- Email: [email protected]
Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems
MBS 7387
No. of Pages 4, Model 5G
6 August 2013 Mathematical Biosciences xxx (2013) xxx–xxx 1
Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 5 6
Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems
3 4 7
Q1
8
C.G. Chakrabarti, Koyel Ghosh ⇑ Department of Applied Mathematics, University of Calcutta, Kolkata 700009, India
9 10
a r t i c l e
1 2 2 5 13 14 15 16 17
i n f o
Article history: Received 31 August 2012 Received in revised form 18 July 2013 Accepted 19 July 2013 Available online xxxx
18 19 20 21 22 23 24
Keywords: Model ecosystem Community matrix Dynamical entropy Stability Complexity
a b s t r a c t In the present paper we have first introduced a measure of dynamical entropy of an ecosystem on the basis of the dynamical model of the system. The dynamical entropy which depends on the eigenvalues of the community matrix of the system leads to a consistent measure of complexity of the ecosystem to characterize the dynamical behaviours such as the stability, instability and periodicity around the stationary states of the system. We have illustrated the theory with some model ecosystems. Ó 2013 Published by Elsevier Inc.
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32 33 34
1. Introduction
35
An ecosystem consisting of a large number of interacting species may be considered as a complex dynamical system [1]. In recent years dynamical system model of complex ecological system has experienced explosive growth. Different mathematical ideas and techniques have been developed in the elucidation of different underlying biological and ecological processes. To understand a complex ecosystem we need some systematic methodology. The differential equations used to model ecosystems are, in general, non-linear. It is very difficult to find out the solution of the system of non-linear equations in closed form. It is customary to study the dynamical behaviours of such systems near the stationary states. The linearized form of the system of equations around a stationary state represents the local dynamical behavior of the system. The community matrix introduced by Levin in the linearization process plays a significant role in the determination of the qualitative nature of the nearby orbits or trajectories [2]. The community matrix represents the mathematical structure of the system near the stationary states [3,4]. The objective of the present paper is to study the dynamical behaviors associated with the community matrix from a different mathematical background. The contribution of the paper is two fold: Firstly, we wish to introduce a measure of dynamical entropy
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
⇑ Corresponding author. Tel.: +91 033 2350 8386/1397x450/451; fax: +91 033
Q2 Q1
2351 9755/2241 3222. E-mail addresses: [email protected] (C.G. Chakrabarti), koyelghosh [email protected] (K. Ghosh).
for the entropic characterization of the time-evolution of the ecosystem. The methodology of the derivation of the dynamical entropy is based on the entropy of a non-random square matrix. Secondly, we wish to study the importance of the dynamical entropy in the characterization of stability, instability and periodicity of the stationary states of the ecosystem with some illustrative model ecosystems. The interrelation between the concept of stability and complexity has been investigated.
57
2. Model ecosystem: dynamical model and entropy
65
We consider a multi-species ecosystem consisting of n species with population densities xi ðtÞ; ði ¼ 1; 2; . . . :; nÞ at any time t. Let us assume the system to be governed by the system of equations
66
x_i ¼ fi ðx1 ; x2 ; . . . :; xn ; aÞ; ði ¼ 1; 2; . . . ; nÞ
ð2:1Þ
where a is a constant parameter. The functions fi are assumed to be continuously differentiable in some open set X ¼ fxi jxi P 0; i ¼ 1; 2; . . . ; ng. The system of model Eq. (2.1) are, in general, non-linear and difficult to find out the solution in closed form. It is customary to study such a system close to some stationary state. Let x ¼ ðx1 ; x2 ; . . . ::; xn Þ be the stationary state of the system for a certain value (or a range of values) of the parameter a. We consider a small deviation about the stationary state: yi ðtÞ ¼ dxi ðtÞ ¼ xi ðtÞ xi ; ði ¼ 1; 2; . . . ; nÞ. Linearizing the system of Eq. (2.1) about the stationary state x ¼ ðx1 ; x2 ; . . . ::; xn Þ we have the system of linear equations
0025-5564/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.mbs.2013.07.016
Q1 Please cite this article in press as: C.G. Chakrabarti, K. Ghosh, Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems, Math. Biosci. (2013), http://dx.doi.org/10.1016/j.mbs.2013.07.016
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y_ i ðtÞ ¼
C.G. Chakrabarti, K. Ghosh / Mathematical Biosciences xxx (2013) xxx–xxx
n X @fi @xj
85
j¼1
86
or in matrix form
87 89
_ yðtÞ ¼ AyðtÞ
yj ðtÞ ði ¼ 1; 2; . . . ; nÞ ¼
x
n X aij ðx Þyj ðtÞ j¼1
ð2:3Þ
90
where the Jacobian
91 93
A ¼ ½aij ðx Þ
94
is the so-called community matrix with elements
95 97
aij ðx Þ ¼
@fi @xj
ð2:2Þ
ð2:4Þ i; j ¼ 1; 2; ::; n
ð2:5Þ
x
100
101 103
yðtÞ ¼ dxðtÞ ¼ eAt dxð0Þ
99
104 105 106 107 108 109 110 111 112 113
