Suppose we are given the three species equations with
the following parameters:
Suppose we wanted to model Monk Seals, Sharks and Fishermen.
Except for simplicity, there is no particular reason
to limit our model to only two species. Let's create
an example of a three species ecosytem! Examples of
three-species ecosystems include: mouse-snake-owl,
and worm-robin-falcon.
The numerical solution matrix returned by rkfixed is
now computed.
References:
Internet Websites (a good source for student
projects!)
The numerical solution matrix returned by rkfixed is
now computed.
Y0
is the Seal population and Y1 is the Shark population
and Y2 represents the Fishermen.
The derivative function that describes the ODE is defined
by:
The number of Fishermen
at time 0.
The number of Sharks at
time 0.
The number of Seals at
time 0.
We can use rkfixed to graph the solution curves as follows.
When the species interact, we assume that the principle
cause of death for the rabbits is related to the size
of the wolf population. This interaction is expressed
as a rate that is proportional to both population sizes.
Since this interaction decreases the number of rabbits
we add to Eq. 1) the term
r is a positive constant.
If there were no rabbits, then our model predicts a
decline in the number of wolves, W.
The exponential growth equation if k > 0.
If there were no wolves and ample food supply then
we might expect the rabbit population R to increase
exponentially, that is
We first consider the situation in which prey are
rabbits with ample food supply, and the predators
are wolves that feed on the prey.
http://ecology.tiem.utk.edu/~king/mam99/index.htm
The Lotka-Volterra model describes interactions
between two species in an ecosystem, a predator and
a prey. This represents a multi-species model. Since
we are have two interacting species, the model requires
two linked differential equations, the first which
describes how the prey population changes and the second
which describes how the predator population changes.
Vito Volterra used a
system of differential equations
to explain the variations in the shark and food-fish
populations in the Adriatic sea.
The population models we have studied thus far (Malthus
and Logistic) are single
species
models. The next type of model that we consider involves
the interaction of two
species.
One of the first models to incorporate interactions
between two speciea, predators and prey, was proposed
in 1925 by the American biophysicist Alfred Lotka and
the Italian mathematician Vito Volterra.
Systems of Differential Equations
Y0
is the rabbit population and Y1 is the wolf population.
The derivative function that describes the ODE is defined
by:
The number of wolves at
time 0.
The number of rabbits at
time 0.
We can use rkfixed
to graph the solution curves as follows.
Suppose we are given the predator-prey equations with
the following parameters:
Graphical Analysis of The Two Species Model
These equations, Eq. 3) and Eq. 4), are known as
the Lotka-Volterra equations, or the predator-prey
equations. A solution of this system of differential
equations is given by a pair of functions, R(t) and
W(t), that describe the sizes of the rabbit and wolf
populations over time.
Similarly, the interaction of wolves and rabbits
increases the size of the wolf population, and we add
to Eq. 2) the term:
