Prey-Predator Work Statement
Context
Students of Environmental Modeling have got some training in mathematical modelling in
Environmental Fluid Mechanics (MFA) where they learned how to solve the diffusion equation
in a one-dimensional geometry and in the Transfer of Energy and Mass (TEM) course where
they added the advection transport processes.
In MFA students studied situations with and without decay (first-order) and with and without
boundary discharges. In this work the effect of dispersion promoted homogenization of the
spatial distribution and the effect of decay promoted elimination of contamination. There was
also a situation with continuous emission where dilution and decay generate a stationary
spatial distribution with maximum concentration at the point of discharge and a spatial
gradient depending on diffusivity.
In TEM students verified the effect of advection in the removal of material from the discharge
zone and studied the effect of heat exchange across the free surface in case of temperature.
They have considered the effect of radiation, convective heat transfer - depending on the
temperature difference between the air and river water and verified the importance of latent
heat loss by evaporation.
The aim of Modelação Ambiental is to add production processes arising from biogeochemical
processes to the advection and diffusion processes. Biogeochemical processes transfer
material between properties by biological and/or chemical processes.
To solve the problem students have software that developed in MFA and TEM, but also a
program provided by the teaching staff that can replace the previous one. This second solution
is important to the students who did not attend the earlier disciplines or for students who
consider this second option more interesting.
Objective
Develop a model for a set of 4 variables (prey, predator, debris and nutrient) and realize
the importance of residence time, i.e. the time that water resides in a system that can be
too short to allow reactions and the effect of model parameters on the solution.
Information available
Students have got:
1. Text on Evolution equation,
2. Text and PPT on discretization of the Evolution equation,
3. PPT's from classes with the model equations to solve
4. Basic program to solve the equation of evolution (that must be adapted to this
case).
Simulations to carry out
The aim is that students present results for situations that illustrate the problem. Globally
those results should put into evidence the consistency between the results obtained
and expected and the sensitivity of the solution to model parameters. The following
results are examples of what can be done:
a) Make the simulation of a discharge of a property with and without decay (of faecal
bacteria type or BOD) and with different intensities of advection and diffusion.
b) Prey-Predator model: make two simulations with two different parameters of the
predator's semi-saturation constant (without transport) and interpret the the
effect of changing the parameter.
c) Assess the importance of the detritus decay rate (mineralizing rate) also without
transport.
d) Repeat the previous simulation with transport and analyse the effect of residence
time.
Students can however include other examples (or different examples), the most interesting
ones being those that surprised them more.
Below it is described the stability issue. However students are not required to implement an
implicit algorithm.
Format of the Report and deadline
The report must be clear and succinct, not requiring more than a 6 pages of text. Must have an
introduction to explain the problem to solve, one chapter with the equations, including the
meaning of the parameters, a short description of the algorithm of solving, presentation and
discussion of results, including a discussion of the expected consequences of variation of
parameters.
The report should be delivered up to March 18th.
Basic Concepts
In Lecture 2 we have seen the equation that describes the dynamics of the population:
Figure 1: Slide from Lecture 2, showing the equation of Population dynamics and a graphical representation in
case of growing (positive k) and decaying (negative k).
We have seen that the permanently positive k is not sustainable. It would generate an
exponential growth. We have seen that the logistics equation is a response to this problem. In
that formulation the value of k tends to zero when we reach the carrying capacity. In this more
complex model the value of k is depends dynamically on the value of other variables and can
be null or negative positive.
Another aspect put into evidence by this four-equations simple model is the stability issue. In
that lecture we have seen that the k is positive when an explicit formulation should be used,
while when k is negative an implicit formulation should be used. Generalizing the concept one
can say that source terms should be computed explicitly and sink terms should be implicitly
computed in order to get the most stable formulation.
In order to minimize instabilities it is common to avoid negative values of the state variables,
just performing a test of the type "if negative then get the minimum positive value". This
simplistic equation can become a source of variable that in some cases can be enough to
create an exponential growth.
Model formulations
Prey-predator models consider only two interacting state variables (a Prey and a Predator)
originating two interdependent temporal evolutions. That model has analytical solution but it
is not realistic because the prey grows from nothing and there is nothing originated from
predator’s activity, i.e. the model is not conserving mass. AT that time we have seen that at
least a third variable would be required to conserve mass. We have called it “Detritus”.
Three state variable’s model
A model with three state variables would generated a system as shown on the Figure below
Predator (r)
mr
G
my
Prey
(y)
Detritus
(d)
μmax
Figure 2: Processes relating state variables in case of a 3 state variables model where detritus include all types of
material lost by living organisms and the prey grows consuming detritus. This model is not realistic but can
preserve total mass.
Equations for this model are described below. The analytical model is conservative. To every
dPy
dt
 (  max  my ).Py  G
dPr
 E .G  mr .Pr
dt
dPd
 my .Py  mr .Pr  (1  E ).G   max .Py
dt
 Py


