Lotka-Volterra Two Species Model
Two Species Models
The models we have discussed so far (Malthus and Logistic) are
single species models. Many of the most interesting dynamics
in the biological world have to do with interactions between species.
Mathematical models which incorporate these interactions are required
if we hope to simulate these dynamics.
One of the first models to incorporate interactions between predators
and prey was proposed in 1925 by the American biophysicist Alfred
Lotka and the Italian mathematician Vito Volterra.
The Lotka-Volterra model describes interactions between two species in
an ecosystem, a predator and a prey. This represents our first
multi-species model. Since we are considering two species, the model
will involve two equations, one which describes how the prey
population changes and the second which describes how the predator
For concreteness let us assume that the prey in our model are rabbits,
and that the predators are foxes. If we let R(n) and
F(n) represent the number of rabbits and foxes, respectively,
that are alive at time period n, then the Lotka-Volterra model is:
R(n+1) = R(n) + a*R(n) - b*R(n)*F(n)
F(n+1) = F(n) + e*b*R(n)*F(n) - c*F(n)
where the parameters are defined by:
a is the natural growth rate of rabbits in the absence
c is the natural death rate of foxes in the absence of food
b is the death rate per encounter of rabbits due to
e is the efficiency of turning predated rabbits into foxes.
The Stella model representing the Lotka-Volterra model will be
slightly more complex than the single species models we've dealt with
before. The main difference is that our model will have two stocks
(reservoirs), one for each species. Each species will have its own
birth and death rates. In addition, the Lotka-Volterra model involves
four parameters rather than two. All told, the Stella representation
of the Lotka-Volterra model will use two stocks, four flows, four
converters and many connectors.
Split the rabbit's difference equation into the births part and the
Do the same for the fox's equation.
Using the following parameter values, write down the difference
equations for the Lotka-Volterra model and find all equilibrium
points. This will involve solving two equations for two unknowns
(namely R(*) and F(*)). HINT: this
model produces two steady states, one of which should be unsurprising.
- a = 0.04
- b = 0.0005
- c = 0.2
- e = 0.1
Optional: Try to find expressions for the
Lotka-Volterra steady states in terms of the parameters. In other
words, try to find formulas for R(*) and F(*)
without plugging in specific values for the parameters.
Create a Stella model for the Lotka-Volterra model. Use the parameter
values given above as values for the four converters in your model.
Try various initial conditions for the rabbits and fox populations;
choose some to be near the equilibria you determined above, and have
some be far away. Use different running
times. Which equilibrium is stable, unstable?
Try both the usual time series graph and the scatter graph to examine
the model output. A scatter plot of rabbit versus fox population is
particularly interesting. To produce such a graph pull down the graph
icon, place it somewhere in the model, double click on the graph when
it appears and select the scatter plot option. This will require you
to choose two quantities to plot. Pick the rabbit and fox
populations. You should get some interesting pictures if you let the
model run long enough.
Analysis of Lotka-Volterra
The Lotka-Volterra model is one of the earliest predator-prey models
to be based on sound mathematical principles. It forms the basis of
many models used today in the analysis of population dynamics.
Unfortunately, in its original form Lotka-Volterra has some
significant problems. As you may have noted in your experiments, neither
equilibrium point is stable. Instead the predator and prey
populations seem to cycle endlessly without settling down quickly. It
can be shown (see any undergraduate differential equations book for
details) that this behavior will be observed for any set of values of
the model's four parameters. While this cycling has been observed in
nature, it is not overwhelmingly common. It appears that
Lotka-Volterra by itself is not sufficient to model many predator-prey
systems. Context specific information must be added.