Lecture Outline: Introduction

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Lecture Outline Introduction
Density-Independent Growth

• What is a population?
• What is population ecology about?
• General approaches in population ecology
• Role of mathematical models
• Density-independent population growth
• Exponential and geometric models
• Assumptions limitations
• Values

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What is a population?
Gotelli (2001). A population is a group of
individuals, all of the same species, that live
in the same place. Although it is difficult to
define the physical boundaries of a population,
the individuals within the population have the
potential to reproduce with one another during
the course of their lifetimes.
Akcakaya et al. (1999). a collection of
individuals that are sufficiently close
geographically that they can find each other and
reproduce Thus, it rests on the biological
species concept In practice, a population is
any collection of individuals of the same species
distributed more or less contiguously.
Begon et al. (1996). A group of individuals of
one species in an area, though the size and
nature of the area is defined, often arbitrarily,
for the purposes of the study being undertaken.
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What is population ecology?

• The study of how and why the distribution,
  abundance, and composition of populations change
  over time and space.

• Historical focus on changes in abundance over
  time
• (e.g., population growth models, population
  cycles).
• Current focus includes aspects of spatial
  structure and dynamics (e.g., dispersal and
  metapopulations).

• Bridge between individual ecology and community
  ecology, and typically includes two-species
  interactions (competition, predation, parasitism).

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Wildlife Population Ecology
Application of principles of population ecology
to conserve, restore, manage or control
non-domesticated vertebrate species.
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Some relevant questions of Wildlife Population
Ecology

• How does demographic and environmental
  stochasticity affect our ability to predict the
  future number of deer based on present number of
  individuals?
• Given data on survival, reproduction, and age
  structure, what are the chances that a population
  of an endangered songbird species persists for
  100 years? What processes are important to
  persistence?

• How does patch structure, dispersal ability, and
  environmental correlation affect extinction
  patterns within a metapopulation?
• 4. Do source-sink dynamics affect our ability to
  evaluate and manage habitat quality for wildlife
  species?
• 5. How does detectability of individuals
  influence field estimates of population size or
  survival?

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General Approaches to Population Ecology
Experimentation
Observational
Modeling

• Observationalincludes statistical analysis of
  time-series data
• Modelingmathematical models of mechanisms
• Experimentationmanipulative field studies with
  controls replication

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Why use models?

• Nature is complex so we need to use simplified,
  abstractions of reality.

• Provide a framework for organizing observations
  and ideas. Alternative is loose collection of
  isolated facts and case studies.

• Generate testable predictions that help us to
  identify mechanisms responsible for observed
  patterns.

• Expose faulty assumptions. That is, many models
  are wrong but the reasons for model failure are
  informative.

• Highlight gaps in field data. Help to direct
  future sampling efforts.

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• Historically, mathematical models and modelers
  were viewed with skepticism by many traditional
  field ecologists (Kingsland 1995).
• Many of the modelers were not ecologists, or
  even biologists (e.g., human demographers,
  theoretical physicists). Relevance of the models
  to practical questions was not initially obvious.

• In contrast, other ecologists welcomed
  mathematical approach and hoped it would help to
  solidify ecology as a hard science
• (aka Physics Envy).

• Today, most ecologists agree that studies in
  population ecology ideally should integrate
  different approaches (strengths weaknesses).

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But tension among ecologists continues
Charles Krebs-Avoid mathematical models. They
are more seductive than useful at this stage of
the subject. If you are addicted to models, at
least do not believe them until all of the
assumptions can be tested and their predictions
verified. There is no such thing in population
dynamics as a reasonable assumption without
data.
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Turchins Approach
Time-series analysis

• Initial stages of investigation
• Quantitative description of patterns of
  population fluctuations
• Potential to reduce number of viable alternatives

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My background and biases
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Density-independent Population Growth

• Simplest model of population growth
• Population processes are not affected by current
  density of population.
• Constant fraction is added to population each
  time step.

