ecology equations

Cohort Life Tables Parameters defined: x = time period or age-class x+1 = next time period or age-class nx = number at each time period or age-class

lx = proportion surviving to each time period or age-class

dx = death rate at each time period or age-class

qx = per capita mortality rate at each time period or age-class

Fx = fecundity = # offspring produced at each time period or age-class

bx = offspring produced/individual at each time period or age-class = age-specific

natality rate lxbx = offspring produced at time period or age-class times proportion surviving to

each time period or age-class

Calculation for cohort life tables lx = nx/n0, l0 = n0/n0 = 1

dx = (nx-nx+1)/n0 = lx-lx+1 (not the same in the book)

qx = dx/lx = (nx-nx+1)/nx bx = Fx/nx lxbx

Ro = Number born at one generation/Number born at next generation = basic

reproductive rate

Ro = ! lxbx = (! Fx)/n0

static lx = lx e-rx

Note: nx-nx+1 is dx in the book

Population Growth Equations:

Ro = basic reproductive rate = ! lxbx G = generation time Gc = cohort generation time

R = fundamental net reproductive rate t = time r = instantaneous rate of population growth R = N1/N0 (1) NG = N0 Ro (2)

Nt = N0 Rt (3)

r = ln R (4)

R = er (5)

r= ln Ro/G (6)

Nt = N0 ert (7)

r " ln Ro/Gc (8)

Gc = ! xlxbx

! lxbx

= ! xlxbx

Ro (9)

1 = ! e-rx lxbx (10)

Population Growth Models K = carrying capacity Continuous Model Exponential Growth dN/dt = rN

or Nt = N0 ert

Logistic Growth Model dN/dt = rN (K - N)/ K Discrete Model

Geometric Growth Model Nt = N0 Rt

Sigmoid Growth Model RNt

1 + (aNt) Nt+1 =

R-1

K a =

Modifications of Discrete Model Compensation Time Lag

RNt

1 + (aNt)b

Nt+1 = RNt

1 + (aNt-1) Nt+1 =

Lotka - Volterra Equations

dN1/dt = r1N1 (K1 - N1 - #12N2)

K1

dN2/dt = r2N2 (K2 - N2 - #21N1)

K2

N1 = K1 - #12N2

N2 = K2 - #21N1

Zero Isoclines