THE most studied topics in theoretical ecology is the

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1 Engineering Letters :3 EL 3_5 Stabilit Nonlinear Oscillations Bifurcation in a Dela-Induced Predator-Pre Sstem with Harvesting Debaldev Jana Swapan Charabort Nadulal Bairagi Abstract A harvested predator-pre sstem that incorporates feedbac dela in pre growth rate is studied Considering dela as a parameter we investigate the effect of dela on the stabilit of the coeisting equilibrium It is observed that there eists a critical length of the dela parameter below which the coeistence equilibrium is stable above which it is unstable A Hopf bifurcation eists when dela crosses the critical value B appling the normal form theor the center manifold theorem we determine the eplicit formulae that demonstrate the stabilit direction of the bifurcating periodic solutions Computer simulations have been carried out to illustrate different analtical findings Our simulation results indicate that the Hopf bifurcation is supercritical the bifurcating periodic solutions are unstable for the considered parameter values The sstem however can be stabilized from its unstable oscillator state onl b lowering the intrinsic growth rate of pre population Inde Terms predator-pre model harvesting dela stabilit oscillations direction of Hopf bifurcation I INTRODUCTION THE most studied topics in theoretical ecolog is the predator-pre interaction its possible outcomes This is because of its all over eistence in the natural world Various mathematical models have been proposed from different biological point of views to stud the interaction between pre predator 1] 3] Harvesting is an old technique used for biological resource eploitation However overeploitation is the most responsible factor to the persistence of harvested stocs causes not onl economic loss but also changes the ecosstem structure functioning 4] 9] Recentl Bairagi et al 1] considered the following predator-pre model with harvesting: d dt = r1 α a+ d dt = αb a+ d qe be+l 1 t t be respectivel the pre predator densities at time t It is assumed that the pre population grows logisticall to its carring capacit with intrinsic growth rate r b d be respectivel the conversion efficienc natural death rate of predator Functional response of predator is assumed to follow ratio-dependent Tpe II form with maimum pre consumption rate α half-saturation constant a Predator is harvested following the Michaelis- Menten tpe catch rate with catchabilit coefficient q Manuscript received August 1 1 This wor was supported b DRS Programme UGC No F51//DRS/1 SAP-1 D Jana N Bairagi are with the Centre for Mathematical Biolog Ecolog Department of Mathematics Jadavpur Universit Kolata-73 INDIA nbairagi@mathjdvuacin S Charabort is with Baruipara High School Halisahar 4 Parganas N West Bengal India harvesting effort E All parameters including b l are assumed to be positive Using Melniov s method the observed that the sstem 1 ehibits heteroclinic bifurcations The also showed that the sstem ma ehibit monostabilit bistabilit tristabilit depending on the initial values of the sstem populations the harvesting effort Dela is frequentl used in a predator-pre model to represent the biological process more accuratel Predatorpre models with discrete dela ehibit different interesting dnamics such as the eistence of Hopf bifurcation Bogdanov-Taens bifurcation 11] 1] An ecellent review wor of predator-pre models with single discrete dela can be seen in 13] Simultaneous effects of harvesting dela on predator-pre sstem were studied b several researchers 14] 18] The general observation is that the time dela maes a stable equilibrium unstable harvesting on the other h helps to regain the stabilit In this stud we consider