CHAOTIC BEHAVIOUR OF COUPLED LOGISTIC EQUATION … · theory, namely the example for the period...

CHAOTIC BEHAVIOUR OFCOUPLED LOGISTIC EQUATION

THROUGH COMPETITION MODEL

M.Sambasivam1,Assistant Professor in Mathematics,

J.N.R.M (Govt. P.G. College),Port Blair.

[email protected]

October 14, 2018

AbstractThe stability of the fixed points of the dynamical system

of two-dimensional is studied by means of Jacobian matrixand its Eigen values. The species fade away for λε (0, 0.245)owing to non stable co-existence. The species survive andthe system settles down in a stationary equilibrium pointas λε(0.245, 0.981). The species oscillates synchronously toa stable non-vanishing fixed quantity as λε (0.981, 1.173).Each one of the species oscillates out of phase and unstablebetween the two fixed values when λε (1.173, 1.185) andthe species turned into a quasi periodic oscillation whenλε (1.185, 1.192). The competition model steps forwardtowards the chaotic region and forms chaotic rings as soonas the value of the parameter λ reaches 1.2.

Key Words: Discrete dynamics, Chaos, Eigen values,Logistic equation, Competition Model, attractor

1 INTRODUCTION

In modern Mathematics, chaotic maps are maps which demonstratevarious kinds of dynamical behaviour. These maps are parameter-

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International Journal of Pure and Applied MathematicsVolume 120 No. 6 2018, 999-1013ISSN: 1314-3395 (on-line version)url: http://www.acadpubl.eu/hub/Special Issue http://www.acadpubl.eu/hub/

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ized by a discrete time or continuous time parameter. These mapsare frequently seen in dynamical system. Logistic map is one of theexemplar for the chaotic map in discrete dynamical system [1]. Italso illustrates how the dynamical system becomes chaos. Malthusstudied the first mathematical model for population dynamic in1798 [2] in the form of equation

Xn+1 = rXn. (1)

He enlightens that the human population was growing geometri-cally while the food supply was growing arithmetically.The complication with the equation (1) is that the population con-tinues to grow unlimited over time. In 1845, Verhulst proved thatthe population growth not only depends on the population size butalso on how far the size is from its upper limit [4]. The equation(1) is redefined as dp

dt= Kp np2. Verhulst called this equation

as Logistic equation. This equation is the source of present chaostheory, namely the example for the period doubling. The discreteversion of Verhulst model for the evolution of the population of aspecies is given by

Xn+1 = rXn(1−Xn) (2)

where Xn represents the population of an isolated species after ngenerations and assume that this variable is bounded in the range0 < Xn < 1. The parameter r lies between 0 < r < 4 in orderto assure that 0 < Xn < 1. The expanding plane is controlledby the term rXn proportional to the current population Xn andto the constant growth rate r. The population dynamics of onespecies can be inspected through one dimensional logistic map andthe study reveals that, the dynamics of one species exhibits chaos[3]. The logistic equation of one species is extended for the purposeof understanding the dynamic behaviour of 2-species.

2 COUPLED LOGISTIC EQUATION

Suppose, if two species (Xn, Yn) are living together, and then eachspecies evolve in the following logistic type dynamics ;

Xn+1 = µx(Yn)Xn(1−Xn);Yn+1 = µY (Xn)Yn(1− Yn) (3)

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International Journal of Pure and Applied Mathematics Special Issue

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The interaction between the species causes the growth rate µx. Theterm µx is a parameter because it varies with time. The parameterµx mostly depends on other population size and on a positive con-stant λ. The positive constant measures the strength of the mutualinteraction.When the two species interact, there could be a chance for linearincreasing and decreasing. This linear increasing and decreasing isbased on the growth rate µ and therefore the parameter µ is givenby the functions µ1 = λ(3X+ 1) and µ2 = λ(−3X+ 4) respectivelyand it lies in the interval [1, 4].The Combination of the functionsµ1 and µ2 gives a way to obtain different models[6].

3 COMPETITION MODEL

Competition is an undeniable force for evolution. It is of two typesnamely intra-specific and inter-specific. The competition amongthe same species is said to be intra-specific. It does not promoteany collective embellishment which would ultimately end up in in-dividual favour. There will be no neutral development among thespecies. One of the best examples is the discriminated human soci-ety. When competition is inter-specific that is among the differentspecies, the diversified utilization of the resource is possible. Inter-specific competition promotes dynamical phenomena by the way ofconsuming same resource in different ways among several species.Therefore the competition need not be a destructive forces insteada force of evolution.If two species (Xn, Yn) share the same ecosystem, then the evolu-tion is given by coupled logistic equation.

