1 Introduction
A "Predator - Prey" system is one such that two or more key species are coexisting, or
attempting to coexist, in the same area while one species survives by predating the other
species. The question to be answered by this report is, "What is an accurate method for
modeling a predator-prey system?" To gain an idea of the future demography of the two
species, or if a certain ecosystem is at equilibrium, models of the current ecosystem are
created and simulations are run. This is done through systems of differential equations.
Beginning in the early part of this century Alfred Lotka and Vito Volterra began studying
systems of equations that could accurately simulate an actual system. From their work
grew a set of equations that have been used as the standard for predator-prey modeling.
These equations provide a relatively good basis for understanding the general relationship
between two species, but it is difficult to produce very accurate results due to problems
encountered when attempting to incorporate the variety of real situations into a
mathematical model.
Accuracy, when using these sets of equations, depends on including and properly
simulating everything that will affect both species throughout the course of the period of
time interested in. This is possible in a controlled laboratory environment, but rather
impossible in an actual ecosystem. Variables such as weather, disease, and other species
interaction are too complicated to accurately model. To overcome this problem the
variables are made into constants that are created from averages of data taken over a
period of years. The idea being that in a sense, history will repeat itself and the system
that has sustained for a number of years will continue to sustain itself into the future.
Again the difficulty arises in trying to predict what will occur and how the system in
equilibrium will react to future changes. This difficulty can be overcome to a point by
using well-researched terms and variables in the equation set, and by using solid
numerical analysis tools.
The procedure for using these sets of differential equations will be studied. This will
include discussion of the representation of the terms and values in the equation sets, the
method for solving the system and displaying the data, the analysis of the data, and
evaluation of the error occurring throughout the process. To begin, a basic form of the
Lotka-Volterra equation will be discussed.
2 Simplified Standard Equations
The standard set of differential equations are broken down with one equation representing
the change in predator population and one representing the change in prey population.
The equations are:
2.1
2.2
dP
 k 3 N (t ) P(t )  k 4 P(t )
dt
dN
 k1 N (t )  k 2 N (t ) P(t )
dt
1
In these equations, and the other equations used in this report "N" will represent the
population of prey, and "P" will represent the population of Predators, in both cases with
respect to time.
For these equations:
k1 = birthrate of prey
k2 = predation rate of prey
k3 = birthrate of predator proportional to the predation rate
k4 = mortality rate of predator
In these basic equations, the representation is essentially birthrate - mortality rate. The
first term in both the prey and predator equations is the birthrate. For the prey, birthrate
is simplified into a ratio of the number of prey alive, and assumes a constant. In the case
of the predator, the number of prey and predators is proportional to the increase in
predator population because of the dependence on the prey as the sole food source. In the
same way, the mortality rate of the prey is connected to the population of the predator as
well as the natural mortality rate. The predator, assuming only natural causes in these
scenarios, dies at a fixed rate relative to current population.
This simplified scheme combines variables and assumptions and creates difficulty in
comprehending the full meaning and coverage of the equations. To increase awareness,
fewer assumptions need to be made and more variables accounted for. A breakdown of
the individual components of these equations and their coefficients will show in more
detail what is included with this type of model, how different variables are grouped, and
what can be done to better model this scenario.
3 Expanded Standard Equations
By breaking down what is included in the coefficients of equations 2.1 and 2.2, and a
clearer picture can be made, and variables can be better evaluated. The expanded
equations are:
3.1
3.2
dP acN (t ) P(t )

 eP(t )
dt 1  at h N (t )
dN
N (t )
aN (t ) P(t )
 rN (t )(1 
)
dt
KN
1  at h N (t )
r=
KN =
a=
c=
e=
th =
Calculated rate of prey increase
Carrying capacity of the prey
Predator vs. prey encounter rate
Predator consumption of prey ratio to predator birthrate
Predator mortality rate
Handling time
2
At first these equations may seem different than equation 2.1 and 2.2, but when studied
closer the original coefficients have simply been expanded. Studying these new
coefficients, there are many more opportunities for accuracy. Where k1 was merely a
proportion before, the expanded coefficient (shown below) now covers more of the prey
demography and brings into account a carrying capacity term. The carrying capacity is
the amount of a species that can inhabit a given area in equilibrium based on space, food,
genetic pool, and other variables. This serves as a correction factor if the level of
predators is not enough to manage the growth of the prey. Though it is a constant in this
case, the carrying capacity related to food is in reality its own predator-prey system with
regards to what the prey will eat.
