Global dynamics and bifurcation of a tri-trophic food chain model

Transcription

1 ISSN 746-7, England, UK World Journal of Modelling and Simulation Vol. 8 ) No., pp Global dnamics and bifurcation of a tri-trophic food chain model Tapan Kumar Kar, Abhijit Ghorai, Ashim Batabal Department of Mathematics, Bengal Engineering and Science Universit, Howrah 7, India Department of Mathematics, Shibpur Sikshalaa, Howrah 7, India Department of Mathematics, Ball Ghoshpara, Howrah 7, India Received June 8, Accepted Ma ) Abstract. This paper deals with the dnamics of a tritrophic food chain model composed of a pre, predator and a super predator. A discrete dela is introduced to the functional response term involved with the growth equation of predator and super predator to allow for a reaction. The main content of this paper is divided into two parts. In the first part we consider the model with densit dependent mortalit terms for the predator and the super predator and in the second part we consider the model without densit dependent mortalit terms. In the first part we analed the global stabilit of the interior equilibrium and the boundar equilibrium points b using a suitable Lapunov function and the Hopf bifurcations of the model with respect to the dela parameter. In the second part of the paper, equilibrium points are obtained and bifurcation is done with respect to two model parameters. Computer simulations are given to verif most of the analtical results. Kewords: food chain, pre-predator, dela, global stabilit, bifurcation Introduction Simple food chain models often displa rich nonlinear mathematical behavior, including numbers and stabilit of equilibrium states and limit ccles, which changes as the model parameters changes. Tritrophic food chain modelling provides challenges in the fields of both theoretical ecolog and applied mathematics. Eamining the parameter space of the model in one or more variables often performs analsis of food chain model. This approach is referred to as bifurcation analsis, and it provides powerful tools for concisel representing a large amount of information regarding both the number and stabilit of equilibrium states stead states) and limit ccles in a model. For simple models analtic techniques and isoclines analsis ma be useful, however, for comple model numerical computation methods are particularl useful. The behavior of the solutions of three level food chain models can be ver complicated. In this paper we focus our attention on a tritrophic food chain model consisting of a pre, predator and a super predator. The pre species obe the logistic law of growth, while the predators and super predators consume biomass according to the Holling tpe I and II response functions respectivel. Here we incorporate the assumption that handling for pre is negligible, whereas the super predator needs sufficient handling for predator. Both predator and super predator ehibit densit dependent mortalit rate. Limit ccles and their bifurcations are interesting dnamical features in a food chain model, but our attention here will be limited to equilibrium states and their bifurcations. The dnamic analsis of the predator-pre model plas an important role in mathematical biolog [, 5, 9,, 8 ]. Though man biologists believe that if unique positive equilibrium point of a predatorpre sstem is locall asmptoticall stable, then it is globall asmptoticall stable, however it is not alwas true. Josef and Hosef [6] found that a unique positive locall asmptoticall stable equilibrium point has at least one limit ccle surrounding the equilibrium point under suitable condition. Thus man mathematicians Corresponding author. address: tkar7@gmail.com. Published b World Academic Press, World Academic Union

2 World Journal of Modelling and Simulation, Vol. 8 ) No., pp tr to use some well known methods to find the conditions for global stabilit for the equilibrium point of predator-pre sstem. Prince et al. [] remarked that the stud of communit behavior with the help of mathematical models must be based on at least three trophic levels and hence more focus should be made to stud the comple behavior ehibited b the deterministic models consists of three and more trophic levels. Hastings and Powel [4] introduced continuous models of three species food chains. Etensions and applications of the Hastings and Powel [4] model were recentl studied b Varriale and Gomes [6] for the case of three species of fish and b Chattopadha and Sarkar [] for microbial sstem. Vaenas and Pavlou [7], and Rai and Sreenivasan [4] also made some variations of the Hastings and Powel model. Tritrophic food chain model also studied b Kunetsov and Rinaldi [8], and McCann and Yodis []. The main objective of this paper is to stud the global stabilit and Hopf bifurcation of the sstem based on some model parameters. Organiation of the paper is as follows. The model formulation and equilibrium of the sstems are given in Section and respectivel. Section 4 describes the global stabilit and Hopf bifurcation of the sstem with respect to the dela as bifurcation parameter. Some numerical simulations are also given in this section. In Section 5, we consider the model sstem having no densit dependent mortalit term and some basic qualitative properties are studied. Some simulations are also given to verif the analtical results. At the end a general conclusion is given in Section 6.. Formulation of the model In this paper we focus our attention on a tritrophic food chain consists of a pre, predator and super predator. The pre species obe the logistic law of growth, while the predator and super predator consume biomass according to Holling tpe I and II response functions respectivel. Here we incorporate the assumptions that handling for pre is negligible, whereas the super predator needs sufficient handling for predator. Both predator and super predator ehibit densit dependent mortalit rate. So our proposed model is: dn dn dn = rn N K ) a N N, = b N + αa N t τ)n a N N m + N b N, ) = b N + βa N t τ)n m + N t τ) b 4N, with N ), N ), N ), where, N t), N t) and N t) represents the densit of the pre, predator and super predator respectivel. Here r is the intrinsic growth rate and K is the carring capacit of the pre, a is inter-species interference coefficient of pre and predator species, b and b are the death rate of the predator and super predator respectivel, a is the predator s searching efficienc of the super predator, b and b 4 are the inter-specific death rate of the predator and super predator respectivel, m is the half saturation co-efficient, α and β are the conversion factors. A discrete dela τ ) is introduced to the functional response term involved with the growth equation of predator and super predator to allow for a reaction. The densit dependent mortalit terms b N and b 4N referred to as either self-limitation or the influence of predation[7, 5]. Equilibriums of the sstem We take the transformation for the state and variable as i = N i /K, µ = rt, i =,, and then the sstem Eq. ) takes the following form still denote µ as t) d = ) a, d = b +c t τ) d +e j, ) WJMS for subscription: info@wjms.org.uk