MULTIPLICATIVE PROCESS

• Deterministic models (vs. stochastic models)
• Parameters are constant do not vary
  unpredictably over time
• No uncertaintygiven starting conditions, models
  always produce the same predictions

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Discrete vs. Continuous Growth Models

• Discrete
• Species with non-overlapping generations (some
  insects)
• Species with pulsed reproduction (many wildlife
  species in seasonal environments)
• Time is modeled in discrete steps (often 1 year)
• Fits well with annual censuses of wildlife
  populations
• Difference equations are used to model population
  growth.

• Continuous
• Species that grow continuously without pulsed
  births deaths (humans, some wildlife species in
  relatively stable environments)
• Time is modeled as a continuous, smooth curve
• Analytically tractable so you can find solutions
  using calculus
• Differential equations are used to model
  population growth.

For density-independent growth, discrete and
continuous models produce qualitatively similar
predictions.
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The BIDE Equation
Nt1 Nt B I - D - E
B number of births per time period D number
of deaths per time period I number of
immigrants per time period E number of
emigrants per time period
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Geometric Growth (discrete model)

• Assume population increases or decreases each
  year by a constant proportion (z)

• If population increases by 25 between years,
  then z 0.25.

Nt 1 Nt zNt
Nt 1 (1 z) Nt
Let 1 z ?.
Lambda is the finite rate of increase.
Nt 1 ? Nt

• If ? 1.25, then population increases 25 per
  year

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Finite rate of increase
Nt 1 ? Nt
Dimensionless ratio, no units
? Nt 1/Nt

• For example, if population size of coyotes in a
  study area is 150 this year and size is 200 next
  year, then lambda equals 1.33
• (200/150).
• If ? 1.33, then population increases 33 per
  time step (year)

? gt 1, exponential increase ? 1, no
changestationary population ? lt 1, exponential
decline
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How do we predict total population size at some
particular time?
For example, how about for two years in the
future?
Nt 2 ? Nt 1
Nt 1 ? Nt
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Note Akcakaya et al. use big R instead of ?
for the Finite Rate of Increase in discrete
models.
Nt Rt N0
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Exponential Growth (continuous model)

• Continuous model is equivalent to a discrete
  difference equation with an infinitely small time
  step.

• We treat time as being continuous so change in
  population size is described by a differential
  equation

r gt 0, exponential increase r 0, no
changestationary population r lt 0, exponential
decline
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How do we predict total population size at some
particular time?
We integrate the differential equation
dN/dt rN
Nt N0ert
where e is 2.718
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Population sizes of muskox on Nunivak Island
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Linear
N
Semi-log
Slope r (intrinsic rate of increase)
ln(N)
(from Gotelli)
Time (t)
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Two predictive equations for population size
Continuous
Discrete
Nt N0ert
Nt ?t N0
Hence,
? er
ln(?) r
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Doubling Time
How long will it take a population to double in
size?
Nt ?t N0
?t Nt/N0
If Nt/N0 2, what is t?
?t 2
t ln(?) ln(2)
t ln(2)/r
t ln(2)/ln(?)
(continuous growth)
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Assumptions of Exponential Model

• Birth and death rates are constant over time
• No competition for limiting resources (no
  density-dependence)
• No random changes over time

2. No age or size structure, and no differences
in birth and death rates among individuals.
3. Population is closed. No emigration or
immigration.
4. No time lags (for continuous model).
5. No genetic structure.
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Where might the exponential model apply?

• In the lab.
• In nature, but typically for relatively short
  time periods.
• Newly established populations, especially with
  few predators.
• Invasive species, pest outbreaks
• Populations recovering from catastrophic
  declines.
• Humans (ability to raise carrying capacity over
  time).

Wildlife populations do not increase without
bound for very long.
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What is the value of exponential growth model?
Gotelli Exponential growth model is the
cornerstone of population biology.
Turchin Exponential growth is the first law of
population dynamics. Exponential law is similar
to laws of physics, such as Newtons law of
inertia.
All populations have the potential for
exponential increase.