POPULATION ANALYSIS IN
WILDLIFE BIOLOGY
Stephen J. Dinsmore1 and Douglas H. Johnson2
1Department of Natural Resource Ecology and
Management, Iowa State University, Ames, IA
and
2USGS- Northern Prairie Wildlife Research Center,
St. Paul, MN
Introduction
Motivating Questions
Animal Populations
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How many individuals?
What are the vital rates of
the population of interest?
Is the population increasing
or decreasing?
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These concepts lack
meaning for lower
(individuals) or higher
(communities) levels of
biological organization.
A population is a group of
organisms of the same
species living in a particular
space at a particular time
Population Analysis
► The
study of population dynamics
 What changes occur over time?
 What are the causes of those changes?
► How
many individuals are in a population?
 What is the survival of those individuals?
 What is their reproductive rate?
 How do individuals move in and out of the population?
A Theoretical Model of Population
Growth
► The
number of individuals (N) in a population at
some future time (time t+1) depends on the number
of individuals present now (time t) and any gains
(births [B] and immigrants [I]) and losses (deaths [D]
and emigrants [E]) that occur between times t and
t+1:
► Nt+1
= Nt + Bt + It – Dt - Et
Constraints on Population Growth?
► What
happens when we place constraints on the
relationship between Nt and Nt+1?
► If population growth is unimpeded it is said to be
density-independent.
► Population growth that depends directly on
population density is density-dependent.
► Under each scenario we can also discuss the rate of
population growth (λ) from time t to time t+1 using
the simple equation λ = Nt+1/Nt.
Population Models
► Modeling
approaches are used to bridge gaps in
knowledge.
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 A model is an abstraction of a real system
 Models can be simple, or complex mathematical exercises
A parameter is something that describes a population, e.g., the
annual survival rate of male Mallards.
Models may be discrete-time (where events such as births only
occur at certain times) or continuous-time (these events occur
continuously).
Some models are deterministic (fixed parameter values
through time) while others are stochastic (parameters vary).
Populations with Unimpeded
Growth
changes by a constant ratio, λ, during each
unit of time (usually a year).
► Thus, Nt+1 = λNt.
► Population
 Population is increasing if λ>1
 Population is stable if λ=1.
 Population is decreasing if λ<1.
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λ is also called the finite rate of population increase.
If N0 is the initial population size in some year, then this
equation can be rewritten as Nt+1 = λtN0.
Another alternative is to replace λ with er (e is the base of
natural logarithms and r is the instantaneous rate of increase)
and now we have Nt = N0ert.
Populations With Densitydependent Growth
► It
is impossible for any population to grow
indefinitely at a constant rate.
► Most
likely, population growth will slow as the
population becomes large and some limiting factor
exerts an influence.
► How
can density-dependence be included in the
model to make it more realistic and useful?
Populations With Densitydependent Growth
► Continuous-time
formulation
► One
approach is to multiply the growth rate by a
factor that has a negligible effect when the population
is small, but reduces the growth rate to zero as the
population approaches some limit, K (which we
might call the carrying capacity).
► The
term (K – N)/K does just that.
 This term is 1 when N is small and converges to 0 as N approaches K.
Populations With Densitydependent Growth
► Continuous-time
formulation, cont.
►
Now, the per capita population growth rate is:
►
𝑑𝑁𝑡
𝑑𝑡
►
The solution, known as the logistic equation, is:
►
𝑁𝑡 =
►
Here, rm, the maximum growth rate, replaces r, and a measures
the size of the population at time 0 relative to asymptotic
population size.
= 𝑟𝑚 𝑁𝑡
𝐾−𝑁𝑡
𝐾
𝐾
1+𝑒 𝑎−𝑟𝑚 𝑡
Populations With Densitydependent Growth
► Discrete-time
formulation
► The
logistic equation specifically applies to
continuously reproducing organisms, although it
works for populations with discrete breeding seasons
if population size is measured at the same time each
year.
► How
can the logistic equation be modified to apply to
species with discrete breeding seasons?
Populations With Densitydependent Growth
► Discrete-time
► The
formulation, cont.
discrete counterpart of the logistic equation is:
► 𝑁𝑡+1
= 𝑁𝑡 + 𝑟𝑚 1
𝑁𝑡
−
𝐾
𝑁𝑡
equation has an implicit time delay – the
population growth rate at time t+1 depends on
population size at time t.
► This
Immigration and Emigration
Dispersal
Estimation Approaches
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The movement an animal
makes from its point of
origin to the place where it
reproduces.
► Immigration (gains) and
emigration (losses) differ
from dispersal.
