Research work
Ecological competition between four species. Some examples.
Professor: Dr. Ezio Marchi
U.N.S.L.
UNIVERSIDAD NACIONAL DE SAN LUIS
FACULTAD DE CIENCIAS FÍSICO-MATEMÁTICAS Y NATURALES
ESCUELA DE FISICA
SAN LUIS
ARGENTINA
Content
1. Introduction ......................................................................................... page 2
2. Volterra’s model .................................................................................. page 2
3. Interaction between four species ……………………………………. page 5
4. Cycle deduction ……………………………………………………... page 10
5. Calculation and estimation of the period ………………………….... page 11
6. Medium values ……………………………………………………….. page 16
7. Example ……………………………………………………………... page 17
8. Generalization to the n-species problem ……………………………. page 17
9. Microcanon assembly ……………………………………………..... page 19
10. Conclusion ………………………………………………………...… page 21
11. Bibliography ………………………………………………………… page 22
1
1. Introduction
There exist many examples of assemblies which contain elements that interact to
each other, in a cooperative or competitive way. Some important cases are: components
of nervous system, populations of several biological species, political parties, physicchemical reactions, etc.
One of the most important problems in theoretical ecology is the description of
the competition between an arbitrary number of species in the same habitat. A lot of
theories have been developed to study this problem, among them the theory introduced
by Volterra, who gave a mathematic model to study the problem, occupies a relevant
place.
A discussion with his friend D’Ancona, who did a statistical analysis of the
fishes in the Adriatic Sea and observed that the population of the two fish species
changed with the same period of time but out of phase, motivated Volterra to research
the competition between species.
In the present work, it is studied a special kind of coexistence between four
species in the same habitat. Ordinary the competition is for the same nutritional resource
or between species. The analyzed problem is referred to the second case, of preypredator type (some of the species serve of food to the others) without imposing strong
restrictions to the problem, thus we obtain as well as Volterra a periodical variation of
the number of individuals of each species in the time. Also the period of that variation is
analyzed and computed. And in order to confirm the results, it is presented an example,
which is solved theoretically and computationally.
Finally it is planed a problem for an arbitrary number of species and it is found a
movement constant. Additionally it is deduced the Liouville’s theorem of the
conservation of the density of individuals in the space of phases.
2. Volterra’s model
To facilitate the analysis, Volterra represents the problem schematically and
studies only the intrinsic phenomenon due to the natality or fertility of the interacting
species.
2
He analyzes two special cases, one is about two isolated species in the same
habitat that compete for the same nutritional resource, and the other is the case of two
species, one seized for the other, where the prey is limited by the predator. These two
cases show different behaviors, in the first the number of individuals of each species
tends asymptotically to one value and in the second case it is obtained the existence of a
periodical fluctuation, under certain conditions, of the two species.
Volterra assumes that the species growth is realized continually and that the
studied species are not affected by the other species, therefore their natality and
mortality coefficients remain constant. In this way, the N density of individuals changes
according to the equation:
dN
 n N m N  (nm)N
dt
( 2 1)
where t is the time, n and m are the natality and mortality coefficients respectively, and
both are constant.
Defining the growth factor as c = n - m, (2-1) is transformed in:
dN
cN
dt
(22)
N  N0 exp( c t )
(23)
and admits solution with the form:
where N0 is the density of individuals for t=0.
If the growth coefficient is positive, the number of individuals of the species
tends to increase and if it is negative tends to disappear.
See now how is transformed the equation (2-2) for the two types of interactions
we mentioned.
A. Two species competing for the same resource: suppose we have two
species in the same habitat and as before N1 and N2 represent the density of individuals
of each species and c1 and c2 the growth index. If the common resource is big enough,
the equations that represent the problem are:
dN1
dN 2
 c1 N1 ;
 c2 N 2
dt
dt
(24)
with c1 and c2 higher than zero. But as the number of individuals of the two species
grows, the available resource for each species falls. Then have h1 N1 as the decrease of
3
the resource due to the first species and h2 N2 the one due to the second species,
therefore the resource drops by h1 N1+h2 N2. This fall affects to each species in different
ways, we can put as growth coefficient due to the interaction to
c1'  c1  a ( h1 N1  h2 N 2 )
c2'  c2  b ( h1 N1  h2 N 2 )
(25)
where a and b are positive parameters that indicate how the fall affects to each species.
The equations system which represents the problem is
dN1
 ( c1  a ( h1 N1  h2 N 2 )) N1
dt
and
dN 2
 ( c2  b ( h1 N1  h2 N 2 )) N 2
dt
( 2 6 )
and the solution of the system is
N1b
 C exp( c1 b  c2 a ) t
N 2a
( 2 7 )
where C is an integration constant. And by the analysis of different cases, it is obtained
that the two species tend always asymptotically to one value.
B. Two species, one food to the other: Be N1 and N2 the density of
individuals of the two species. If the second species is not present, the first one has a
growth coefficient higher than zero (prey). If we suppose the second species is alone,
that goes to the extinction because does not has food, then it has a growth index lower
than zero (predator). Be c1 and c2 the growth coefficients of each species. Therefore, as
each species is alone, the equations are
dN1
dN 2
 c1 N1 ;
 c2 N 2
dt
dt
(28)
where we have taken c1 and c2 higher than zero. But if we consider the interaction
between the two species, we see that because of the presence of the second, the first one
will see reduced its growth coefficient in a proportional way to the number of
individuals of the second species, so
c1'  c1  g N2
(29)
4
inversely the presence of the first species increases the growth coefficient of the second
one, and it is given by
 c2'  c2  f N1
( 2  10 )
then the differential equations system which represents the problem is
dN1
 ( c1  g N 2 ) N1
dt
dN 2
 ( c2  f N 1 ) N 2
dt
( 2  11 )
where all the coefficients are positive. Here g measures the susceptibility of the species
1 to the predation and f the predatory ability of the species 2.
We saw that to the previous case, the solution was of asymptotic type, depending
on the growth coefficients and the constants h1, h2, a and b. But by the case of preypredator, the solution of (2-11), called Lotka-Volterra‘s differential equations system,
that was reached by Volterra, expresses a functional relation between the variables N1
and N2 that is
 n1 


