Population Growth Reading Krebs Chapter 11

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Population Growth Reading Krebs Chapter 11
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Population Growth

• A central process of living organisms
• No population grows forever
• Growth can be positive or negative
• A function of births - deaths
• What regulates populations?
• Species interactions, e.g., competition,
  predation, herbivory, disease
• Changes in environment - alteration of micro-site
  or habitat conditions that influence intrinsic
  capacity for increase

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Population growth Outline

• Population growth models
• I. Discrete generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• Equilibrium, Oscillations, Stable limit
  cycles, Chaos
• II.Overlapping generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• III. Alternatives
• 1. Time-lag models
• 2. Stochastic models
• 3. Leslie matrices

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I. Growth equation for discrete generations

• Nt1 R0Nt
• Nt is the number of individuals at time t
• N0 is the number of individuals at time 0
• R0 is the net reproductive rate i.e.number of
  female offspring per female per generation
  (averaged across all ages)

e.g., If there are 1500 female spiders now, and
the net reproductive rate is 1.1 (i.e., on
average 1 in 10 females give birth to young
(female) that survive at least 1 year, then next
year there will be 1500 1.11650 females.
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• Constant multiplication rate independent of
  population size
• Geometric or exponential population growth,
  discrete generations, reproductive rates
  constant. Starting population size, N 10.
• If Ro lt 1 population size decreases

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2. Multiplication rate dependent on population
size

• Unlike previous example, in most populations,
  growth rate decreases as population increases
• Competition for food/nutrients/water/light
  increases
• Disease increases
• Predation increases
• Birth rate falls
• Death rate rises

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Net reproductive rate R0 can be modeled as a
linear function of population density N at time t
(simplest assumption).
R01
Birth rate death rate
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Net reproductive rate R0 modeled as a linear
function of population density N at time t
(R0 when N0)
Ro-1/N-Neq -B Ro-1 -B (N-Neq) R0 1-B(N-Neq)
Slope -B
R0
In this example
R0-1
R01
N-Neq (-ve)
Neq
N
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Ro-1/N-Neq -B Ro-1 -B (N-Neq) R0
1-B(N-Neq) But Nt1 R0 Nt So Nt1 1-B
(Nt-Neq) Nt

• Results of this model depend on the values of Neq
  and B
• Let LNeq B
• Interesting results with different values of L.

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Population growth with discrete generations and
multiplication rate a linear function of
population density. Starting density10,
equilibrium density 100.
Neq 100 L 1.8
L between 1 and 2, convergent oscillation
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Neq 100 L 2.5
L between 2 and 2.57, stable limit cycle
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Neq 100 L 2.9
L greater than 2.57, chaotic irregular
fluctuations
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Simple model gives rise to surprising,
non-intuitive results (LBNeq) If 0ltLlt1
equilibrium approached without oscillations
1ltLlt2 oscillations of decreasing amplitude
(convergent oscillations) 2ltLlt2.57 stable
limit cycles Lgt2.57 chaotic oscillations,
depending on starting conditions
Note Nt1 1-B(Nt-Neq) Nt is the discrete
version of the better-known logistic equation
that is used for overlapping generations
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Population growth Outline

• Population growth models
• I. Discrete generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• Equilibrium, Oscillations, Stable limit
  cycles, Chaos
• II.Overlapping generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• III. Alternatives
• 1. Time-lag models
• 2. Stochastic models
• 3. Leslie matrices

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II. Discrete vs. Overlapping Generations

• I. Discrete generations a female reproduces
  once and dies so only a single generation exists
  at any one time e.g., annual plants,
  grasshoppers
• II. Overlapping generations a female can
  reproduce multiple times, so more than one
  generation exists at the same time e.g.,
  perennial polycarpic plants, many animals, fish.

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II. Population growth model continuous
generations1. Rate of growth fixed

• dN / dt r N (b d) N
• Where
• b birth rate
• d death rate
• r per-capita rate of growth b d
• Under optimum conditions, r rm
• Where
• rm the maximum value for r

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Population doubling times
Nt/N0 ert If population doubles, Nt/N0 2
ert so Loge 2 rt 0.69315 and
t O.69315/ r
r t 0.01 69.3 0.02 34.7 0.03 23.1 0.04 17.3 0.05 1
3.9 0.06 11.6
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Example of geometric growth - 1
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Example of geometric growth - 2
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2. Multiplication rate dependent on population
size

• In most populations, growth rate decreases as
  population increases
• Competition for food/nutrients/water/light
  increases
• Disease increases
• Predation increases
• Birth rate falls
• Death rate rises

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Example of population density control - 1
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Example of population density control - 2
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Terminology for logistic growth
Logistic curve
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Contrast of logistic and geometric growth
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Formal demonstration of effect of density on
realized population growth rate
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II. Population growth model

• Geometric 1. dN / dt r N
• Logistic 2. dN / dt r N (K N)/K
• Where K is the upper asymptote or maximum value
  of N, the i.e. Karrying capacity
• Note (1 N/K) (K N) / K
• (K-N)/K is the unutilized opportunity for
  population growth.
• This is the differential form of the logistic
  curve
• (so cannot be used to calculate N in this form
  - has to be integrated).