3 68 T. Kar & A. Ghorai & A. Batabal: Global dnamics and bifurcation of a tri-trophic food chain model where a = Ka r, b = b r, c = αka r d = f + g t τ) t τ)+e h,, d = a r, e = m K, f = b r, g = βa r, h = b 4K r, j = nk r. Possible equilibriums of the sstem Eq. ) are,, ),,, ), E = j+ab)/j+ac), c b)/j+ac), ) and E,, ), where = + a, = + a, = Here = / is a real root of the equation where g f + ae)) fe b c) j + ab) )e + + ae) ), or = he + + ae) ) d + a ). S + S + S + S 4 =, ) S = a dg f + ae)) + hj + ab) + ae), S = adg f + ae)) a def hc d) + ae) + hej + ab) + ae), S = dg f + ae)) adef + he j + ab) ehc d) + ae), S 4 = he c d) def. If g > f + ae), then S > and if df + hec > beh then S 4 <. Then the Eq. ) has at least one positive real root. Now if > fe/g f + ae)), then the sstem Eq. ) has an interior equilibrium point E,, ). Let us now investigate the global stabilit of the sstem Eq. ) about the interior equilibrium point E,, ) and the boundar point E j + ab)/j + ac), c b)/j + ac), ). Global stabilit and hopf bifurcation Before proving global stabilit we state the following lemma: Lemma. [] Let f be a nonnegative function define on [, ) such that f is integrable on [, ) and is uniforml continuous on [, ). Then lim t ft) = Before proving global stabilit we define m = c, m = ) jge min + e dge g + e) = + e) + e) m = m = ac, m =m = min dge + e) + e) = m 5 = m 5 = min ge t τ) + e) + e) je d + e ), m = h, dg + e), m = m 4 = cge + e), ) g = + e), m 44 = c, m 55 = E, m = m =, m 4 = m 4 =, m 5 = m 5 =, m 5 = m 5 =, m 4 = m 4 =, m 45 = m 54 =. We denote the matri M = m ij ) 5 5 whose elements are defined above. Theorem. Let m, m m m, m m m m ) m m, m m m m 44 m m 44, m m m m 4, m m m m 44 m 55 +m m m 5 m 44 m m m 44m 55 m m m 4 m 55 m m 4 m 5 m m 44 m 55 m m 44m m 5 are all positive, then the interior equilibrium point of the sstem Eq. ) is globall asmptoticall stable. WJMS for contribution: submit@wjms.org.uk

4 World Journal of Modelling and Simulation, Vol. 8 ) No., pp Proof. Define a Lapunav function V,, ) such that V,, ) = cp η) p )) dη + p η) + D t t τ s) ) ds + E gp η) p )) p η) t t τ dη + + log ) s) ) ds, 4) where p ) = a and p ) = +e. Here D and F are positive constants to be chosen later. Calculating the rate of change of V along the solution of the sstem Eq. ), we get dv = cp ) p )) ẋ + gp ) p )) ẋ + + p ) p ) )ẋ +D ) D t τ) ) + E ) E t τ) ) = c ) a ) + ge + e ) + e) b + c t τ) d + e j )+ ) f + g t τ) t τ) + e h )+D ) D t τ) ) +E ) E t τ) ) E jge + e + dge = D c) ) + + e) + e) dge ac ) ) + e) + e) + ) + ) + t τ) ) + Choose D = c/ and E = ) jge min + e) dge + e) = + e) ) ) h ) cge + e) ) ge t τ) + e) + e) ) t τ) ) D t τ) ). g je d ) + e) + e = c ) E ) h ) dge ac ) ) + e) + e) ) )+ cge + e) ge ) t τ) ) + t τ) + e) + e) ) t τ) ) D t τ) ) E t τ) ) m ) m ) m ) m m Therefore m 4 t τ) m 5 t τ) m 44 t τ) ) m 55 t τ) ). dv XT MX, where X T = {,,, t τ), t τ) }. The matri M is positive definite iff all the principle minors of M are positive. The principle minors of M are m, m m m, m m m m ) m m, m m m m 44 m m 44 m m m m 4 m m 44 m, m m m m 44 m 55 +m m m 5 m 44 m m m 44m 55 m m m 4 m 55 m m 4 m 5 m m 44 m 55 m m 44m m 5. From the assumption of the theorem it follows that is positive definite. Therefore we have dv XT MX υ s) + s) + s), 5) WJMS for subscription: info@wjms.org.uk