exp(
n
)
1 

c2
 n2 

 C 
exp(
c
)
2 

 c1
( 2  12 )
where
N
 f  1
g
n1    ; n2   
 c2 
 c1 
N2
For the relation (2-12), it is found the existence of the cyclical variation between
n1 and n2 that, except the constants, it is the same existent fluctuation between the
populations of each species N1 and N2.
3. Interaction between four species
Following the same process as Volterra, we analyze a particular case of
interaction between four species with one absolute predator and other which is relative
and two absolute preys.
The equations system which represents the problem is
dx
 ( c1  h x  j y  a z )x
dt
5
dy
 ( c2  k x  l y  b z ) y
dt
dz
 ( c3  c x  e y  m z  f u ) z
dt
du
 ( c4  g z  n u ) u
dt
where x and y are the absolute preys since that they serve of food to z and u, z is a
relative predator because it is fed of x and y, and at the same time it is a prey of u, and
finally u is an absolute predator. The ci are the growth coefficients in the equations. The
coefficients which are preceded by a minus sign represent the susceptibility to the
predation with regard to the species that accompanies it and those of positive sign can
be interpreted as the predator’s ability with the species.
If we consider the food for what the two first species compete is unlimited, that
is to say the quantity do not change appreciable with the number of individuals of x and
y, and supposing that there is not competition between the same species individuals, the
coefficients: h, j, k, l, m and n are zero, then the equations system is reduced to
dx
 ( c1  a z ) x
dt
( 3 1)
dy
 ( c2  b z ) y
dt
(32)
dz
 ( c3  c x  e y  f u ) z
dt
(33)
du
 ( c4  g z ) u
dt
(34)
We will solve the equations system, for that we multiply (3-1) by
by
b
and (3-2)
x
a
and we obtain
y
b
d ln x
 ( b c1  b u z )
dt
(35)
a
d ln y
 ( a c2  a b z )
dt
(36 )
and
and subtracting (3-6) from (3-5), we have
6
b
d ln x
d ln y
a
 c1 b  c2 a
dt
dt
( 3 7 )
that we can write it as
d (ln x b  ln y a )
 c1 b  c2 a
dt
(38)
 xb 
ln  a   ( c1 b  c2 a ) t  K1
y 
(39)
and by integration, we have
where K1 is an integration constant.
If we impose the condition c1 b - c2 a = 0 to the problem we find the existent
relation between the two first species, from the previous relation we have that
b
c2 a
c1
( 3  10 )
and (3-9) is reduced to
xb
 exp( K1 )
ya
or what is the same, taking into account (3-10)
xy
c1
c2
exp(
K1 c1
)
a c2
( 3  11 )
from this equation we can obtain the first conclusion: “The variation of x and y have the
same phase.”
We will look for a relation between y and u, for that we will multiply (3-2) by
g
b
and (3-4) by
y
y
g
d ln y
 c2 g  g b z
dt
( 3  12 )
b
d ln u
 c4 b  g b z
dt
( 3  13 )
adding (3-12) and (3-13) we obtain
g
d ln y
d ln u
b
 c2 g  c4 b
dt
dt
( 3  14 )
by integration, after to write it in the same way that in the previous case, we have