a discrete dela in the specific growth rate of pre to incorporate the effect of densit dependence feedbac mechanism which taes τ units of time to respond to changes in the pre population 19] Thus the model sstem 1 reduces to the following dela-induced predator-pre model with harvesting: d dt d dt The initial condition is = r1 tτ = αb a+ d α a+ qe be+l ] φ = ψ 1 φ φ = ψ φ φ τ ] Dela in the sstem is based on the assumption that in the absence of predators the pre satisfies the delaed logistic equation ] The objective is to stud the dnamic behavior of the sstem determine the direction stabilit of the bifurcating periodic solutions if an The organization of the paper is as follows: In Section II we perform the local stabilit of the model sstem Section III deals with the direction stabilit of bifurcating periodic solutions Rigorous numerical simulations of the model sstem are performed in the Section IV Finall a brief discussion is presented in the Section 4 II LOCAL STABILITY Coeistence of species are given utmost importance in a predator-pre interaction we are therefore onl interested in the coeistence equilibrium of the sstem The coeistence equilibrium point of the above sstem is given b E the equilibrium pre densit is given Revised online publication: 3 August 1

2 Engineering Letters :3 EL 3_5 TABLE I PARAMETER VALUES USED IN THE MODEL SIMULATION 6 Parameter Default value r 1 1 α 7 a 1 b 45 d 4 q 1 b 1 E 1 l 4 4 Predator Isocline E * λ 1 = 931 λ = 34 E Pre Isocline λ 1 = 9878 b the positive root of the cubic equation with p p + p 3 + p 4 = p 1 = ablr ] p = r αb d l ab r l + abe abb p 3 = r E ar α + abder + b lar α ] + d l + aqe p 4 = E abb rar α qar α bd ar α α bb ] Observe that p 1 is alwas positive p 4 is negative if ar > α Thus the above cubic equation has at least one positive root when ar > α The equilibrium predator densit is given b r 1 = α ar 1 Note that will be feasible if ar ar α < < To observe the number of feasible coeistence equilibrium points we plot the pre predator isoclines of the sstem 1 in the Fig 1 This figure confirms that there eists two positive equilibrium points E E for the parameter set given in the Table 1 Eigenvalues evaluated at these equilibrium points show that the equilibrium point E has two negative eigenvalues thus stable The eigenvalues corresponding to the equilibrium point E 1 are of opposite signs therefore unstable The rest of the paper deals with the stable equilibrium point E Let Xt = t Y t = t are the perturbed variables Linearizing the sstem at E we get the linear sstem as follows: dx dt = α X a + α Y a + r Xtτ dy dt = αab X a + + qel be+l αab a + ]Y 3 The characteristic equation of the corresponding variational matri is given b λ + Aλ + B + Cλ + De λτ = 4 λ = Fig 1 The pre predator isoclines of the model sstem 1 There eists two equilibrium points shown b bullets pre predator isoclines intersect Equilibrium values the corresponding eigenvalues are also mentioned Parameters are as in the TABLE 1 A = ] α1ab a + + qel be+l B = α qel a + be+l αab C = r ] D = r qel be+l αab a + a + ] We now discuss two following cases + α ab a + 4 Case 1: Instantaneous feedbac mechanism τ = In the case of instantaneous feedbac response of pre population the value of τ becomes zero In this case the equation 4 becomes λ + A + Cλ + B + D = 5 All roots of the equation 5 have negative real parts if onl if H 1 A + C > B + D > Therefore the equilibrium point E is locall asmptoticall stable when H 1 holds Then the following theorem is true Theorem 1 The coeistence equilibrium E of the sstem 1 is locall asmptoticall stable in absence of dela if H 1 holds Case : Delaed feedbac mechanism τ We first reproduce some definitions given b 1] ] Definition 1 The equilibrium E is called asmptoticall stable if there eists a δ > such that sup τ θ ψ 1 θ ψ θ ] < δ Revised online publication: 3 August 1

