__Presentation
of the phenomenon__

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 natural growth rate of the prey in the absence of predation ( a )
- the natural death rate of predators in the absence of food ( prey ) ( b )
- the death rate per encounter of prey due to predation ( c )
- the efficiency of turning predated prey into predators ( d )

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

and

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
of a.

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 .
It
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

- the coordinates of the point for the begining of the calculation ( number of rabbits then of foxs )
- the number of iterations
- the step where results are used to be ploted
- the step of the calculation ( time step )
- the values of a,b,c,d ( parameters of the system )

__Results
obtained and interpretations__

The paramaters which permit a realistic simulation of the model are :

a=0.04

b=0.0005

c=0.2

d=0.1

The phenomenon has a periodical oscillation of its population.

Influence
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 :

From the point of view of biology, the rise of a represent an increase of the population of rabbits and as a consequence the development of foxs.

a=0.004 | a=0.4 |

The parameters b has the opposite role of the parameters a.Paramaters b :

The result obtains is here logical because when the death rate of rabbits rises, the consequence for the rabbits population and then as a consequence for foxs are negative .

b=0.005 | b=0.05 |

As c increases, the population is more important.Paramaters c :

When c is less important, the population of foxs decreases less ( in the left part of the graphe, when number of rabbits is very small ) than when c is more important. Consequently, the population of rabbits grows slower and so the development of population is too very slowest.

c=0.02 | c=0.2 |

The decrease of d makes the population rises more before the stabilisation.

Parameters d :

d=0.01 | d=0.1 |

An interesting simulation mistake:

This evolution is ploted for :

a=0.04Initially, there are 4 rabbits and 1 fox.

b=0.05

c=0.2

d=0.1

100 iterations |
1000 iterations |

\
1500 iterations |
5000 iterations |

5000 iterations |
40000 iterations |

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.