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Observations of marked
individuals
Genetic markers
Studies using radio
telemetry or satellite
markers
Modeling approaches such
as robust design or multistate models
Birth and Death Models
► Population
growth is the net result of births and
deaths (ignoring immigration and emigration)
► Recall
the equation Nt+1 = Nt + Bt + It – Dt – Et
► From
this we can define the birth rate (b = Bt/Nt) and
death rate (d = Dt/Nt) as simple proportions
► Distinguish between
closed and open populations
Estimating Birth Rates
Some Terms
Estimation
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Fertility is the number of
live births per unit time
(usually a year)
Fecundity is the potential
level of reproductive
performance of a population
Recruitment is the addition
of new individuals through
reproduction
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Counts of live births, eggs,
etc.
Age ratios from direct
counts
Mark-recapture methods
Indirect measures such as
clutch size and nest success
in birds
Estimating Survival Rates
► Two
common measures of survival:
 True survival – the real probability of living
 Apparent survival – the product of true survival and fidelity
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If fidelity is 100%, true survival = apparent survival
Relative to true survival, apparent survival will be biased low
when emigration is permanent
Five general approaches to estimating survival:
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Observed survival
Ratios of population size or indices
Change-in-ratio methods
Mark-recapture approaches
Methods based on tag recoveries
Estimating Survival Rates
Observed Survival
Ratios of Population Size
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Difficult to use in practice!
Deaths in the population
must be known:
 Captive populations
 Instances where radio
telemetry or other technology
provides information on all
deaths
For a closed population, the
mortality between times t
and t+1 is the population at
time t minus the population
at time t+1.
 If survivors can be
distinguished from young,
survival can be computed
directly.
 Indices can be substituted for
direct counts.
Estimating Survival Rates
► Change-in-Ratio
Methods
 Is usually applied to estimating population size
 Can also be used to estimate rate of mortality due to
exploitation
 Requires two distinguishable types of animals (male and
female, adult and young, etc.)
 Can then estimate the proportion of each type in the
population before, in, and after the harvest.
 This approach has strong assumptions!
Estimating Survival Rates
► Mark-recapture
Methods
 The most widely used approach to estimate survival
 Many models available, some for survival and others for a combination
of survival and other parameters
 “Recapture” can be physical recapture, live resightings, etc.
►
General Sampling Framework:
 Consider a study with J occasions on which animals are captured,
marked, and returned to the population
 Assume all animals are alike and have the same probability of capture
on each occasion, and that they all have the same probability of
surviving between occasions
 On each occasion animals are captured, of which some are already
marked and some will be newly marked
Estimating Survival Rates
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Definitions:
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Let Si be the probability of surviving from occasion i to i+1
Define Ni to be the number of animals in the population at occasion i
Suppose that Mi of these animals had been previously marked
On the ith occasion, ni of these animals are captured, of which mi are
already marked and ui are unmarked
The survival rate, Si, can be estimated from the equation:
𝑆𝑖 =
𝑀𝑖+1
𝑀𝑖 −𝑚𝑖 +𝑅𝑖
where 𝑀𝑖 =
𝑅𝑖 𝑧𝑖
𝑚𝑖 +
𝑟𝑖
Estimating Survival Rates
► Equation
Terms (from previous slide):
 Ri is the number of the ni animals that are released after the
ith occasion
 ri is the number of Ri that are released at i and captured
again
 zi is the number of animals that were captured before i, not
captured in i, but captured again later
Estimating Survival Rates
► More
Detailed Models:
 Robust design – a combination of open and closed
population models to estimate survival rate and population
size in the presence of temporary emigration
 Reparameterized Jolly-Seber model – can also estimate
seniority, recruitment, and rate of population change
 Multi-state or multi strata models – also useful for
estimating survival, but more appropriate for estimating
transition probabilities between stages (e.g., life stages, age
classes, etc.)
Estimating Survival Rates
► Methods
Based on Tag Recoveries
► Also called tag or band recovery models, most often
used for waterfowl studies
► General Sampling Framework:
 Capture and uniquely mark large numbers of individuals
each year
 Birds are harvested by hunters and the band number is
reported
 Many occasions, but only a single recovery is possible for
an individual bird
Estimating Survival Rates
► Methods
based on tag recoveries, cont.
► Band recovery models estimate:
 Recovery rate – the probability that a bird is shot and its
band is reported
 Survival rate – the probability that a bird survives from the
beginning of the first hunting season to the beginning of the
second
► Can
also ask questions about effects of age, sex, and
other covariates of interest
► Parameters can be estimated using the recoveries only
model in program MARK
Life Tables
►A
summary of the mortality schedule of a population,
or the pattern of deaths by age class.
► Can estimate survival and mortality rates.
Example of a life table based on known deaths of 42 gray squirrels born in 1954 (from Downing
1980:256).