3 Engineering Letters :3 EL 3_5 implies that lim t t t = t t is the solution of the sstem which satisfies the prescribed initial condition Definition The equilibrium E is called absolutel stable if it is asmptoticall stable for all delas τ conditionall stable if it is stable for τ in some finite interval For the dela-induced sstem the interior equilibrium E will be asmptoticall stable if all the roots of the corresponding characteristic equation 4 have negative real parts But the classical Routh-Hurwitz criterion cannot be used to discuss stabilit of the sstem since equation 4 is a transcendental equation has infinitel man eigenvalues To determine the nature of the stabilit we require the sign of the real parts of the roots of the equation 4 We start with the assumption that E is asmptoticall stable in the case of non-delaed sstem Case 1 then we find conditions for which E is still stable in presence of dela 3] B Rouche s Theorem 4] the continuit in τ the transcendental equation 4 has roots with positive real parts if onl if it has purel imaginar roots From this we shall be able to find conditions for all eigenvalues to have negative real parts Let λτ = ξτ + iωτ ξ ω are real As the interior equilibrium E of the non-delaed sstem is stable we have ξ < B continuit if τ > is sufficientl small we still have ξτ < E is still stable The change of stabilit will occur at some values of τ for which ξτ = ωτ ie λ will be purel imaginar Let τ be such that ξτ = ωτ = ω so that λ = iωτ = iω In this case the stead state loses stabilit eventuall becomes unstable when ξτ becomes positive Now iω is a root of 4 if onl if ω + iaω + B + icω + De ωτ = Equating the real imaginar parts of both sides we get This leads to Dcosω τ + Cω sinω τ = ω B Cω cosω τ Dsinω τ = Aω 6 ω 4 C A + B]ω + B D = 7 This equation has no positive roots if the following conditions are satisfied: H A C B > B D > Hence all roots of the equation 7 will have negative real parts when τ if conditions of the Theorem 1 H are satisfied Let H 3 B D < If Theorem 1 H 3 hold then the equation 7 has a unique positive root ω Substituting ω into 6 we have τ n = 1 ] D ACω cos BD ω D + C ω + nπ n = 1 ω Let H 4 C A + B > B D > C A + B > 4B D If Theorem 1 H 4 hold then 7 has two positive roots ω+ ω Substituting ω ± into 7 we get τ ± = 1 D ACω cos ] ± BD ω ± D + C ω± + π = 1 ω ± If λτ be the root of 4 satisfing Reλτ n = respectivel Reλτ ± = Imλτ n = ω respectivel Imλτ ± = ω ± we get d dτ Reλ] τ=τ ω=ω = A C Bω +ω 4 C ω +D > b H Similarl we can show that d dτ Reλ] > τ=τ + ω=ω+ d dτ Reλ] < τ=τ ω=ω From Corollar 4 in Ruan Wei 5] we have the following conclusions Theorem Assume τ conditions of the Theorem 1 are satisfied Then the following conclusions hold: i If H hold then the equilibrium E is asmptoticall stable for all τ ii If H 3 hold then the equilibrium E is conditionall stable It is locall asmptoticall stable for τ < τ unstable for τ > τ Furthermore the sstem 1 undergoes a Hopf bifurcation at E when τ = τ τ = 1 ] D ACω cos BD ω D + C ω iii If H 4 holds then there is a positive integer m such that the equilibrium is stable when τ τ + τ τ+ 1 τ m τ+ m unstable when τ τ + τ τ+ 1 τ 1 τ+ m τ m τ+ m Further more the sstem undergoes a Hopf bifurcation at E when τ = τ + = 1 III DIRECTION AND STABILITY OF HOPF BIFURCATION In the previous section we obtain the conditions under which a famil of periodic solutions bifurcate from the coeistence equilibrium E at the critical value τ = τ As pointed out in Hassard et al 6] it is interested to determine the direction stabilit period of the periodic solutions bifurcating from the positive equilibrium Following the ideas of Hassard et al we derive the eplicit formulae for determining the properties of the Hopf bifurcation at the critical value τ b using the normal form the center manifold theor Throughout this section we alwas assume that the sstem undergoes a Hopf bifurcation at the positive equilibrium E for τ = τ then ±iω is the corresponding purel imaginar roots of the characteristic equation at the positive equilibrium E Let 1 = = i = i τt τ = τ + µ Dropping the bars for simplification of notations sstem Revised online publication: 3 August 1