5 7 T. Kar & A. Ghorai & A. Batabal: Global dnamics and bifurcation of a tri-trophic food chain model where υ is the smallest eigen value of we have from Eq. 6): V,, ) + υ s) + s) + s) ds V ), ), )). 6) It follows from definition of V and Eq. 6) that i t), i =,, are uniforml bounded on [, ), impling the uniform boundedness of d /, d / and d / on [, ). B lemma and Eq. 6) we conclude that This completes the proof. lim it) i =, i =,,. t Again let us take m = c/, m = j, m = c/, m = m = ac/, m = m = ac/, m = m =, and m m m c A = m m m ac = ac m m m j c c Now we show that E is globall asmptoticall stable in the -plane. Theorem. The matri is positive definite if j > c+a ), and then the point E is globall asmptoticall stable in the -plane. Proof. Define a Lapunav function V, ) such that V,, ) = cp η) p )) p η) dη + log ) t + F s) )ds, t τ where p ) = a and F is a positive constant to be chosen later. Calculating the rate of change of V along the solution of the sstem Eq. ), we get dv = cp ) p )) ẋ + p ) )ẋ + F ) F t τ) ) = c ) a ) + ) + e) b + c t τ) j ) + ) + F ) F t τ) ) = F c) ) j ) ac ) ) + c ) t τ) ) F t τ) ). Let us choose F = c/ then dv = c ) j ) ac ) )+c ) t τ) ) c t τ) ) c ) j ) + ac + c t τ) c t τ) ) m ) m ) m t τ) ) m m t τ). c Therefore where dv XT AX, X T = {,, t τ) }. WJMS for contribution: submit@wjms.org.uk

6 World Journal of Modelling and Simulation, Vol. 8 ) No., pp The matri A is positive definite iff all the principle minors of are positive. From the theorem it follows that is positive definite. Therefore we have dv XT AX γ s) + s) 7) where γ are the smallest eigen values of A. We have from Eq. 7) V, ) + γ s) + s) ds V ), )). 8) It follows from definition of V and Eq. 8) that i t), i =, are uniforml bounded on [, ) impling the uniform boundedness of d / and d / on [, ). B lemma and Eq. 8) we conclude that This completes the proof. lim t i t) i =, i =,,. We now stud the Hopf bifurcation for the sstem Eq. ), using dela τ as bifurcation parameter. Let i = X i t) + i, i =,,, where X i t), is small perturbation about E,, ). Then from the sstem of Eq. ), we obtain The characteristic equation is where dx = X a X, dx = c X t τ) j d + e) )X d + e X, dx = ge + e) X t τ) h X. A = + h + j, B = h + C = h j d + e) Let us put λ = iωω > ) in Eq. 9) and we get Separating real and imaginar part it gives Squaring and adding, we get where λ + Aλ + Bλ + C + Dλ + E)e λτ =, 9) j d ) + e) + h ), ), D = ac + ged + e), E = ach + ged + e). ) C Aω ) + i ω + Bω) + E + idω)cos ωτ i sin ωτ). E cos ωτ + Dω sin ωτ = Aω C, Dω cos ωτ E sin ωτ = ω Bω, ) ω 6 + Q ω 4 + Q ω + Q =, ) Q = A B, Q = B AC D, Q = C E. The Eq. )) has a unique positive solution under an one of the three cases given below. Case : Q >, Q >, and Q <. WJMS for subscription: info@wjms.org.uk

