Models in Ecology: Population growth and competition

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Models in EcologyPopulation growth and
competition

• David Hansen
• Dept. of Biology
• Pacific Lutheran University

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Linear Growth
dn/dt c
Where c is the number of individuals added each
time unit
The integrated form
Nt ct N0
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Assumptions for linear growth

• Constant number of individuals added each time
  unit
• Number added not proportional to population size

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Exponential Growth
dn/(dtN) r
Where r is the instantaneous rate of change.
The integrated form
Nt N0ert
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Assumptions for exponential growth

• Reproductive rate constant per individual
• Number of individuals reproducing proportional to
  population size
• Unlimited environment

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Realistic growth

• Resources are limited
• Birth rates change
• Death rates change

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Deriving equation for changing birth and death
rates
given
dn/dt rN
if
r b - d
where b birth rate d death rate
then
dn/dt (b - d)N
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Change in birth and death rates with increased
numbers
b0
b b0 - kbN
dn/(dtn)
d d0 kdN
d0
Neq
numbers
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Deriving equation for changing birth and death
rates
in
1. dn/dt (b - d)N
substituting for birth and death rates in
equation 1, the equations for changing birth and
death rates
2. dn/dt (b0 - kbN) - (d0 kdN)N
Rearranging and grouping terms
dn/dt b0 - kbN - d0 - kdNN
dn/dt b0 - d0 - kbN - kdNN
And finally
3. dn/dt b0 - d0 - N(kb kd)N
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Determining values for (kb kd)
For no change in population size, r 0, and
since r b d, then b d. At this condition,
N Neq
thus
b0 - kbNeq d0 kdNeq
solving for (kb kd)
b0 - d0 kbNeq kdNeq
(kb kd)Neq (b0 - d0)
and finally
(kb kd) (b0 - d0)/Neq
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Substituting for (kb kd) in equation 3
dn/dt (b0 - d0) - N(kb kd)N
becomes
dn/dt (b0 - d0) - N((b0-d0)/Neq)N
rearranging
dn/dt (b0 - d0)(1 - N/Neq)N
If b0 - d0 is redefined as r0 (intrinsic rate
of growth) and Neq is defined as K (carrying
capacity), then
dn/dt r0 N(1 - N/K)
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Plot of logistic growth
the differential equation
dn/dt r0N(1 - N/K)
Zone of low or no growth
the integrated form
Zone of rapid growth
Nt K/1 (K/N0 -1)e-rt
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Reproductive strategies derived from a
populations location on logistic growth curve
r - selection
K - selection

• occurs at low pop. density
• little competition
• high no. of offspring
• low energy/offspring
• little or no parental care
• rapid development
• often semelparous
• good colonizers

• occurs at high pop. density
• resources limiting
• large parental investment/offspring
• parental care
• slow development
• low no. of offspring
• iteroparous
• good competitors

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Interactions between two species

• Lotka - Volterra Equations

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Measurement of intraspecific and interspecific
competition
In the logistic equation, the term N/K measures
the effect on the growth of a population by
addition of a new member of the same species
(intraspecific competition).
dn/dt rN(1 - N/K)
The effect of a second species on the growth of
species one can be modeled by adding a second
term measuring the effect of adding individuals
of a second species (interspecific competition).
dn1/dt r1N1(1 - N1/K1 - N2/K1)
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Measuring ecological equivalence
Since the effect of one species on the growth of
another would not likely be identical to a
species effect on its own growth, an equivalence
adjustment must be made such that
N1 ?12N2
Where ?12 measures the effect on species1 by
species2.
and
N2 ?21N1
Where ?21 measures the effect on species2 by
species1. .
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Lotka - Volterra competition equations
Thus for a two species system, the equation for
each species becomes
for species1
dn1/dt r1N1(1 - N1/K1 - ?12N2/K1)
for species2
dn2/dt r2N2(1 - N2/K2 - ?21N1/K2)
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Graphical analysis of Lotka-Volterra equations

• Searching for equilibrium conditions

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Solving equations for zero growth
Set equation for each species equal to zero (no
growth)
r1N1(1 - N1/K1 - ?12N2/K1) 0
and
r2N2(1 - N2/K2 - ?21N1/K2) 0
Dividing by riNi, multiplying through by Ki and
rearranging yields the following pair of linear
equations
N1 K1 - ?12N2
and
N2 K2 - ?21N1
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Finding endpoints for linear equations
For each equation, substituting 0 for each Ni
gives the end points on a graph of N1 vs N2,
defining an isocline for each species.
For species1
In absence of sp2, sp1 will reach carrying
capacity
N1 K1
and
It will require K1/?12 of sp2 to eliminate sp1.
N2 K1/?12
For species2
In absence of sp1, sp2 will reach carrying
capacity
N2 K2
and
It will require K2/?21 of sp1 to eliminate sp2.
N1 K2/?21
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Graphing isoclines
K1
dn1/dt 0
K2/?12
N1
dn2/dt 0
K1/?21
K2
N2
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Graphical solutions
Sp1 always wins
Sp2 always wins
K1
K2/?12
K2/?12
K1
N1
K1/?21
K2
K1/?21
K2
Unstable equilibrium
Stable equilibrium
K1
K2/?12
K2/?12
N1
K1
K2
K2
K1/?21
K1/?21
N2
N2
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Conditions for outcomes
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Further analysis of inequalities
Assuming the carrying capacity Ki is equal for
both species and inverting the inequalities, the
following conditions occur.
For unstable equilibrium
For stable equilibrium
?12 gt 1
?12 lt 1
and
and
?21 gt 1
?21 lt 1
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Biological interpretations

• The ?s measure the ability of a species to
  contain the other species relative to itself.
• If both ?s are lt1, then each species has greater
  effect on its own growth than on the other
  species, ie. They are using different resources.
• If both ?s are gt1, then each species is able to
  exclude the other from resources either by
  out-consuming or defense.

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Principle of competitive exclusion

• Ecologically equivalent species cannot coexist.
  One will go extinct in the area of competition or
  switch resources.

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The End