G  gz Pr 
P K 
y
s
y


Equation 1
Equation 2
Equation 3
Equation 4
positive term in one equation correspond a negative term in another equation. However the
model is too simple because the prey is generating detritus and is consuming also detritus. At
least another equation. The parameters in these equations are given in Table 1. The values
proposed are in the typical order of magnitude for phytoplankton and zooplankton, but can
vary by 200% or 300%.
Table 1: Parameters used in Equation 1 to Equation 4 and some typical values
Description
µmax
Ksy
my
mr
E
G
Maximum Growth Rate
Half-Saturation Constant
for Prey
Mortality Rate Prey
Mortality Rate Predator
Grazing Efficency
Grazing Rate
0.11 (d-1)
1 (mg/L)
0.06 (d-1)
0.08 (d-1)
0.8
0.3 (d-1)
The units are the same in all equations (e.g Nitrogen). In this case, if the Prey were “Primary
Producers”, e.g. Phytoplanktor the biomass would be accounted as Organic Nitrogen in the
form of Phytoplankton. In that case, the Predator could be Zooplankton and the biomass
would be Organic Nitrogen in the form of Zooplankton and on the same way for Detritus.
Four Equations model
Four state variable (Equation 5 to Equation 8) increase the realism of the model. In these
conditions the prey is consuming a nutrient that is regenerated by other state variables. This
nutrient would be mineral Nitrogen if mass was quantified through nitrogen, as above. The
total mass of each state variable should be accounted knowing the ration between Nitrogen
and other atoms (e.g. using the Redfield ratio, C:N:P = 116:16:1).
Again, this model would consider that there is a single nutrient controlling growth. If more
than one nutrient could limit it, than an extra equation should be added for that nutrient. Also
the temperature and the solar radiation should be added to obtain a seasonal variation.
Without that effect we cannot expect this model to follow the reality.
Predator (r)
mr
G
e
my
Prey (y)
Detritus (d)
μ
Nutrients
(n)
ϕ
Figure 3: schematic representation of processes in case of a four equations model.
Equations for this model are described below. The analytical model is conservative. To every
dPy
dt
 (  max  my ).Py  G
dPr
 E.G  mr .Pr
dt
dPd
 my .Py  mr .Pr  (1  E ).G  .Py
dt
dPn
 .Pd  e.Pr  .Py
dt
 Py


G  g z Pr 
P K 
sy 
 y


Pn

 Pn  K s n 
   max 
Equation 5
Equation 6
Equation 7
Equation 8
Equation 9
Equation 10
The actual growth rate of the Prey depends on a maximum growth rate and on the availability
of nutrients using a “Michaelis-Menton” formulation based on a “semi-saturation constant”.
The larger is the value of that constant, the larger is the concentration required by
phytoplankton to grow.
Temporal discretization
Temporal discretization requires the time derivatives to be described by the rate of
infinitesimal differences and the appointment of a time to calculate the term on the right
equations’ member. Using an explicit discretisation method the Prey equation becomes:
t  t
Py
 Py
t
 (   my ).Pyt  G t
t
t  t
t
Py
 Py .(1  t .(   my )  tG t
This equation can generate negative values if the parenthesis becomes negative of if G is too
large. Using an explicit formulation G is computed as:
 Pyt


G  gz P  t
P K 
sy 
 y
t
t
r
Replacing this equation in the equation above one gets:
Py
t  t



 
1
t
 
 Py 1  t .  my  g z Prt  t
.
 P  K  



y
s
y

 


When mortality plus grazing is larger than production it gets a negative value, that multiplied
by the time step very easily generates negative values for the Prey. An implicit calculation of
the decaying terms manages this problem.
In that case one would write:
Py
t  t
 Py
t
 Pyt  my Pyt  t  G t  t
t
G
t  t
 Pyt  t 

 gz P  t
P K 
sy 
 y
t
r
Replacing G in the Prey Equation one gets:
Py
t  t
 Py
t
Py
t  t

t



1

 Pyt  .Pyt  t my  g z  t


P

K
s y 
 y

tPyt



1

1  t my  g z  t
 P  K 

y
s
y


And in this case the negative values will never be obtained. Doing on a similar way to every
property one gets a more stable model. This is however not enough to get a stable solution. In
fact in order to preserve mass the source terms in this equation (that do not generate
instabilities in this equation) are negative terms somewhere. It is the case of tPyt that is a
negative term in the nutrients equation.
A straight forward solution is to perform a test to the final values obtained and when a
negative value is obtained to push it to the minimum reasonable value. In that case we are in
fact creating a mass of nutrients equal to the difference between the minimum value assumed
and the negative value to be corrected. Doing so we are implicitly assuming that it is a sporadic
situation and the error is not important. If it is s systematic situation the error can be very
large and we can end up with a huge biomass. This can be avoided comparing the value of
tPyt with the nutrients available before updating the Py. The following algorithm avoids the
problem:
PyGrowth  tPyt
if (PyGrowth P nt .)Then
PyGrowth P nt
EndIf
Using this value of PyGrowth in the equation for Py
t  t
We will preserve the mass and will
get a model that is unconditionally stable, supporting any time step, which will be limited only
by accuracy and not by stability.
Transport Processes
The numerical issues for solving the transport processes are described in the Statement
provided to support the TEM practical work. Diffusion is always computed using central
differences. This algorithm is physically correct and has second order accuracy. Upwind or
Central-Differences are simple algorithms to solve advection. Central differences are physically
correct when the grid Reynolds number is smaller than 2 and upwind is the most adequate
above that value. The accuracy of the results depends on the grid size, requiring spatial steps
much smaller than the size of the plumes being transported.
Stability is limited also by the Courant number that must be smaller than 1 in case of explicit
algorithms. In implicit algorithms there is no such limitation.