Population growth &

Consumption of resources

1. Exponential growth model2. Logistic growth model

Chapter 3: Masters & Ela

Out of 7 billions

½ Poverty 1/5 in bad health

Cannot Stand Already!!!

Beh Tahan !!!

Exponential growth model

• Exponential growth is the growth of a system in which the amount being added to the system is proportional to the amount already present: the bigger the system is, the greater the increase.

• Exponential function is a very useful mathematical tool used in environmental studies, e.g. population growth, resource consumption, pollution accumulation and radioactive decay.

• It is a first-order rate process and overall the exponential growth can be given as N = N0ert

Exponential growth model

N0=initial number of species, Nt = number of species after time t, and r = growth rate,

General forms:

Nt = N0(1 + r)t

N = N0ert

Doubling time

• A quantity that is growing exponentially requires a fixed amount of time to double in size, regardless of the starting point.

• It takes the same amount of time to grow from N0 to 2N0 and from 2N0 to 4N0.

• 2N0 = N0ertd , where td is doubling time

Exponential DecayWhen the rate of decrease of a quantity is proportional to the amount present, exponential growth becomes exponential decay. N = N0e-kt k= reaction rate constant (time-1)

Half-life

•Half-life, t½ ,is the time required for half of a substance to decay

into other elements. Example : if half life is 1 year and initial mass

of the substance is 100 grams, then after 1 year 50 grams will

remain; 25 g after another year and so on.

•This concept is especially useful for radioactive isotopes (Table 2.6,

Masters & Ela)

• Exponential decay rate can be described using a reaction rate

coefficient (k, time-1) or a half-life (t ½).

Disaggregated Growth Rates; a product of a number of individual factors:

Affluence is energy demand per person

Technology is the carbon emissions per unit of energy

If each factor grows exponentially, Pi=pierit , then the total rate of growth is the sum of the individual rates (r=r1 + r2 + r3…+ rn). Final estimating growth: P=Poert

GDP = Gross domestic product

Resource consumption

If resource production, Q, follows exponential growth:

Then time required to produce amount of resource ,Q, can be estimated:

According to this model, if resource production continues to grow exponentially for a long period of time, the number becomes unrealistically large.

Logistic growth

Such growth indicates that initially the rate is exponential followed by slower rates as the population reaches its carrying capacity (K). It is a common successful method in biological and microbiological studies to reflect growth of living organisms.

The logistic curve is derived from the following differential equation

where N=the population size, K=the carrying capacity, r=the

exponential growth rate

(1-N/K) = environmental resistance

tN (

For projections of population growth, a logistic or S-shaped (sigmoidal) growth curve is normally used.

•The factor (1 – N/K) = the environmental resistance, as the population grows, the resistance to further population growth continuously increases.

•To calculate the population at time t

•To find ‘r’, a factor of instantaneous rate constant, R0, is introduced:

•If carrying capacity is known then it is possible to estimate time required for certain population increase.

For logistic growth, the maximum sustainable yield is obtained when the population is half the carrying capacity, N* = K/2. The maximum sustainable yield is the maximum rate that individuals can be harvested (removed) without reducing the population size.

Eq3.29

Eq3.30

Express in term of the current growth rate, Ro and current size, No:

Human population growth, definitions:• Crude birth rate, b, which is the number of live births

per 1000 population in a given year. In the developing countries this rate reaches 30-40, and in developed countries it is about 10.

• Crude Death Rate, d, which is the number of deaths per 1000 population per year.

• Infant mortality rate, the number of deaths to infants (under one year old) per 1000 live births in a given year. One of the best indicators of poverty in a country.

• Rate of natural increase, r is the difference between crude birth rate and crude death rate.

r = b – d

• Net migration rate, m, is the difference between immigration and emigration

r= b – d + m

• Total Fertility Rate (TFR) is the average number of children that would be born alive to a woman, assuming that current age-specific birth rates remain constant through the woman’s reproductive years (how many children each woman is likely to have in her lifetime).

Human population growth, definitions:

• Replacement level fertility is the number of children that a woman must have, on the average, to replace herself with daughter in the next generation. It accounts for differences in the ratio of male to female births as well as child mortality rates

• Age structure reflects a country’s population trends (a population pyramid showing year of the births and number of women and men born in those years). A graphical presentation of the data indicating numbers of people (or % of the population) in each age category is called age structure or population pyramid.

Human population growth, definitions:

Age structures:

Factors limiting growth of population:

• Short-term factor is the disruption of food distribution in a country, commonly caused by drought, energy shortage for food transportation.

• Intermediate-term factors include desertification, dispersal of toxic pollutants; disruption of energy supplies;

• Long-term factors include soil erosion, a decline in groundwater supplies, and climate change.

• Natural disasters of high magnitude (e.g. tsunami, earthquake) can result in significant sudden decrease of population.

Actual situation on human population growth

Example: Human population in 1960 was 3 billion with a growth rate of 1.2%. Estimate population size in 2010.N0 = 3x109; r=0.012; t=50 yrs

N = N0xert = 5.5 x109

Use the same data and assume that human population growth follows a logistic model where K=150 billions.