Cap.3 – pg.50 - cycle in abundance

Intrinsic rate of increase or innate capacity for increase

Pg.50. : 3.1.6.An equation for density – independent population growth in continuous time.

Pg.52 : Changing exponentially : N_t = e^t (3.1)

dN_t = e^t = N_t a taxa de variação é igual à população.
dt

por outro lado se t=0 à N_t = 1.

Para evitar este defeito faz-se :

N_t = N₀ e^t (3.2)

Mas de novo a taxa de variação ( em t) é igual à população (em t) :

embora agora a população no tempo t=0 seja igual a N₀ = N_o e⁰

Para evitar a limitação de a taxa de variação ser igual própria população ( ou seja, dobra a população a cada instante....(é mesmo??) (verificar isto com a máquina de calcular!), it would be more helpful to express the rate of population increase as a multiple of the existing population size. To achieve this we can replace equation 3.2 by eqn.3.4 :

N_t = N₀ e^rt (3.4)

And now dN / dt = r N₀ e^rt = r N_t (3.5)

This is a more useful equation than Eqn 3.3 because we now have the r term on the RHS ( right hand side) of equation . Therefore, the rate of population change at time t is equal to the population size of that time (N_t) multiplied by r which is referred to as the intrinsic rate of increase or innate capacity for increase. Strictly we should reserve the term intrinsic or innate for the maximum value of r (r_m) which occurs under optimum conditions of temperature, light, food supply and so on. The actual instantaneous rate of increase, r, will always be lower than r_m and may vary from year to year or even day to day for a particular species (r or r_m is also called the malthusian parameter).. Equation 3.5 is an important equation, much used in ecological applications.

For a closed population, from and into which no migration occurs, Eqn.3.5 can be rewritten so that the parameter r is replaced by b – d:

DN/dt = (b – d) N_t ( 3.6)

Where b is the instantaneous birth rate (per individual) and d is the instantaneous death rate ( per individual).

To illustrate the estimation of r we will use the example of population change in the United States from 1790 to 1910. Although these data were presented by Pearl and Reed (1920) to illustrate a different point, it is interesting to use them to contrast with their analysis, which is considered in the next section (3.2). To estimate the parameter r, we linearize Eqn 3.4 by taking the natural log (ln) :

Ln(N_t) = ln(N₀) + rt