7
ln y g ub  ( c2 g  c4 b ) t  K2
( 3  15 )
soliciting that c2 g - c4 b = 0, the equation (3-15), taking into account the previous
relation, is reduced to:
 c4
uy
c2
K c 
exp  2 4 
 c2 g 
( 3  16 )
replacing now the (3-11) and (3-16), we simplify the four differential equations system
in three systems of two equations, which have analogous solutions. We will work in the
phase space y, z but it is possible to do the same in the space of phases x, z or u, z.
The equation (3-3) results
c1
K c 
K c 
dz
 ( c3  c exp  1 1  y c2  e y  f exp  2 4  y
dt
 a c2 
 g c2 
 c4
c2
) z
( 3  17 )
with the equation (3-2) it forms a two equations system.
It is observed that this system is similar to (2-11), therefore can be considered as
a generalization of the Lotka-Volterra’s equations for two species. The two additive
terms appear in the second equations due to the others two intervening species.
To find the solution of the system formed by (3-2) and (3-17), call
p1 
c1
c2
p2 
c 
r  3
e
p1
c4
c2
K c 
c
exp  1 1 
c3
 a c2 
and
c 
s 3
e
 p2
K c 
f
exp  2 4 
c3
 g c2 
and defining the new variables Y and Z in the following way
Y 
Z
e
y
c3
( 3  17' )
b
z
c2
we can write the equations system in such a way that
dY
 c2 ( 1  Z ) Y
dt
( 3  18 )
8
dZ
 c3 ( 1  Y  r Y p1  s Y  p2 ) Z
dt
( 3  19 )
Multiplying (3-18) by c3 ( r Y p1 1  s Y  p2 1 ) and (3-19) by c2, and adding the
two equations, we obtain
c3 ( r Y p1 1  s Y  p2 1 )
dY
dZ
 c2
 c2 c3 ( r Y p1  s Y  p2  Z  Z Y ) ( 3  20 )
dt
dt
If we multiply (3-18) by c3 and (3-19) by
c2
, subtracting the previous one it
Z
results
c2 dZ
dY
 c3
 c2 c3 ( 1  r Y p1  s Y  p2  Z Y )
Z dt
dt
( 3  21 )
If we order the equations (3-20) and (3-21) it is obtained that the second terms
are equals, then we can put
c3 ( r Y p1 1  s Y  p2 1 )
dY
dZ c2 dZ
dY
 c2

 c3
 c2 c3 ( 1  Z ) ( 3  22 )
dt
dt Z dt
dt
and replacing (1-Z) from the (3-18) equation, we have
c3 ( r Y p1 1  s Y  p2 1 )
dY
d ln Y
dY
d ln Z
dZ
 c3
 c3
 c2
 c2
dt
dt
dt
dt
dt
( 3  23 )
that we can write as follows
d(
c3 r p1 c3 s  p2
Y 
Y
 c3 ln Y  c3 Y  c2 Z  c2 ln Z )
p1
p2
0
dt
and by integration
Z c1 exp( c2 Z ) 

1  c3
r
s  p2 

Y
exp c3 Y  Y p1 
Y
C
p1
p2


( 3  24 )
where C is an integration constant.
As we will see in the next section, this function represents a cycle in the space of
phases Y, Z, and if we compare it with the equation (2-12) that was obtained by
Volterra, we can consider to (3-24) as a generalization of the two species problem. The
 r

s
term: exp Y p1  Y  p2  can be considered as a perturbation due to the presence of
p2
 p1

two new species, which is not considered by Volterra, in the solution of prey-predator
problem.
9
4. Cycle deduction
We will show that the equation (3-24) represents a cycle in the space of phases
Y, Z. For that, equalize this equation to a new variable  :
  Z c1 exp( c2 Z ) 

