Lecture Support for Lab 1. Population Growth and Numerical Integration
Population Growth Models
Let’s begin with a mass balance expression for microbial biomass (X) in lakes:
V
dX
 Q  X in  Q  X  Reaction
dt
where Reaction refers to all of the processes relating to organism growth and death.
Assuming that the flow terms can be ignored for now (a batch reactor) and that the
reaction term is well described by first order kinetics, the equation becomes:
V
dX
V k  X
dt
and canceling the Vs:
dX
kX
dt
Here, we will explore three variations on ‘k’ (population growth models): unlimited
growth (the exponential model), space-limited growth (the logistic model) and resourcelimited growth (the Monod model).
Exponential or Unlimited Growth
dX
 X
dt
where µ is the specific growth rate coefficient (d-1), a special case of the first order rate
constant, k. Note that the units for µ could also be expressed as organisms∙organism-1∙d-1.
This equation can be integrated with respect to time yielding what is termed the closed
form or analytical solution:
X t  X 0  e(  t )
The output from the exponential model, a J-shaped curve, and the impact of changes in
the magnitude of µ, is illustrated in Figure 5-4 of Mihelcic (1999).
Note, however, that after some time biomass levels predicted by the exponential model
become unrealistic (2  1043 in 100 days at µ = 1 d-1). Populations do not continue to
increase exponentially due to space and/or resource limitations.
Logistic Growth: The Effect of Carrying Capacity
Here population growth becomes limited by carrying capacity or the ability of the
environment to sustain growth. We conceptualize this as limitation by non-renewable
resources (i.e. density-dependent phenomena such as space limitation, with the effect
being one of environmental resistance; Milhelcic 1999, Figure 5-5). Mathematically, the
model takes the form:
dX
 X
 max  1    X
dt
 K
where µmax is the maximum specific growth rate coefficient (d-1) and K is the carrying
capacity (units of biomass). The analytical solution to this equation is:
Xt 
K
 K  X 0    max t  
1  

e
 X 0 

The output from the logistic model, an S-shaped curve, is illustrated and compared with
the exponential growth model in Figure 5-7 of Mihelcic (1999).
Example: carrying capacity in Daphnia, Mihelcic (1999, Figure 5-6)
Only rarely, however, do populations approach their carrying capacity because they run
out of (renewable) resources first.
Resource Limited Growth: The Monod Model
Resource or nutrient limitation is accommodated using the Monod model:
 S 
dX
 max  
 X
dt
K

S
 s

where S is the nutrient (substrate, S) concentration and Ks is the half saturation constant
for that nutrient, i.e. the nutrient concentration at which one-half the maximum specific
growth rate is achieved.
The definition of Ks and the relationship between µ and S for two values of Ks are
illustrated in Mihelcic (1999, Figure 5-8). Physically, Ks reflects the affinity of enzymes
for substrate (nutrients) where organisms having a low Ks value are more competitive at
low nutrient levels.
Example: competition in protozoa, Mihelcic (1999, Figure 5-9)
Relationship to Substrate: The Yield Coefficient
We can model the temporal dynamics of substrate as well as biomass if we can quantify
the yield, i.e. mass of organisms produced per unit substrate consumed. This is
accomplished through the yield coefficient:
Y
X
S
and thus the change in substrate is given by:
dS
1 dX
 
dt
Y dt
where the appropriate model is substituted for the dX/dt term.
Respiration: The Death Coefficient
Up to this point, nothing has died. We are aware that ‘death’ is going on continuously in
the form of respiration which is a ‘sink’ in growth models:
dX
  kd  X
dt
where kd is the first order rate coefficient for endogenous respiration.
Putting it All Together
These models can be combined into a pair of equations useful in simulating substrate and
biomass in batch culture.

dX 
 X  S 
  max  1    
  kd   X
dt 
 K   Ks  S 


dS
1 
 X  S 
    max  1    
  kd   X
dt
Y 
 K   Ks  S 

This is illustrated in the classic batch growth curve (Mihelcic 1999; Figure 5-10).
Consider the significance of including kd in the substrate equation.
Numerical Integration
The field of mathematical modeling was developed on the backs of those who could
derive solutions for the ordinary differential equations that constitute these models. As
the models become more complex, the development of analytical solutions becomes more
challenging and the achievement of such solutions has, in the past, been sufficient to earn
the person working on it a doctorate! Consider for example the analytical solution for the
system of ordinary differential equations describing pollutant fate in four lakes connected
in series as presented by Chapra (1997; p. 91-92).
Other limitations, as described by Chapra (1997, p. 120) include:

non-idealized loading functions: analytical solutions require the idealized loading
functions (impulse, step, etc.) developed in Chapter 3. Actual loads behave in a
much more arbitrary fashion can only be accommodated numerically.

variable parameters: an analytical solution requires coefficient values remain
constant over the course of the simulation. This is certainly not the case, as the
specific growth rate for microbial populations, for example, may vary over the
course of the simulation due to changes in environmental forcing conditions (e.g.
light, temperature and nutrients).

multi-segment systems: the development of 1, 2, and 3-D models for requires
spatial segmentation that is inefficient to handle with analytical solutions (see
Chapra 1997, p. 92, Eq. 5.21).

non-linear kinetics: while linear (e.g. first order) kinetics are quite useful, there
are several water quality applications that require non-linear kinetics (e.g. the
Monod model). Analytical solutions cannot be obtained for these.
Thus we turn to numerical methods, applying computer-based techniques in solving
differential equations. This represents a major breakthrough for the field of water quality
modeling as the methods are trivially simple (arithmetic), but prohibitively lengthy were
it not for the availability of computing power.
The Euler Method
The Euler Method is the simplest of several similar methods used in solving ordinary
differential equations. Other, more involved, approaches (e.g. the Heun or 4th Order
Runge-Kutta methods) seek to achieve acceptable accuracy with a reduction in
computation time.
In the Euler method, we transform the differential equation (here the exponential growth
model) as follows:
dX
 X
dt
dX    X  dt
such that we can calculate a change in X and then add it to the original value of X to get
an updated value at the end of the time interval dt:
X new  X old  dX
This approach is implemented in a computer program in the form of a loop, calculating X
over an interval 0 to tmax:
For t  0 to tmax
dX    X  dt
X  X  dX
Next t
The selection of a value for dt is critical to the success of the process (Spain 1982, Figure
5.1). We find as dt0, the numerical solution approaches the analytical solution. The
trade-off becomes one of computation time – as we decrease the time step (dt) the
solution becomes more accurate, but the number of computations that must be performed
increases, thus the value of more advanced numerical methods (e.g. Heun or 4th Order
Runge-Kutta).
Output
The most straightforward approach for implementing the time step is to include it
explicitly in the code, e.g.
For t  0 to tmax Step dt
dX    X  dt
X  X  dX
Next t
and the code would simply move along from 0 to tmax in steps of the specified dt.
However, we typically wish to include an output statement in the code:
For t  0 to tmax Step dt
dX    X  dt
X  X  dX
Print t , X
Next t
and this would result in 1/dt lines of output – not a good idea for very small dt values.
We can get around this by using a nested loop:
For t  0 to t max
For k  1/ dt
dX    X  dt
X  X  dX
Next k
Print t , X
Next t