## 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.