Grand-daughters, Great Granddaughters, Daughters. : Σ lx m x e r (Tmax - x )

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1 Basic reproductive rate, R o = Σ l x m x umber of offspring produced by an individual female in her lifetime, can be used as multiplier to compute population growth rate if generations don t overlap. If they do overlap, total number of descendents left by an average female at the end of her life will be her daughters, and the expected grand-daughters, great grand-daughters, etc., produced when female was age x until her death at T max : Σ lx m x e r (Tmax - x ) Daughters Grand-daughters, Great Granddaughters, etc.

2 Basic reproductive rate, R o = Σ l x m x umber of offspring produced by an individual female in her lifetime, can be used as multiplier to compute population growth rate if generations don t overlap. If they do overlap, total number of descendents left by an average female at the end of her life will be her daughters, and the expected grand-daughters, great grand-daughters, etc., produced when female was age x until her death at T max : Σ lx m x e r (Tmax - x ) Fundamental net reproductive rate, R t+1 / t λ e r = Tmax ( o = 1) e rtmax = Σ l x m x e r ( Tmax -x)

3 Fundamental net reproductive rate, R t+1 / t λ e r 1 =R 0, 2 = R 1 =R(R 0 ),. t = R t 0 t / o e r t Tmax = ( o = 1) e rtmax = Σ l x m x e r (Tmax - x ) Divide both sides by e rtmax to get Lotka s equation that can be solved iteratively for r: 1 = Σ l x m x e - rx How biologists derive the intrinsic rate of natural increase from life tables for species with overlapping generations.

4 d/dt = b d = (b d) = r (closed population) b = per capita birth rate (number of births individual -1 time -1 ) = (time -1 ) if = 1000 and there were 34 births in a year, b = year -1 d = per capita death rate (number of deaths individual -1 time -1 ) = (time -1 ) if = 1000 and there were 14 deaths in a year, d = year -1 r = b d = per capita rate of population growth = intrinsic rate of natural increase (time -1 )

5 d/dt = b d = (b d) = r (t) = (0) e rt (e = = base natural logarithm) If r > 0, population grows exponentially. If r < 0, population declines exponentially. If r = 0, population is in a stable equilibrium (zero population growth), although individuals turn over (some die, and are replaced by new births). r max is the per capita population growth under the most favorable of environmental circumstances, and probably at low density. d/ dt = r max exponential growth what keeps the world from being smothered in elephants, E. coli, or us?

6 d dt d T 0 ln ln T 0 T d = = = T = T 0 r rdt = e 0 rt e T t= 0 ln = rt rdt 0 rt = rt r, intrinsic rate of natural increase for exponential growth. r0 If r = years -1, what is the doubling time for the population?

7 Human Population Growth Billions of people:

8 Exponential population increase observed to stop at high population densities

9 Population density death birth Per capita birth or death rate

10 K, carrying capacity of environment for a population = population density at which no net change occurs (births = deaths if population closed, or B+I = D+E if it s open)

11 Intraspecific competition Intraspecific competition is a mutually adverse interaction between conspecific individuals brought about by a shared requirement for a limiting resource, resulting in reduced survivorship, growth, or reproduction. (-,-) Asymmetric: some competitors suffer more than others Exploitative: mediated indirectly through depletion of shared resource (similar to scramble ), often produces overcompensating density dependence + + (destabilizing) - - Interference: involves direct interactions of competitors (e.g. territoriality, or poisoning with allelochemicals) (similar to contest competition, often leads to perfectly compensating density dependence (stabilizing) - -

12 Smooth approach to K overshoot oisy (unstable), or density independent factors?

13 Change in limiting factor e.g. speed limit, versus regulation by enforcement of minimum and maximum speed Period of looser regulation Time

14 Analogy: speed limit (60 mph) regulated strictly (55-65) or more loosely (50-70). Fast speed (high rate of natural increase (r) ), oversteering (strong density-dependent feedback), or distracted drivers (time lags in feedbacks ) all can destabilize population growth. So can sharp bends in the highway (environmental fluctuation). Time

15 Logistic difference equation: t+1 = t R 1 + ( a t ) b a = (R-1)/K b < 1: undercompensating dd b > 1: overcompensating dd b = 1: perfectly compensating dd b = 0: density independence

16 T+1 Superimposition of redds Later hatching fry have poorer survival T

17 Size structured stock recruitment curves (Paulik, G. J. 1973)