Bifurcation Analysis for the Delay Logistic Equation with Two Delays

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1 IOSR Jounal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 5 Ve. IV (Sep. - Oct. 05), PP Bifucation Analysis fo the Delay Logistic Equation with Two Delays Mohammed O.Mahgoub, Mohammed A.Gubaa (Mathematics Depatment, College of Science / Majmaa Univesity, Saudi Aabia) (Mathematics Depatment, College of technology of mathematical sciences/ Alnelain Univesity, Sudan) Abstact : Time-delay dynamical systems ae utilized to descibe many phenomena of physical inteest. Special effot of this pape is given to Hopf bifucation of time-delay dynamical systems. Linea analysis is caied out to detemine the citical conditions unde which Hopf bifucation can occu. Delays being inheent in any biological system, we seek to analyze the effect of delays on the gowth of populations govened by the logistic equation. The local stability analysis and local bifucation analysis of the logistic equation with two delays is caied out consideing one of the delays as a bifucation paamete. Keywods: Logistic equation, delay, stability, Hopf bifucation. I. Intoduction Time delays have been incopoated into biological models to epesent esouce egeneation times, matuation peiods, feeding times, eaction times, etc. by many eseaches[],[],[3] fo discussions of geneal delayed biological systems. In geneal, delay diffeential equations ehibit much moe complicated dynamics than odinay diffeential equations since a time delay could cause a stable equilibium to become unstable and cause the populations to fluctuate. The logistic equation is a simple model of population gowth in conditions whee thee ae limited esouces. It was fomulated by Vehulst in 838 to descibe the self limiting gowth of a biological population. Accoding to the logistic equation, the gowth ate of a population is popotional to the cuent population and the availability of esouces in the ecosystem. The logistic equation model is then eads d ( ) () d t K Whee is the population at any instant, ( > 0 ) is the intinsic gowth ate and K(> 0) is the caying capacity of the population. In pactice, the pocess of epoduction is not instantaneous. Fo eample, in a Daphnia a lage clutch pesumably is detemined not by the concentation of unconsumed food available when the eggs hatch, but by the amount of food available when the eggs wee foming, some time befoe they pass into the boad pouch. Between this time of detemination and the time of hatching many newly hatched animals may have been libeated fom the bood pouches of othe Daphnia in the cultue, so inceasing the population. Hutchinson (948) incopoated the effect of delay into the logistic equation and assumed egg fomation to occu units of time befoe hatching and poposed the following moe ealistic logistic equation d ( t ) [ ( t ) ] ( ) d t K Many topological changes in the population size like damped oscillations, limit cycles and even chaos may occu as a consequence of delay [4]. The bifucation analysis of a system with a single delay has been pefomed in [5]. The impotance of two delays in the logistic equation can be found in [6]. In this pape, the effect of such delays will be analyzed, we will investigate local stability, and local bifucation phenomena taking into account one of the two delays as a bifucation paamete. The pape is oganized as follows; in section II we discuss model fomulation and the accompanying hypothesis. Section III is devoted to discuss the stability analysis and section IV fo the Hopf bifucation analysis. A bief conclusion is given in section V. II. Model Fomulation Milind and Peetish [7], in thei pape descibe the model employed with the accompanying assumptions fo a single delay system and a two delays system. A. Logistic equation with single delay The model which has been employed to chaacteize population dynamics is as follows: The equation is nomalized. DOI: / Page