4 Engineering Letters :3 EL 3_5 is transformed into an functional differential equation FDE in C = C ] R as ẋt = L µ t + fµ t 8 t = 1 T R L µ : C R f : R C R are given b a11 a L µ φ = τ + µ 1 φ1 a 1 a φ 9 b11 φ1 + τ + µ φ a 11 = a 1 = c1 fµ φ = τ + µ c α a + α a + a 1 = αab a + a = qel be+l αab a + b 11 = r c 1 = rφ1φ1 αφ1φ aφ +φ 1 c = αbφ1φ aφ +φ 1 qlφ b E 1 B Riesz representation theorem there eits a function ηθ µ of bounded variation for θ ] such that L µ φ = dηθ µφθ for φ C 11 In fact we can choose a11 a ηθ µ = τ + µ 1 δθ a 1 a b11 τ + µ δθ δ is the Dirac delta function For φ C 1 ] R define dφθ θ Aµφ = dθ dηµ sφs θ = Rµφ = Then sstem 8 is equivalent to θ fµ φ θ = ẋt = Aµ t + Rµ t 13 t θ = t + θ for θ ] For ψ C 1 ] R define A dψs s 1] ψs = ds dηt t ψt s = a bilinear inner product ψs φθ = ψφ θ ξ= ψξ θdηθφξdξ 14 ηθ = ηθ Then A A are adjoint operators B the discussion in section II we now that ±iω τ are eigenvalues of A Thus the are also eigenvalues of A We first need to compute the eigenvalues of A A corresponding to iω τ iω τ respectivel Suppose that qθ = 1 β T e iωτθ is the eigenvector of A corresponding to iω τ then Aqθ = iω τ qθ It follows from the definition of A that iω a τ 11 b 11 e iωτ a 1 q = a 1 iω a Thus we can easil obtain β = iω q = 1 β T αab a + qel be+l + αab a + Similarl let q s = D1 β T e iωτs is the eigenvector of A corresponding to iω τ B the definition of A we can compute q s = D1 β e iωτs = D 1 α a + iω + qel be+l αab a + e iωτs In order to assure q s qθ = 1 we need to determine the value of D From 14 we have q s qθ = D1 β 1 β T θ ξ= 1 β T e iωτξ dξ = D1 + β β = D D1 β e iωτξθ dηθ 1 β θe iωτθ dηθ1 β T } 1 + β β τ r ] e iωτ Thus we can choose D as 1 D = 1 + β β τr e iω τ In the remainder of this section we use the same notations as in 7] We first compute the coordinates to describe the center manifold C at µ = Define zt = q t Wt θ = t θ Reztqθ} 15 On the center manifold C we have Wt θ = Wzt zt θ Wz z θ = W θ z + W 11θz z + W θ z + W 3θ z z z are local coordinates for the center manifold C in the direction of q q Note that W is real if t is real We onl consider real solutions For solution t C of 13 since µ = we have żt = iω τ z + q f Wz z + Rezqθ} def iω τ z + q f z z Revised online publication: 3 August 1

5 Engineering Letters :3 EL 3_5 We rewrite this equation as żt = iω τ zt + gz z gz z = q f z z = g z + g 11z z + g z + g 1 z z + 17 It follows from that t θ = Wt θ + Reztqt} = W θ z + W 11θz z + W θ z + 1 βt e iωτθ z + 1 β T e iωτθ z + 18 It follows together with 1 that gz z = q f t = τ D1 β r 1t 1t = τ Dr α1tt a t+ 1t αb 1t t a qlt t+ 1t b E + W 1 z + z + z + W 1 z } + W 1 z + W 1 11 z z + W 1 z + } + W 1 11 z z ze iωτ + ze iωτ + β b 1τ Dα a z + z + W 1 z + W 1 1 z 11 z z + W } + τ Dβ d βz + β z + W z } + W z 11 z z + W + τ Dβ ql b βz + E β z + W z + W 11 } z z z + W + 19 Comparing the coefficients with 17 we have g = τ D reiωτ α a W 1 }] + β αb a W 1 d W β qlβ b E + e g 11 = τ D reiωτ iωτ α a W β αb a W 1 11 d W qlβ β 11 b E g = τ D reiωτ α a W 1 + β αb a W 1 d g 1 = τ Dr W W 1 1 W + eiωτ }] W β ql β b E }] } + W 1 11 τ Dβ ql eiωτ b E W 11 W β + β } Since there are W θ W 11 θ in g 1 we still need to compute them From we have Ẇ = t ż z q AW Re q = f q θ} θ AW Re q f q θ} + f θ = def AW + Hz