114 116
ð2:6Þ
where dxð0Þ is the initial deviation from the stationary state x . Let us now find the explicit form of the solution (2.6) in terms of the eigenvalues ðk1 ; k2 ; . . . :; kn Þ of the community matrix A. If the eigenvectors corresponding to the eigenvalues ðk1 ; k2 ; . . . :; kn Þ are linearly independent then the matrix A can be converted to the form of a diagonal matrix with diagonal elements same as ðk1 ; k2 ; . . . :; kn Þ. Under the mathematical conditions of linearly independence of the eigenvectors and the distinct eigenvalues ðk1 ; k2 ; . . . :; kn Þ, the explicit form of the solution (2.6) are given by Lakshmanan and Rajasekar [6] and Rosen [7]
dxi ðtÞ ¼ dxi ð0Þeki t ;
ði ¼ 1; 2; . . . ; nÞ
ð2:7Þ
126
All the basic criteria of stability, instability and periodicity of the system follow from the solution (2.7). Let us now try to develop an entropic theory of time-evolution of the ecosystem on the basis of the linearized model Eq. (2.3). Entropy plays a significant role in the study of evolution of a thermodynamic system [8]. Like in thermodynamics we need an expression of dynamical entropy for the study of time-evolution of the ecosystem described by the dynamical model Eq. (2.3). According to Eq. (2.3) the time-evolution of the system from the initial state to the current state is given by
127 129
dxðtÞ ¼ eAt dxð0Þ ¼ BðtÞdxð0Þ
117 118 119 120 121 122 123 124 125
ð2:8Þ
At
137
where BðtÞ ¼ e is the matrix of evolution dxð0Þ!dxðtÞ. The evolution matrix BðtÞ ¼ eAt characterizes the time-evolution of the system. To find out a dynamical entropy for the time-evolution dxð0Þ!dxðtÞ we need a measure of entropy of the non-probabilistic square matrix BðtÞ ¼ eAt . Following Jumarie the entropy (of order 1) of a n n square matrix A consistant with Shannon classical entropy and Von Neumann quantum entropy is given by Jumarie [9]
140
Pn jki j ln jki j Pn H1 ½A ¼ i¼1 i¼1 jki j
130 131 132 133 134 135 136
138
ð2:9Þ
145
148
Pn ki t i¼1 tki e H1 ½BðtÞ ¼ H1 ½eAt ¼ P n ki t i¼1 e
142 143 144
146
149 150
ð2:11Þ
151 152 153
154 156
The expression (2.11) can be written as
157
( ) n X d HðtÞ ¼ log eki t dt i¼1
158
ð2:12Þ 160
where the sum-function i¼1 eki t is the trace of the diagonal evolution matrix BðtÞ ¼ eAt . This is analogous to the canonical partitionfunction in statistical mechanics. The dynamical entropy HðtÞ is a real quantity inspite of the feasibility of complex or imaginary eigenvalues. Jumarie [9] has used the expression of dynamical entropy (2.12) to measure the complexity of a dynamical system. We shall consider both the terms dynamical entropy and dynamical complexity to be synonymous. In the next section we shall study the significance of the dynamical entropy (2.12) in the analysis of stability, instability and periodicity of some model ecosystems.
161
3. Model ecosystem: analysis of stability, periodicity and complexity
171
In this section we shall study the role of the measure of dynamical entropy (2.12) in the characterization of different dynamical behaviours such as stability, instability and periodicity etc. of the system around a stationary state. Let us illustrate these with a few simple model ecosystems. (A) Let us first consider the prey-predator model system [4,5]
173
x_1 ¼ x1 ð2 x1 x2 Þ;
x_2 ¼ x2 ðx1 x2 Þ
162 163 164 165 166 167 168 169 170
172
ð3:1Þ
It has three stationary states: ð0; 0Þ; ð2; 0Þ; ð1; 1Þ. (i) : Stationary point ð0; 0Þ; Eigenvalues ð0; 2Þ; Then the dynamical complexity is given by
174 175 176 177 178
179 181 182 183 184
185
2e2t HðtÞ ¼ 1 þ e2t
ð3:2Þ
which is positive and tends to 2 as t!1. The positive value of the dynamical complexity HðtÞ indicates that the stationary point is non-attractive and unstable. The stationary point ð0; 0Þ is thus a fixed-point repeller. (ii) : Stationary point ð2; 0Þ; Eigenvalues ð2; 2Þ; The dynamical complexity now is 2t
187 188 189 190 191 192 193
194
2t
2e 2e HðtÞ ¼ 2t e þ e2t
ð3:3Þ
which is positive and tends to 2 as t!1. The positivity of the dynamical complexity indicates that the stationary point ð2; 0Þ is non-attractive and unstable. The stationary point ð2; 0Þ is then a fixed-point repeller. (iii) : Stationary point ð1; 1Þ; Eigenvalues ð1 iÞ;
196 197 198 199 200 201 202
This entropy H1 ½A given by (2.9) provides an entropic measure of complexity (structural) of the community matrix A. The evolution matrix BðtÞ ¼ eAt is dependent on both the community matrix A and the time t. Following (2.9) the entropy (of order 1) of the evolution matrix BðtÞ ¼ eAt is given by
141
Pn H1 ½BðtÞ ki eki t HðtÞ ¼ ¼ Pi¼1 n ki t t i¼1 e
Pn
The elements faij ðx Þg of the community matrix A play significant role in the dynamical behaviours of the ecosystem [4,5]. The solution of the matrix Eq. (2.3) is given by
98
dynamical entropy (analogous to Kolmogorov-Sinai dynamical entropy or simply K.S. entropy) as the rate of change of the entropy H1 ½BðtÞ or the entropy-production rate [10]
ð2:10Þ
The entropy H1 ½BðtÞ given by (2.10) then takes care of both the aspects of the community matrix A and the time t. We define the
We use the formula (2.12) to find out the dynamical complexity. The sum-function is given by
203 204
205 2 X eki t ¼ eð1þiÞt þ eð1iÞt ¼ et ½eit þ eit ¼ 2et cos t
207
i¼1
