Population growth - the cohort biomass

The science of population dynamics emerged from demographic studies of human populations and is closely related to classical economics. The Malthusian law of exponential growth is well known and inspired later modellers. Individual net growth is determined by the metabolic processes of anabolism (growth) and catabolism (degradation). The upper graph to the right shows the widespread individual growth model of von Bertalanffy, individual weight as a function of age (time). Below is the mortality represented by the decline in cohort numbers over time (the Baranov equation). At bottom the product of individual weight at time and number of individuals at time gives the bell shaped curve of cohort biomass.

Population biomass - the sum of cohorts

Consider the case of constant recruitment to a stock. Each year a cohort of the type described above is recruited to the stock. The total biomass of the whole stock will then be the sum of biomasses of all the cohorts. The very young and the old cohorts hold small biomasses, while some larger exist in between. If the recruitment period is reduced towards zero the total stock biomass will equal the cohort biomass integral shown by the red curve to the right.

Types of population growth

We may categorise population growth into three types: Compensation, Depensation and Critical depensation. The three types are illustrated by graphs on the right hand side.

Compensation growth is a growth type where population decline is compensated by increased growth rate. Depensation is the opposite case, while the extreme opposite is Critical depensation where a critical population level is identified. Population levels below the critical will lead to extinction.

The type of population growth we find based on a von Bertalanffy type of individual growth, Baranovs mortality equation and constant recruitment, is compensation growth, as indicated in the graph to the left.

In this course we will use the following notation:
Stock biomass at time t:

Change in stock biomass per unit of time: 

The natural net growth as a function of stock biomass:

Population harvest

Net growth of the population is now shown to be a function of population size (stock biomass). Other factors will also effect growth and/or mortality. One of the factors influencing the latter is population harvest.

Consider a fishery. Production of harvest (H) is obtained by the use of two input factors: Fishing effort (E) and a fish stock biomass (X):

The fishing effort could be regarded as output from another production process where capital and labour are input factors. The cost of capital and labour constitute the cost of fishing effort, while the stock biomass in most cases are freely accessible.

When fishing activity adds fishing mortality to the stock and the net change in stock biomass over an unit period of time is now

The change in stock may be positive, negative or zero, the latter representing a stock biomass equilibrium situation.

Biological equilibrium

Biological equilibrium is defined by

which implies that the harvest equal the natural net growth in stock biomass:

Recalling that Harvest is produced by the input factors of Effort and Stock biomass, the equilibrium stock biomass is a function of fishing effort and the euilibrium catch the same.