z θ 1 Here Hz z θ = H θ z + H 11θz z + H θ z + Substituting the corresponding series into 1 comparing the coefficients we obtain A iω τ W θ = H θ AW 11 θ = H 11 θ From 1 we now that for θ Hz z θ = q f qθ q f θ = gz zqθ ḡz z qθ Comparing the coefficients with we have 3 4 H θ = g qθ ḡ qθ 5 H 11 θ = g 11 qθ ḡ 11 qθ 6 From 3 5 the definition of A it follows that Ẇ θ = iω τ W θ + g qθ + ḡ qθ Notice that qθ = 1 β T e iωτθ hence W θ = ig ω τ qe iωτθ + iḡ 3ω τ qe iωτθ + E 1 e iωτθ 7 E 1 = E 1 1 E 1 R is a constant vector Similarl from 3 6 we obtain W 11 θ = ig 11 qe iωτθ + iḡ 11 qe iωτθ + E ω τ ω τ 8 E = E 1 E R is a constant vector In what follows we shall see appropriate E 1 E From the definition of A we obtain dηθw θ = iω τ W H 9 dηθw 11 θ = H 11 3 ηθ = η θ B 1 we have H = g q ḡ q + τ re iωτ α a W 1 31 Revised online publication: 3 August 1

6 Engineering Letters :3 EL 3_5 H 11 = g 11 q ḡ 11 q + τ reiω τ +e iω τ α a W 1 11 Substituting 7 31 into 9 noticing that iω τ I e iωτθ dηθ q = iω τ I e iωτθ dηθ q = we obtain iω τ I eiωτθ dηθ E 1 re iωτ = τ α a W 1 This leads to iω a 11 b 11 e iωτ a 1 E a 1 iω a 1 re iω τ = α a W 1 It follows that E 1 = A E 1 1 = A reiω τ α a W1 a 1 iω a 3 iω a 11b 11e iω τ reiω τ α a W1 a 1 A = iωa11b11eiω τ a 1 a 1 iω a Similarl substituting 8 3 into 3 we get a11 b 11 a 1 It follows that a 1 a = E 1 = B re E = B E α a W 1 11 reiω τ +e iω τ iω τ +e iω τ α a W1 11 a 1 a a 11b 11 reiω τ +e iω τ α a W1 11 a 1 B = a11b11 a1 a 1 a Thus we can determine W θ W 11 θ from 7 8 Furthermore g 1 in can be epressed b the Population a Time b Fig These figures show that the coeistence equilibrium point E is locall asmptoticall stable when τ = Fig a is the time evolutions of the model sstem Fig b is the corresponding phase plane Parameters are as in the TABLE 1 parameters dela Thus we can compute the following values: c 1 = i g g 11 g 11 g + g 1 ω τ 3 µ = Rec 1} Reλ τ } β = Rec 1 T = Imc 1} + µ Imλ τ } ω τ which determine the qualities of bifurcating periodic solutions in the center manifold at the critical value τ we state the following theorem Theorem 3 µ determines the direction of the Hopf bifurcation: if µ > µ < then the Hopf bifurcation is supercritical subcritical the bifurcating periodic solutions eit for τ > τ τ < τ β determines the stabilit of the bifurcating periodic solutions: the bifurcating periodic solutions are stable unstable if β < β > T determines the period of the bifurcating periodic solutions: the period increases decreases if T > < IV NUMERICAL RESULTS In this section we give some numerical simulations to illustrate the analtical results observed in the previous sections For illustration purpose we consider the parameter values as in the TABLE 1 Initial value is considered as 1 5 for each simulation it is indicated b empt circle the coeistence equilibrium E is shown b bullet When τ = the parameter set satisfies all conditions of the Theorem 1 therefore trajectories eventuall reach to the coeistence equilibrium point E indicating stabilit of the sstem around the coeistence equilibrium Fig When τ one can compute from Theorem ii ω = 941 τ = Therefore the coeistence equilibrium E is locall asmptoticall stable for τ < τ = unstable for τ > τ = Time evolutions of the sstem for τ = 15 is shown Revised online publication: 3 August 1

