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.
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)
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.
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 = 0.034 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 = 0.014 year -1 r = b d = per capita rate of population growth = intrinsic rate of natural increase (time -1 )
d/dt = b d = (b d) = r (t) = (0) e rt (e = 2.71828 = 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?
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 = 0.023 years -1, what is the doubling time for the population?
Human Population Growth Billions of people: 1650 0.5 1850 1 1930 2 1975 4 2000 6
Exponential population increase observed to stop at high population densities
Population density death birth Per capita birth or death rate
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)
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) - -
Smooth approach to K overshoot oisy (unstable), or density independent factors?
Change in limiting factor e.g. speed limit, versus regulation by enforcement of minimum and maximum speed Period of looser regulation Time
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
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
T+1 Superimposition of redds Later hatching fry have poorer survival T
Size structured stock recruitment curves (Paulik, G. J. 1973)