7 7 T. Kar & A. Ghorai & A. Batabal: Global dnamics and bifurcation of a tri-trophic food chain model Case : Q <, Q <, and Q <. Case : Q >, Q <, and Q <. From Eq. ), we get tan ωτ = AD E)ω + BE CD)ω Dω 4 + AE BD)ω EC. and it gives τ n = ω arctan [ AD E)ω ] + BE CD)ω Dω 4 + AE BD)ω + nπ, EC ω where n =, ±, ±, ±,. We have now to show that dreλ) τ=τ >, dτ This will signif that there will be at least one eigenvalue with positive real part for τ > τ. Moreover, the condition for Hopf bifurcation is then satisfied ielding the required periodic solution. Now differentiating Eq. 9) with respect to τ, we get {λ + Aλ + B) τdλ + E)e λτ + De λτ + De λτ } dλ dτ ) dλ λ + Aλ + B = dτ λλ + Aλ + Bλ + C) + D λdλ + E) τ λ. = λdλ + E)eλτ Thus sign of { } dreλ) = sign of{re dλ dτ λ dτ ) } λ=iω ω = sign of B)ω B) + Aω C) ω Bω ) + Aω C) D D ω { +E D ω 6 = sign of +E +D A B))ω 4+E A B)ω +B E C D ACE } ) {ω Bω ) + Aω C) }D ω +. E ) dreλ) dτ τ=τ >, if A B > and B E C D ACE >. Therefore the transversalit condition holds, hence Hopf bifurcation occurs at ω = ω, τ = τ. Theorem. If E,, ) eists with A B >, B E C D ACE > and an one of Q >, Q >, Q <, or Q <, Q <, Q <, or Q >, Q <, Q <, holds then as τ increases from ero, there is a value τ such that the interior equilibrium E,, ) is asmptoticall stable when τ [, τ ) and unstable when τ > τ. Further, Eq. ) undergoes Hopf-bifurcation at E,, ), when τ = τ. To illustrate the result numericall we take a =.5, b =.5, c =, d =.8, e =., f =., g =, h =., j =.6. With these parameter s values sstem Eq. ) has an equilibrium point.7456,.85696, ). For the eistence of the positive equilibrium point E,, ), we must have > fe/g f+ae)). Here =.574 and fe/g f+ae)) = , thus > fe/g f+ae)) is tenable. Let us take another set of values of the parameters of the sstem Eq. ) as a =.5, b =, c = 5, d =, e =.9, f =., g =, h =, j =. Then we see that all the principle minors of the matri are positive. Thus the conditions of the Theorem hold, hence the sstem Eq. ) is globall stable under the above set of values of the parameters see Fig. ). Phase portrait of the sstem Eq. ), corresponding to different initial levels. It clearl indicates that the interior equilibrium point.7456,.85696, ) is globall asmptoticall stable. WJMS for contribution: submit@wjms.org.uk

8 h=, j=. Then we see that all the principle minors of the matri M are s the conditions of the Theorem hold, hence the sstem.) is globall Again if we take a=., b=.5, c=, j=, the equilibrium point the above World set Journal of values of Modelling of and the Simulation, parameters Vol. 8 ) see No., figure pp. 66-8). 7 becomes globall asmptoticall stable see figure ) portrait of Fig. the. Phase sstem portrait Fig. of.), the. sstem Shows.), that corresponding the equilibrium to to Fig. different. Showspoint thatinitial the , equilibrium levels. point , It.777, ) is glo different initial levels. It clearl indicates that the interior.777, ) is globall asmptoticall stable tes that the equilibrium interior point equilibrium asmptoticall.7456,.85696, point stable.7456, ) is.85696, ) is ptoticall stable. globall asmptoticall stable.4 Again if we take a =., b =.5, c =, j =, the equilibrium point ,.777, ) becomes globall asmptoticall stable see Fig. ). Fig. Shows that the equilibrium point ,.777, ) is globall asmptoticall stable. Now for the same values of parameters as for the first figure,.8 it is seen from the Theorem. that there eists a critical value of τ = τ = and losses its stabilit as τ crosses the critical value τ see Fig. 5). We have also.6given some graphical support in favors of our numerical results Sstem without densit dependent mortalit rate A ver natural and significant question arises: if predator and super predator do not have densit dependent mortalit rate, does the model still ehibit coeistence and have Fig.. a stablesolution positive equilibrium? curves. For this let us put j = and h = in Eq. ) and the sstem reduces to Now for the same values of parameters as for the first figure, it is seen f = b +c t τ) d. that there eists a critical value + e, d = f + g t τ) of τ = τ = t τ) + e. ) and E *, stabilit as τ crosses the critical value τ. We have also given some gra favors g f of + our ae) numerical results. geg f)c d) acef) d = ) a, d Possible equilibrium points are E,, ), E,, ), E b/c, c b)/ac, ) and E,, ) where = g f, = fe g f and = focus on local stabilit which is much more important foractual application dg f) So there eists unique positive equilibrium point if 5 ) g > f + ae), ) g f)c b) > acef. 4 Although global stabilit insures the survival of species, it is ver strong recruitment and b contrast, we 4. Case : τ = where The characteristic Eq. ) changes to λ + t λ + t λ + t =, 4) Fig. 4. This figure shows the cclic behavior of the sstem.) for τ = WJMS for subscription: info@wjms.org.uk