Dynamical complexity is then given by
208
209 2 X d d HðtÞ ¼ log e ki t ¼ log½2et cos t ¼ ½1 þ tan t dt dt i¼1
ð3:4Þ
Q1 Please cite this article in press as: C.G. Chakrabarti, K. Ghosh, Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems, Math. Biosci. (2013), http://dx.doi.org/10.1016/j.mbs.2013.07.016
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The dynamical complexity consists of two parts. The first part is the contribution of the negative real part of the eigenvalue, the second part is a negative trigonometric function resulting from the imaginary part of eigenvalue representing oscillatory behaviour. The negative value of the resultant dynamical complexity indicates the attractive character and asymptotically stable nature of the stationary point (1,1). The stationary point ð1; 1Þ is a fixed point attractor in view of the negative real part of the complex eigenvalues. (B): Lotka–Volterra model system [11]. The model equations are
x_ ¼ ax bxy;
234 235 236 237 238 239
y_ ¼ cy þ dxy
ð3:5Þ
where x and y are the prey and predator population respectively. Two stationary points are: ð0; 0Þ and ðdc ; abÞ (i) The stationary point ð0; 0Þ; Eigenvalue: ða; cÞ. The dynamical complexity is given by
aeat cect HðtÞ ¼ at e þ ect
232 233
ð3:6Þ
which tends to a as t!1. The dynamical complexity (3.6) may be positive or negative depending on the values of the parameters a and c. This implies that the solution may approach the stationary point (0,0) along one direction and recede from it along another. This represents a saddle point [11]. pffiffiffiffiffi (ii) The stationary point ðdc ; baÞ has the eigenvalues ð aciÞ. The dynamical complexity is given by
240
(
2 X d HðtÞ ¼ log eki t dt i¼1
)
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi d ¼ log½2 cos act ¼ ac tan act dt
242
ð3:7Þ
243
The stationary point ðdc ; baÞ having purely imaginary eigenvalues represents an oscillatory closed orbit. The dynamical complexity is a negative tangent function. Let us find the time-average of the dynamical complexity over the complete cycle from t ¼ 0 to t ¼ 2p. We have then
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pffiffiffiffiffi Z 2p Z 2p pffiffiffiffiffi ac 1 HðtÞdt ¼ tan act ¼ 0 2p 0 2p 0
ð3:8Þ
This is an important characteristic behaviour of a periodic phenomena for which the time-average of the dynamical entropy or complexity along a closed orbit is zero. The same result also hold good for the thermodynamic entropy production rate [12]. This is a significant result characterizing the analogy between the dynamical and thermodynamical entropy-production rates. (C): Limit-Cycle model system [13,14]. Let us consider the prey-predator system
x_1 ¼
x21 ð1
x1 Þ x1 x2
x_2 ¼ kðx1 lÞx2
ð3:9Þ
where the parameters k; l > 0. The stationary points of the system are ð0; 0Þ; ð1; 0Þ and ðl; lð1 lÞÞ. The parameter l lies within ð0; 1Þ. Instead of carrying out a complete analysis of the system (3.9), we consider only the stationary point ðl; lð1 lÞÞ. The eigenvalues corresponding to the stationary point ðl; lð1 lÞÞ are
k1;2 ¼
1 ½lð1 2lÞ fl2 ð1 2lÞ2 4kl2 ð1 lÞg 2 1 2
ð3:10Þ
When l ¼ 12, the eigenvalues (3.10) are purely imaginary. When l passes through the value 12 on going from higher to lower values, the real part of the eigenvalues passes from negative (stable focus) to positive (unstable focus) values. It can be shown that for l ¼ 12 there is a supercritical bifurcation and for l < 12 (which corresponds
to unstable focus) the Hopf-bifurcation predicts the existence of a stable limit-cycle [14]. Taking k ¼ 1 and l ¼ 12 we have the eigenvalues
278 279 280
281
k1;2
i ¼ pffiffiffi 2 2
ð3:11Þ
So the sum-function is given by
284
285
2 X 1ffi t 1ffi t t i p i p eki t ¼ e 2 2 þ e 2 2 ¼ 2 cos pffiffiffi 2 2 i¼1
ð3:12Þ
The dynamical entropy or complexity is then given by
283
288
289
2 X d d t 1 t HðtÞ ¼ log eki t ¼ log 2 cos pffiffiffi ¼ pffiffiffi tan pffiffiffi dt dt 2 2 2 2 2 2 i¼1
ð3:13Þ
The time-average of the dynamical complexity or entropy along the limit-cycle is then given by
Z 2p Z 2p 1 1 t HðtÞdt ¼ pffiffiffi tan pffiffiffi ¼ 0 2p 0 4p 2 0 2
287
291 292 293
294
ð3:14Þ
296
characterizing a periodic orbits. We may have difficult situations with the analysis of the character of the stationary point ðl; lð1 lÞÞ. We may not provide deep analysis using linear model. We have to take higher-order terms in the right hand sides of the model Eq. (2.2) and go to the non-linear analysis leading to different number of limit-cycles resulting from bifurcation transitions [3]. On the basis of the above analysis with some model ecosystems we make the following observations:
297
(i) The negative value of the dynamical complexity (2.12) implies the attractive nature and stability of the stationary state. This is the case when all the eigenvalues have negative real parts. (ii) The positivity of the dynamical complexity (2.12) implies the repulsive character and instability of the stationary state. If one or more eigenvalues have positive real parts, all we can say with certainty is that there is not a stable stationary state. (iii) The periodic character of the stationary state (i.e. a center) is characterized by the vanishing of the time-average (over a period) of the dynamical entropy or complexity.
