A Theoretical Study of Biological Lotka-Volterra Ecological Model Using Comprehensive Thermodynamic Theory of Stability of Irreversible Processes (Cttsip)

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International Journal of Knowledge Engineering ISSN: 0976-5816 & E-ISSN: 0976-5824, Volume 3, Issue 1, 2012, pp.-91-94. Available online at http://www.bioinfo.in/contents.php?id=40

A THEORETICAL STUDY OF BIOLOGICAL LOTKA-VOLTERRA ECOLOGICAL MODEL USING COMPREHENSIVE THERMODYNAMIC THEORY OF STABILITY OF IRREVERSIBLE PROCESSES (CTTSIP)

RAWAT S.G.1, BHALEKAR A.A.2 AND TANGDE V.M.3*

1Department of Applied Chemistry, Priyadarshini College of Engineering, Nagpur, MS, India. 2Department of Chemistry, Rashtrasant Tukadoji Maharaj Nagpur University, Nagpur, MS, India. 3Department of Applied Chemistry, Smt. Bhagwati Chaturvedi College of Engineering, Nagpur, MS, India. *Corresponding Author: Email- [email protected]

Received: February 28, 2012; Accepted: March 06, 2012

Abstract- The dynamical relationship between predator and prey (Lotka-Volterra model or host-parasitoid system)[1] is one of the dominant themes in ecology. It was observed from the population data that interaction between a pair of predator-prey influences the population growth of both the species. This paper presents the study of thermodynamic stability of periodic Lotka-Volterra system against Prey population perturbation. The thermodynamic stability of representative model of Lotka-Volterra ecosystem has been investigated using proposed thermodynamic Lyapunov function in CTTSIP [2, 3, 4, 5] which follows the steps of Lyapunov's second method (also termed as direct method) of stability of motion[6, 7]. The thermodynamic Lyapunov function, used herein is the excess rate of entropy production in thermodynamic perturbation space that conforms well with the dictates of second law of thermodynamics[8, 9]. Moreover, present study reveals the regions of stability, asymptotic stability and instability. Keywords- Lotka-Volterra model, perturbation, predator, prey, Lyapunov function, irreversible thermodynamics

Citation: Rawat S.G., Bhalekar A.A. and Tangde V.M. (2012) A Theoretical Study of Biological Lotka-Volterra Ecological Model Using Com- prehensive Thermodynamic Theory of Stability of Irreversible Processes (CTTSIP). International Journal of Knowledge Engineering, ISSN: 0976-5816 & E-ISSN: 0976-5824, Volume 3, Issue 1, pp.-91-94.

Copyright: Copyright©2012 Rawat S.G., et al. This is an open-access article distributed under the terms of the Creative Commons Attribu- tion License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cred- ited.

Introduction an appropriate thermodynamic space using the entropy source Lyapunov's second method of stability of motion [6, 7, 10, 11, 12] strength function, the latter is a positive definite quantity as per has a generality element, which is very much akin to that of ther- the second law of thermodynamics that appears in Clausius- modynamics. In this direct method there is no need to rigorously Duhem inequality[8, 9, 15], that reads as, solve the differential equation of motion. Indeed, this course is ds adopted in Lyapunov's first method [6, 10, 11, 12], wherein one rs+divJ ss = ³ 0 tackles every case individually. This then allows the investigator to dt (1) map the regions of stability and instability (if any). In essence,

Lyapunov's second method involves effecting of a sufficiently r J s small perturbations and then observing the response in the corre- where, is the mass density, s is per unit mass entropy, sponding perturbation space. Previously, the stability aspects of s s some elementary chemical reactions including autocatalytic ones is the entropy flux density, is the entropy source strength [13] and industrial sulphur dioxide oxidation [14] has already been and t is the time. investigated using this generalized framework of CTTSIP[5]. Here 0 in the present case we will be using “Comprehensive thermody- Yti () namic theory of stability of irreversible processes(CTTSIP)” which Let 's be the thermodynamic coordinates of the given uses the fabric of Lyapunov's direct method of stability of motion. Yti () CTTSIP[2, 3, 4, 5] defines a thermodynamic Lyapunov function in process on the unperturbed trajectory and 's be that on

International Journal of Knowledge Engineering ISSN: 0976-5816 & E-ISSN: 0976-5824, Volume 3, Issue 1, 2012 Bioinfo Publications 91

