1

Environmental BiologyPopulation Dynamic

2

• Population dynamics is the study of the changes in the numbers and composition of individuals in a population within a study unit and the factors that affect these numbers.

• The population can be Escherichia coli, moose, otters, humans, or any other unit.

3

As environmental scientists and engineers, evaluating population dynamics is critical to

(1) Understanding how environmental perturbations affect populations,

(2) Predicting human populations so as to determine water resource needs,

(3) Predicting bacteria populations in engineered systems, and

(4) Using populations as indicator of environmental quality.

4

Resource development specialists and wildlife biologists

(1) Estimating how many animals can be harvested,

(2) Predicting when a species or population is threatened or endangered with extinction, and

(3) Understanding how one population might affect another.

5

• Factors that cause populations to change may be related to or independent of the number of organisms in the study area.

• The decrease in available food and other resources along with the production of waste will result in a decline in the health of the individual organisms and an increase in the death rate along with a decrease in the rate of replication or reproduction.

• Typical density-independent causes of mortality are weather, accidents, and environmental catastrophes.

6

Bacterial Population Growth• Bacteria have been included in this

section because they have simple reproductive scheme, allowing us to more easily study the fundamentals of population dynamics and to develop the models used in this field.

• The dynamics of bacterial populations are also relevant to environmental scientists and engineers because of their importance in wastewater treatment and water quality.

7

Bacteria Growth Requirements1. A terminal electron acceptor2. Macronutrients3. Micronutrients4. Appropriate environment

8

Growth in Pure Cultures

Growth is shown as L = log (numbers) where numbers is the number of colony forming units per ml, versus T (time.)

9

Example 1If the initial density of bacteria is 104 cells

per liter at the end of the accelerated growth phase, what is the number of bacteria after 25 generations?

10

Units of Expression for Population Size

• It is possible to characterize individual populations through direct enumeration, the communities typical of natural or engineered systems comprise many populations of various sizes and forms, and thus are difficult to describe in his way.

• An alternative approach involves the use of a surrogate parameter to represent all living material or biomass.

11

• For example, in the activated sludge reactor of a wastewater-treatment plant, total suspended solids and volatile suspended solids can be provide measures of biomass.

• Particulate organic carbon and chlorophyll are surrogate parameters commonly applied in natural system.

12

Model of Population Growth•A physical model is defined as a small object, built to scale, that represents another often larger object and that can be used to examine the response of an entity to perturbation.•Consider the case of the algal or bacterial community of a lake or river, or the community of microorganisms in a waste-treatment system.•The mass balance on biomass may be written as:

(1)where X is biomass (mg/L), V is volume (L), Q is flow (L/day), t is time (days), and “Reaction” refers to all the kinetic processes describing the growth or death of the organisms.

actionQXQXdtdXV in Re±−=

13

•Note that in this “mass balance” each term in Equation 1 has units of mass per time (mg/d).•To simplify the conceptual development of the models that follow, the flow terms will be ignored here.•Assuming that first-order kinetics adequately describe the reaction term, Equation 1 can be written as:

(2)VkXdtdXV =

14

•Dividing Equation 2 by V results in:(3)

where k is the first-order rate coefficient (time-1).•Three models are introduced here describing unlimited, space-limited, and resource-limited growth.

kXdtdX

=

15

Exponential or Unlimited Growth• The dynamics of many organisms, from bacteria

to humans, can be described using a simple expression termed the exponential growth model:

(4)• This relationship is identical to that presented

previously as Equation 3, with µ, the specific growth-rate coefficient being a special case of the first-order rate constant, k.

• Equation 4 can be integrated to yield:(5)

Where Xt is the biomass at some time t, and X0 is the initial biomass, reported as numbers or as a surrogate concentration, for example, mg DW/L.

XdtdX µ=

)(0

tt eXX µ=

16

Example 1: Exponential Growth and Effect of Specific Growth Rate on the Rate of Growth

Consider a population or community with an initial biomass (X0) of 2 mg DW/L and a specific growth rate (µ) of 1 day-1. Determine the biomass concentration (mg DW/L) over a time period of ten days.

