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Global stability of two-species competing ecological model.

1. Preliminaries

The basic equations for growth rates of two competitive species [S.sub.1] & [S.sub.2] are given by

[d[N.sub.1]/dt] = [N.sub.1]([a.sub.1] - [a.sub.11][N.sub.1] - [a.sub.12][N.sub.2]}

[d[N.sub.2]/dt] = [N.sub.2]([a.sub.2] - [a.sub.22][N.sub.2] - [a.sub.21][N.sub.1]}

A. Notations

[N.sub.1], [N.sub.2]: Population sizes of species [S.sub.1] & [S.sub.2] respectively

[a.sub.1], [a.sub.2]: Natural growth rates

[a.sub.11], [a.sub.22]: Rates of decrease of species due to limitation of natural resources

[a.sub.12], [a.sub.21]: Inhibition coefficients (Rates of decrease of each because of inhibition of other)

All these community coefficients [a.sub.1] [a.sub.2], [a.sub.11], [a.sub.22], [a.sub.12], [a.sub.21] > 0

B. Equilibrium states

The competing system under investigation has the following four equilibrium states.

I. [bar.[N.sub.1] = 0, [bar.[N.sub.2]] = 0 (Fully washed out state)

II. II. [bar.[N.sub.1]] = 0, [bar.[N.sub.2]] = [a.sub.2]/[a.sub.22] ([S.sub.1] washed out state)

III. [bar.[N.sub.1]] = [a.sub.1]/[a.sub.11], [bar.[N.sub.2]] = 0 ([S.sub.2] washed out state)

IV. [bar.[N.sub.1]] = [[a.sub.1][a.sub.22] - [a.sub.2][a.sub.12]/[a.sub.11][a.sub.22] - [a.sub.12][a.sub.21]]; [bar.[N.sub.2]] = [[a.sub.2][a.sub.11] - [a.sub.1][a.sub.21]/[a.sub.11][a.sub.22] - [a.sub.12][a.sub.21]] Co-existent state)

This state is realizable when [[a.sub.12]/[a.sub.22]] < [[a.sub.1]/[a.sub.2]] < [[a.sub.11]/[a.sub.21]] co-existent state can also be referred as "Normal steady state".

2. Construction of Liapunov's Function for Global Stability in case of Co-existent equilibrium state of the Two-Species Competitive Ecosystem

By putting [N.sub.1] = [bar.[N.sub.1]] + [u.sub.1] and [N.sub.2] = [bar.[N.sub.2]] + [u.sub.2] Linearized Perturbed Basic Equations are

[d[u.sub.1]/dt] = [a.sub.11][bar.[N.sub.1][u.sub.1] - [a.sub.12][bar.[N.sub.2]][u.sub.2]

[d[u.sub.2]/dt] = -[a.sub.21][bar.[N.sub.2][u.sub.1] - [a.sub.22][bar.[N.sub.2]][u.sub.2]

The characteristic equation is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore the conditions for Liapunov's function are satisfied.

We now define E([u.sub.1], [u.sub.2]) = 1/2(a[u.sub.1.sup.2] + 2b[u.sub.1][u.sub.2] + c[u.sub.2.sup.2])

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

D = pq = {[a.sub.11][bar.[N.sub.1]] + [a.sub.22][N.sub.2]}{[a.sub.11][a.sub.22] + [a.sub.12][a.sub.21]}[bar.[N.sub.1]][bar.[N.sub.2]]

It is clear that D > 0 and a > 0. Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[therefore] (ac-[b.sup.2] [??] 0. So the function E([u.sub.1], [u.sub.2]) is positive definite.

By Substituting the values of a, b, and c from above equations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which is clearly negative definite

So E([u.sub.1], [u.sub.2]) is a Liapunov function for the linear system.

Next we prove that E([u.sub.1], [u.sub.2]) is also a Liapunov's function for the non linear system.

