Based on the above, the model under
consideration is governed by the following set of nonlinear ordinary
differential equations.
(i) Equation for the growth
rate of first species (N1):
(1.1)
(ii) Equation for the growth
rate of second species (N2):
(1.2)
(iii) Equation for the growth
rate of third species (N3):
(1.3)
with the following notations:
Ni’s
are the strengths of the i th species.
ri’s
are the intrinsic growth rates of Ni’s; i=1,2,3.
θi’s are the death
coefficients of the three species.
α is
the rate of decrease of N1 due to inhibitions by the predator
N3.
α1 is
the rate of increase of N3 due to successful attacks on N1.
β is the rate of decrase of N2
due to inhibitions by the predator N3.
β1 is the rate of
increase of N3 due to successful attacks on N2.
Li’s are the carrying
capacities of Ni’s.
2. THE INTERIOR EQUILIBRIUM STATE:
The system under study has eight
equilibrium states and our focus is on the interior equilibrium point i.e., the
point at which all the three species exists, given by:
This state
exists only when,
and 
3. LOCAL STABILITY OF THE INTERIOR EQUILIBRIUM STATE:
Let 
= 
Where U = 
is the
perturbation over the equilibrium state. 
.
The basic equations are linearized to
obtain the equations for the perturbed state.
(3.1)
Where 
With the characteristic equation
(3.2)
Let
Using Routh-Hurwitz criteria, on
simplification, we have, 
,
and 
Hence the interior equilibrium point is
locally asymptotically stable.
4. GLOBAL
STABILITY:
Statement: The
interior equilibrium point is globally asymptotically stable.
Proof: Let 
be the
interior equilibrium point.
Consider the following Lyapunov function
for the interior equilibrium point:
(4.1)
Then calculate 
which is obtained
as follows
(4.2)
Substituting, 
, 
, 
We get
(4.3)
Hence the system is globally
asymptotically stable.
5. DYNAMICS OF THE STOCHASTIC MODEL:
By a Wiener process ξ(t), we
construct the stochastic differential equation whose solutions are continuous
sample paths of a wide class of Markov processes as follows:
Where σi, i=1,2,3
are real constants, ξti, (i=1,2,3);
independent from each other is a standard Wiener process[10].
Let 
, then the linearized stochastic
differential equations take the form:
(5.1)
Where 
and
Let W be the set 
Hence 
is a continuously differential function
with respect to t and twice continuously differentiable function with
respect to u.
With
reference to the book by Afanasiev et al[11], the following theorem holds:
and 
i,j=1, 2 and T means
transposition.
Theorem 1:- Suppose there exists
a function 
satisfying
the following inequalities:
then the
trivial solution of (5.1) is exponentially p-stable 
Theorem 2:- Suppose that 
, then the
zero solution of (5.1) is asymptotically mean square stable.
Proof:- Consider the Lyapunov
function 
(5.2)
Choose 
and 
Then we have,
< 0 (5.3)
Hence the zero solution of the
linearized stochastic differential equation is asymptotically mean square
stable, which proves the theorem.
6. NUMERICAL SIMULATIONS:
Case(i): r1=0.5;
r2=1.65; r3=5.5; L1=10; L2=20; L3=10;
α=0.01; α1=0.05; β=0.05; β1=0.05;
θ1=0.2; θ2=0.2; θ3=0.2.
Fig. 1a:
Fig. 1b:
Fig. 1c: Fig.
1d:
When the initial strengths of first
prey, second prey and predator are 10,12,15 respectively and the interaction
coefficient between second prey and the predator is 0.05, initially the
strength of the predator falls down to a certain stage very quickly and
stabilizes and continues to dominate first and second prey further. It is also
evident from fig. 1a and 1b that all the species are in asymptotic equilibrium
state which shows that the state is asymptotically stable. The stochastic
behavior of the model is shown in fig. 1c and 1d.
Case(ii): r1=0.5;
r2=1.65; r3=5.5; L1=10; L2=20; L3=10;
α=0.01; α1=0.05; β=0.095; β1=0.05;
θ1=0.2; θ2=0.2; θ3=0.2.
Fig. 2a: Fig.
2b:
Fig. 2c: Fig.
2d:
Fig. 2a and 2b
shows the variation in the second prey which decreases rapidly when compared to
the first prey at the beginning. It is increasing and stabilizing slowly after
some time period by a small change in the interaction coefficient β
from 0.05 to 0.095.
Case(iii): r1=0.5;
r2=1.65; r3=5.5; L1=10; L2=20; L3=10;
α=0.01; α1=0.05; β=0.15; β1=0.05;
θ1=0.2; θ2=0.2; θ3=0.2.
Fig. 3a: Fig.
3b:
Fig. 3c: Fig.
3d:
For β
≥ 0.15, the second prey is completely extinct and the system does not
exist.
Case(iv): r1=0.5;
r2=1.65; r3=5.5; L1=10; L2=20; L3=10;
α=0.025; α1=0.05; β=0.05; β1=0.05;
θ1=0.2; θ2=0.2; θ3=0.2.
Fig. 4a: Fig.
4b:
Fig. 4c: Fig. 4d:
Similarly, keeping β=0.05,
and changing the interaction coefficient α from 0.01 to 0.025, the above
graphs shows the variations again in all the three species in both the
deterministic and stochastic cases.
7.
CONCLUSIONS:
A three species ecological model with
prey-predation and neutralism is taken up for stability analysis and stochastic
response. As a particular case study, we have incorporated death coefficients
for all the three species and the effects are visible particularly in the
populations of the prey species since the survival of preys is mainly dependent
on their respective intrinsic growth rates. The interior equilibrium point is
identified for the model and its local stability is obtained using
Routh-Hurwitz criterion. The global stability of the system is established by
constructing a suitable Lyapunov function. Also the exponential and mean
square stability of the stochastic model corresponding to the system is
derived. With changes in the interaction coefficients α and β
the graphical analysis shows that the predator is benefitting from the preys in
spite of the alternate food resource available. Numerical simulations using
MATLAB in both the deterministic and stochastic cases further justifies that the
system is globally asymptotically stable.
8.
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Volterra
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WJ. Concepts of Mathematical Modelling. Mc Graw-Hill, 1985.
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JM. Integro-differential equations and Delay models in Population Dynamics, Lecture
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JD. Models for interacting populations. Chapter 3, pp. 79-118.
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(1.1)
(1.2)
(1.3)