2 Bifucation Analysis fo the Delay Logistic Equation with Two Delays The gowth ate is which is finite, positive and time independent. The delay is also finite and positive. With these assumptions, the single delay logistic equation simplifies to d ( t ) [ b ( t )] (3) B. Logistic equation with multiple delays The same assumptions as made in the single delay logistic equation may be applied hee as well. In the case of a system with n delays, the equation takes the fom n d ( t ) [ b ( t )] ( 4 ) i i i The case of two delays is investigated in this pape taking into consideation one of the two delays as a bifucation paamete. III. Stability Analysis As it is well known the delay logistic equation is non-linea. We fist lineaize it then poceed to detemine conditions fo the stability of both the single delay logistic equation as well as the logistic equation with two delays. A. Stability analysis of logistic equation with single delay Let the equilibium point (whee the gowth ate is zeo) be denoted by. Then So, thee ae two equilibium points 0 (descibing the initial inteval) and ( t ) ( t ) b (descibing the satuating intevals). Now afte the lineaization of the delay logistic equation about the fist equilibium point we eveal that 0 is unstable and this equilibium point is not of much inteest. The second equilibium point dy y ( t ) Its chaacteistic equation is theefoe, e 0 (5) The system will be stable if ( < 0 ) values of lie on the imaginay ais. Substituting i into (5) we find, i [co s( ) i sin ( ) ] 0 Afte equating eal pats in both sides, we get ( n ), n 0,,,... (6 ) And afte equating imaginay pats in both sides, we get s in ( ) 0 To find the fist solution we eplace n 0 0 is of geate inteest and afte lineaizing about it we get b.bifucation point is the value of the paametes fo which the in (6), so Fo 0 the chaacteistic equation is, which implies that <0 because is assumed positive and the system is stable. i.e., fo 0 the oots of the chaacteistic equation ae still negative (< 0 ).Theefoe, necessay and sufficient condition fo stability is, < ( 7 ) DOI: / Page

3 Bifucation Analysis fo the Delay Logistic Equation with Two Delays It is vey clea that if the system has no delays o 0, the system is always stable. B. Stability analysis of logistic equation with two delays At the fied points, ( t ) ( t ) ( t ), so we have 0 o b b We have seen in the pevious subsection that the fist equilibium is unstable and is not of much inteest. We investigate now the second value, and afte lineaizing the delay logistic equation about it we get dy [ b y ( t ) b y ( t )] (8) If b b delay system., then the second delay tem in (8) can be neglected and the analysis educes to the case of a single Conside the case when b b, the chaacteistic equation theefoe is e e 0 (9 ) The system will be stable if ( < 0 ) values of lie on the imaginay ais. Substituting i into (9), we find i [c o s ( ) i s in ( ) ] [c o s ( ) i s in ( ) ] 0 ( 0 ) Afte equating eal pat to zeo, we get ( C ), C Z ( ).Bifucation point is the value of the paametes fo which the And afte equating imaginay pat to zeo and using the value of obtained in (), we get the bifucation point at ( ) ( ) co s[ ] ( ) ( ) ( ) co s[ ] 0 ( ) Whee And ( ) co s[ ] ( ) ( ) ( 3) On simplifying (3) we get the bifucation point at sin [ ] ( ) ( 4 ) Theefoe, necessay and sufficient condition fo stability is, ( ) ( ) co s[ ]<,, > 0 ( 5 ) ( ) Substituting, we get the familia condition (7) DOI: / Page

4 Bifucation Analysis fo the Delay Logistic Equation with Two Delays IV. Bifucation Analysis In this section, it is shown that the delay logistic equation undegoes a Hopf bifucation at a citical value of the paametes gowth ate ( ) o delay ( ) fo the single delay case [7], and at a citical value of the paametes gowth ate ( ) o one of the two delays ( ) o ( ) fo the two delays case. A. Hopf bifucation analysis fo the single delay case Fom (7), at the citical point fo the single delay case,. Diffeentiating chaacteistic equation (5) with espect to the gowth ate, we get: d e d e Evaluating at citical point, d R e{ } > 0 ( 6 ) d 4 Similaly diffeentiating (5) with espect to the delay, we obtain: d e d e Evaluating at citical point, d R e { } > 0 ( 7 ) d 4. The tansvesality condition of the Hopf bifucation with espect to the gowth ate and time delay is satisfied in (6) and (7) espectively. Thus, logistic equation with single delay undegoes a Hopf bifucation at the citical point given by B. Hopf bifucation analysis fo the two delays case B. the bifucation paamete is the gowth ate ( ): Conside the logistic equation with two delays given by (9). Diffeentiating (9) with espect to the gowth ate we get, d e e d e e Evaluating at citical point ( ) ( ) co s[ ] ( ), we get d ( ) R e{ } > 0 ( 8 ) d Hence the tansvesality condition of the Hopf bifucation is satisfied with espect to the gowth ate. It is impotant to ealize that in this pape we assume that both of the delays ae equally impotant, so we cannot scale time to obtain one of the delays equal to. B. the bifucation paamete is the fist delay ( ): Diffeentiating (9) with espect to the fist delay we get, d e d ( e e ) Recall that fom equality of eal and imaginay pats to zeo in (0) we found, ( 9 ) DOI: / Page