A Theoretical Study of Biological Lotka-Volterra Ecological Model Using Comprehensive Thermodynamic Theory of Stability of Irreversible Processes (CTTSIP)

 i ()t i the perturbed one. The perturbation coordinates are de- punov’s theory of stability of motion20-24 ’s need to be fined as, sufficiently small the higher order terms in the expansion of 0  (t ) ( Y ( t )  Y ( t )); ( i  1,2,  , n ) LtS () i i i (2) can be ignored and hence the expressions for it get and hence the equation of unperturbed trajectory is obtained as, restricted to,  0  0. 00 i 0  SS  L  (3) LS S   S >    i     i iiii     ()t (10) Thus i 's determine the thermodynamic perturbation space. 0 The constitutive equations of motion within the perturbation space LSi/   in general are described as , In Eq. (10) each is either positive or negative d L i f( , , ,  , t ) ( i  1,2,  n ). The total time derivative of S then reads as, dt i 111 (4) dL L L d S S S i LS  The thermodynamic Lyapunov function, is defined as, dt ti i dt (11) 0 LSSS( ( t ) ( t )) 0 0 (5) Si/0   On the other hand, if and hence   0 Where S and S are the corresponding entropy 0 LSi/0    source strengths. , and as stated above in view of the Obviously, Eq. (26) leads to the following two possibilities,  either requirement of Lyapunov’s theory of stability of motion i ’s L L( ,  ,  ,  , t )  0 need to be sufficiently small the third and higher order terms in the SS1 2 3 ( 6 ) Lt() or expansion of S can be ignored and hence the expressions LSS L(1 ,  2 ,  3 ,  , t )  0 L ( 7 ) for S , get reduced to,

0 and on the unperturbed trajectory we have, 2 0 1   S LS() S   S    i  j LSS L(0,0,0,  , t )  0 2  ( 8 ) ij, ij 0 L 2 S 1  L where, , is a differentiable function. In the generalized setup  S  ˆ 0 ij (12)  2 ij, ij of CTTSIP we expand the entropy source strength, S , and

L thermodynamic Lyapunov function, S , in a Taylor expansion in L The total time derivative of S then reads as, i 00 terms of the perturbation coordinates 's as, dL1  22     d  S S S i i  j     i   0 0 dt2  t           dt 1 2 ii j  j i  i j  L ()  0 sS     (13) S S S i i j ii2 i, j    i   j where we have used the following expression for the local time 0 3 1   LS S      derivative of , namely: 6       i j k i,, j k i j k 0 (9 2 LSS1   Thus, even the bilateral perturbations can also be treated by this  ij  tt2     0  ij, ij (14) Si/0   formalism. However, if and hence Thermodynamic stability is then determined by finding out the sign 0 dL/ dt LSi/0    S , and as per the requirement of Lya- of on the lines of Lyapunov’s direct method of stability

International Journal of Knowledge Engineering ISSN: 0976-5816 & E-ISSN: 0976-5824, Volume 3, Issue 1, 2012 Bioinfo Publications 92

Rawat S.G., Bhalekar A.A. and Tangde V.M.

of motion[6,7,11,12]. Thus the unperturbed trajectory is said to be dx=- x x stable if , Perturbation in Prey population ( 0 ) The Thermodynamic stability of above model is then analyzed dLS LS .0 using CTTSIP under the perturbation effect in Prey population ( dt (15) The asymptotic stability is obtained if the result is either, ). Let us consider a case when the concentra- dL x L 0, S    0, tion of prey is perturbed by a suffciently small amount say, S dt (16) dx . We then have by the definition of perturbation coordinate, dL S dx=- x x 0 ˜ 0 LS 0,   0, (20) Or dt (17) Now, we investigate the effect of such perturbation on the thermo-  dynamic stability of process below. Let us consider the population where, , is a strictly positive number that van- rate equation of prey as ishes only at the origin. The stability under constantly acting small disturbances, as per Malkin’s theorem [6], is obtained if in addition dx =-ax bxy dt LSi/   (21) to Eq. (16) or (17) each is finite. Following the track of CTTSIP let us directly move for deriving a

