Brief review of previous lecture

1
Brief review of previous lecture

• Density dependence
• Logistic equation for continuous growth
• Time lags in continuous model can produce damped
  oscillations and stable limit cycles
• Nonlinear density-dependent effects
  (theta-logistic model).
• Discrete models of density dependence Ricker
  (scramble), Beverton-Holt (contest), Ceiling.
• Chaotic population dynamics
• Allee effects or Positive Density Dependence

dN/dt rN(1 N/K)
dN/dt rN1 (N/K)?
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Lecture Outline Population Regulation

• Historical background
• Utility of the logistic equation
• Long-lived debate about density dependence
  population regulation

• Density dependence vs. density vagueness

• Issues regarding detection of density dependence

• Population regulation vs. population limitation

• Tentative synthesis

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The Logistic Equation

• First used by a Belgian mathematician, Verhulst,
  in 1838.
• Independently rediscovered by Raymond Pearl (with
  Lowell Reed) in 1920. Pearl was professor of
  Biometry at John Hopkins University.

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• Alfred Lotka saw logistic as useful empirical
  formula but cautioned against extrapolation into
  future.
• Viewed logistic equation similar to
  Hardy-Weinberg Law in genetics
• Overly simplified
• Built on assumptions that can often be falsified.
• But by examining how population departs from
  law, we get more realistic idea of actual
  mechanisms underlying population growth and how
  to refine assumptions accordingly.

Lotka
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The Big Debate Density Dependence and Population
Regulation

• Heated debate among ecologists that peaked in
  1950s, but never completely resolved, regarding
  whether population dynamics were controlled
  mainly by biotic or abiotic interactions.

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Nicholsons View

• Governing reaction induced by density change
  holds populations in a state of balance in their
  environments.
• the mechanism of density governance is almost
  always intraspecific competition, either amongst
  the animals for a critically important requisite,
  or amongst natural enemies for which the animals
  concerned are requisites.

• Even under extreme influence of
  density-independent factors density governance
  is merely relaxed from time to time and
  subsequently resumed, and it remains the
  influence which adjusts population densities in
  relation to environmental favourability.

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Andrewartha and Birchs View

• Population size can be limited by
• 1. shortage of resources
• 2. inaccessibility of resources (movement and
  searching constraints)
• 3. shortage of time when rate of increase (r)
  is positive

• Number 1 is probably least important in nature.
• Number 3 is probably most important in nature.
  Fluctuations in r may be caused by weather,
  predators, or any other component of the
  environment.

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Thrip abundances over time

• Weather variables explained 78 of variation in
  peak thrip numbers.
• Not only did we fail to find a density
  dependent factor but we also showed that there
  was no room for one. (Andrewartha Birch)

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Some Definitions of Population Regulation

• 1. The process by which a population returns to
  its equilibrium density. (Sinclair 1989)
• Notion of single equilibrium (K) embodied by the
  logistic equation.
• The idea of a stable equilibrium point seems
  unrealistic to many field ecologists who see
  nature as non-equilibrium.

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2. A tendency in a population for some factor to
cause density to increase when it is low and to
decrease when it is high. (Begon et al. 1996)
3. The ability to act on a very wide range of
starting densities, and bring them to a much
narrower range of final densities. (Begon et al.
1996)

• These suggest a looser relationship between
  population growth rate and density and invoke the
  idea of boundedness rather than a single
  equilibrium density.

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Density-Vague Population Change

• Don Strong suggested the concept of density
  vagueness was a realistic alternative to the
  mathematical convenience of explicit density
  dependence.

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Density-Vague Population Change
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Issues in detecting density dependence from time
series
1. Length of time series

• Woiwod and Hanski examined 5715 time series of
  annual abundances of 447 species of moths and
  aphids in UK.

• Detection of density dependence increased with
  length of series.

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2. Time lags

• Peter Turchin analyzed time series for 14 forest
  insects.
• Detected density dependence in only 5 of 14.

• However, 7 of 9 that did not show direct density
  dependence exhibited delayed density dependence.

• Conclusion time lags might be common and
  studies that do not consider lag effects will
  underestimate frequency of population regulation.
  

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3. Spurious correlations
Rate of increase
Density-independent growth
(Nt1/Nt)
Density-dependent growth
Nt

• Problem is that the ratio (rate of increase) is
  NOT independent of Nt.
• A false correlation is induced by the fact that
  Nt occurs in both Y and X.

• Even if Nt1 and Nt are unrelated, there will be
  a substantial negative correlation between
  (Nt1/Nt) and Nt.

• So, using this naïve approach could suggest
  presence of density dependence in a time series
  when it does not really occur.

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Population Regulation vs. Population Limitation
Population regulation process by which a
population returns to its equilibrium density (or
range of densities).
Population limitation process that sets the
equilibrium density.
Population dynamics analysis of causes for
change in population density, including both
regulation and limitation.
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• Charles Krebs suggested that there are two
  paradigms of population regulation
  density-dependent paradigm and mechanistic
  paradigm.

• Populating density is not the relevant variable
  to analyze because density is not a mechanism, it
  is only used as a surrogate for factors such as
  food supply, territoriality, and predation.

(Krebs. 1995. Two paradigms of population
regulation. Wildlife Research 221-10.)
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• Charles Krebs suggests that population dynamics
  addresses three questions

• What stabilizes population density?
• What prevents population growth?
• What limits population density in good or poor
  habitats?

• In general, there has been little practical
  progress in understanding real populations by
  asking what stabilizes population density, and
  this question by itself is the least interesting
  of the three

(Krebs. 2002. Beyond population regulation and
limitation. Wildlife Research 291-10.)
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Small Group Discussion
1) Should wildlife management include a focus on
population regulation? If so, what research
approaches should be used?
2) What might be gained by focusing efforts on
understanding population limitation instead of
(or in addition to) population regulation?
3) Can problem-solving in wildlife conservation
benefit from ecological theory?
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Tentative Synthesis

• All wildlife populations are regulated in the
  very broadest sense, or else extinctions and huge
  irruptions would be more common than what we
  observe.

• However, strict density-dependence with a stable
  equilibrium (carrying capacity) depicted by
  simple math models is unlikely in nature.
  Density-vagueness, boundedness, and a
  stochastic K is more likely than a simple
  equilibrium.

• Growth rates and population sizes are related to
  endogenous and exogenous factors, such as
  weather. The relative importance of these
  processes will differ among species, but perhaps
  generalizations exist to aid conservation efforts.

• Ultimately, sound management of wildlife
  populations depends on understanding mechanisms
  of population dynamics.

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Density Dependence and Population Regulation
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Krebs, CJ. 1995. Two paradigms of population
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