5 Bifucation Analysis fo the Delay Logistic Equation with Two Delays c o s c o s s in s in Using these togethe with i, (9) becomes, d d i (c o s i s in ) [ ( c o s i ( s in )) (c o s i s in )] And on simplifying (0), we get d i d ( ) ( i )(co s ) ( i )(sin ) So the eal pat of () is, d ( co s sin ) R e{ } ( ) d [ ( ) co s sin ] [ co s sin ] Accoding to Hopf classical theoem, the system undegoes a Hopf bifucation at the citical value given by (4) if the value of d R e{ } d ( 0 ) ( ) evaluated at is not equal to zeo. It is vey clea that the denominato of () is always positive whateve the value of, so we confine ouselves to the numeato only. So eplacing the value of fom () and the citical value of the bifucation paamete fom (4), the numeato becomes c sc( ) c sc( ) c o s ( ) sin ( ) ( ) ( ) ( 3) Now it is clea that the denominato of (3) is always positive and in addition if, csc( ) 0 < < ( 4 ) ( ) The numeato of (3) will also be positive. Recall fom (4) that c sc( ) ( ) 0 < < ( ) ( 5), so inequality (4) becomes So () will be positive at the citical value of the bifucation paamete if the condition (5) is satisfied. Hence the tansvesality condition of the Hopf bifucation is satisfied with espect to the fist delay if (5) is hold. Thus, the two delay logistic equation undegoes Hopf bifucation with espect to the fist delay at the citical point given by (4). B.3 the bifucation paamete is the second delay ( ): Fom (4) the citical value of as a bifucation paamete is sin [ ] ( ) ( 6 ) Following the same manne, () will be positive at the citical value of the bifucation paamete given by (6) if the net condition is hold 0 < < ( ) (7 ) DOI: / Page

6 Bifucation Analysis fo the Delay Logistic Equation with Two Delays Hence the tansvesality condition of the Hopf bifucation is satisfied with espect to the second delay if (7) is hold. Thus, the two delay logistic equation undegoes Hopf bifucation with espect to the second delay at the citical point given by (6). V. Conclusion We have pefomed local stability analysis of the delay logistic equation with and without multiple time delays. Sufficient conditions to aid design wee also etacted. While the ideal system (without delay) is always pefectly stable, the actual system that we have consideed using delays undegoes a Hopf bifucation fo both paametes, the gowth ate and each of the two delays. The pape can be consideed as a base fo futue wok concening delay logistic equation with moe than two delays. Refeences [] E. H. Ait Dads and K. Ezzinbi (00), Boundedness and almost peiodicity fo some state- dependent delay diffeential equations, Electon. J. Diffeential Equations 00, No. 67, pp.-3. [] W. C. Allee (93), Animal Aggegations: A Study in Geneal Sociology, Chicago Univesity Pess, Chicago. [3] O. Aino and M. Kimmel (986), Stability analysis of models of cell poduction systems, Math. Modelling 7, [4] S. H. Stogatz, Nonlinea Dynamics and Chaos. Peseus Books, 994. [5] C. Sun, M. Han and Y. Lin, Analysis of stability and Hopf bifucation fo a delayed logistic equation, Chaos, Solitons and Factals, vol. 3, pp , 007. [6] R. L. Kitching, Time, esouces and population dynamics in insects, Austalian Jounal of Ecology vol., pp. 3-4, 997. [7] M. M. Rao and Peetish K. L., Stability and Hopf Bifucation Analysis of the Delay Logistic Equation, Conell univesity libay, axiv:.70v [q-bio.pe] 9 Nov 0. DOI: / Page

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