1  c3
r
s  p2 

Y exp c3 Y  Y p1 
Y
C
p
p
1
2


( 4 1)
and draw  vs. Z, that it was done in the drawing on the left of the Figure 1, which has
a maximum in Z=1 with the value of   exp( c2 ) , that coincides with that was
obtained by Volterra from the equation (2-12).
In the same way, on the right of the Figure 1, it was drawn the full line  vs. Y.
This graphic presents a minimum to the value of Y equal to the Y0 radix, that the
equation  1  Y  rY p1  sY  p2 has in the interval (0, 1). Besides, it was drawn in
traces lines the graphic that corresponds to Y for the equation (2-12), that permit us to
observe how the solution is altered when a perturbation term appears due to the other
two species.
By the two graphics of the Figure 1, we can obtain the curve which corresponds
to the space of the phases. For that we operate in the following way
The right line   exp( c2 ) cuts to the second graphic into two points, A2 and
B2, which correspond to the values a2 and b2 for Y, respectively.
In the same way, the parallel right to the abscissa axis that passes through the
minimum of the graphic which is located on the right, determines the points A1 and B1
in the one that is on the left. The values a1 and b2 for Z correspond to the last one.
In the space of the phases, if we take into account the previous analysis we
obtain the following points which are indicated in the Figure 2
R1 cuando Y  Y0 and Z  a1
S1 cuando Y  Y0 and Z  b1
R2 cuando Z  1 and Y  a2
S 2 cuando Z  1 and Y  b2
these points are which determine the close points of the cycle, if we take intermediate
rights between the minimum and the maximum of  , we obtain the corresponding
points to complete the graphic of the Figure 2.
10
In that way, in the space of the phases we find the cyclical fluctuation we had
predicted.
In the Figure 2, the traces graphic marks the result that was obtained by Volterra
with two species for the prey-predator case.
Using the same procedure, we can prove the existence of a cycle in the space of
phases (X, Z) and (U, Z).
5. Calculation and estimation of the period
Once the existence of the cycle was proved, it is important and very interesting
to try to calculate the period of the fluctuation around the equilibrium point.
Using the first equality of the equation (4-1), we conclude that
ln   c2 (ln Z  Z )
( 5 1)
and by derivation the last equation respect to the time
1 d
1
 dZ
 c2   1
 dt
Z
 dt
and replacing
(52)
dZ
for its equivalent of the equation (3-19), it is obtained
dt
1 d
1

 c2 c3   1 ( 1  Y  r Y p1  s Y  p2 ) Z
 dt
Z

(53)
and clearing away dt
dt 
d
c2 c3  ( 1  Z ) ( 1  Y  r Y p1  s Y  p2 )
(54)
To calculate the period we must integrate the equation (5-4), for that we will use
as guide the Figure 2.
For the calculation of the integral, following the same procedure as Volterra, we
can divide the total trajectory into four segments: R2 S1, S1 S2, S2 R1 and R1 R2.
The four points R1, R2, S1 and S2 are singular, according to the equation (5-4).
Then adding the four segments contribution, we will calculate the period.
First, consider small fluctuations for Z around the equilibrium point, which
corresponds to the maximum of the curve on the left of the Figure 1.
That is to say
11
Z  1  V1
(55)
replacing this equation in the first equality of (4-1), we have that
1
c2
e
 ( 1  V1 ) exp( V1 )
and developing the exponential function we have
1 e 
1
c2


V2 V3
 ( 1  V1 )  1  V1  1  1  
2'
3'


and operating we get
1 e 
1
c2
 V12
1

V
V2
  2 1  3 1  
 2'

3'
4'


and from the last equation we conclude that
1  Z  V1  1  e 
1
c2
1
S ( V1 )
(5 6 )
where we called
S ( V1 ) 
1
V
V2
 2 1  3 1 
2'
3'
4'
We will do the same analysis for Y, named
Y  1  V2
( 5 7 )
then
Y  r Y p1  s Y  p2  ( 1  V2 )  r ( 1  V2 ) p1  s ( 1  V2 ) p2
and developing that conveniently it is obtained
V22
V2
 s  s p2 V2  p2 ( p2  1 ) s 2  
2
2
2
2
regrouping and clearing away the terms bigger than V2 since that we said V2 is a small
Y  r Y p1  s Y  p2  1  V2  r  p1 r V2  r p1 ( p1  1 )
perturbation, we have
 1  Y  r Y p1  s Y  p2  ( r  s )  V2 ( 1  p1 r  p2 s )  
(58)
Taking the second equality of (4-1), we have that
( C )
1
c3

