Population limitation: history

1
Population limitation history background

• Both geometric, exponential growth rare in
  nature--because populations are limited (by
  amount of resources, by predators, parasites,
  competitors, etc.)
• Thomas Malthus understood this idea, expressed in
  his 1798 book (An essay on the principle of
  population as it affects the future improvement
  of society)
• Darwin picked up on Malthuss idea in his theory
  of evolution by Natural Selection
• Population limitation involves any factor that
  keeps a population from growing

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• Density-independent limiting factors are not
  proportional to population size (e.g.,
  catastrophic weather events)

• Density-dependent limitation is proportional to
  population size, and has a special name
  population regulation

3
Population regulation

• Implicit in the concept of regulation is
  intraspecific competition
• decreased per capita growth, reproduction, and/or
  survival within a population or species due to
  interactions among individuals over limited
  resources
• Strength of intraspecific competition
  proportional to population size

4
Yeast begin growing exponentially...
5
Modeling population regulation

• Assumptions of logistic model
• Relationship between density and rate of increase
  is linear
• Effect of density on rate of increase is
  instantaneous (well relax this assumption later)
• Environment is constant (i.e., r is constant, as
  is K carrying capacity)
• All individuals are identical (i.e., no sexes,
  ages, etc.)
• No immigration, emigration, predation,
  parasitism, interspecific competition, etc.
• Purpose of such a heuristic, deterministic model
  is to include just the essential idea of
  regulation, and nothing else

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Logistic model of population regulation

• First, an intuitive mathematical derivation of
  logistic population model
• Also known as sigmoid population growth model
• Developed by Pearl and Reed, based on work of
  Verhulst others (early 1900s)
• Let dN/dt r(N)N
• This is just the exponential model, except that r
  is now a function of N ( population size)
• Specifically, r declines with population size
• Define r(N) as r(1-(N/K)) notice that this
  function r(N) goes from r as N--gt0, to 0 as
  N--gtK
• K defined as population carrying capacity.
• Model in full dN/dt rN(1-(N/K))
  rN(K-N)/K

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Graphic of logistic model
per capita growth rate
(dN/dt)(1/N)
Exponential model
r
r
(dN/dt)(1/N)
Population size (N)
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How does logistic model behave?

• Heres the model again dN/dt rN(K-N)/K
• When N approaches K, right-hand expression
  ((K-N)/K) approaches 0. Thus, dN/dt approaches
  0, which means that N does not change with time
  Population is stable!
• Alternatively, when N approaches 0, right-hand
  expression ((K-N)/K) approaches 1. Thus dN/dt
  approaches rN1, i.e., dN/dt is approximately
  equal to rN Population grows exponentially!
• Graph of N versus time (t) is sigmoidal in shape,
  starting out like exponential growth, but
  approaching a line with slope of zero.

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Logistic population growth in Lynx recall that r
b - d!

• Solution to logistic model (involves solving a
  differential equation, using methods of
  differential calculus)
• N(t) K/(1 be-rt), where b K-N(0)/N(0)
• This equation can be used to plot N vs. t

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More about behavior of logistic population model

• How does the slope of the logistic curve (N as a
  function of t) vary with N? This can be seen
  intuitively--goes from 0 (at low N) to maximum
  (at intermediate N), back to 0 at N K (i.e.,
  hump-shaped curve, with maximum at N K/2).

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What exactly does regulation mean?

• Regulation means the tendency for a population to
  remain dynamically stable, no matter where it
  starts (assuming it is non-zero)
• Thus N approaches K, the carrying capacity, from
  both N lt K, and N gt K
• K is thus an equilibrium point of the model,
  because of negative feedback on r as N gets
  larger
• We can show this idea of dynamic equilibrium
  graphically

deaths
deaths
deaths
births
births
births
K
K
K
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How do ecologists test for population regulation,
density-dependence?

• Laboratory Study population growth in
  controlled, simple environment with limited
  resources
• Look for evidence for carrying capacity
  (population stays at, or returns to fixed
  abundance)
• Field look for evidence of density-dependence
  of demographic variables
• Lets look at some evidence of these three
  types...

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Example of logistic growth yeast population
growth in lab
14
Sheep population on island of Tasmania leveled
off after initial exponential growth
15
Ringed necked pheasants on Protection Island
again Population growth rate declines away from
exponential, approaching constant population
because of limited resources! (from G.E.
Hutchinson, 1978, An Introduction to Population
Ecology, Yale University Press.)
16
Density-dependent fecundity in Daphnia pulex,
lab cultures
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Density-dependent survival probability in Daphnia
pulex, lab cultures
18
Density-dependent Lambda for populations of
Daphnia pulex, lab cultures Note linear decline
in lambda with density!
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Regulatory density-dependence in Mandarte Island
(British Columbia) song sparrow population
(Melospiza melodia) (a) size of floater
non-territorial individuals, (b) no. young
fledged per female, (c) proportion of juveniles
surviving one year
20
Density-dependence of larval migration and
mortality in grain beetle (Rhizopertha dominica)
21
Density-dependence of plant dry weight in flax
(Linum) plants in greenhouse
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Returning to real world, how important are
density-dependent versus density-dependent (
regulatory) population limitation?

• Controversy erupted among ecologists in 1950s on
  relative importance of these two kinds of
  limitation
• Andrewartha and Birch challenged primacy, and
  even necessity of, density-dependence in
  population limitation
• The distribution and abundance of animals
  (1954)
• Work based primarily on Thrips imaginis (rose
  pest)
• Their argument Weather alone is sufficient to
  control (regulate?) these insect populations

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Example of one years thrips population sizes
(1932)
24
78 of variability in Thrips imaginis population
(just prior to peak abundance in December)
attributable to weather variables (e.g., rainfall
in Sept. Oct.)
25
Other ecologists championed primacy of
density-dependence in populations

• Scientists in this density-dependence school
  Lotka, Gause, Nicholson, David Lack
• Lacks (1954) book particularly influential The
  natural regulation of animal numbers
• These scientists argued that even in the kinds of
  insects that Andrewartha and Birch studied,
  density-dependence is important
• Fred Smith (1961) pointed out that even in
  Andrewartha and Birchs Thrips imaginis data,
  population change is density-dependent

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Density-dependence inThrips imaginis Change in
population size from November to subsequent
October decreases with increasing size of
previous October population
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Resolution of debate on density-dependence versus
density-independence?

• Ecologists today recognize that the dichotomized
  positions of scientists in 1950s were
  unnecessarily extreme
• Most, if not all, populations are limited at
  least to some extent by density-dependent factors
• Density-independent factors are also usually
  important (e.g., weather, disease)
• Weather does not act just in density-independent
  way (e.g., proportionately more individuals
  occupy refuges in smaller population)
• Thus DD and DI factors interact in complex ways
• We still do not understand regulation in most
  pops.

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Conclusions

• In nature, most populations are limited by
  resources, predators, etc.
• We developed a model for population growth in a
  limited environment, using linear decrease in r
  with population size--the logistic population
  model
• Logistic population growth is also known as
  sigmoid growth, because population approaches an
  equilibrium size (K) in an s-shaped manner
• Lots of examples of limitation, regulation,
  density-dependence seen in nature
• Debate about prominence of density-dependent
  versus density-independent factors in nature
  resolved both are important, but in complex ways

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Acknowledgements Most illustrations for this
lecture from R.E. Ricklefs and G.L. Miller.
2000. Ecology, 4th Edition. W.H. Freeman and
Company, New York.