Density Dependent Models

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Density Dependent Models
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Modification of Exponential Growth Model

• Density affects populations through
• Lower (decreased) per capita birth rates as
  populations increase
• bb-aN bper capita birth rate, b is ideal
  birth rate under uncrowded conditions a is a
  Constant variable that measures the strength of
  density dependence
• Note that when a 0, there is no density
  dependence and the model is the exponential model
  (bb)

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Density effects

• Density increases the death rate through
  competition for limited resources
• dd cN d is per capita death rate, d is
  idealized death rate under no density dependence
  and c is a constant that measures the strength of
  density dependence

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Measuring Density Dependence

• Simplest models use a linear effect (straight
  line) of density.
• This is not the only possibility. Some
  populations might be better modeled by a
  threshold model
• Allee effects might be important at small
  population sizes (small populations show density
  effects due to not finding mates)

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Logistic Model

• If we substitute our new birth and death rate
  terms into our original exponential model, we
  have the logistic model
• See Gotelli for details on equations
• dN/dtrN1-(ac)/(b-d)N
• K(ac)/(b-d)
• K is the carrying capacity, the maximum
  population density the environment can support
  indefinitely

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Logistic Model continued

• Growth rate under density dependence is
• dN/dt rN(1-N/K)
• 1-N/K is the unused portion of the carrying
  capacity
• Growth rate is 0 when either r0, N0, or most
  importantly when 1-N/K 0 this occurs when NK

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Population Projection of the Logistic Model

• NtK/1(K-N0)/N0e-rt
• This generates a S-shaped curve when N is small
  compared to the K (with a positive r)
• Population grows until it reaches K
• When NgtK, population growth is negative and the
  population declines to K

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Logistic Model-Projecting in Time

• dN/dt is a parobola. The highest growth occurs at
  the midpoint of the growth curve when NK/2
• In the exponential growth model, dN/dt is a
  straight line

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Logistic Model-Projecting in Time

• dN/dtN in the logistic model is a negative
  straight line while in the exponential model it
  is a flat straight line

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Discrete Model of Logistic Growth

• N t1NtrdNt(1-Nt/K)
• Analogous to the continuous model
• Growth rate is rd, the discrete growth factor
• Has a built in time lag of 1 in that the
  population size 1 time step in the future depends
  on the current population size
• This time lag can vary

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Time Lags

• Time lags between the effects of the density on
  the growth rate can create very interesting
  dynamics in both the continuous and discrete
  logistic models

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Continuous Time Lags

• dN/dtrN(1-Nt-?/K)
• The time lag ? is the time at a past population
• Three possibilities
• Small r ? (0lt ?lt0.368), smooth approach to K
• Medium r ? (0.368lt ?lt1.57), damped oscillations
• Large r ? (gt1.57), stable limit cycle

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Red line r? is small, regular approach to K Blue
line r? is medium, damped oscillation Green line
r? is large, stable limit cycle
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Periodicity in Continuous Lagged Models

• Amplitude increases with r?
• Period is always about 4? regardless of r
• For example, a one year lag will produce a 4 year
  pattern of peaks and valleys, 2 years will
  produce 8 yr peaks

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Discrete Time Lags

• Time lag is fixed at 1 (discrete model)
• Only value that influences dynamics is r
• R is very small, regular approach
• Small r (rlt2), damped oscillations
• Less small r (2ltrlt2.45), stable 2 point limit
  cycle
• Medium r (2.45ltrlt2.57), variable limit cycle (2,
  4, 8,16,32,64, etc) as r increases
• Large r (rgt2.57), no pattern (chaos). Chaos is
  not the same as stochasticity. The same chaotic
  pattern results each time the model is run
  (sensitive to initial conditions)

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Red line-r is lt2, blue r is 2ltrlt2.45 (2 limit
cycle), green line r is 2.45lt ltrlt2.57 (more
complex limit cycle), pink line is chaos
(rgt2.57), no pattern
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Random Variation in Carrying Capacity
Average population size is less than the average
carrying capacity when the carrying capacity
varies randomly. This is due to the fact that the
population decreases faster when above the
carrying capacity than it increases when below
the carrying capacity. More variable the
environment, smaller the average population. Also
depends on r (large or small), small r is
sluggish, large r shows more variation
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Periodic Variation in Carrying Capacity

• What if K varies periodically, not randomly

Describes the periodic K by mean carrying
capacity (k0), kt is the amplitude of the cycle,
and c is the length of the cycle
Effect depends on rc, if small (lt1), Averages
out the fluctuations
If rc is large (gt1), population tracks environment
In both types, average N is less than actual K.
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Stochastic and Periodic Variation in Carrying
Capacity-Summary

• Tends to reduce population size below K
• More variable environment, lower the average
  population size
• Populations with large r (insects) track
  environment
• Populations with small r (mammals) average the
  environment and show little response.