305
Z 2p 1 HðtÞdt ¼ 0 2p 0
298 299 300 301 302 303 304
306 307 308 309 310 311 312 313 314 315 316
317
ð3:15Þ
(iv) From the above analysis of the model ecosystems we see that the dynamical complexity is positive ðHðtÞ > 0Þ for instability and negative ðHðtÞ < 0Þ for stability. The dynamical complexity as a measure should, however, be positive. So we reserve the positive value of HðtÞ for the complexity implying instability and the negative value for the simplicity (opposite to complexity) implying stability of the system.
319 320 321 322 323 324 325 326 327
4. Conclusion
328
The object of the present paper is to introduce a measure of dynamical entropy or complexity of a model ecosystem and to study its importance in the characterization of dynamical behaviours such as stability and periodicity of the ecosystem near the stationary states. The stability of the system can be studied easily by the traditional methods of ODEs. Our problem is not this. Our main objective is to introduce some new more or less abstract mathematical concepts to the characterization of the dynamical behaviours around the stationary states of the ecosystem. The main results and characteristic features are as follows:
329
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(i) We have first explained the concept of community matrix obtained in the linearization process of a system of non-linear model equations describing the ecosystem. The present analysis based on community matrix is confined to the local analysis of the system around the stationary states. The states far from equilibrium have been excluded. (ii) We have introduced a measure of dynamical entropy or complexity of the ecosystem on the basis of the dynamical model of the system. The measure of dynamical entropy is based on the measure of entropy of a non-probabilistic square matrix consistent with classical Shannon entropy. A different measure dynamical entropy or complexity based on Boltzmann-like entropy function has been used earlier in the study of evolution of a complex biological system [15]. The dynamical entropy dependent on the eigenvalues of the community matrix makes the study of the dynamical behaviour of the system very easy. (iii) The dynamical entropy or complexity characterizes both qualitatively and quantitatively the dynamical behaviours of the system. Entropic characterization of the dynamical behaviours of stability, instability and periodicity of the system have been explained clearly with some illustrative model ecosystems. (iv) The stability and complexity are two vital concepts in ecology [3,16,18]. The relationship between the concepts of stability and complexity is a long standing well-debated problem. In spite of a great deal of controversies about the unique relation between stability and complexity the present analysis is based on dynamical entropy without consideration of any genomic and environmental aspects, still gives indication of the fairly general and important results that the complexity usually results instability rather than stability [3,16].
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5. Uncited references
373
Acknowledgments
375
The authors thank the learned referees whose valuable comments help the better exposition of the paper. The work was done under major research project sanctioned by U.G.C. (India).
376
References
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[1] M. Higachi, T.P. Burns, Theoretical Studies of Ecosystem: Network Perspective, Camb. Univ. Press, Cambridge, 1991. [2] R. Levins, Evolution in Changing Environment, Princeton Univ. Press, Princeton, 1968. [3] R.M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1974. [4] C. Jeffries, Mathematical Modelling in Ecology, Birkhäunser, Boston, 1989. [5] Yu.M. Svirezhev, D.O. Logofet, The Mathematics of Community Stability, in: B.C. Patten (Ed.), In Complex Ecology, Prentice Hall PT 12, Eaglewood cliff, New Jersey, 1995, p. 343. [6] N. Lakshmanan, S. Rajasekar, Non-linear Dynamics, Springer, Berlin, 2003 (pp. 85). [7] R. Rosen, Dynamical System Theory in Biology, Wiley and Sons, New-York, 1978 (p. 173). [8] G. Nicolis, I. Prigogine, Exploring Complexity, An Introduction, W. H. Freeman and Co., New-York, 1989. [9] G. Jumarie, Maximum-Entropy, Information without Probability and Complex Fractals, Kulwer Academic Publ., Dordrecht, 2000 (p. 92, 99). [10] R.C. Hilborn, Chaos and Non-linear Dynamics, Oxford Univ. Press, Oxford, 1994 (p. 335). [11] L. Edelstein-Keshet, Mathematical Models in Biology, The Random House, Birkhanser, New-York, 1988. [12] K. Ishida, S. Matsumoto, Non-equilibrium of temporally oscillating chemical reaction, J. Theor. Biol. 52 (1975) 343. [13] G.N. Odel, Appendix A3 , in: L.A. Segal (Ed.), Mathematical Models in Molecular and Cellular Biology, Cambridge Univ. Press, Cambridge, 1980. [14] R. Serra, M. Andretta, M. Compiani, Introduction to Physics of Complex Systems, Pergamon Press, Oxford, 1986. [15] C.G. Chakrabarti, Koyel Ghosh, Biological evolution: entropy, complexity and stability, J. Mod. Phys. 2 (2011) 621 (Special Issue:Thermodynamics of life and evolution). [16] J.M. Smith, Models in Ecology, Camb. Univ. Press, Cambridge, 1978 (p. 85). [17] J.D. Murray, Mathematical Biology, Springer-verlag, Berlin, 1989 (p. 68). [18] R. Solé, J. Bascompte, Self-Organization in Complex Ecosystems, Princeton Univ. Press., Pinceton, 2006 (p. 220).
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[17].