9 Now for the same values of parameters as for the first figure, it is seen fro. *. that there eists a critical value of τ = τ = and E, stabilit as τ crosses the critical value τ. We have also given some graph that the equilibrium 74 T. point Kar & , A. Ghorai & A. Batabal: favors of our numerical.777, Global dnamics results. ) is and globall bifurcation of a tri-trophic food chain model stable Fig.. Fig. Solution. Fig. Solution 4. This curves curves. figure shows Fig. the 4. The cclic cclic behavior of the of sstem the sstem.) for τ.) = for τ = Fig. 5. For τ =.96495> τ sstem.) become unstable. ame values of parameters as for the first figure, it is.5seen from the Theorem.5 * eists a critical value of τ = τ = and E,, ) losses its rosses the critical value τ. We have also given some graphical support in umerical results > τ = Fig. 5. For sstem τ = ) Fig. become > 6. τ For sstem unstable. τ.) = become < τ, population approach to their equilibrium unstable Sstem without densit dependent mortalit rate.4 t.6 = d +, t e) = dge + d..8.4 e) + + e) ac, t = dge +. e).6.8 Since,, and all the constants are positive, t >. Routh-Hurwit criteria state that all the roots of the A ver natural and significant question arises: If predator and super preda characteristic Eq. 4) have densit negativedependent real part if t > mortalit, t >, t rate, t t does >. We the nowmodel show that still t t ehibit t > coeisten ure shows where the t cclic, t and t behavior are instable their simplest of positive the form sstem equilibrium? as below.) for For τ = this let us put. j = and h = in.) reduces to.5 t = f g c b), t = f g c )g f ) + ac, t = f d = ) g g f)c b)., a Let P ) = c b, then the above epression changes to d t d = 4 f 6 g P 8 ), t = f g P 4 6 )g f 8 = ) b + ac + c, t = τ ) f g g f)p ),. + e Now τ = < τ, population approach to their d equilibrium values. g t τ ) t t t = = f. f g P )) f g P )g f + ac t )) f τ ) g + g e f)p ) ithout densit dependent mortalit rate = f g Possible g f))p f g + ac )P + ac. equilibrium points are E,,), E ), ral and significant question arises: If predator and super predator do not have,, E b c * * * endent mortalit WJMS rate, for contribution: does and submit@wjms.org.uk the Emodel *, still, ) ehibit coeistence and have a ive equilibrium? For this let where us put j = and h = in.) and the sstem Fig. 6. For τ = < τ, population approach to their equilibrium values

10 World Journal of Modelling and Simulation, Vol. 8 ) No., pp Thus we have ) t > iff f g P ) >, ) t t t > iff AP )) >. Observed that for the quadratic form AP )), we have A = f g {4acg f) + ac ) }. Therefore A >, since g > f. Also A can be written as Now A = f g { + ac ) 4ac g f))}. A g f ) = g f) <, since g > f. The roots of the quadratic equation AP )) are P± g [ ] = f + ac g f)) ) ± + ac ) 4ac g f)). Here we see that if > g f) >, then both the roots of AP )) are positive. Therefore t t t > if and onl if P ) < P or P ) > P +. Theorem 4. Assume that the positive equilibrium E,, ) eists, then if f g P ) > and P ) < P or P ) > P + the equilibrium is locall asmptoticall stable. Pre-predator models with constant parameters are often found to approach a stead state in which the species coeist in equilibrium. But if parameters used in the model are changed, other tpes of dnamical behavior ma occur and the critical parameter values at which such transitions happen are called bifurcation points. From ecological point of view bifurcations endanger the eistence of particular species in a food chain. When a stable stead state goes through a bifurcation will in general either lose its stabilit or disappear entirel. Even if the sstem ends up in another stead state the transition to that state will often involve the etinction of one or more levels of the food chain. On the other hand the entire sstem ma survive in a non-stationar state, but further bifurcation ma lead to local etinction of species. In order to preserve the sstem under consideration in its natural state, crossing bifurcation should be avoided and in doing so it is of great importance to determine the critical parameter values at which bifurcation occur. However in order to understand the general mechanisms leading to bifurcations in ecosstems much simpler models are needed. Bifurcation for the parameter a The Eq. 4) has two purel imaginar roots if and onl if t t = t for some value of a sa ā. There eist a unique ā such that t t = t. Therefore there is onl one value of a at which we have a Hopf bifurcation. Thus in the neighborhood of ā the characteristic Eq. 4) can t have real roots. For a = ā, we have λ + t )λ + t ) =. This equation has two purel imaginar roots and a real root as The roots are in general of the form λ = i t, λ = i t, λ = t. λ ā) = pā) + iqā), λ ā) = pā) iqā), λ ā) = t i ā). To appl the Hopf bifurcation theorem as stated in Marsden and Mckracken 976) [] we need to verif the transversalit condition dp da a=ā. Substituting λ ā) = pā) + iqā), in Eq. 4) and differentiating the resulting equation w.r.t. a, and then setting pā) = and qā) = t = q, we get WJMS for subscription: info@wjms.org.uk

