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Population Growth
Not so stable populations
Limits to Growth
• We are going to work now on adding some realism to growth • Reasonably, growth will be influenced by intrinsic (or finite) rates
– Figure 9.1
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Limits to Growth
• We are going to work now on adding some realism to growth • Reasonably, growth will be influenced by intrinsic (or finite) rates
– Figure 9.1
• But, also reasonably, growth can be influenced by conditions such as population size
Z = N – Neq (equilib pop size where R0 = 1)
(‐0.02 is slope = B)
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Conclusion
• The behavior of the population will depend on the interaction of the slope (sensitivity) of the response to population size, and the equilibrium population size
– If L (= B*Neq) is between 0 and 1, then things go pretty smoothly
– Between 1 and 2, less smoothly
– 2 and 2.57, indefinite but regular oscillations
– Over 2.57, mayhem
• So?
– So the interplay between B and Neq can cause a variety of effects
– In real populations you could see all of these interactions
Switching to Overlapping Generations
• So far we have been looking at discrete generations
– What about when they are overlapping?
• For overlapping generations we will need the instantaneous rate r
dN/dt = rN = (b‐d)N
Nt/N0 = ert
Doubling Time
• When Nt/N0 = ert = 2, t = 0.69315/r • See how r affects t: 4
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Adding “Carrying Capacity” K
• K is the maximum sustainable population
• To get an effect of population size on growth:
dN/dt = rN(K‐N/K)
K‐N/K has been called the “unutilized opportunity for population growth”
dN = r(K‐N/K)
Dt*N
It can be positive
or negative
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A Not So Good Fit
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K can Vary Seasonally or Between Years
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Don’t Worry About Theta Logistic
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