(2.1)
with the following notation
(t) :
Population of the ith species at time “ t”, i = 1, 2, 3.
:
Natural growth rate of the species ,
i=1, 2, 3.
:
Rate of decrease of species due to insufficient resources, i= 1, 2, 3.
:
Rate of decrease of the third species due to attacks of ith species,
i= 1, 2.
Further
the variables 
are
non-negative and the model parameters 
are
assumed to be non negative constants.
3. EQUILIBRIUM POINT:
The equilibrium point is identified by solving
(3.1)
and 
(3.2)
The system under investigation has eight equilibrium points. Among these, the interior equilibrium point is
(3.3)
This
state would exists only when 
(3.4)
4. STABILITY OF THE INTERIOR EQUILIBRIUM POINTS:
To
examine the stability of the equilibrium state (
) we
consider a small perturbation 
such
that
,
,
(4.1)
After linearization we get
(4.2)
Where
A=
(4.3)
And
. (4.4)
The
characteristic equation for the system is
. (4.5)
The equilibrium state is stable when the roots of the equation (4.5) are negative if they are real or have negative real parts are negative if they are complex.
Linearized equations for the equilibrium state (That is co-existing state or normal steady state) are
(4.6)
The characteristic equation for the co-existing state or normal steady state is
(4.7)
And the Eigen values for the characteristic equation (4.7) are
(4.8)
Here clearly all the Eigen values of the characteristic equation (4.8) are negative. Hence the equilibrium state is asymptotically stable by using Routh-Hurwitz criterion.
The solutions of perturbed equations are
(4.9)
NUMERICAL EXAMPLE:
Let a1=0.03, a2=0.8, a3=0.8, u10=0.15, u20=0.4, α11= 0.4, α22= 0.5, α31= 0.06, α32= 0.05, α33= 0.8.
Fig (1.A) Fig (1.B)
Figures (1.A) and (1.B) shows that the linearized state is asymptotically stable
5. GLOBAL STABILITY:
Theorem: The Interior equilibriumState is Globally Asymptotically Stable.
Proof: Let us consider the Lapunov function for the interior equilibrium state is
(5.1)
Here
(5.2)
Differentiate equation (5.1) with respect to‘t’, we get
(5.3)
(5.4)
(5.5)
By the proper choice of
(5.6)
Substitute
and 
in (5.6),
we get
(5.7)
(5.8)
(5.9)
Since we have
(5.10)
Then
(5.11)
Substitute these all in equation (5.9), we get
(5.12)
(5.13)
(5.14)
(5.15)
Since
all
are
positive
Therefore the System is Globally Asymptotically Stable
6. NUMERICAL EXAMPLES:
EXAMPLE 1:
Let a1=0.03, a2=0.05, a3=0.08, α11=0.4, α22=0.5, α33=0.06, α31=0.06, α32=0.06, N10=5, N20=10, N30=15.
Fig (2.A) Fig (2.B)
In this figures2.A and2.B are trajectories and phase portrait in which the interacting coefficients of two species i.e., α31 and α32 are same (0.06) and the equilibrium point is (0.500812, 0.443787, 2.843224).
EXAMPLE 2:
Let a1=0.03, a2=0.05, a3=0.08, α11=0.4, α22=0.5, α33=0.06, α31=0.06, α32=0.6, N10=5, N20=10, N30=15.
Fig (6.2.A) Fig (6.2.B)
Figures 6.2.A and6.2.B represents where α31 is kept at same and α32 is increased (0.6) and the equilibrium point is (0.500812, 0.443787, 0.159545), which shows this case is more asymptotic stability than previous case when we increased the interacting coefficient between third species and first species then third species will be weekend .
EXAMPLE 3:
Let a1=0.03, a2=0.05, a3=0.08, α11=0.4, α22=0.5, α33=0.06, α31=0.6, α32=0.06, N10=5, N20=10, N30=15.
Fig (6.3.A) Fig (6.3.B)
Figures 6.3.A and6.3.B show that the deflections when α32 is increased to (0.6) and α31 is kept at same value (0.06) and the equilibrium point is (0.500812, 0.443787, 1.091639). That is when we increased the interacting coefficient between third species and second species then third species will be weekend.