Y
 r

s  p2
exp  Y p1 
Y
 Y 
p2
 p1

(59)
12
using again (5-7) and calling
a
r
s
and b 
p1
p2
the equation (5-9) can be written as
(C )
1
c3

1  V2
exp 1  V2  a ( 1  V2 ) p1  b ( 1  V2 ) p2


developing the binomials, regrouping and clearing away the terms bigger than V22 , the
previous equation is reduced to
1
c3
( 1  V2 )
exp(V2 ( a p1  1  b p2 )   )
( C  ) exp( 1  a  b ) 
( 5  10 )
calling
K C
1
c3
exp( 1  a  b )
and subtracting (5-10) from 1, we have
1 K 
1
c3
 1  ( 1  V2 ) exp( V2 ( a p1  1  b p2 )   )
developing the exponential function, the previous equation is
1 K 
1
c3
 1  ( 1  V2 ) ( 1  V2 ( a p1  1  b p2 )   )
operating and clearing away some terms, we obtain
1 K 
1
c3
 V2 ( a p1  b p2 )
( 5  11 )
clearing away V2 we have
V2 
1
c3
1 K 
a p1  b p2
( 5  12 )
but taking into account the meaning of a and b, V2 results
V2 
1  K
rs
1
c3
( 5  12 )
Replacing ( 5-12) in (5-8) we have
13
 1  Y  r Y p1  s Y  p2
1

 1  K  c3
 (r  s) 
 rs



 ( 1  p1 r  p2 s )   ( 5  13 )


Replacing (5-6) and (5-13) in the equation (5-4), we obtain the following
expression which gives us approximately the period
1
1
T 
4
2

1
c2 c3  1  e 
1
c2
12
d
2
1 



c
1  K  3 


( 1  r p1  s p2 )
( r  s )  
 rs 






where T is the period, 1 and  2 represent the coordinates of the two subsequent
singular points.
Naming
B  rs
and distributing and ordering the terms, the previous integral is
1
12
d
2
2
1
T 
1
4
c2 c3  1  e 
1
c2
1
1
1 

c
c
c
3
3

K
r p1 r p1 K 
s p2 s p2 K  3 
B







B
B
B
B
B




( 5  14 )
defining
P  B
r p1 s p2

B
B
and
G
K  r p1  s p2
B
the integral (5-14) is reduced to
14
1
12
d
2
1
2
T 
1
1 
1 
4
c3 
c2 
c2 c3  1  e   P  G  


( 5  15 )
If we realize the following change of variables
t  1 e
1
c2
the integral (5-15) takes the form
 4
T  
 2c3




dt
t2
t
( 5  16 )
c2
 c2 



c3
t ( 1  t )  P  G e ( 1  t ) c3 


1
If we make a new change of variables and call
u t
the integral results
 8
T  
 2c3




du
u2
u
1
c
1 2
c3
2
p ( 1  u2 )  G ( 1  u )
( 5  17 )
c2
e c3
The integral (5-17) is a general expression that gives us a period with a good
approximation.
To evaluate this integral we compute the particular case where c2 is equal to c3.
With that condition the integral remains
 8 

T  

2
c
3

u2
u
1
du
(  Pu  P  G e )
2
And the solution to the last integral is the following
u
 8 

T  

2
c
3

2
( 4 P ( P  G
1
2
e)