From 2.1 to 3.1 the k coefficients have expanded to this:
k1  r (1 
N (t )
)
K
k2 
a
1  at h N (t )
The prey mortality term from 2.1 now shows more of a functional response by the
predator as opposed to just a mortality rate of the prey. The encounter rate and handling
time constants take into account that as the prey population falls predators take more time
in hunting prey; as well as the fact that there are less prey overall.
Studying the predator equations we see that from 2.2 to 3.2 the k coefficients have
expanded to this:
k3 
k4  e
ac
1  at h N (t )
The predator increase term is very similar to the prey decrease term with the added
constant, "c" which is a proportion that approximates how the predator converts the
consumption of prey into an increase in population. While the predator mortality rate in
this simplified system remains constant and was not expanded, it is important to note that
like the prey, the predator is presumably a part of another predator-prey system in which
it is the prey.
4 Methods
With the differential equations for the system defined the focus shifts to extracting useful
data and information. This can be done with a variety of numerical analysis tools ranging
from Taylor Series to Runge-Kutta. For this study, the Runge-Kutta Method for systems
of differential equations will be used to analytically solve the differential sets. This
method is described in the text Numerical Analysis (pp.320-321) as algorithm 5.7.
This Runge-Kutta method for systems of differential equations utilizes the fourth-order
Runge-Kutta method for initial value problems, and can be used for an nth order system.
3
The Runge-Kutta method for systems differs from the normal Runge-Kutta fourth-order,
in that it is written to solve the system of equations simultaneously with respect to time.
In modeling the ecosystems, the coefficients must be calculated from years of research.
The averages of the research become the values for the differential equations. A sample
set of values is shown below.
r = 0.50
K = 70
a = 0.05
h = 0.02
c = 0.50
e = 0.50
Using these sample coefficient values, and the Runge-Kutta method, the populations of
species of this predator prey scenario are normally graphed in two types of axes. The
first type is a plot of predator and prey population beginning with an initial count of the
two populations, and continuing over a period of years.
Stable Equilibrium
Population
50
40
30
Prey
20
Pred.
10
0
4.5
9.5
.5
14
.5
19
.5
24
.5
29
.5
34
Time
.5
39
.5
44
.5
49
.5
54
.5
59
Table 1
This graph shows an initial value of 40 for the prey and 14 for the predator. This leads to
a period of time when the balance is not stable, but is slowly settling down to a stable
equilibrium of about 21 and 7 prey and predators respectively.
Population
This type of graph can also show another type of equilibrium that is not stable.
Unstable Equilibrium
4500
4000
3500
3000
2500
2000
1500
1000
500
0
Prey
Pred.
T
2.4
5.5
9
12.5 16 19.5 23 26.5 30 33.5 37 40.5 44 47.5
Time
Table 2
4
Table 2 shows unstable equilibrium for a larger population of species. The initial values
are at 1000 and 500 for prey and predator respectively, but in this case, there is no stable
equilibrium of population for the species. The values cycle around an average of about
1500 for both the predator and prey. For the range of values shown in Table 2 it is not
conclusive that the system ultimately does not reach stable equilibrium. It is possible that
the dampening effect inherent in the equations is small, and needs more time before a
stable equilibrium is reached. Another method of graphing that can quickly determine
the types of equilibrium is needed.
An easy way of showing equilibrium is to use a field plot. A field plot does not use time,
but plots the two differential equations against each other to create a field of vectors.
These field plots were created by using the fieldplot command in Maple V with the set
of equations. This field of vectors will show the growth or decline of the population of
each species relative to the other for any value. The field plot can also be read to show
whether the population is stable or unstable by following the path of the vectors. If the
vectors lead to one point, that is the stable population. If they lead around in a circle, that
is an unstable population revolving around a central population value where the system is
most stable. Here is the stable equilibrium system graphed on to a field plot.