Xn+1 = λ(−3Yn + 4)Xn(1−Xn)Yn+1 = λ(−3Xn + 4)Yn(1− Yn)

}(4)

where λ, expresses the strength of the mutual competing interac-tion. The system above can be represented by the applicationTλ : [0, 1][0, 1]→ R2Tλ[Xn, Yn][Xn+1, Yn+1]where λ is a positive constant which has sense in the range 0 < λ <1.21. Here competition is a constructive force in the model. A weakcompetition can lead the system to extinction whereas a stronger

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interaction makes the system to visit different states viz., stabilizein a fixed population, periodic oscillations and chaotic oscillations[5] [8][7]

4 FIXED POINTS, PERIOD TWO CY-

CLES AND PERIOD FOUR ATTRAC-

TOR

Finding the fixed point is an important task to analyze the be-haviour of the dynamical system. The fixed point of the equation(4) is restricted to the axes and to the diagonal. The restrictionof the equation (4) to the axes is the logistic map. Therefore theequation (4) written as

Xn+1 = 4λXn(1−Xn)Yn+1 = 4λYn(1− Yn)

}(5)

The solution for Xn+1=Xn and Yn+1 = Yn are the fixed points of theequation, Therefore

• Xn+1 = 4λXn(1−Xn) is written asXn = 4λXn(1−Xn)Xn = 4λ−1

4λ

• Yn+1 = 4λYn(1−Xn) is written asYn = 4λYn(1− Yn)Yn = 4λ−1

4λ

Thus the fixed points of the equation (5) areP2 =

(4λ−14λ

, 0)(Since Yn = 0)

P2 =(0− 4λ−1

4λ

)(Since Xn = 0)

The equation (4) is restricted to the diagonal, therefore (4) writtenas

Xn+1 = λ(−3Yn + 4)Xn(1−Xn)Yn+1 = λ(−3Xn + 4)Yn(1− Yn)

}(6)

The solution for Xn+1 = Xn and Yn+1 = Yn are the fixed points ofthe equationTherefore the equation (6) is written as

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Xn = λ(−3Yn + 4)Xn(1−Xn)⇒ 3λX2

n − 7λXn + 4λ− 1 = 0⇒ Xn = 1

6(7±

√1 + 12λ)

Yn = λ(−3Yn + 4)Yn(1− Yn)⇒ 3λY 2

n − 7λYn + 4λ− 1 = 0⇒ Yn = 1

6(7±

√1 + 12λ)

Thus the fixed points of (6) areP3 = 1

6(7 +

√1 + 12λ, 7 +

√1 + 12)

P4 = 16(7−

√1 + 12λ, 7−

√1 + 12)

In competition model the fixed points P0, P1, P2 are on the axesand P4, P5 lies on the diagonal. The stability of the fixed pointsof 2D dynamical system is analyzed through Jacobian matrix Jand using Eigen values r. If the fixed point is stable then |r| < 1otherwise the point is unstable.

5 EIGEN VALUES FOR THE FIXED

POINTS OF COMPETITION MODEL

The Jacobian matrix for 2D dynamical system is

The Jacobian Matrix for (4) is

Let J0 = J at P0

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The Eigen values are derived from the characteristic equation for|J0r0I| = 0

The Eigen values of the fixed point P0 is r0 = 4λ

The Eigen values are derived from the characteristic equation —J1r0I—=0

The Eigen values of P1 are r1 = 2(1− 2λ), r′1 = 14(3 + 4λ)

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The Eigen values are derived from the characteristic equation |J2r2I| =0

The Eigen values of P2 are r2 = 2(1− 2λ), r′2 = 14(3 + 4λ)


The Eigen values of P3 are

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The Eigen values of P4 are

with these Eigen values, the range at which the existence of attrac-tor, repellor and bifurcation of fixed points are enlightened in thesubsequent lines. For 0 < λ < 0.245, P0 is an attractive node. Forall the rest of parameter values, P0is a repelling node. The pointsP1 and P2 exist for every parameter value and both are unstablefor every value of λ. Since the modulus of the Eigen values of P1

and P2 are greater than one.The point P3 is located outside the square [0, 1] x [0, 1] and is alsounstable for every value of λ. For λ=0.2, P0 and P4 exchange sta-bility in a transcritical bifurcation and P4 becomes a stable node.When λ=0.981, P4 loses stability by a flip bifurcation on the diag-onal that gives rise to a period two orbit, P5 and P6. This cyclebecomes unstable by a transversal flip bifurcation for λ=1.185 andit gives rise to an off-diagonal period four orbit. Its co-ordinates aregiven by (P51, P52, P61, P62) = ((0.34, 0.16), (0.16, 0.34), (0.48, 0.94),(0.94, 0.48)). Further increasing value of λ=1.1912 the off-diagonalperiod four orbit destabilizes by a Neimark-Hopfbifurcation andmoves toward a period four invariant curve. This invariant closedcurve grows when the parameter interval increases and it transitstowards a period 4-chaotic bands.