 2
2Pu


arctg 
1
 ( 4 P )( P  G E ) 2 

 u1
15
6. Medium values
To the problem of four species that we have developed, we find as conclusion
that the number of individuals of each species oscillates periodically around the
equilibrium point.
In this point we will calculate the respective medium values of the populations.
Consider firstly, the first equation of the system, the (3-18), from that we can see that
d ln Y
 c2 ( 1  Z )
dt
(6 1)
by integration of the equation between the times t1 and t2
t2
Y 
ln  2   c2 ( t1  t2 )   c2 Z dt
t1
 Y1 
(6  2 )
If t2  t1  T , the oscillation period, so Y2  Y1 and we can write the integral
( 6  2 ) using the relations (3-17’) as
c2 1

b T
T
0
(6 3)
z dt
As we see, the last equation gives the medium value of the predator’s number of
the individuals, z.
In the same way, if we consider the equation (3-2) and operate in the same form,
we have
d ln Z
 c3  c x  e y  f u
dt
(6 4 )
and by integration
t2
t2
t2
t2
z 
ln  2   c3  dt  c  x dt  e  y dt  f  u dt
t1
t1
t1
t1
 z1 
and again if t2  t1  T this integral results
c3 
T
1  T
 c 0 x dt  e 0 y dt  f
T 
T
0
u dt 

(6 5 )
and similar relations are obtained for the other two species.
16
7. Example
We will consider in this point an example for the four species case that we
developed early.
In the example, the constants take the following values
a1
c1  1
c2  2
c3  2
c4  2
b  0.5
c1
e  0.5
f 1
g 1
And considering the following initial values
x( 0 )  1
y(0)  2
z(0)  0.2
u(0)  0.5
using these values, the integration constants result
K1  0.6931
K 2  0.6466
r2
s  0.1139
In the graphic of the Figure 3, it is shown the variation of the number of the
individuals, except constants, for the four species respect to the time.
The graphics of the Figure 1 were obtained with the same values, as well as the
cycle that we showed in the Figure 2.
To obtain all of the graphics it was solved computationally the differential
equations system and it was proposed as solution a potencies series in t for each species.
From that we obtained a formula of recurrence.
8. Generalization to the n-species problem
The equations proposed by Volterra to describe the behavior in the time of the
interaction of n-biological species, having populations N1, N2, … Nn, are
dN r
1
 cr N r 
dt
Br
 asr N r N s
( 8 1)
s
The first term on the right indicate us how a particular species is propagated in
the time when is isolated. It gives a solution of exponential type, increasing or
decreasing, depending on the sign of cr.
17
The remaining term expresses the interaction of the species r with the others s,
the growth or drop of the Nr per unit of time is proportional to the number of meetings
per second between r and s, being measured by the product NrNs. If in the meeting r
wins, s must lose, that indicates the parameters are asymmetric, that is to say
ars   asr . The positive quantities Br1 are the Volterra’s equivalent numbers, such as
in a r, binary meeting s, the reason of the number of won (or lost) s per second, to the
number of lost (or won) r per second is
Bs1
.
Br1
An interesting particular case occurs in the biological association steady state,
that is when
dN r
 0 , where each species has the value qr which is defined by
dt
cr Br   asr qs  0
(82)
s
We assume the equation has a unique solution with all qr that are positive. The
solution is possible only if all of cr do not have the same sign.
Volterra showed that Nr are variables between two finite positive limits, all of
them fluctuate without decay and their averages are the values qr and are independent of
the initial values.
Also we can write the equation (8-2) in the following way
Br dN r
 cr Br   asr N s
N r dt
s
(83)
where we introduced the expression cr Br given by the equation (8-2) and the following
new variables
N 
Vr  ln  r 
 qr 
N r  qr eVr
N 
Vs  ln  s 
 qs 
then the equations of movement are
Br Vr   asr qs ( eVs  1 )
(84)
s
dV
with V 
. We will use these equations.
dt
An omission of the previous equations is the auto-interaction term arr N r2 ,
because the treatment would have been very difficult if it had been considered.
18
9. Microcanon assembly
The purpose to introduce the variables Vr is to assure the fulfillment of Liuville’s
theorem. Consider a big number of biological associations assemblies, each one with the
same characteristics and controlled by the same equations system (8-4), but having a
whole variety of initial values Vr.
In the cartesian space Vr (space of phases), the configuration of each assembly is
represented by one point and the assembly by one assembly of points. The points are
propagated in the space of phases according to the equation (8-4).
When we take a big enough quantity of points, we constitute a fluid of density
 ( V1 ,V2 , ,Vn ) of the points ( V1 ,V2 ,,Vn ) and velocity v  ( V1 ,V2 , ,Vn ) to those
points. Therefore if neither the fluid is increased nor is destroyed (there exist neither
sources nor sewers), the hydrodynamic continuity equation must be fulfilled

 (  Vr )