If [F.sub.1] and [F.sub.2] are defined as

[F.sub.1]([N.sub.1], [N.sub.2]) = [N.sub.1]{[a.sub.1] - [a.sub.11][N.sub.1] - [a.sub.12][N.sub.2]}, [F.sub.2] ([N.sub.1], [N.sub.2]) = [N.sub.2] {[a.sub.1] - [a.sub.22][N.sub.2] - [a.sub.21][N.sub.1]}

We have to show that [[partial derivative]E/[partial derivative]][u.sub.1]] [F.sub.1] + [[partial derivative]E/[partial derivative][u.sub.2]] is negative definite.

By putting [N.sub.1] = [bar.[N.sub.1]] + [u.sub.1] and [N.sub.2] = [bar.[N.sub.2]] + [u.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [f.sub.1]([u.sub.1], [u.sub.2]) = -[a.sub.11][u.sub.1.sup.2] - [a.sub.12][u.sub.1][u.sub.2]

Similarly [F.sub.2]([u.sub.1], [u.sub.2]) = [d[u.sub.2]/dt] = [-[a.sub.22][bar.[N.sub.2]][u.sub.2] - [[a.sub.21][bar.[N.sub.2]][u.sub.1] + [f.sub.2]([u.sub.1], [u.sub.2]])

where [f.sub.2]([u.sub.1], [u.sub.2]) = -[a.sub.22][u.sub.2.sup.2] + [a.sub.21][u.sub.1][u.sub.2]

We have [partial derivative]E/[partial derivative][u.sub.1] = a[u.sub.1] + b[u.sub.2] and [partial derivative]E/[partial derivative][u.sub.2] = b[u.sub.1] + c[u.sub.2]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

By introducing polar co-ordinates we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Denote largest of the numbers [absolute value of a], [absolute value of b], [absolute value of c] by M

Our assumptions and [absolute value of [f.sub.1]([u.sub.1], [u.sub.2])] < [r/6M] and [absolute value of [f.sub.2]([u.sub.1], [u.sub.2])] < [r/6M]

So, [[partial derivative]E/[partial derivative][u.sub.1]] [F.sub.1] + [[partial derivative]E/[partial derivative][u.sub.2]] [F.sub.2] < - [r.sup.2] + [4K[r.sup.2]/6M] = -[r.sup.2]/3 < 0 [for all] sufficiently small r > 0,

Thus E([u.sub.1], [u.sub.2]) is a positive definite function with the property that [[partial derivative]E/[partial derivative][u.sub.1]] [F.sub.1] + [[partial derivative]E/[partial derivative][u.sub.2]] [F.sub.2] is negative definite. [therefrore] The equilibrium point is "asymptotically stable".

References

[1] Bhaskara Rama Sarma .B & N. Ch. Pattabhiramacharyulu: 'Stability Analysis of a two-species competitive eco-system-, International Journal of Logic Based Intelligent Systems, Volume-2, No. 1, 2008,PP. 79-86.

[2] Kapur, J. N.: 'Mathematical Models in Biology and Medicine', Affiliated East-West, 1985.

[3] Meyer, W.J.: 'Concepts of mathematical modeling', Mc-Grawhill, 1985

[4] Simmons .G.F.: Differential equations with applications and historic notes McGraWHill Inc: Newyork, 1972

B. Bhaskara Rama Sarma (1) and N.Ch. Pattabhi Ramacharyulu (2)

(1) Director, BRS Classes, #2-284, Vivekananda Street, Hanumannagar, Ramavarappadu, Vijayawada, A.P.-521108, India

E-mail:bbramasarma@yahoo.co.in

(2) Formerly Faculty, Department of Mathematics & Humanities, N.I.T, Warangal, A.P.-506004, India
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Author:Sarma, B. Bhaskara Rama; Ramacharyulu, N.Ch. Pattabhi
Publication:Global Journal of Pure and Applied Mathematics
Article Type:Report
Geographic Code:9INDI
Date:Aug 1, 2010
Words:1128
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