Q1 Please cite this article in press as: C.G. Chakrabarti, K. Ghosh, Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems, Math. Biosci. (2013), http://dx.doi.org/10.1016/j.mbs.2013.07.016
No. of Pages 4, Model 5G
6 August 2013 Mathematical Biosciences xxx (2013) xxx–xxx 1
Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs 5 6
Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems
3 4 7
Q1
8
C.G. Chakrabarti, Koyel Ghosh ⇑ Department of Applied Mathematics, University of Calcutta, Kolkata 700009, India
9 10
a r t i c l e
1 2 2 5 13 14 15 16 17
i n f o
Article history: Received 31 August 2012 Received in revised form 18 July 2013 Accepted 19 July 2013 Available online xxxx
18 19 20 21 22 23 24
Keywords: Model ecosystem Community matrix Dynamical entropy Stability Complexity
a b s t r a c t In the present paper we have first introduced a measure of dynamical entropy of an ecosystem on the basis of the dynamical model of the system. The dynamical entropy which depends on the eigenvalues of the community matrix of the system leads to a consistent measure of complexity of the ecosystem to characterize the dynamical behaviours such as the stability, instability and periodicity around the stationary states of the system. We have illustrated the theory with some model ecosystems. Ó 2013 Published by Elsevier Inc.
26 27 28 29 30 31
32 33 34
1. Introduction
35
An ecosystem consisting of a large number of interacting species may be considered as a complex dynamical system [1]. In recent years dynamical system model of complex ecological system has experienced explosive growth. Different mathematical ideas and techniques have been developed in the elucidation of different underlying biological and ecological processes. To understand a complex ecosystem we need some systematic methodology. The differential equations used to model ecosystems are, in general, non-linear. It is very difficult to find out the solution of the system of non-linear equations in closed form. It is customary to study the dynamical behaviours of such systems near the stationary states. The linearized form of the system of equations around a stationary state represents the local dynamical behavior of the system. The community matrix introduced by Levin in the linearization process plays a significant role in the determination of the qualitative nature of the nearby orbits or trajectories [2]. The community matrix represents the mathematical structure of the system near the stationary states [3,4]. The objective of the present paper is to study the dynamical behaviors associated with the community matrix from a different mathematical background. The contribution of the paper is two fold: Firstly, we wish to introduce a measure of dynamical entropy
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
⇑ Corresponding author. Tel.: +91 033 2350 8386/1397x450/451; fax: +91 033
Q2 Q1
2351 9755/2241 3222. E-mail addresses: [email protected] (C.G. Chakrabarti), koyelghosh [email protected] (K. Ghosh).
for the entropic characterization of the time-evolution of the ecosystem. The methodology of the derivation of the dynamical entropy is based on the entropy of a non-random square matrix. Secondly, we wish to study the importance of the dynamical entropy in the characterization of stability, instability and periodicity of the stationary states of the ecosystem with some illustrative model ecosystems. The interrelation between the concept of stability and complexity has been investigated.
57
2. Model ecosystem: dynamical model and entropy
65
We consider a multi-species ecosystem consisting of n species with population densities xi ðtÞ; ði ¼ 1; 2; . . . :; nÞ at any time t. Let us assume the system to be governed by the system of equations
66
x_i ¼ fi ðx1 ; x2 ; . . . :; xn ; aÞ; ði ¼ 1; 2; . . . ; nÞ
ð2:1Þ
where a is a constant parameter. The functions fi are assumed to be continuously differentiable in some open set X ¼ fxi jxi P 0; i ¼ 1; 2; . . . ; ng. The system of model Eq. (2.1) are, in general, non-linear and difficult to find out the solution in closed form. It is customary to study such a system close to some stationary state. Let x ¼ ðx1 ; x2 ; . . . ::; xn Þ be the stationary state of the system for a certain value (or a range of values) of the parameter a. We consider a small deviation about the stationary state: yi ðtÞ ¼ dxi ðtÞ ¼ xi ðtÞ xi ; ði ¼ 1; 2; . . . ; nÞ. Linearizing the system of Eq. (2.1) about the stationary state x ¼ ðx1 ; x2 ; . . . ::; xn Þ we have the system of linear equations
0025-5564/$ - see front matter Ó 2013 Published by Elsevier Inc. http://dx.doi.org/10.1016/j.mbs.2013.07.016
Q1 Please cite this article in press as: C.G. Chakrabarti, K. Ghosh, Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems, Math. Biosci. (2013), http://dx.doi.org/10.1016/j.mbs.2013.07.016
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y_ i ðtÞ ¼
C.G. Chakrabarti, K. Ghosh / Mathematical Biosciences xxx (2013) xxx–xxx
n X @fi @xj
85
j¼1
86
or in matrix form
87 89
_ yðtÞ ¼ AyðtÞ
yj ðtÞ ði ¼ 1; 2; . . . ; nÞ ¼
x
n X aij ðx Þyj ðtÞ j¼1
ð2:3Þ
90
where the Jacobian
91 93
A ¼ ½aij ðx Þ
94
is the so-called community matrix with elements
95 97
aij ðx Þ ¼
@fi @xj
ð2:2Þ
ð2:4Þ i; j ¼ 1; 2; ::; n
ð2:5Þ
x
100
101 103
yðtÞ ¼ dxðtÞ ¼ eAt dxð0Þ
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104 105 106 107 108 109 110 111 112 113
114 116
ð2:6Þ