0.
t
Vr
Then


Vr
  Vr
 
0
t
Vr
Vr
but according to (8-4), the last sum is equal to zero.
This is the Liuville’s theorem of the conservation of density in the space of
phases.
Now if we multiply (8-4) by qr ( eVr  1 ) and we add over r, we have
Br Vr
 qr ( eV
r
 1 )   asr qs qr ( eVr  1 ) ( eVs  1 )  0
r
s
r
Then
G   Tr ( eVr  1 )  c t e
( 9 1)
r
with Tr  Br qr
The equation (9-1) is the universal (one to one) movement constant for the
system. It is a sum of terms that specify each species separately. This is an advantage
because it permits us to talk in the sense of system “components” from the mechanicstatistic point of view.
19
A surface of constants G in the space of phases has all the mathematic properties
that we need, it comprehends a determinate finite region and it is enough smooth.
Remember that an assembly in statistic equilibrium (steady assemblies) is that
for

 0.
t
The assembly properties are the same for all the times.
The medium value of any function f ( V1 ,V2 ,,Vn ) is defined by
f 
  f dT
  dT
(92)
where the integral is extended all over the space of phases and is the density of
probability.
Now we must introduce the “a priori” principle of probabilities equality, that is
to say to all the possible copies of the system that are compatible with the information
we have about it, we assign them the same probability.
Suppose that we only know of a biological association that G is a constant of
value G0. Then in equal regions over the surface G = G0 the probability is the same, we
choose
  0  ( G  G0 )
where  is the Dirac’s delta function, and 0 is a not important constant.
That defines a Gibbs’ microcanon assembly.
We represent an element of volume dr as dS.dn, which is the area element over
the surface of constant G by the increase in the length in the normal, the last one can be
written as
dG
, the equation (9-2) can be written as
G
G
G
f 
dS dG
 ( G  G0 ) G
 f ( G  G0 ) dS
(93)
and if we integrate respect to dG, we obtain
20
f 
G
f
0
G
0
dS
G
dS
G
(94)
as an application and example, it is interesting to calculate the average of


N

G  Tr ( eVr  1 )

 Tr  r  1
Vr
Vr
 qr

(95)
to make the calculation, observe that
G  
G
V̂r  G û
Vr
and
G
Vr
n̂  V̂r 
G
then we can write
Vr
G dS
 Vr ( n̂  V̂r ) dS  n̂  V̂r dS
Vr G
where Vr denotes the vector that corresponds to the point ( 0 , ..0,Vr , 0 ..,0) .
With these substitutions and calling 0 to the denominator of (9-4), the equation
(9-5) results
G
1
1

n̂  V̂r dS 
div V̂r dr  0

Vr 0
0 
(9 6 )
which indicates that the medium value of Nr is qr.
10. Conclusion
According to that was developed in the present study, we found a complete
solution for a particular case of four competitive species. We can add that apart from
this important result, and as well as it is followed a similar analysis to the Volterra’s one
in his model for two species, it can be concluded that also it was found a generalization
to his problem, seeing that it is clear from the equations (3-18) and (3-19) it is possible
to solve the problem prey-predator, aggregating a perturbation term in one of them. This
21
term, for example, could be a potencies series, without changing the method that was
developed in the study to solve the problem. That offers the possibility to attack the
problem if we consider the effects that have the influence of, for example, other species,
the man, the climatic reasons, etc.
In spite of the competitive treatment between species, there exist numerous
examples of assemblies which contain a determinate number of elements, whose
influence among them is competitive. The examples can be solved in the same way we
solved the problem in this study. The components of nervous system, the elemental
problems of fluids excitation, chemical reactive components in the atmosphere, physicchemical reactions, etc., are examples of these assemblies.
11. Bibliography
VOLTERRA VITO: Variazioni e fluctuazioni del numero d’individui da specie
animali conviveni. R. Comitato Talassografico Italiana, Memoria
121: 1-142. (1947).
SCUDO F. M. and ZIEGLER J. R.: The Golden age of Theoretical Ecology: 19231940. Lecture Notes in Biomathematics Nº 22. Springer-Verlag.
(1978).
GOEL N. S., MAITRA S. C. and MONTROLL E. W.: On Volterra and other Nonlinear Models of Interacting Population. Rev. Mod. Phis. Vol. 43.
M2. Part 1. 231-276. (1971).
22