Table 3
With this format, the fact that the system converges to a single equilibrium point is
quickly observed. The initial value of 40 prey and 14 predators is marked on the Table,
and the path of convergence can be followed around the loop.
5
Table 4 shows the unstable system, but using the field plot it is certain that the system is
unstable.
Table 4
As the Table displays, the data is unstable, but it is caught in a repeatable orbital pattern
that creates equilibrium. These elliptical circuits confirm that the system will never
converge to equilibrium, and also show where the unstable equilibrium point is; in this
case, around 900 for the prey and 1500 for the predator. The initial value of 1000 prey
and 500 predators is marked on Table 4. Even though the prey was close to the
equilibrium point, because the predators were so far away from the stability point a large
instability was created.
5 Analysis
The data from above can be used and interpreted in a variety of different ways depending
on the information available and the results that are needed. By asking what the goal of
the study is, the correct tools can be incorporated to gain the necessary results. Typical
questions asked when studying a predator prey system are the following: Is this system
in equilibrium, what are the populations at the equilibrium point, and when does the
system reach equilibrium?
To answer these questions, it is easy to use the Runge-Kutta method to calculate values
for the population vs. time graph. With those values, the populations cycle up and down
for an unstable equilibrium, or dampen to a stable equilibrium. Because the Runge-Kutta
method simultaneously solves both equations with respect to time, the time values at any
point in the range are readily available. Using the Runge-Kutta alone may be the best
way to find the equilibrium time for an initial value problem relating to the introduction
of a species to the system or an event which suddenly modifies the relative populations of
predator and prey; however, if other information is needed the Runge-Kutta can be used
with other tools to more quickly find a solution.
6
In order to find the relative stability or instability of a system in equilibrium quickly
using Runge-Kutta, a good set of initial values is needed. By using the field plot , the
equilibrium can be instantly categorized, if there is one, and the equilibrium point can be
read from the graph. Using the field plot to as a guide, points can be chosen from any
where in the plot and used as the initial values in the Runge-Kutta algorithm to plot the
exact dynamic of the populations and calculate the actual population values.
Another tool that can be used to find the exact solution for the equilibrium point, and tell
if the system goes to equilibrium is an isocline. An isocline is the solution of one of the
differential equations in the set with respect to the other species when the change in
population over the change in time is set equal to zero. This represents the population of
one species if the population of the other species is constant. Here are the isoclines for
the basic equations.
5.1
5.2
P(t ) 
k1
k2
N (t ) 
k4
k3
Substituting the numerical values in for the k variables, the isoclines for the system are
found to be 1500 for the predator population, and 833 for the prey population (Note that
in cases of larger prey, the prey population will be smaller than the predators').
Below are the isoclines for the expanded equations.
5.3
5.4
at
r
1
P(t )  ( )[1  (at h  ) N (t )  ( h ) N (t ) 2 ]
a
K
K
N (t ) 
e
(ac  eat h )
Again, the two equations are solved, and the system isoclines are calculated for the
predator to be 7 and 20 for the prey. When the isoclines are plotted against each other
and they intersect, it means that the system is in or goes to equilibrium. The intersection
point of the isoclines is the equilibrium point. Here are the field plots again, this time
with the isoclines superimposed.
Table 5
Table 6
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Using all of these tools in combination, the needed information can be quickly extracted
from the data generated by the system. How likely is it, however, that the analysis
correctly predicts the actual number of species, and the precise fluctuations? The
accuracy of the computational method is very good with respect to what is needed; the
deviation lies in the variance of the coefficients, and the ability of the equation to
realistically model the population dynamics of the ecosystem.
6 Error Analysis
To analyze and quantify the error involved in this report two levels must be studied; the
first is the inner level with the computational error, and the second is the outer level and
the error produced by the equations used to create the models. The computational error
covers the accuracy of the data calculated from the set of differential equations. The
modeling error covers the accuracy of the set of differential equations in modeling the
actual populations of species in the system. The data calculated from the equation set
included the population versus time plot, the field plot, and the isoclines.