where dxð0Þ is the initial deviation from the stationary state x . Let us now find the explicit form of the solution (2.6) in terms of the eigenvalues ðk1 ; k2 ; . . . :; kn Þ of the community matrix A. If the eigenvectors corresponding to the eigenvalues ðk1 ; k2 ; . . . :; kn Þ are linearly independent then the matrix A can be converted to the form of a diagonal matrix with diagonal elements same as ðk1 ; k2 ; . . . :; kn Þ. Under the mathematical conditions of linearly independence of the eigenvectors and the distinct eigenvalues ðk1 ; k2 ; . . . :; kn Þ, the explicit form of the solution (2.6) are given by Lakshmanan and Rajasekar [6] and Rosen [7]
dxi ðtÞ ¼ dxi ð0Þeki t ;
ði ¼ 1; 2; . . . ; nÞ
ð2:7Þ
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All the basic criteria of stability, instability and periodicity of the system follow from the solution (2.7). Let us now try to develop an entropic theory of time-evolution of the ecosystem on the basis of the linearized model Eq. (2.3). Entropy plays a significant role in the study of evolution of a thermodynamic system [8]. Like in thermodynamics we need an expression of dynamical entropy for the study of time-evolution of the ecosystem described by the dynamical model Eq. (2.3). According to Eq. (2.3) the time-evolution of the system from the initial state to the current state is given by
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dxðtÞ ¼ eAt dxð0Þ ¼ BðtÞdxð0Þ
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ð2:8Þ
At
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where BðtÞ ¼ e is the matrix of evolution dxð0Þ!dxðtÞ. The evolution matrix BðtÞ ¼ eAt characterizes the time-evolution of the system. To find out a dynamical entropy for the time-evolution dxð0Þ!dxðtÞ we need a measure of entropy of the non-probabilistic square matrix BðtÞ ¼ eAt . Following Jumarie the entropy (of order 1) of a n n square matrix A consistant with Shannon classical entropy and Von Neumann quantum entropy is given by Jumarie [9]
140
Pn jki j ln jki j Pn H1 ½A ¼ i¼1 i¼1 jki j
130 131 132 133 134 135 136
138
ð2:9Þ
145
148
Pn ki t i¼1 tki e H1 ½BðtÞ ¼ H1 ½eAt ¼ P n ki t i¼1 e
142 143 144
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149 150
ð2:11Þ
151 152 153
154 156
The expression (2.11) can be written as
157
( ) n X d HðtÞ ¼ log eki t dt i¼1
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ð2:12Þ 160
where the sum-function i¼1 eki t is the trace of the diagonal evolution matrix BðtÞ ¼ eAt . This is analogous to the canonical partitionfunction in statistical mechanics. The dynamical entropy HðtÞ is a real quantity inspite of the feasibility of complex or imaginary eigenvalues. Jumarie [9] has used the expression of dynamical entropy (2.12) to measure the complexity of a dynamical system. We shall consider both the terms dynamical entropy and dynamical complexity to be synonymous. In the next section we shall study the significance of the dynamical entropy (2.12) in the analysis of stability, instability and periodicity of some model ecosystems.
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3. Model ecosystem: analysis of stability, periodicity and complexity
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In this section we shall study the role of the measure of dynamical entropy (2.12) in the characterization of different dynamical behaviours such as stability, instability and periodicity etc. of the system around a stationary state. Let us illustrate these with a few simple model ecosystems. (A) Let us first consider the prey-predator model system [4,5]
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x_1 ¼ x1 ð2 x1 x2 Þ;
x_2 ¼ x2 ðx1 x2 Þ
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ð3:1Þ
It has three stationary states: ð0; 0Þ; ð2; 0Þ; ð1; 1Þ. (i) : Stationary point ð0; 0Þ; Eigenvalues ð0; 2Þ; Then the dynamical complexity is given by
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2e2t HðtÞ ¼ 1 þ e2t
ð3:2Þ
which is positive and tends to 2 as t!1. The positive value of the dynamical complexity HðtÞ indicates that the stationary point is non-attractive and unstable. The stationary point ð0; 0Þ is thus a fixed-point repeller. (ii) : Stationary point ð2; 0Þ; Eigenvalues ð2; 2Þ; The dynamical complexity now is 2t
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2t
2e 2e HðtÞ ¼ 2t e þ e2t
ð3:3Þ
which is positive and tends to 2 as t!1. The positivity of the dynamical complexity indicates that the stationary point ð2; 0Þ is non-attractive and unstable. The stationary point ð2; 0Þ is then a fixed-point repeller. (iii) : Stationary point ð1; 1Þ; Eigenvalues ð1 iÞ;
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This entropy H1 ½A given by (2.9) provides an entropic measure of complexity (structural) of the community matrix A. The evolution matrix BðtÞ ¼ eAt is dependent on both the community matrix A and the time t. Following (2.9) the entropy (of order 1) of the evolution matrix BðtÞ ¼ eAt is given by
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Pn H1 ½BðtÞ ki eki t HðtÞ ¼ ¼ Pi¼1 n ki t t i¼1 e
Pn
The elements faij ðx Þg of the community matrix A play significant role in the dynamical behaviours of the ecosystem [4,5]. The solution of the matrix Eq. (2.3) is given by
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dynamical entropy (analogous to Kolmogorov-Sinai dynamical entropy or simply K.S. entropy) as the rate of change of the entropy H1 ½BðtÞ or the entropy-production rate [10]
ð2:10Þ
The entropy H1 ½BðtÞ given by (2.10) then takes care of both the aspects of the community matrix A and the time t. We define the
We use the formula (2.12) to find out the dynamical complexity. The sum-function is given by
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205 2 X eki t ¼ eð1þiÞt þ eð1iÞt ¼ et ½eit þ eit ¼ 2et cos t
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i¼1
