Population Biology:Demographic Models

Wed. Mar. 2

Ideal Models of Population Growth

Basic Population Parameters

B = # births/time

I = # immigrants/time

D = # deaths/time

E = # emigrants/time

These units are used in our more advanced model. Note that all are rates. Moreover, they are rates based on changes over very tiny moments in time -- the smallest possible change, called dt.

This will allow us to express our next equations as changes in N over very small times, dN/dt.

b = # births/(time * N) = B/N

i = # immigrants/(time * N) = I/N

d = # deaths/(time * N) = D/N

e = # emigrants/(time * N) = E/N

ALL FOR AN "INSTANT" OF TIME

Rates of Changes in Population Size

Clearly, the rate of change of a population over time is the sum of all of the rates given on the last page.

Thus: dN / dt = B + I - D - E

Change of population size per time =

additions per time (births + immigrants)

minus

losses per time (deaths + emigrants)(notice that everything has the same units)

We can also state the same relationship using b, d, e & i:

dN / dt = (b + i - d - e) * N

The Rate of Increase, rIf we assume that i = e (no net migration) then the last equation becomes:

dN / dt = (b - d) * N

We now define the rate of change of the population, r, as: r = b - dr has units of individuals / (time * N)

r > 0, the population is increasingr < 0, the population is decreasingr = 0, the population is steady

And so, the equation becomes:dN / dt = r * N

The Intrinsic Rate of Increase, r0If we assume that resources (including mates in sexual

populations) are not limiting in any way on population growth; i.e., that:

• there are enough resources so that every female reproduces to the maximum extent possible in the species and that the death rate is as low as it can be for a given genetic structure and environment and

• no density dependent increases in mortality due to disease, starvation, etc.

then the rate of population growth is at its maximum. This theoretical concept is called the intrinsic rate of increase or intrinsic growth rate, r0:

r0 =bmax −dmin

The Equation for Exponential Growth

dN

dt=r0N

dNN

=r0dt

integrate rate over all times

dNNt=0

∞

∫ = r0 dtt=0

∞

∫ =r0 dtt=0

∞

∫ln(N) + c1 =rt+ c2ln(N) =r0t+ c2 −c1Let c=c2 −c1Substitute

ln(N) =rt+ cGet rid of the natural logs by exponentiation

eln(N) =er0tec

N ==erotec

Setting time to zero (starting conditions)

N0 =ec

Therefore:

Nt ==er0tN0

Growth rate:dN

dt=r0N

Population size: Nt =er0tN0

Population Growth Predicted by the

Exponential Model

0

10

20

30

40

50

60

0 0.5 1 1.5 2

Time

Population Size (N) r=2.0

r=1.0

Whooping Crane Recovery

What Happens at High Population Densities?

Density Dependence and Density Independence

Density dependence -- when an effect is proportional to the population density. Example: death by starvation, disease -- the proportion that die increase as the density itself facilitates the problem. Note density dependence may also work so that greater density, up to a point, helps a population.

Density independence -- not related to population density. Example -- increased death rate due to extreme weather conditions (the proportion that are susceptible die, regardless of the pop. density).

Population

Equilibrium and

Density FactorsThese are theoretical

lines; in fact the slopes can be much more complex than shown.

The equilibrium density is termed the carrying capacity, K.

Reduction of population growth rate with increasing population

size (N)

Modeling Density Dependent Growth

dN

dt=r0 (

K −NK

) =r0 (1−NK)

The rate of growth for any population, where b and d are not constant but vary with population size or density can be given by:

dN

dt=rN

where r =b−dand b and d are variables

A useful approximation can be made by assuming that the dimensionless expression (K-N)/K approximates the effects of density (N) on (b-d)

Comparison of Exponential and Logistic Growth

About Carrying Capacity, K

K is a density (or number N of individuals in an area) at population equilibrium.

It is NOT a constant but it varies with environmental conditions.

Growth of Various Natural Populations

The Growth of Human Populations