of the phenomenon
of the study
As the name of the model explains it, it has been created
by the American biologist Alfred
Lotka and the Italian mathematician Vito
Volterra. Proposed in 1925, its objective is to simulate interactions
between predators and prey in an ecosystem.
It take account by the use of 4 parameters of :
The model is discribed by two equations.
In order to be more concretened, we will use rabbits as prey and foxs as predators.
R represents the population of rabbits and F for foxs.
It is easy to find two different points whose coordinates are :Studies of the critical point
In order to study the stablity of these points, we represents R and F by Re+R* and Fe+F* :
the system is equivalent to :For the point (0,0),
It shows that this point is unstable due to the positivity
The problem is less easy to solve:For the point (a/b,c/(d*b)),
By using the value of Re and Fe,
and by taking into account that can be neglected in comparison with and , this system is equivalent to :
The eigen values are .
is impossible to conclude for the stability of the system
The method of discretisation is a uncentred scheme :
The system becomes equivalent to :
Matlab is used to make the simulation of the phenomemon; the program is in the file named volterra.m :
It allows to choose
obtained and interpretations
Realistic case :
The paramaters which permit a realistic simulation of the model are :
a=0.04Graph of the simulation :
The phenomenon has a periodical oscillation of its population.
of the parameters and interpretation in biology:
For this study, I use a too important time step, which
is responsible for th growth of the population. Neverthless, I decided
to let these results because they show that an error in numerical simulation
is easy to make. (this error is presented at the end of the report)
In spite of this, interpretations on parameters are good.
The augmentation of this parameters makes increase the population.Parameters a :
The parameters b has the opposite role of the parameters a.Paramaters b :
As c increases, the population is more important.Paramaters c :
The decrease of d makes the population rises more before the stabilisation.
Parameters d :
An interesting simulation mistake:
This evolution is ploted for :
a=0.04Initially, there are 4 rabbits and 1 fox.
With a time step too important, the population.of rabbits
and fox continue to grow. Why ? Because of the unprecision of the calculation.
During the simulation,the calculation near the singular points aren't
accurate, which provock a small error and, consequently,the graph doesn't
close itself as it done it fot the perfect case.
This study was interesting. It gives me the possibility
to discover a simple application of hydrodynamics instability and problem
of time step that can appears in a simulation. Neverthless, I didn't success
in finding the caracter stable or unstable of the second critical point.
From the point of view of Biology, the model is quite interested too and it is an original application of a mathematical model.