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# Population change

## Theory

The rate of population growth (or decline) of a closed population depends on the combined effects of the birth rate and the death rate. If the population is not closed, we must also include immigration with births and emigration with deaths. Providing we use instantaneous rates, there is a very simple relationship between the growth rate and birth and death rates, namely:

 r   =   b   -   d Where: r is the instantaneous rate of increase per unit time, b is the instantaneous birth rate, d is the instantaneous death rate

Using the instantaneous rate of increase we can describe exponential population growth with the following equation.

#### Algebraically speaking -

Nt   =   Noe(r t)

Where:
• Nt is the population size at time t,
• No is the population size at time zero (0),
• r is the instantaneous rate of increase per unit time,
• t is the number of time units elapsed since time 0.

If birth and death rates remain constant, r will also be constant, and a stable age distribution will develop. The maximum rate of increase that is possible for a species in an unlimited environment, for given climatic conditions, is known as the intrinsic rate of natural increase (rm). This measure is used extensively by applied ecologists as a means of describing the growth potential of an organism.

We can readily convert the instantaneous rate of increase to a finite rate. For a single time period we can rewrite the equation above as follows:

#### Algebraically speaking -

Nt+1   =   Nter
so   Nt+1/Nt   =   er   =   λ

Where:
• Nt+1 is the population size at time t+1,
• λ ('lambda') is the finite rate of increase of the population, per unit time,
• all other symbols are as above ## Estimating rate of growth/decline

### From reproductive parameters

We can predict the maximum potential rate of increase (rm) from the age specific birth and death rates.

• #### Approximate method

Life table values are obtained from laboratory data, and used to calculate the net reproductive rate and the generation time, as shown in the More Information page. An approximate estimate of rm (known as the innate capacity for increase, rc) is then obtained from :

#### Algebraically speaking -

 rc   = ln[Ro] Tc
Where:
• rc is the innate capacity for increase,
• Ro is the net reproductive rate,
• ln is the natural logarithm (loge), and
• Tc is the cohort generation time.

• #### Precise method

Life table values are obtained from laboratory data. The value of rm is then determined directly from the Euler equation. #### Algebraically speaking -

Σ[e-(rm x) li mi]  =  1

Where:
• li is the number surviving at the beginning of an age class,
• mi is the number of living females born per female during the age interval,
• x is time at the mid-point of the age interval, and
• rm is the intrinsic rate of population increase.

This has to be done iteratively - in other words different values for rm are substituted on the left hand side of the equation, until it gives the required value of 1.

### By direct observation

We often want to assess the actual rate of increase (or decrease) of a population, for comparison with its potential rate of increase.

This is done by simply plotting the natural log (loge) of the number of individuals against time. The slope of the line is equal to the instantaneous rate of increase per unit time. 