17

0

10000

20000

30000

40000

50000

60000

0 2 4 6 8 10

Time (days)

Bio

mass,

X (

mg

/L)

18

0

20

40

60

80

100

120

0 2 4 6 8 10

Time (days)

Bio

mass,

X (

mg

/L)

0.20.30.4

19

Logistic Growth: The Effect of Carrying Capacity

• The logistic growth model provides a framework that is more in tune with our concept of how populations and communities of organisms behave, that is, the size of a population or community is limited by the carrying capacity of the environment to sustain growth.

• Carrying capacity refers to an upper limit to population or community size imposed through environmental resistance.

• In nature, the resistance is related to the availability of renewable and nonrenewable resources as they impact biomass through reproduction, growth, and survival.

20

• A companion approach treats limitation by renewable resources.

• The logistic growth model is developed by modifying the maximum specific growth rate to account for carrying-capacity effects:

(6)

where µmax is the maximum specific growth-rate coefficient, K is the carrying capacity, that is the maximum sustainable population biomass.

−=

KX10µµ

21

• Substituting Equation 6 into Equation 4 yields:

(7)

• Equation 7 can be integrated to yield: (8)

XKXX

dtdX

−= 1maxµ

( )

−+

=− t

t

eX

XKKX

max

0

01 µ

22

Example 2: Logistic Growth

Consider the population from Example 1 (X0= 2 mg DW/L; µmax = 1 day-1), but with a carrying capacity (K) of 5,000 mg DW/L. Determine the population biomass over a time period of 10 days.

23

Resource Limited Growth: The Monod Model

• The relationship between nutrients and the population or community growth rate can be described using the Monod model, a conceptual framework important in natural and engineered systems.

• The Monod model, is written as:(9)

here S is the nutrient or substrateconcentration (mg S/L) and Ks is the half-saturation coefficient (mg S/L).

SKS

s += maxµµ

24

• Substrate may be either a major food source for the organism or a growth-rate-limiting micronutrient.

• The half-saturation constant, Ks, is defined as the substrate concentration at which the growth rate is one-half its maximum, that is, µ = µmax/2.

25

• The value of Ks reflects the ability of an organism to acquire resources.

• Organisms with a small Ks approach the maximum specific growth rate (µmax) at comparatively low substrate concentrations, while those with high Ksvalues require more abundant substrate levels to achieve the same level of growth.

26

• The Monod model (Equation 9) can be substituted into Equation 4 to yield:

(10)XSK

SdtdX

s += maxµ

27

The Yield Coefficient: Relating the Rate of Growth to the Rate of Substrate Utilization

• While attention has been largely devoted here to tracking biomass, substrate fate may be of more interest in many engineering applications.

• The capability to model substrate concentrations is gained through application of the yield coefficient, defined as the quantity of organisms produced per unit substrate consumed:

(11)SXY∆∆

=

28

• In Equation 11, Y is the yield coefficient. • The yield coefficient is also commonly

applied to relate the rate of substrate utilization (dS/dt) to the rate of organism growth (dX/dt):

(12)dtdX

YdtdS 1

−=

29

• Substituting the Monod growth-limitation model (Equation 10) for dX/dt in Equation 12 results in

(13)

an expression used in engineering applications, for example, to develop the mass balances on organism growth and substrate utilization that support waste-treatment facility design and operation.

XSKS

YdtdS

s +−= max1 µ

30

Example 3: Use of the yield coefficient

The organic matter present in municipal wastewater is removed in an aerated biological reactor, part of the activated sludge process at waste-treatment plants. The organic matter is removed at a rate of 25 mg BOD5/L-h.

Use the yield coefficient to compute the mass of microorganisms (measured as volatile suspended solids, VSS) produced daily due to the consumption of organic matter by microorganisms in the aeration basin. Assume the aeration basin has a volume of 1500 m3 (4.6 m deep by 9 m wide by 36 m long) and that the yield coefficient, Y, equals 0.6 mg VSS/mg BOD5.