11 76 T. Kar & A. Ghorai & A. Batabal: Global dnamics and bifurcation of a tri-trophic food chain model dp da q + t ) + dq da t q ) = t q t, dp da t q ) + dq da q + t ) = t q. where t, t, t is function of the bifurcation parameter a and Solving for dp dq da and da we have dp da = t, da = t, da = t. da a=ā = t t t t + t t ) t + t t. ) To establish Hopf bifurcation at, we need to show that Here dp da a=ā i.e. t t t + t t. t = c ), t = g f)c b) acef) + cg f) aeg g f) ), gg f) t = def [ ] + gec e) + dg f). Let us consider a set of values of the parameters as a =.5, b =, c =.7, d =.5, e =., f =., g =.9. Substituting these values, we get t t t + t t = Therefore Hopf bifurcation occurs for the parameter a at a = ā, make sense. Bifurcation for the parameter b To appl the Hopf bifurcation theorem for the parameter b we need to show that dp db b= b that is t t t + t t, where t, t, t are functions of b and db = t, db = t and db = t. Now t = f g, t = f g g f)) and t = g f)f g. Therefore, whenever interior equilibrium point eists and > g f) > holds then t t t + t t >. So, dp db b= b for > g f) >. Hence Hopf bifurcation occurs at b = b. 4. Case : τ = where For τ the characteristic equation becomes λ + Aλ Bλ + C + Dλ)e λτ =, 5) A = d + e), B = d + e), C = dge + e), D = ac + dge + e). Putting λ = iωω > ) in Eq. 5), we get Aω + C cos ωτ + Dω sin ωτ) + i ω Bω C sin ωτ + Dω cos ωτ) =. Equating real and imaginar parts, we get Squaring and adding, we get C cos ωτ + Dω sin ωτ = Aω, C sin ωτ Dω cos ωτ = ω Bω. 6) WJMS for contribution: submit@wjms.org.uk

12 equilibrium * * * E,, ) is locall asmptoticall stable when τ [,τo) * when τ > τ o. Further, sstem 5.) undergoes Hopf-bifurcation when τ = τ o. World Journal of Modelling and Simulation, Vol. 8 ) No., pp To illustrate the analtical results numericall for the sstem of equa the values ω 6 + of P ωthe 4 + parameters P ω + P = are, a=.5, b=, c=.7, d=.5, e=., f=., g interior equilibrium point.9769,.84654,.58) eist a where P = A + B, P = B D, P = C. The characteristic Eq. 5), has a pair of imaginar roots of the form ±iω. From Eq. 6) figure7). we get τ on corresponding to ω as τ n = [ AD C)ω tan ].4 BC ω BD + AC)ω + Dω + nπ, ω where n =, ±, ±, ±,. Now the { } { } dreλ) Redλ).8 sign = sign dτ λ=iω dτ λ=iω.6 { D ω 6 =sign + C + BD + A D )ω 4 + 4BC + A C )ω + B C } ω + Bω ) + A ω 4.4 )D ω +. C ) the transversalit condition holds, Hopf bifurcation occurs atτ =. τ. 5. If E ) eists, then as τ increases from ero, the interior The* term* in * the second bracket is positive. Therefore dreλ) dτ τ=τ >. Therefore the transversalit condition *,, holds, Hopf bifurcation occurs at τ = τ. * * * E*,, Theorem ) is locall 5. If E asmptoticall stable when τ [,τ,, ) eists, then as increases from ero, o) and unstable Fig.7. Population the interior converges equilibriumto Etheir, equilibrium, ) is value. o. Further, sstem locall asmptoticall 5.) undergoes stablehopf-bifurcation when τ[, o) and unstable when τ when = τ o. τ > τ o. Further, sstem Eq. ) undergoes * * * * It is interesting to observe that, the positive equilibrium E Hopf-bifurcation when τ = τ o.,, ) los strate the analtical To illustrate results the analtical numericall and results for a Hopfnumericall the sstem bifurcation for the of sstem equations occurs at of Eq. 5.), the critical ), let let value a = Now i the values of the parameters * * * of the parameters are a = are.5, a=.5, b =, c b=, =.7, c=.7, d increase = d=.5, e the = e=., value f f=., = of a, g g=.9. keeping =.9. Then other the parameters interior equilibrium fied, point then E, uilibrium point E.9769, E *.9769,.84654,.84654,.58) stabilit eist.58) from and stable instabilit see eist Fig. as and 7). a crosses stable see its critical value a = se ). E * Fig.7. Population Fig. 7. Population converges to their equilibrium value Fig.8. value. 8. Bifurcationfor for the the critical value ā a = = * * * * sting to observe that, the positive equilibrium E,, ) losses its stabilit f- bifurcation occurs at the critical value a = Now if we graduall It is interesting to observe that, the positive equilibrium E * * *, *, ) losses its stabilit and a Hopfbifurcation a, keeping occurs other at theparameters critical value ā fied, = then ENow if, we e value of, graduall ) achieves increase the value of a, keeping om instabilit otheras parameters a crosses fied, its then critical E value,, a ) achieves = stabilitsee fromfigures instabilit 8, as 9, a crosses its critical value ā = see Figs. 8 ). Fig. 9 shows that when a = < ā, the equilibrium point becomes unstable. Fig. shows that when a = > ā, the equilibrium point becomes stable. Similarl for8 the death rate B of the predator when > g f), then a Hopf-bifurcation occurs. We see that E,, ) achieves stabilit from instabilit as b crosses its critical value b = The Fig. shows that when b =.5896 < b, the equilibrium.5 point becomes unstable. The Fig. shows that when b = > b the equilibrium point becomes stable. The numerical stud presented here shows that, using parameter a or b as control parameter, WJMS for subscription: info@wjms.org.uk.5