7 Engineering Letters :3 EL 3_5 a 3 d a 6 b 1 1 Population b e 1 Time c Time c Time f 65 E * E * τ 1 5 Unstable region Stable region d 64 r 6 3 e Fig 3 Conditional stabilit of the sstem Figs in the first panel a-c in the second panel d-f depict the time evolutions phase portraits of the sstem for τ = 15< τ = τ = 17> τ = respectivel Figs a c show that the sstem is stable when the dela is less than its critical value Figs d f depict that the sstem is unstable when the dela eceeds its critical value Parameters are as in the TABLE 1 6 c r 35 Fig 5 Behavior of the sstem with respect to r Figs 5a 5b are respectivel the time series phase diagram of the sstem with τ = 17 r = 9 These figures indicate that the sstem returns to its stable coeistence equilibrium when the value of r is lowered Fig 5c is the stabilit region of the sstem in r τ parameter plane Bifurcation diagrams of pre predator populations with respect to the intrinsic growth rate r are depicted in the Figs 5d 5e respectivel All other parameters are as in the Fig 3f r b τ τ 15 Fig 4 Bifurcation diagrams of the sstem with respect to τ Fig a is drawn in the three-dimensional space τ Fig b is its projection on the two-dimensional plane τ Fig c is the same on the τ plane These figures show that the coeistence equilibrium is stable for τ < unstable for τ > a Hopf bifurcation eists at τ = τ = Other parameters are as in the TABLE 1 a in the Figs 3a 3b for pre predator populations τ respectivel The corresponding phase diagram is depicted in the Fig 3c These figures confirm that the sstem is locall asmptoticall stable when the length of dela remains below its critical value τ Similar figures for the dela τ = 17 higher than its critical value are plotted in the Figs 3d- 3f These figures show that populations fluctuate around the coeistence equilibrium when length of dela eceeds its critical value depicting the instabilit of the sstem Sstem behavior can be demonstrated more prominentl if we plot the bifurcation diagram in the three-dimensional space τ Fig 4a shows that the coeistence equilibrium is stable for τ < shown b blac solid line but the instabilit sets in when τ > shown b colored ccles Observe that the amplitude of oscillations increases with the length of dela The behavior of each population with varing τ can be observed if we tae the projection of the three-dimensional figure on the two-dimensional plane Figs 4b 4c depict the bifurcation diagrams of the pre predator populations in the τ τ planes respectivel Using Theorem 3 one can determine the values of µ β T as µ = β = 34 T = 63 Since µ > β > the Hopf bifurcation is supercritical unstable Also period of the bifurcating periodic solution increases with τ as T > To verif the sensitivit of the sstem to other parameters Revised online publication: 3 August 1

8 Engineering Letters :3 EL 3_5 we perform etensive numerical simulations with respect to each parameter It is observed that the sstem is sensitive onl to the parameter r the intrinsic growth rate of pre population Oscillations can be reduced completel the sstem returns to stable state if the value of r is reduced Figs 5a 5b show that trajectories converge to the coeistence equilibrium E when r is lowered to 9 from 1 eeping τ above its critical value To observe the interrelationship between r τ we have determined the stabilit region in r τ parameter plane Fig 5c The line separating the stable unstable regions is the Hopf bifurcation line This figure indicates that the sstem can tolerate longer dela for lower value of r Bifurcation diagrams of pre population Fig 5d predator population Fig 5e indicate that both populations go to etinction when r = Population densities then sta in stable state for long higher values of r before it becomes unstable at τ = 985 V DISCUSSION In this paper we have studied the effects of dela on an predator-pre interaction with predator harvesting The dela is proposed in the densit-dependent pre growth term on the assumption that a fied time τ is elapsed b the pre population before it affects on the per capita growth rate This stud in fact is an etension of the wor of Bairagi et al 1] the studied the sstem in absence of dela We have obtained sufficient conditions on the sstem parameters