Age
No. in
No. of
Mortality
Survival
(years)
pop’n
deaths
rate
rate
(x)
(nx)
(dx)
(qx)
(sx)
0-1
1-2
2-3
3-4
4-5
42
20
10
3
1
22
10
7
2
1
22/42 = 0.52
20/42 = 0.48
10/20 = 0.50
10/20 = 0.50
7/10 = 0.70
3/10 = 0.30
2/3 = 0.67 1/3 = 0.33
1/1 = 1.00 0/1 = 0
Stable Age Distribution
► The
age distribution of a population is the number of
individuals of each age class in the population at a
particular time.
 If age-dependent survival and fertility rates remain constant
for a long period, the proportion in each age class will
stabilize
 This leads to the stable age distribution
 The fraction of the population in each age class x is called
Cx and is equal to:
 𝐶𝑥 =
𝑒 −𝑟𝑥 𝑙𝑥
−𝑟𝑖 𝑙
𝑖
𝑖𝑒
Leslie Matrices
►
A useful way to “project” the population:
 The population is age structured with M age classes and breeds
seasonally
 Assumes the population has reached a stable age distribution
 The age-dependent survival and fertility rates are known
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Let nx,t be the number of individuals of age x in year t
The number of 1-year-olds in year t+1 (n1,t+1) will be the
number born in year t (n0,t) times the survival rate of 0-yearolds (S0).
So, n1,t+1 = S0n0,t
Leslie Matrices
► Next,
consider the age specific births
► The number of 0-year-olds (births) in year t+1 (n0,t+1)
represents the number of 1-year-olds in that year
(n1,t+1) times their fertility rate (m1), plus the number
of 2-year olds in that year (n2,t+1) times their fertility
rate (m2), etc.
► Thus, n0,t+1 = m1n1,t+1 + m2n2,t+1…mMnM,t+1
► If we replace (m1n1,t+1) with (m1S0n0,t) and then use gi
= mi+1Si to simplify the notation, we are left with the
components of a Leslie Matrix
Leslie Matrices
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The Leslie Matrix, L, is then:
𝑔0
► 𝑆0
0
►
𝑔1
0
𝑆1
𝑔𝑀
0
𝑆𝑀
We can use this to predict the number of individuals in each
age class at time t+1 using matrix multiplication:
𝑛0
► 𝑛1
𝑛𝑀
𝑡+1
► So, nt+1
𝑔0
= 𝑆0
0
= L ˣ nt
𝑔1
0
𝑆1
𝑔𝑀
𝑛0
0 × 𝑛1
𝑛𝑀
𝑆𝑀
𝑡
Parameter Estimation
► Modeling
is an iterative process where multiple
models (a model set) to explain the same
phenomenon are compared, and then one (or more)
are used to describe the process
► Key steps
 Develop a priori biological hypotheses about the process;
these should strive to answer “Why?”
 Use computers and approaches like the method of
maximum likelihood to estimate parameters of interest
(e.g., a survival rate)
 Use model selection procedures (e.g., AIC model selection)
to select a model or models for inference
Modeling Considerations
► Parameter
estimates themselves are important, but
biologists should seek to understand why a parameter
may vary and what that means.
 Factors that affect a parameter estimate, called covariates,
should be incorporated into an analysis if possible
 Examples of covariates are gender (male and female), sites
(A and B), day of nesting season, measures of weather, or
attributes of individual animals (e.g., body condition,
genetic characteristics, etc.)
Computer Programs
► Computer
programs to estimate population
parameters are becoming more sophisticated.
 Older programs like JOLLY and SURVIV may be outdated
and the software is no longer supported
 Program MARK is a powerful and flexible program
designed to meet the needs of a population analyst:
► Can
model group effects and covariates
► Model selection by AIC or other approaches
► Model averaging can be performed
► Goodness-of-fit testing for some models
►A Bayesian modeling tool is included
Population Viability Analysis
► Population
Viability Analysis (PVA) seeks to make
predictions about future population status using
population data.
► Two general approaches:
 Estimate the probability that a population of a specified size will persist
for a certain time period (PVA)
 Estimate the Minimum Viable Population (MVP) needed for a
population to persist for a specified time period
► Predictions
are made using computer programs such
as RAMAS Metapop 5.0 or VORTEX.
Inference
► Once
a population analysis is complete we often
desire to draw conclusions from the data.
 Controlled, manipulative experiments are desirable, but
often difficult to implement
 Increasingly, we model one or more parameters of interest,
and different hypotheses about how the population behaves
are embedded in different (competing) models.
 Model selection procedures are used to arrive at
conclusions
SUMMARY
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What is population analysis?
Introduction to theoretical models of population growth
 Exponential and logistic models
 Birth and death processes
 Age effects
Estimating survival and mortality
Making predictions about growth – Leslie matrices
General approaches to parameter estimation
Predicting the future viability of a population
Making inference from a population analysis