13 figure shows that when a=.79978< a, the equilibrium.5 point becomes 78 T. Kar & A. Ghorai & A. Batabal: Global dnamics and bifurcation of a tri-trophic food chain model s figure shows that when a=.79978> a, the equilibrium. point becomes This figure shows that when a=.79978>.8 a, the equilibrium point becomes l for Fig.. the death Fig.. Bifurcation rate b of for for the Fig.. the predator criticalthis value when value b figure =.5896 * b > = shows Fig. that. when This figure b=.5896<b shows that when b, the = equilibrium.9 g f ), then a Hopf-bifurcation.8 unstable < b, the equilibrium.5 point becomes unstable.7 * We see that E *.6 * *.5,, ) achieves stabilit from instabilit as b crosses its alue b = see figures, and)..6 dela, remains.5 stable under certain conditions when the dela is less than the threshold value, otherwise Fig.9. This figure shows that when a=.79978< a, the equilibri unstable. critical value e b = see figures, and). *. Similarl for the death rate b of the predator when > g f ), then.8 * occurs. We see that E * * *,, ) achieves stabilit from instabilit to keep the population levels at a required state using the above control g =.9. For the above choices of parameters E,, ) is locall asmptoticall stable in the absence value of τ.5 = τ = and E,, ) losses its stabilit as crosses the critical value τ. We have also given some graphical support in favors of our.9 numerical results s see Figs. 4 6). The Fig shows the cclic behavior of the sstem Eq..7 ) for τ.6 = The Fig 5, for τ = the.5 equilibrium point.9769, stable ,.58) becomes stable. The Fig 6, for τ = > τ his figure shows that when a=.79978< * r the death rate Fig. 9. bthis of figure the predator shows that when Fig.. when a = This figure a <, the ᾱ, shows equilibrium that when point a= becomes > g f ) Fig., then. a This Hopf-bifurcation figure shows.79978> that when a a =, the equilibri. * the equilibrium point becomes stable. unstable > ā, the equilibrium point becomes sta- see that E * * *,, ) achieves stabilit from instabilit as b crosses its ble.4. Fig.. Bifurcation The for the numerical critical value stud b = presented here shows that, using parameter a o WJMS for contribution: submit@wjms.org.uk parameter, is possible to break unstable behavior of the sstem 5.) a stable state. Also it is possible to keep the population levels at a required above control. b = see figures, and) Fig...Bifurcation for the critical value b =.589 it is possible to break unstable behavior of the sstem Eq. ) and drive it to a stable state. Also it is possible Now we like to mention that the stabilit criteria of the sstem.8 without dela do not necessaril guarantee the stabilit of the sstem with dela. It has been shown that the positive equilibrium which is stable without the stable equilibrium become unstable. To illustrate the results numericall, let us choose a =.5, b =, c =.7, d =.5, e =., f =., of dela. Now for the same values of parameters, it is seen from the Theorem., that there eists a critical.4 Fig.. This figure shows that when b=.65896>b the equilibrium the equilibrium point.9769,.84654,.58) becomes unstable. 9

14 Fig.4. This figure shows the cclic behavior of the sstem 5.) for τ.9789 is figure shows that when b=.5896<b, the equilibrium point becomes = World Journal of Modelling and Simulation, Vol. 8 ) No., pp This figure shows Fig.. that This figure when shows b= that Fig.4. Fig >b when b This For = figure τ = the shows > equilibrium b the cclic < τ the point behavior equilibrium becomes of the sstem point Fig. 4. This figure shows the cclic behavior of 5.).9769, the for τ = the equilibrium point becomes.58) stable becomes stable. sstem 5.) for τ = his figure shows the cclic behavior of the sstem 5.) for τ rical stud presented here shows that, using parameter a or b as control it is possible to. break unstable behavior of the sstem 5.) and drive it to a.8 e. Also it is possible to keep the population levels at a required state using the.6 trol we like to mention that the stabilit criteria of the sstem. without dela do not.6 guarantee the.6 stabilit of the sstem with dela. It has been shown that the quilibrium which is stable without dela, remains stable.4.8 under certain.4.6 when the dela is less than the threshold value, otherwise the stable become unstable lustrate the results numericall, let us choose a=.5, b=, c=.7, d=.5, e=., * * * * =.9. For the above choices of parameters E,, ) is locall all For stable τ = in the absence < τ of dela. Fig.5. Now For for the τ = same values < of τ parameters, the equilibrium it point.9769, the equilibrium point.9769,.84654, Fig. 5. For τ = Fig.6. < τ thefor equilibrium τ = point Fig. > 6. τ For the τ equilibrium = point > τ.9769, the equilibrium.84654,.5 8) m becomes the Theorem stable..9769,., that.84654, there.58) eists becomes a critical becomes unstable. becomes value stable. of point τ =.9769, τ = , and.58) becomes unstable * ) losses its stabilit as τ crosses the critical valueτ. We have also given hical support 5 in Conclusion favors of our numerical results s s 4,5 and6)..8.8 as control,.6 it is possible to maintain the sstem in its stable state..8 = A tri-trophic food chain model is considered, with main regard to eistence of equilibria, their stabilit.6.4 and bifurcations. In most of the ecosstems, population of one species does not respond instantaneousl to.4 the interactions. with other species. To incorporate this idea in modeling approach, the dela models. have been developed. It is observed that the stabilit criteria of the sstem without dela do not necessaril guarantee the stabilit of the sstem with dela. It is also observed that using some other biological parameters From.4 the point of view of ecological managers, it ma be.6desirable to have a unique positive equilibrium which is. globall asmptoticall stable in order to keep sustainable development of the ecosstem. Keeping.4 this view in mind we have established the criteria of the global stabilit of the sstem Bifurcation is an important qualitative behavior to be studied with respect to some main parameter of the sstem. To preserve the sstem under consideration in its natural state, crossing 5 bifurcation 5 should 5 be 4avoided 45 5 or τ = > τ the equilibrium point.9769,.84654,.58) and in doing so it is ver much important to determine the critical parameter values at which bifurcation occur. unstable. There are still tremendous Fig.6. amount For ofτ work = need to be done > τ in the this equilibrium area. Parameters point are.9769, rarel constant,.84654, because the depend on environmental becomes unstable. conditions. We do not know, however, the detailed relationship between these parameters and environmental conditions. We assume these parameters are constant. Also it can be.4 WJMS for subscription: info@wjms.org.uk