Dynamical complexity is then given by
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209 2 X d d HðtÞ ¼ log e ki t ¼ log½2et cos t ¼ ½1 þ tan t dt dt i¼1
ð3:4Þ
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The dynamical complexity consists of two parts. The first part is the contribution of the negative real part of the eigenvalue, the second part is a negative trigonometric function resulting from the imaginary part of eigenvalue representing oscillatory behaviour. The negative value of the resultant dynamical complexity indicates the attractive character and asymptotically stable nature of the stationary point (1,1). The stationary point ð1; 1Þ is a fixed point attractor in view of the negative real part of the complex eigenvalues. (B): Lotka–Volterra model system [11]. The model equations are
x_ ¼ ax bxy;
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y_ ¼ cy þ dxy
ð3:5Þ
where x and y are the prey and predator population respectively. Two stationary points are: ð0; 0Þ and ðdc ; abÞ (i) The stationary point ð0; 0Þ; Eigenvalue: ða; cÞ. The dynamical complexity is given by
aeat cect HðtÞ ¼ at e þ ect
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ð3:6Þ
which tends to a as t!1. The dynamical complexity (3.6) may be positive or negative depending on the values of the parameters a and c. This implies that the solution may approach the stationary point (0,0) along one direction and recede from it along another. This represents a saddle point [11]. pffiffiffiffiffi (ii) The stationary point ðdc ; baÞ has the eigenvalues ð aciÞ. The dynamical complexity is given by
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(
2 X d HðtÞ ¼ log eki t dt i¼1
)
pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi d ¼ log½2 cos act ¼ ac tan act dt
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ð3:7Þ
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The stationary point ðdc ; baÞ having purely imaginary eigenvalues represents an oscillatory closed orbit. The dynamical complexity is a negative tangent function. Let us find the time-average of the dynamical complexity over the complete cycle from t ¼ 0 to t ¼ 2p. We have then
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pffiffiffiffiffi Z 2p Z 2p pffiffiffiffiffi ac 1 HðtÞdt ¼ tan act ¼ 0 2p 0 2p 0
ð3:8Þ
This is an important characteristic behaviour of a periodic phenomena for which the time-average of the dynamical entropy or complexity along a closed orbit is zero. The same result also hold good for the thermodynamic entropy production rate [12]. This is a significant result characterizing the analogy between the dynamical and thermodynamical entropy-production rates. (C): Limit-Cycle model system [13,14]. Let us consider the prey-predator system
x_1 ¼
x21 ð1
x1 Þ x1 x2
x_2 ¼ kðx1 lÞx2
ð3:9Þ
where the parameters k; l > 0. The stationary points of the system are ð0; 0Þ; ð1; 0Þ and ðl; lð1 lÞÞ. The parameter l lies within ð0; 1Þ. Instead of carrying out a complete analysis of the system (3.9), we consider only the stationary point ðl; lð1 lÞÞ. The eigenvalues corresponding to the stationary point ðl; lð1 lÞÞ are
k1;2 ¼
1 ½lð1 2lÞ fl2 ð1 2lÞ2 4kl2 ð1 lÞg 2 1 2
ð3:10Þ
When l ¼ 12, the eigenvalues (3.10) are purely imaginary. When l passes through the value 12 on going from higher to lower values, the real part of the eigenvalues passes from negative (stable focus) to positive (unstable focus) values. It can be shown that for l ¼ 12 there is a supercritical bifurcation and for l < 12 (which corresponds
to unstable focus) the Hopf-bifurcation predicts the existence of a stable limit-cycle [14]. Taking k ¼ 1 and l ¼ 12 we have the eigenvalues
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k1;2
i ¼ pffiffiffi 2 2
ð3:11Þ
So the sum-function is given by
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2 X 1ffi t 1ffi t t i p i p eki t ¼ e 2 2 þ e 2 2 ¼ 2 cos pffiffiffi 2 2 i¼1
ð3:12Þ
The dynamical entropy or complexity is then given by
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2 X d d t 1 t HðtÞ ¼ log eki t ¼ log 2 cos pffiffiffi ¼ pffiffiffi tan pffiffiffi dt dt 2 2 2 2 2 2 i¼1
ð3:13Þ
The time-average of the dynamical complexity or entropy along the limit-cycle is then given by
Z 2p Z 2p 1 1 t HðtÞdt ¼ pffiffiffi tan pffiffiffi ¼ 0 2p 0 4p 2 0 2
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291 292 293
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ð3:14Þ
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characterizing a periodic orbits. We may have difficult situations with the analysis of the character of the stationary point ðl; lð1 lÞÞ. We may not provide deep analysis using linear model. We have to take higher-order terms in the right hand sides of the model Eq. (2.2) and go to the non-linear analysis leading to different number of limit-cycles resulting from bifurcation transitions [3]. On the basis of the above analysis with some model ecosystems we make the following observations:
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(i) The negative value of the dynamical complexity (2.12) implies the attractive nature and stability of the stationary state. This is the case when all the eigenvalues have negative real parts. (ii) The positivity of the dynamical complexity (2.12) implies the repulsive character and instability of the stationary state. If one or more eigenvalues have positive real parts, all we can say with certainty is that there is not a stable stationary state. (iii) The periodic character of the stationary state (i.e. a center) is characterized by the vanishing of the time-average (over a period) of the dynamical entropy or complexity.
