The Allen Curve (Allen Plot) - Basics		  
 
The Allen curve is a graphical approach to the somatic production of one cohort developed by Allen (1950). It is equivalent to the Increment Summation Method.

Data required are:
A series of subsequent samples of abundance Nt and mean individual body mass Mt of a cohort during the period of investigation.

Nt and mean Mt at time t (t = 1, 2, 3,...n)
 
 
The Allen curve consists of a plot of abundance Nt (Y axis) against mean individual body mass Mt (X axis), i.e. the two curves we looked at separately up to now are combined into one single curve.
 

 
 
From this curve production or elimination of one cohort can be estimated graphically. Let's look at the interval between two subsequent sampling dates t=1 and t=2. The area under the curve consists of four "boxes" (see figure above). The area of each box is defined by the values of N1, M1, N2 and M2, and each box represents a distinct part of the ongoing production process:
OBL
  Original Biomass Lost
= biomass present at t=1 but lost during t=1 -> t=2
= (N1 - N2) * M1
OBR
  Original Biomass Remaining
= biomass present at t=1 and still present at t=2
= N2 * M1
NPL
  New Production Lost
= biomass newly produced as well as lost during t=1 -> t=2
= [(N1 - N2) * (M2 - M1)]/2
NPR
  New Production Remaining
= biomass newly produced during t=1 -> t=2 and still present at t=2
= N2 * (M2 - M1)

Now, production P and elimination E between t=1 and t=2 are easy to compute;

Production P1->2 = NPL + NPR
Elimination E1->2 = NPL + OBL

More generally formulated,

Production Pt->t+1
 = NPL + NPR
= [(Nt - Nt+1) * (Mt+1 - Mt)]/2 + Nt+1 * (Mt+1 - Mt)
= (Nt + Nt+1) / 2 * (Mt+1 - Mt)
Elimination Et->t+1
 = NPL + OBL
= [(Nt - Nt+1) * (Mt+1 - Mt)]/2 + (Nt - Nt+1) * Mt
= (Mt + Mt+1) / 2 * (Nt - Nt+1)

and cohort production Pc and elimination Ec over the investigation period are

Cohort
Production Pc
 = (Pt->t+1); t = (1, 2, 3,...n)
Cohort
Elimination Ec
 = (Et->t+1); t = (1, 2, 3,...n)

whereas population production P and elimination E of a population with z cohorts are

Population
Production P
 = (Pc); c = (1, 2, 3,...z)
Population
Elimination E
 = (Ec); c = (1, 2, 3,...z)

Production-to-biomass ratios (P/B) on the cohort or population level are computed by dividing production or elimination by the corresponding average biomass of the observation period.

  
 
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