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Net Reproductive Rate versus Intrinsic Capacity for Increase
• Net Repro Rate = R0 = Daughters produced in Gen t + 1 Daughters produced in Gen t
• To do this for multiple generations:
R0 = ∑ lxbx
• Intrinsic rate is dN/dt = rN
or Nt = N0ert (for the math savvy)
R <> r
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Log vs. Arithmetic Scale
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Per Generation – what’s that?
• The mean period elapsing between the production of parents and production of offspring
• Gx = ∑ lxbxx/∑ lxbx
= ∑ lxbxx/R0
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Deriving r (finally)
• The instantaneous rate of increase is r
– We haven’t figured out how to get r before this…
• r = loge(R0)/G
– for our example, r = loge(3.0)/1.33
• And the finite rate of increase λ (Lambda) is er
– This will come in handy later
Keeping Rr Terms Straight • Net rate R0 is per generation – It is calculated cumulatively over all age classes
– If R0 is 1 then the population is not growing
– It is a finite rate • Intrinsic rate r is the potential rate of growth under a certain set of conditions
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It is a single value and not calculated cumulatively
It is an instantaneous (short time interval) rate
If r is 0 (zero), then the population is not growing λ is a conversion of r back into a finite rate
“Interestingly,” finite rate = einstantaneous rate and instantaneous rate = loge finite rate (geeks please see Appendix II)
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Why Intrinsic Matters
Two scenarios, Same # of Offspring
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Reproductive Value
• Reproductive Value (for a stable population)
Vx = ∑ ltbt/lx
t=x
(Q for you: Why is Vx = R0 when t = 0?)
Age Distribution
• If Cx is the proportion of organisms in a particular age category, then:
Cx = λ‐xlx/∑ λ‐ili
(λ‐x is the same as 1/λx) • Returning to our make‐believe example in Figure 8.11:
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Back to Our Example (p 129)
• Follow calculations on p 133 using above:
– C0 = [(1.0)*(1.0)]/1.6572 = 0.6035
– C1 = [(0.4144)*(1.0)]/1.6572 = 0.25
– C3 = 0.104 and C4 = 0.043
• It worked!
– And we have a stable age distribution
Stationary Age Distribution
• In the previous example, survivorship was high and the population grew rapidly
• As mortality and natality balance each other out, the population becomes stable • If conditions remain stable, then the age distribution will also become stable, and you get a stationary age distribution • This can only occur if population size and the distributions of lx and bx are also unchanging
– In reality, these things are always in flux
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Stable vs. Stationary Age
Distributions
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Evolution of Demography
An Interesting Outcome
• It only takes a small amount of additional reproduction for annual to catch perennial
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Reproductive Considerations
• If risk of dying is high for adults, it might pay to put energy into more offspring sooner
• If chances of success in a particular year are poor, then it might be better to stay alive and keep trying
• If reproduction is taxing (it is) then this may influence when to reproduce (i.e., delay)
• But can’t wait for too long (won’t be here)
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