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Z 2p 1 HðtÞdt ¼ 0 2p 0
298 299 300 301 302 303 304
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317
ð3:15Þ
(iv) From the above analysis of the model ecosystems we see that the dynamical complexity is positive ðHðtÞ > 0Þ for instability and negative ðHðtÞ < 0Þ for stability. The dynamical complexity as a measure should, however, be positive. So we reserve the positive value of HðtÞ for the complexity implying instability and the negative value for the simplicity (opposite to complexity) implying stability of the system.
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4. Conclusion
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The object of the present paper is to introduce a measure of dynamical entropy or complexity of a model ecosystem and to study its importance in the characterization of dynamical behaviours such as stability and periodicity of the ecosystem near the stationary states. The stability of the system can be studied easily by the traditional methods of ODEs. Our problem is not this. Our main objective is to introduce some new more or less abstract mathematical concepts to the characterization of the dynamical behaviours around the stationary states of the ecosystem. The main results and characteristic features are as follows:
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(i) We have first explained the concept of community matrix obtained in the linearization process of a system of non-linear model equations describing the ecosystem. The present analysis based on community matrix is confined to the local analysis of the system around the stationary states. The states far from equilibrium have been excluded. (ii) We have introduced a measure of dynamical entropy or complexity of the ecosystem on the basis of the dynamical model of the system. The measure of dynamical entropy is based on the measure of entropy of a non-probabilistic square matrix consistent with classical Shannon entropy. A different measure dynamical entropy or complexity based on Boltzmann-like entropy function has been used earlier in the study of evolution of a complex biological system [15]. The dynamical entropy dependent on the eigenvalues of the community matrix makes the study of the dynamical behaviour of the system very easy. (iii) The dynamical entropy or complexity characterizes both qualitatively and quantitatively the dynamical behaviours of the system. Entropic characterization of the dynamical behaviours of stability, instability and periodicity of the system have been explained clearly with some illustrative model ecosystems. (iv) The stability and complexity are two vital concepts in ecology [3,16,18]. The relationship between the concepts of stability and complexity is a long standing well-debated problem. In spite of a great deal of controversies about the unique relation between stability and complexity the present analysis is based on dynamical entropy without consideration of any genomic and environmental aspects, still gives indication of the fairly general and important results that the complexity usually results instability rather than stability [3,16].
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5. Uncited references
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Acknowledgments
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The authors thank the learned referees whose valuable comments help the better exposition of the paper. The work was done under major research project sanctioned by U.G.C. (India).
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References
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[1] M. Higachi, T.P. Burns, Theoretical Studies of Ecosystem: Network Perspective, Camb. Univ. Press, Cambridge, 1991. [2] R. Levins, Evolution in Changing Environment, Princeton Univ. Press, Princeton, 1968. [3] R.M. May, Stability and Complexity in Model Ecosystems, Princeton Univ. Press, Princeton, 1974. [4] C. Jeffries, Mathematical Modelling in Ecology, Birkhäunser, Boston, 1989. [5] Yu.M. Svirezhev, D.O. Logofet, The Mathematics of Community Stability, in: B.C. Patten (Ed.), In Complex Ecology, Prentice Hall PT 12, Eaglewood cliff, New Jersey, 1995, p. 343. [6] N. Lakshmanan, S. Rajasekar, Non-linear Dynamics, Springer, Berlin, 2003 (pp. 85). [7] R. Rosen, Dynamical System Theory in Biology, Wiley and Sons, New-York, 1978 (p. 173). [8] G. Nicolis, I. Prigogine, Exploring Complexity, An Introduction, W. H. Freeman and Co., New-York, 1989. [9] G. Jumarie, Maximum-Entropy, Information without Probability and Complex Fractals, Kulwer Academic Publ., Dordrecht, 2000 (p. 92, 99). [10] R.C. Hilborn, Chaos and Non-linear Dynamics, Oxford Univ. Press, Oxford, 1994 (p. 335). [11] L. Edelstein-Keshet, Mathematical Models in Biology, The Random House, Birkhanser, New-York, 1988. [12] K. Ishida, S. Matsumoto, Non-equilibrium of temporally oscillating chemical reaction, J. Theor. Biol. 52 (1975) 343. [13] G.N. Odel, Appendix A3 , in: L.A. Segal (Ed.), Mathematical Models in Molecular and Cellular Biology, Cambridge Univ. Press, Cambridge, 1980. [14] R. Serra, M. Andretta, M. Compiani, Introduction to Physics of Complex Systems, Pergamon Press, Oxford, 1986. [15] C.G. Chakrabarti, Koyel Ghosh, Biological evolution: entropy, complexity and stability, J. Mod. Phys. 2 (2011) 621 (Special Issue:Thermodynamics of life and evolution). [16] J.M. Smith, Models in Ecology, Camb. Univ. Press, Cambridge, 1978 (p. 85). [17] J.D. Murray, Mathematical Biology, Springer-verlag, Berlin, 1989 (p. 68). [18] R. Solé, J. Bascompte, Self-Organization in Complex Ecosystems, Princeton Univ. Press., Pinceton, 2006 (p. 220).
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[17].
Q1 Please cite this article in press as: C.G. Chakrabarti, K. Ghosh, Dynamical entropy via entropy of non-random matrices: Application to stability and complexity in modelling ecosystems, Math. Biosci. (2013), http://dx.doi.org/10.1016/j.mbs.2013.07.016