31

Respiration: The Decay Coefficient

• The models developed to this point have not included the term that accounts for respiration, “the cost of doing business,” or in the terminology of wastewater engineering, “endogenous decay”.

• The rate of decay of a population of organisms can be written as:

(14)

where kd is the respiration-rate coefficient (time-1). In some situations, the definition of the term kd is expanded to include other losses, for example, settling and predation.

XkdtdX

d−=

32

• Integration of Equation 14 yields the analytical solution for a first-order decay:

(15))(0

tkt

deXX −=

33

Biokinetic Coefficients• The terms µmax, Ks, Y, and kd are

commonly referred to as biokinetic coefficients because they provide information about the manner in which substrate and biomass change over time.

Coefficient Range of Values Typical Value

µmax

Ks

Ykd

0.1 – 0.5 hr-1

25 – 100 mg BOD5/L0.4 – 0.8 mg VSS/mg BOD5

0.0020 – 0.0030 hr-1

0.12 hr-1

60 mg BOD5/L0.6 mg VSS/mg BOD5

0.0025 hr-1

34

Batch Growth: Putting It All Together

• Respiration and the growth-mediating mechanisms introduced earlier may be integrated into a single expression describing population growth in a batch culture:

(15)XkSK

SKX

dtdX

ds

−

+

−= 1maxµ

35

• Substrate utilization can then be related to Equation 16 through the yield coefficient:

(17)XkSK

SKX

YdtdS

ds

−

+

−−= 11

maxµ

36

Example 4: Simplified calculations of substrate and biomass

The differential equations that describe biomass and substrate dynamics contain nonlinear terms that require specialized numerical methods for their solution. By applying certain simplifying assumptions, however, quite a bit can be learned about the dynamics of microbial populations and communities. Consider a population of microorganisms with the following characteristics growing in batch culture: initial biomass, X0 = 10 mg DW/L; maximum specific growth rate, µmax = 0.3 day-1; half-saturation constant, Ks = 1 mg/L; carrying capacity, K = 100,000 mg DW/L; respiration-rate coefficient, kd = 0.05 day-1; initial substrate concentration, S0 = 2,000 mg S/L; and yield coefficient, Y = 0.1 mg DW/mg S.

37

1. Determine whether this population will ever approach its carrying capacity.

2. Calculate the population biomass after the first three days of growth, assuming no lag phase.

3. Calculate the substrate concentration after the first three days of growth.

4. If the population peaks at 100 mg DW/L when the substrate runs out, calculate the biomass 10 days after the peak.

38

Growth Models and Human Population

• You should ask yourselves several questions:(1)What population model best fits these data?(2)What does this model suggest will happen in

the future?(3)Are humans subject to the same limiting

factors as other organisms?(4)If so, what are the implications for food,

energy, waste disposal, and interactions with other organisms as human approach the carrying capacity of their environment?

39

Figure 6: Human Population Growth over Time

40

• The human population growth of the last century has been truly phenomenal.

• It required only 40 years after 1950 for the population to double from 2.5 billion to 5 billion.

• This doubling time is less than the average human lifetime.

• The world population passed 6 billion just before the end of the 20th century.

• Present estimates are for the population to reach 8-12 billion before the end of the 21st century.

• During each lecture hour, more than 10,000 new people enter the world, a rate of ~3 per second!

41

• Of the 6 billion people, about half live in poverty and at least one fifth are severely undernourished. The rest live out their lives in comparative comfort and health.

42

• The factors affecting global human population are very simple.

• They are fertility, mortality, initial population, and time.

• The current growth rate of ~1.3% per year is smaller than the peak which occurred a few decades ago (~2.1% per year in 1965-1970), but since this rate acts on a much larger population base, the absolute number of new people per year (~90 million) is at an all time high.

• The stabilization of population will require a reduction in fertility globally.

• In the most optimistic view, this will take some time.