15 8 T. Kar & A. Ghorai & A. Batabal: Global dnamics and bifurcation of a tri-trophic food chain model checked weather the Hopf bifurcating periodic solution is stable or not b using Centre manifold theorem. We leave it for future works. References [] I. Barbalat. Sstem d equations differentielle d oscillations non linearies. Revue Roumaine De Mathématiques Pures ET Appliquées, 959, 4: [] J. Chattopadha, R. Sarkar. Chaos to order preliminar eperiments with population dnamics models of three trophic levels. Ecological Modelling,, 6/): [] H. Freedman, S. Raun. Uniform persistence in functional differential equations. Journal of Differential Equations, 995, 5: 7 9. [4] A. Hastings, T. Powell. Chaos in a three species food chain. Ecolog, 99, 7: [5] S. Hsu, T. Huang. Global stabilit for a class of predator-pre sstems. The SIAM Journal on Applied Mathematics, 995, 55: [6] H. Josef, W. Josef. Multiple ccles for predator-pre models. Mathematical Biosciences, 99, 99: [7] C. Kohlmeier, W. Ebenhoh. The stabiliation role of cannibalism in a predator-pre sstem. The Bulletin of Mathematical Biolog, 995, 57: 4 4. [8] Y. Kunetsov, S. Rinaldi. Remarks on food chain dnamics. Mathematical Biosciences, 996, 4:. [9] W. Liu. Criterion of hopf bifurcation without using eigenvalues. The Journal of Mathematical Analsis and Applications, 994, 8: [] J. Marsden, M. McCracken. The Hopf Bifurcation and Its Applications. Springer, New York, 976. [] K. McCann, P. Yodis. Bifurcation structure of a three species food chain model. Theoretical Population Biolog, 995, 48: 9 5. [] A. Nindjin, M. Ai-Alaoui, M. Cadivel. Analsis of a predator-pre model with modified leslie-gower and holling tpe ii schemes with dela. Nonlinear Analsis, 6, 7. [] P. Prince, C. Bouton, et al. Interactions among three trophic levels: influence of plants on interactions between insect herbivores and natural enemies. The Annual Review of Ecolog, 98, : [4] V. Rai, R. Sreenivasan. Period doubling bifurcations leading to chaos in a model food chain. Ecological Modelling and Sstems Ecolog, 99, 69/): [5] J. Steele, E. Henderson. The role of predation in plankton models. Journal of Plankton Research, 99, 4: [6] M. Varriale, A. Gomes. A stud of a three species food chain. Ecological Modelling and Sstems Ecolog, 998, ): 9. [7] D. Vaenas, S. Pavlou. Chaotic dnamics of a microbial sstem of coupled food chains. Ecological Modelling and Sstems Ecolog,, 6/): [8] L. Wang, R. Xu, X. Tian. Global stabilit of a predator-pre model with stage structure for the predator. World Journal of Modelling and Simulation, 9, 5: 9. [9] W. Wang, Z. Ma. Harmless delas for uniform persistence. The Journal of Mathematical Analsis and Applications, 99, 58: [] R. Xu, M. Chaplain. Persistence and global stabilit in a delaed predator-pre sstem with michaelis-menten tpe functional response. Applied Mathematics and Computation,, : [] X. Zang, R. Xu, Q. Gan. Periodic solution in a delaed predator pre model with holling tpe III functional response and harvesting term. World Journal of Modelling and Simulation,, 7: 7 8. WJMS for contribution: submit@wjms.org.uk