Basic reproductive rate, Ro = Σ lx mx

Number 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 Tmax

: Σ lx mx e r (Tmax - x )

Daughters Grand-daughters, Great Grand-daughters, etc.

Basic reproductive rate, Ro = Σ lx mx

Number 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 Tmax

: Σ lx mx e r (Tmax - x )

Fundamental net reproductive rate,R�N t+1 / N t � λ �e r

N Tmax = (No = 1) e rTmax= Σ lx mx e r (Tmax - x )

Fundamental net reproductive rate, R�N t+1 / N t � λ�e r

N1 = R N0, N2 = R N1 = R(R N0), …. Nt = R t N0

N t / N o � e r t

N Tmax = (No = 1) e rTmax = Σ lx mx e r (Tmax - x )

Divide both sides by e rTmax to get Lotka’s equation that can be solved iteratively for r:

1 = Σ lx mx e -rx

How biologists derive the intrinsic rate of natural increase from life tables for species with overlapping generations.

dN/dt = b N – d N = (b – d) N = r N (closed population)

b = per capita birth rate (number of births individual-1 time-1) =(time-1)

if N = 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 N = 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 )

dN/dt = b N – d N = (b – d) N = r N N(t) = N(0) ert (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.

dN/ dt = r max N exponential growth—what keeps the world from being smothered in elephants, E. coli, or us?

rT

rT

N

N

T

t

eNN

eNN

rTNN

rrTNN

rdtNdN

rdtNdN

rNdtdN

T

T

T

T

T

0

0

0

0

0

ln

0lnln0

=

=

=⎟⎠⎞

⎜⎝⎛

−=−

=

=

=

∫ ∫=

r, intrinsic rate of natural increase for “exponential growth”.

If r = 0.023 years-1, what is the doubling time for the population?

Human Population Growth

1650 – 0.51850 – 1 1930 – 2 1975 – 4 2000 – 6

Billions of people:

Exponential population increase observed to stop at high population densities…

Population densityPer

capi

ta b

irth

or

deat

h ra

te death

birth

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

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)

Intraspecific competition

+ +- -

- -

overshoot

Smooth approach to K…

“Noisy” (unstable), or density independent factors?

N

Time

Change in limitingfactor

Period of looser

regulation

e.g. speed limit, versus regulation by enforcement of minimum and maximum speed

Analogy: speed limit(60 mph) regulatedstrictly (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).

N

Time

N t+1 = N t R

_______

1 + ( a Nt )b

b < 1: undercompensating dd

b > 1: overcompensating dd

b = 1: perfectly compensating dd

b = 0: density independence

Logistic differenceequation:

a = (R-1)/K

N T+1

N T

Superimposition of redds

Later hatching fry have poorer survival

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