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6 RESULTS AND DISCUSSION

The competition modelmoves in the direction of the chaotic regionwhen the value of the parameter λ reaches1.2. Here I summarizethe dynamical behaviour of this coupled logistic system throughcompetition model when λ is modified.

Case (i) λε (0, 0.245)

The extinction of species took place from 3162nd iteration whenλ = 0.2 for the initial values x(0) = 0.54 and y(0)=0.8. The resul-tant output is

Fig 1.

As competition is an vital force for existence in a pure competitiveinteraction and its absence or its weakness can cause extinction ofspecies. Consequently the small value of λ provokes the extinctionof species.

Case (ii) λε (0.245, 0.981)

The species settles down in a stationary equilibrium point (0.441898,0.441898) at 42nd iteration as soon as λ = 0.67 for the initial valuesx(0) =0.7 and y(0)=0. 3. The equivalent output is

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Fig 2.

Here both the populations can survive and therefore the systemsettles down in a stationary equilibrium point.

Case (iii) λε (0.981, 1.173)

The species oscillates between the points (0.333302, 0.333302) and(0.779988, 0.779988) at 382nd iteration at λ = 1.17 for the initialvalues x(0) =0..45 and y(0)=0. 8. The output for the above is

Fig 3.

For λ=0.981 the destabilization of the fixed point gives rise to astable period two cycle located on the diagonal. The populationsoscillate synchronously between the two values of that orbit.

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Case (iv) λε (1.173, 1. 185)

The species oscillates between period four points (0.941433, 0.485418),(0.166201, 0.348005), (0.485418, 0.941433), (0.348005, 0.166201)at194th iteration at λ = 1.185 for the initial values x(0) =0.81 andy(0)=0. 6. The equivalent output is

Fig 4.

At this range, the period two cycles becomes unstable and an off-diagonal period four orbit appears.

Case (v) λε (1.185, 1.192)

The period of four variant closed curves occurred at λ = 1.1912for the initial values x(0) =0.349 and y(0)=0. 858 The resultantoutput is

Fig 5.

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In this range, a Neimark-Hopf bifurcation happens and the periodfour orbits gives rise to a period four invariant closed curve.

Case (vi) λε (1.192, 1.201)

The dynamic of the species become complex by forming four chaoticrings when λ = 1.2 for the initial values x(0) =0.81 and y(0)=0. 56The output for the above is

Fig 6.

For the value of λ in this range, the dynamics becomes complexbecause the population visit the attractor which is formed by fourchaotic rings.

Case (vii) λε (1.201, 1.206)

The species become to the state of chaos by forming two chaoticbands when λ = 1.204 for the initial values x(0) =0.61 and y(0)=0.86. The corresponding output is

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Fig 7.

Lastly in this range, the big chaotic bands are formed.

7 CONCLUSION

In competition model the interaction between the two species isthe constructive force i.e two species strive for same resources toexist. Suppose weak competition exists between the two species,it makes system to extinct.For stronger interaction, the value of λincreases and the population attains different states and they arestabilization of fixed population, origin of periodic oscillation andchaotic oscillation. The competition model has dynamical richnessin the range 1.1912 < λ < 1.2 because co-existence of high periodorbits and their period doubling cascades exists in that range.Fromthe above conversation, the competition model lead to chaos beyondcertain range. Therefore extension of logistic map also exhibitschaos.This discrete two dimensional logistic systemis considered asa fundamental step to build more complex systems with n-species.

8 ACKNOWLEDGEMENT

I would like to thank Dr. R. Nandikeswari for her kind Cooperationand continuous encouragements. I also thank Dr. N. Gururajanand S. Navin for their valuable suggestions and support.

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References

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[2] Eckmann, J.P., Roads turbulence in dissipative dynamicalsystems Review of modern physics, 53, 643, (1981).

[3] Feigenbaum, M.J., Quantitative universality for a class of non-linear transformations, J. Stat. Phys. 19, 25-52,(1978).

[4] Foruzakis, C., Gardini, L., Keverkidis, I., Millerioux, G., Mira,C., On some properties of invariant sets of two-dimensionalnoninvertible maps. International journal of bifurcation andchaos, 7, 1167-1194, (1997).

[5] Huisman, J. and Wessing, F. J., Biological conditions for os-cillations and chaos generated by multispecies competition:,Ecology 82, 2682-2695, (2001).

[6] Lopez-Ruiz. R and Fournier Prunaret, D., Periodic and chaoticevents in a discrete model of logistic type for the com-petitive interaction of two Species, In Arxiv, preprint nlin.CD/0504028, (2006).

[7] May R.M., Simple Mathematical Model with very complicateddynamics, Nature 261, 459-467, (1976).

[8] May. R.and Leonard, W.J., Nonlinear aspects of competi-tion between three species, SIAM J. App. Math., 29, 243-253,(1975).

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