The values for the population versus time plot were generated by the Runge-Kutta
method. To find the error between what was calculated to be the equilibrium points of
the stable system, the last value of the set can be compared to the exact answer found by
the isoclines. The stable value from the time plot, at time 74.5 years, was 20.50 for the
prey, and 7.19 for the predators. Actual isocline values were 20.41 and 7.23 for the prey
and predator. The error would thus be within 0.09 species for the prey and only 0.04
species for the predator. For this case, that amount of error is excellent. The reason is that
in reality there is not only 0.04 of any live animal (There may be a case for fractions of
populations in terms of vegetation as a prey, but that is not covered in this report). In this
situation, only integers are acceptable and the error would be within half of an integer, so
the calculations are well within the limits needed. Past year 53.5 the calculations were
consistently within a half of an animal of the isocline population. During the next 20
years, the population slowly dampened to the given values. Whether the actual
population is actually 20 or 21, or 7 or 8 at time 53.5 years or 74.5 years is not the
responsibility of the computation. Though finding the equilibrium error is
straightforward for the stable system, calculating the error for the calculations for the
unstable equilibrium is difficult because the exact values are not known.
The major source of error is initiated by the constraints placed on the system. A system
with only a predator and a prey is very unlikely. The models used for these calculations
would have to be expanded to incorporate all relevant species in the system. Almost all
animals compete against each other in some way, be it for food, shelter, water, or other
limited resources. To accurately capture the all the interactions in a system through a
mathematical model would be unlikely, so only with a controlled study would these
models be truly effective.
8
Equations used in the models should try to not only capture the interactions of the
predator and prey, but be mathematically manipulated to fit the model. An example of
this is with the stable equilibrium case. Is the population at 53.5 years actually 20 or 21?
Accuracy of the structure of the equations and selection of variables play a role in this.
Is the equation sufficiently damped to correctly model the population dynamics? How
many animals are there at year 53? Is this population truly in equilibrium? These
questions could only be answered by a yearly census of the population, and are not in the
scope of this report.
7 Conclusions
At the beginning of this report the question was asked, "What is an accurate method for
modeling a predator-prey system?" The most complete answer that is available begins
with the Lotka-Volterra model. Using that fundamental set of differential equations
expanded to incorporate as much information about the ecosystem as possible to
mathematically model a fairly simple scenario works very well. Together with the
analysis tools, an accurate predator prey model is created.
The numerical analysis tools available to solve these systems, specifically the RungeKutta Method for Systems of Differential Equations quickly calculate values that show
the population dynamics. The Runge-Kutta fourth order method simultaneously solves
the equations with accuracy that is compatible with what is needed. The Maple V
fieldplot routine solves the system graphically and displays an equilibrium solution.
Using these tools together with the isoclines, it is possible to accurately portrait the
results of the equation set.
The question to whether the equation set is accurate or not is where error will enter the
solution. To try to model nature accurately for a period of time in an uncontrolled
environment is very difficult. There are too many variables to solve, too many unknowns
to account for. But, as all natural ecosystems are in some kind of equilibrium, though
consistent accuracy can not be depended on; the trends and dynamics of the ecosystem
are consistent, and that is what these systems have modeled.
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References
Richard L. Burden and J. Douglas Faires, Numerical Analysis (Brooks/Cole Publishing,
1997)
Manfred Peschel and Werner Mende, The Predator-Prey Model: Do We Live In A
Volterra World? (New York: Springer-Verlag Wien, 1986)
fisher.teorekol.lu.se/simulation_server/predator_prey_model.html
gumby.syr.edu/s3_survey/survd3.htm
www.cds.caltech.edu/~hinke/courses/CDS280/predprey.html
www.frog.biology.yale.edu/ginger/java/iso.html
www.gypsymoth.ento.vt.edu/~sharov/popecol/lec10/lotka.html
www.stolaf.edu/people/mckelvey/envision.dir/lotka-volt.html
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