suitable thermodynamic Lyapunov function for the present case, System under investigation as In this study, we investigate the thermodynamic stability of Lotka- Volterra ecosystem model (Prey-Predator interaction). A L()=-a by0 d x In ecosystem, one species do not occur in isolation of another. s T They compete for resources, assists, excludes and kills one an- (22) other. In ecology, predation describes a biological interaction and on solving the steps through-out we finally reach to the total where a predator feeds on its prey. The predator-prey relationship L is important in maintaining the balance between the different spe- time derivative of s as cies. Without prey, there would be no predators and vice-versa. e. d (L ) A g. Cheetahs and gazelles, foxes and rabbits, lion and foxes, foxes s =-2 (a by02 ) d x and goat, birds and insects,goat and grass, etc. The Lotka- dt T Volterra Prey-Predator model involves coupled equations, one (23) which describes how the prey population changes and second Conclusion which describes how the predator population changes, namely: Analytical conclusions regarding the thermodynamic stability of above system against the perturbation in prey population have put forward in following different situations, namely: dx =-ax bxy dt 1. Perturbation is positive enough ( > 0) while product of (18) predator population with the predation rate coefficient and 0 dy (bg y< a) =-pxy cy is less than the intrinsic rate of prey population dt (19) 2. Perturbation is positive enough ( > 0) while product of The dynamics of the two populations depends on the parameters predator population with the predation rate coefficient a b p c is greater than the intrinsic rate of prey population , , , where, x (bg y0 > a) = prey population y dx < 0 = predator population 3. Perturbation is negative ( ) while product of predator population with the predation rate coefficient is less than the intrin- = Intrinsic rate of prey populations increase (bg y0 < a) = predation rate coefficient sic rate of prey population

= reproduction rate of predators(after eating preys) 4. Perturbation is negative ( ) while product of predator population with the predation rate coefficient is greater than the = death rate of predators

International Journal of Knowledge Engineering ISSN: 0976-5816 & E-ISSN: 0976-5824, Volume 3, Issue 1, 2012 Bioinfo Publications 93

A Theoretical Study of Biological Lotka-Volterra Ecological Model Using Comprehensive Thermodynamic Theory of Stability of Irreversible Processes (CTTSIP)

0 References (bg y> a) [1] Pistorius C.W.I. Utterback, James M. (1958) Lotka-Volterra intrinsic rate of prey population model for multi-mode technological interaction: modeling com- petition, symbiosis and predator prey modes. Ls For stability discussions based on prediction of the signs of [2] Bhalekar A.A. (2001) Far East J. Appl. Math., 5, 199-210. d (L ) [3] Bhalekar A.A. (2001) Far East J. Appl. Math., 5, 381-396. s [4] Bhalekar A.A. (2001) Far East J. Appl. Math., 5, 397-416. and dt , we now consider each case individually, as fol- [5] Bhalekar A.A. (2004) J. Indian Chem. Soc., 81 1119-1126. lows: [6] Malkin I.G.(1952) Theory of stability of motion, US Atomic Energy Commission [ACE-tr-3352 physics and mathematics]. 0 dx > 0 bg y< a [7] Malkin I.G. (1952) American Mathematical Society., 5. 1. Lyapunov function analysis when , [8] Callen H.B.(1960) Thermodynamics, Wiley. : [9] DeGroot S.R., Mazur P.(1962) Nonequilibrium thermodynam- Under these conditions equations (22) and (23) analytically leads ics. [10] Chetayev N.G.(1961) The stability of motion. [11] Elsgolts L.(1970) Di_erential equations and calculus of varia- to > 0 and > 0 showing the instable region of the tions. process. [12] LaSalle J., Lefschetz S. (1961) Stability by Lyapunov's direct method with applications. bg y0 > a 2. Lyapunov function analysis when , [13] Tangde V.M., Bhalekar A.A. and Venkataramani B. (2008) : Bull. Cal.,Math. Soc., 100(1), 47-66. Under these conditions equations (22) and (23) analytically leads [14] Tangde V.M., Bhalekar A.A. and Venkataramani B. (2007) Phys. Scr., 75, 460-466. [15] Haase R.(1969) Thermodynamics of irreversible processes, Addison-Wesley, Reading (MA). to < 0 and > 0 showing the stability of the process. 0 dx < 0 bg y< a 3. Lyapunov function analysis when , : Under these conditions equations (22) and (23) analytically leads

to < 0 and < 0 showing the instability of the process and finally

4. Lyapunov function analysis when , : Under these conditions equations (22) and (23) analytically leads

to > 0 and < 0 showing the stability of the process. Above discussion reveals a very fundamental aspect of the ecosystem that one species do not occur in isolation of another. With- out prey there would be no predators and vice-versa. If product of predator population with the predation rate coefficient is less than the intrinsic rate of prey population irrespective of perturbation is positive or negative, the ecosystem balance would get disturbed and prey-predator interaction collapse. Subsequently both the species of system will face adverse circumstances to survive and eventually lead to succumb. There are so many examples in the forehand we have seen so vital species especially in metro cities like sparrows, myphans and vultures disappeared existing.

International Journal of Knowledge Engineering ISSN: 0976-5816 & E-ISSN: 0976-5824, Volume 3, Issue 1, 2012 Bioinfo Publications 94