Population Viability Analysis
Pedro F. QuintanaAscencio,
Office: Biology Bldg. 401 E
Phone: 823-1662
pquintan@mail.ucf.edu
http://biology.ucf.edu/~pascencio/
Office hours : Tuesday & Thursday: 9:00-11:00
What is population viability
analysis?
• The use of quantitative methods to
predict the likely future status of a
population or collection of populations
of conservation concern
Population viability analysis
• Provide assessments of
population persistence based
on a combination of empirical
data and modeling scenarios
Advantages of modeling
• Analyze and synthesize data
• Make assumptions and data limitations
transparent
• Formulate the logic of the research
problems in a consistent and unambiguous
way
Potential products and uses of PVA
Assessment of Risk
• Assessing the extinction risk of populations
• Anticipate demographic changes
• Compare relative risks of several
populations
Potential products and uses of PVA
Guiding management
• Identify key life stages or demographic
processes as management targets
• Determine how large a population needs to
be to avoid extinction
• Determine how many individuals to release
• Setting limits to harvesting compatible with
persistence
• Deciding how many populations to protect
Other uses and products?
• Spatial partitioning of individuals among
populations
• Genetic partition of individuals
• Role of corridors
• Spatial architecture of networks
• Multiple species approach (community viability?)
• Environmental contextual dynamics
• Assessing social structure
How to describe demographic change?
• Number of individuals
• Population structure
• Individual characteristics
• Population number and structure
Types of PVAs
• Count-based
Data needed: exhaustive counts or
estimates of the total number of
individuals, or of a subset of
individuals in the population
Assumptions: All individuals are
identical
Types of PVAs
• Demographic models
Data needed: rates of demographic
processes separately for each type of
individual in the population
Limitations: require more and data
Types of PVAs
• Multi site models
Data needed: measures of local occupancy,
and estimates of rates of movement between
populations and of rates of local extinction.
Information on spatial structure and
environmental correlation
Limitations: They may require extensive data
Types of PVAs
• Individual based models
Data needed: information on actual local
and behavior of all individuals
Limitations: They require the most
detailed data
Quantitative Conservation Biology
Williams F. Morris y Daniel F. Doak
(2002). Sinauer Associates
A model philosophy
Morris y Doak recommend:
• “Keep it simple”
• “Let the available data tell you which type
of PVA to perform”
• “Make sure you know what your model is
doing”
Tinkering !!!!
Authors urge you to:
• Understand the structure of their programs
as a way to truly know how the underling
models work
• Combine and modify them to suit your
needs
Mark Shaffer and collaborators
Their seminal work used computer
simulations to evaluate if the population of
“grizzly bear” (Ursus arctos) in the Greater
Yellowstone ecosystem had at least a 0.95
chance of surviving for different periods in
the future.
The Grizzly Bear
100,000 were estimated
in 1800
Today there are only 1000
They estimated that local
populations needed 100
bears to have a 95 %
chance of survival
Their results indicated a
high probability of
persistence for the 100
years but a more uncertain
future during the next 300
years. Their conclusions
affected the management
within the park and had
legal consequences.
Russell Lande and collaborators
• Studied changes in the populations of the
Northern Spotted Owl (Strix occidentalis) in
the context of the logging of old growth
forests on which the owl depends.
•Their results suggest
that populations may
be declining but the
results were not
enough to eliminate
other scenarios
including stable
populations.
Deborah Crouse and collaborators
• They evaluated the relative effect of two
human activities on the persistence of the
loggerhead sea turtle (Caretta caretta) in
the southeastern coast of the United States:
trampling of eggs and hatchlings on beaches
and drowning of older-aged turtles in
fishing nets were hypothesized to underline
their declining numbers.
Crouse et al. 1987
•Their results showed that the use of
mechanisms that reduce the mortality of
adult turtles is much more effective to
increase these populations.
The African elephant
• Lande and collaborators
analyzed how large
should be the national
paks in Africa to
maintain viable
populations of
•Their conclusions suggest that
elephants (Loxodonta at least 2500 km2 are necessary
africana)
An example:
An example: Population change
Change in
population size
during time interval
N
t
=
=
Births during _ Deads during
time interval
time interval
B
_
Nt+1= Nt + B - D
D
Per capita birth rate
• Is the number of offspring produced per unit
time by an average member of the population
If there are 34 births per year in a
population of 1000 the annual per
capita birth rate is
34/1000=0.034
Use of the rates
• b=0.034, N=500
• B=bN
• B=0.034*500
• B= 17 per year
Instantaneous rate of population size change
N
t
N
t
=
rN
D
bN
_
mN
=
N
t
r
B
_
=
=
b
_
m
dN
dt
=
rN
This equation can be solved by integration to
give the familiar equation for exponential
population growth
Nt= No e(b-d)t
Nt= No ert
e = the base of the natural logarithms
r = intrinsic rate of natural increase
We can replace
Nt= No λt
λ = finite rate of increase
r
e
with λ
The exponential model describes
population growth in a idealized,
unlimited environment
Useful terminology
• Let A x B be a product set consisting of all
ordered pairs (a, b) where a is a member of
A and b is a member of B. Then any subset
of A x B is called a relation
Useful terminology
• Function: is a relation that can uniquely
identify an element in the domain for each
element in the range
• Domain: The set A of the variable is called
the domain
• Range: The set B is defined in such a way
that all members of B are associated with
members of A
Model
• Is an equation describing the relationship
between the independent variables and the
dependent ones
Useful terminology
• Dependent variable: the thing in the model
you want to estimate
this entity depend on other factors
• Independent variable: these other factors
• Parameters: those components that
mediate the relationship between
independent and dependent variables
Then??
Nt= No λt
λ = finite rate of increase
Model assumptions
• There are no density dependent effects
• Births and Deaths are mutually independent
• B and D are also independent of the age of
the individuals
• B and D are constant in time
• There is no uncertainty in the prediction
Parameters and initial conditions
• 57 rhinoceros (45 adults, 4 yearlings + 8 juveniles)
• Another assumption: Females are usually the
limiting sex in reproduction
• Birth rate =0.14 per year
• Death rate= 0.08 per year
• r= (0.14-0.08) = 0.06
• In 1986 there were 35 females
• Please predict the number the individuals under
the assumption of 1:1 ratio after 50 years
Conway and Goodman observed a
50 % death rate in the transition
from yearling to juvenile
• It is not strictly important that rates are
different in different age classes if the
proportion of the population within each
age class remains more or less constant
• We will demonstrate this later
• By now we should assume this so
For the purpose of the management plan
we will evaluate the next 50 years
• Our starting date will
be 1986
• The deterministic
prediction:
Nt= 35e0.06 * t
Nt= 35*1.061837 t
Deterministic prediction
Adding realism: using integer
numbers
•
Algorithm 2.1
1.
2.
3.
4.
5.
6.
7.
For each time step from1 to t, do steps 2 to 7
Let N(t+1) take the value of the current pop size N(t)
For each animal from 1 to N(t+1), do steps 4 to 7
Choose a uniform random number U1
Choose a uniform random number U2
If U1 is less than d then decrease N(t+1) by 1
If U2 is less than b, then increase N(t+1) by 1
Carrying capacity (K)
• Maximum population
size that a particular
environment can
support
The logistic
Growth Model
dN
dt
=
(K-N)
rmaxN
(K)
dN
dt
=
(K-N)
rmaxN
(K)
K-N = the additional number of
individuals that the environment can
accommodate
(K-N)/K = the proportion of K that is
still available for population growth