second law of populations:
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SECOND LAW OF POPULATIONS:. Population growth cannot go on forever. - P. Turchin 2001 (Oikos 94:17-26). !?. The Basic Mathematics of Density Dependence: The Logistic Equation. How does population growth change as numbers in the population change?. We can start with the equation for - PowerPoint PPT PresentationTRANSCRIPT
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SECOND LAW OF POPULATIONS:
Population growth cannot go on forever
- P. Turchin 2001 (Oikos 94:17-26)
!?
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The Basic Mathematics of Density Dependence:The Logistic Equation
We can start with the equation for exponential growth….
How does population growth changeas numbers in the population change?
dN/dt = rN
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Recall the definition of r
dN/dt = rN … r is a growth rate, or thedifference between birth and death rate
So, we can write, dN/dt = (b-d)N
For the exponential equation, these birth
and death rates are constants…What if they change as a function of
population size?
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We can rewrite the exponential rate equation
Let b’ and d’ represent birth and death ratesthat are NOT constant through time
So, dN/dt = (b’- d’)N
Modeling these variables…. The simplestcase is to allow them to be linear
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Linear relationship:
Let b’ = b - aN and Let d’ = d - cN
N
b’
The intercept: b
The slope: a
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What happens if...
The values of a or c equal zero?
This demonstrates that the exponentialequation is a special case of thelogistic equation….
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Rearranging the equation...
The Carrying Capacity, K
•Definitions
•Issues
Assumptions of the logistic model
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Logistic Growth, Continous Time
0
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-5 5 15 25 35
Time
Po
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Siz
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dN/dt
Rate of change and actual population size
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What if we do have time lags?
Biologically realistic, after all (consider thestage models we’ve been working with)
What happens?
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Other forms of density dependence
•Ricker model
•Beverton-Holt model
Both of these originally developed for fisheries;many more possibilities exist.
Nt+1 = Ntexp[R*(K-Nt)/K]
Nt+1 = Nt * (1 + R) 1+(R/K)*Nt
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The Allee EffectMinimum density required to maintain the
population
•Defense or vigilance
•Foraging efficiency
•Mating
James F. ParnellGary Kramer
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SUMMARY
•When density affects demographic rates,“density dependence”
•Many ways to model this mathematically:Logistic (linear)RickerBeverton-Holt
•In constant environment, population willstabilize at carrying capacity
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SUMMARY, continued
•Unlike with density-independent models, thediscrete and continuous forms are NOTequivalent in behavior
•Discrete form can exhibit damped oscillations,stable limit cycles, or chaos
•Carrying capacity has multiple definitions,biological reality must be considered
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SUMMARY, continued
Allee effect: important implications formanagement and conservation
SECOND LAW OF POPULATIONS:they can’t grow forever…