, death (D)
“BIDE”
Emigration (E)
Age-specific birth rates
A fecundity schedule for Chamois from New Zealand.
Age (yrs)
N
#
Female births per
pregnant
female
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0
0.000
1
60
2
0.017
2
36
14
0.194
3
70
52
0.371
4
48
45
0.469
5
26
19
0.365
6
19
16
0.421
7
6
5
0.417
>7
10
7
0.350
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Survivorship curves for male &
female moose on Isle Royale
females
Survivors (lx)
males
2 4 6 8 10 12 14 16 18 20
Age at Death (years)
Emigration and Immigration
Juvenile dispersal: movement from place of
birth to place of breeding
Breeding dispersal: movement by adults from
one place of breeding to another
Birds: Female dispersing sex
Mammals: Male dispersing sex
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American Robin
post-fledging
movements
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500
1000
1500
2000
2500
3000
3500
4000 Meters
Population Growth
Carrying capacity (k)
Exponential
N
k
N
Logistic
time
Classic growth curve,
unlimited resources
time
Classic growth curve,
limited resources (k)
The Simplest Quantification of Population
Growth Assumes Exponential Growth
Nt=N0ert-----let t = 1 year
N1=N0er
er=N1/N0===Lambda, Finite rate of Increase
Lambda goes from 0 (extinction) to 1 (stable growth)
to positive infinity (Exponential growth of various
magnitude)
Exponent Indicates the
Magnitude of Change
er=N2/N1---Take ln (natural log, loge) of both sides to
get:
r = ln(N2/N1)
varies from negative infinity (decrease) to 0 (Stable) to
positive infinity (increase)
r, the exponential multiplier, = Intrinsic
(instantaneous) rate of increase
Exponents provide consistent
quantification of magnitude of
change
Doubling and halving of population produces same
exponent multiplier of change----sign of multiplier
changes
N1=50 ---N2=100--doubling
er = (lamda) = 100/50 = 2
r = ln (2) = .693
N1=100--N2=50---halving
er = (lamda) = 50/100 = 0.5
r = ln (0.5) = -.693
Units of r and lambda
Units of lambda are obvious
numbers per unit time
restricted to the unit it was calculated over
t = 1 year, then rate is change per year
Units of r not obvious
it is a multiplier, not a rate
“growth multiplier of ln(#s) per unit time”
not restricted to unit it was calculated over
r from 1 year can be transformed to r for each day by
dividing by 365, etc.
Lambda and r
Both present the same information in varying
formats
Population increases at lambda per unit time or r at
any instant in time
r is useful because it can be transformed to fit time
interval of interest, lambda is more intuitive
Unlimited Growth
Australian rabbit (European hare)
• 1859: 24 hares introduced (for human food?)
• 1865: over 20,000 hares were harvested, actual
population much greater.
• Mid-1800’s to mid-1900’s: major problem with too
many hares; caused habitat destruction and
reduction in native mammals
• 2000: still present, local problems
Carrying capacity
Rabbits exceeded k
Rabbit-proof fence
No rabbits
Carrying capacity
Carrying capacity (k): the number of
organisms that can be supported by a given
area; the actual number of organisms
fluctuates near this
# of
Animals
(N)
k
time
Adding A Limit to
Population Growth
More Realistic than Exponential Growth
Growth is adjusted as population approaches carrying
capacity (K) of the environment
Population growth simply stops at K
Population crashes after resource is consumed
Population growth is under negative feedback as it
approaches K and gradually reaches K
Population Growth is Gradually Reduced as
Carrying Capacity is Reached; Resources
Renew Independently of Population Size
Logistic Growth
Inflection Point
#s
K
simple favorite in
wildlife management
Rate of Increase is
only a function of
Population Density
Assumes resources
are not damaged by
large populations
Time
Wildebeast don’t affect
grass roots
Logistic Math
Verhulst (1838) and Pearl & Reed (1920) independently
derived equation
Verhulst-Pearl Equation (Sigmoidal Growth)
dN/dt = derivative form of change in N with respect to
time
dN/dt = rmN(1-N/K)
dN/dt = rmN = exponential growth
As N approaches K, N/K approaches 1. Therefore rmN(1N/K) approaches 0
With K and Typical Seasonal
Patterns of Reproduction, There
is Often A “Doomed Surplus”
Good Sites
Poor Sites
High Pops
Low
Pops
Deep
Water
(Errington 1946)
Dry
Upland
Mink control distribution of
muskrats
those in poor sites
including dispersers are
eaten
Predators often take the
young, homeless, sick, injured,
dispersing, or old individuals
so effect on species or
community is less
Logistic Growth Model May be
Used to Calculate Harvest
K
#s
Maximum
Yield
=1/2 K
Time
Maximum
Sustainable Yield is
at Inflection Point
Growth is Maximum
and Population is at
Largest Size
Larger Populations
Start to Have Slower
Growth
Another View of Logistic
Growth
Inflection Point
Max Sustainable
Yield
Growth rate starts
slow, peaks, and
ends slow
dN/dt
K
N
Maximum
Sustainable Yield is
at rate of fastest
population growth
Assumptions of Logistic
Growth
All individuals contribute equally to population
growth--equal reproduction regardless of age or sex
Growth rate is constant regardless of environmental
variation
K is constant--not affected by growth
Reduction in growth as population approaches K is
linear and instantaneous (no time lags)
Populations fluctuate due to
Density dependent factors
Ex: Predation, competition, habitat availability
change population growth in predictable ways
N is driven by population density
Density independent factors
Random or Stochastic events
Ex. Weather, accidents
Breeding
14 aug 2007
Reindeer
(caribou)
# young
produced
Bighorn sheep
Population density (top) or size (bottom)
Population regulation: food
High food addition
Low food
addition
Townsend’s vole
No food
added
Shaded area is winter
Population regulation: food
Population cycles: Ex. peaks in lynx populations show
time lag behind peaks in snowshoe hare populations
Population size
Snowshoe hare
Lynx
Time (years)
Population regulation: climate
Population regulation: competition
Competition – demand by 2 or more individuals of the same
or different species for a common resource
Between 2 individuals of same species: Intraspecific
Between 2 individuals of different species: Interspecific
Limited supply of resource: Exploitation
Not limited but interaction detrimental: Interference
Inter- or Intraspecific competition?
Exploitation or Interference competition?
Population regulation: competition