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Integral form of the logistic equation

• Nt K / ( 1 e a-rt)
• Where a is an empirically derived parameter
  defining the position of the curve relative to
  the origin.
• Attractive model because of simplicity (only 2
  constants, r and K) and agreement with
  observations from many populations.

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Laboratory culture 1 Same number of bacteria
added each day as food. Constant pH 8, and 26C
Gause changed the water every few days, added
constant quantity of food (bacteria) each day and
maintained a constant environment
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Laboratory culture - 2
Drosophila melanogaster fruit fly
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Laboratory culture - 3
Tribolium castaneum Flour beetle

• After initial logistic behavior population size
  fluctuates.
• Later variations caused by competition - not
  considered in logistic model.
• Even laboratory cultures with artificial controls
  do not always follow model.

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Field data - 1
Brent geese Branta bernicla bernicla
Rise in population correlated with reduced
hunting Slight reduction of rate of increase by
1990s will population size stabilize?
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Field data - 2
Reindeer. Pribilof islands, Bering Sea, Alaska.
Winter food shortage associated with population
crash, but no obvious cause of the differences
between the islands
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Grus americana Whooping crane, counted at Aransas
Texas during migration
10 yr frequency of declines possibly related to
declines in snowshoe hares and switching of
predators (coyote, lynx) to crane nests.
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Daphnia rosea (Cladoceran) Strong seasonality
disrupts simple logistic growth. Growth restarts
each yr, sometimes exhibits logistic behavior
(e.g. 1983) before the annual crash.
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Human population of USA from 1790 to 1990
Geometric growth rather than logistic
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Example of application of logistic growth model
to optimal fish catch size To maintain maximum
yield, keep population at about 53 maximum
density Not at maximum density
dN/dt
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Population growth Outline

• Population growth models
• I. Discrete generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• Equilibrium, Oscillations, Stable limit
  cycles, Chaos
• II.Overlapping generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• III. Alternatives
• 1. Time-lag models
• 2. Stochastic models
• 3. Leslie matrices

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Time-lag population growth

• In some situations, the population growth in one
  generation depends on density conditions present
  during the previous generation, e.g., there is a
  time lag in population growth

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Time-lag population growth

• Rewrite the discrete generation model to make
  reproduction dependent on the size of the
  population in the previous generation.
• The term L B Neq drives the levels of
  oscillation in the population growth curve.
• When L is between 0 and 1, then the population
  approaches the equilibrium point with out
  oscillation.
• When L gt 1, then oscillations will occur.

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Model result with time lags
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Example - laboratory culture
Daphnia magna
18C
25C
Mechanism related to utilization of stored oil
that allows continued reproduction after food
supply is exhausted, followed by crash.
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Population growth

• Population growth models
• I. Discrete generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• Equilibrium, Oscillations, Stable limit
  cycles, Chaos
• II.Overlapping generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• III. Alternatives
• 1. Time-lag models
• 2. Stochastic models
• 3. Leslie matrices

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Deterministic versus stochastic models

• In deterministic models, there is one outcome
  based on the input parameters
• In reality, biological systems are probabilistic,
  e.g., there are varying degrees of probability of
  a specific outcome
• Probabilities of biological outcomes, in turn,
  are driven by the various environmental factors
  that influence specific processes
• Because of this, stochastic or probabilistic
  models most commonly used

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Use coin toss to decide whether 1 offspring
(heads) or 3 (tails). N0 6, average birth rate
2
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Population growth Outline

• Population growth models
• I. Discrete generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• Equilibrium, Oscillations, Stable limit
  cycles, Chaos
• II.Overlapping generations
• 1. Growth rate determined only by
  multiplication rate
• 2. Growth rate determined by population size
  also
• III. Alternatives
• 1. Time-lag models
• 2. Stochastic models
• 3. Leslie matrices

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3. Leslie Matrix models (Population Projection
Matrices) Leslie Matrix models allow age and
size to be taken into account. Read details
pp173-176, Krebs Ecology
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Population density of youngest class at t1
FoNoF1N1F2N2F3N3..FkNk
Population density of all others at t1
PoNoP1N1P2N2P3N3..PkNk