for which the dela-induced sstem is asmptoticall stable around the coeistence equilibrium for all values of the dela parameter if the conditions are not satisfied then there eists a critical value τ of the dela parameter below which the sstem is stable above which the sstem is unstable B appling the normal form theor the center manifold theorem the eplicit formulae which determine the stabilit direction of the bifurcating periodic solutions have been determined Our analtical simulation results show that when τ passes through the critical value τ the coeistence equilibrium E losses its stabilit a Hopf bifurcation occurs ie a famil of periodic solutions bifurcate from E Also the amplitude of oscillations increases with increasing τ Thus the quantitative level of abundance of sstem populations depends cruciall on the dela parameter if the feedbac dela eceeds the critical value τ For the considered parameter values it is observed that the Hopf bifurcation is supercritical the bifurcating periodic solutions are unstable From management point of view it is important to maintain the sstem population at stable state for ecosstem-based sustainable harvesting Our simulation studies show that the unstable oscillator sstem caused due to higher feedbac dela can be made stable onl when the biological growth rate of pre r is lowered The sstem is not sensitive to other parameters including the harvesting effort E Thus our stud sas against the traditional concept 14] 16] that time dela maes a stable equilibrium unstable harvesting helps to regain the stabilit REFERENCES 1] M L Rosenzweig R H MacArthur Graphical representation stabilit conditions of predator-pre interactions American Naturalist vol 47 pp ] R Arditi L R Ginzburg Coupling in predator-pre dnamics: ratio-dependence Jr Theor Biol LR vol 139 pp ] D Alstad Basic Populas Models of Ecolog Prentice Hall Inc NJ 1 4] R A Mer B Worm Achieving sustainable use of renewable resources Nature vol 43 pp ] FAO Fisheris Development The state of the world fisheries aquaculture FAO Rome 4 6] C W Clar Mathematical Bioeconomics: The Mathematics of Conservation John Wile & Sons New Jerse 1 7] N Bairagi S Chaudhur J Chattopadha Harvesting as a disease control measure in an eco-epidemiological sstem -a theoretical stud Mathematical Biosciences vol 17 pp ] S Charabort S Pal N Bairagi Predator-pre interaction with harvesting: mathematical stud with biological ramifications Appl Math Analsis vol 36 pp ] S Charabort S Pal N Bairagi Management analsis of predator-pre fisher model Nonlinear Studies vol 19 no pp ] N Bairagi S Charabort S Pal Heteroclinic Bifurcation Multistabilit in a Ratio-dependent Predator-Pre Sstem with Michaelis-Menten Tpe Harvesting Rate Lecture Notes in Engineering Computer Science: Proceedings of The World Congress on Engineering 1 WCE Jul 1 London UK vol1 pp ] D Xiao S Ruan Global dnamics of a ratio-dependent predator-pre sstem J Math Biol vol 43 pp ] M MacDonald Biological Dela Sstems: Linear Stabilit Theor Cambridge Universit Press Cambridge ] S Ruan On nonlinear dnamics of predator-pre models with discrete dela Math Model Natural Phenom vol 4 pp ] F Brauer Stabilit of some population models with dela Math Biosci vol 33 pp ] A Martin S Ruan Predator-pre models with dela pra harvesting J Math Biol vol 43 pp ] D Xiao S Ruan Bogdanov-Taens bifurcations in predator-pre sstems with constant rate harvesting Fields Inst Commun vol 1 pp ] J Xia Z Liu R Yuan S Ruan The effects of harvesting the dela on predator-pre sstems with holling tpe II functional response SIAM J APPL Math vol 7 no 4 pp ] T K Kar Selective harvesting in a pre-predator fisher with time dela Mathematical Computer Modelling vol 38 pp ] H I Freedman Deterministic Mathematical Models in Population Ecolog HIFR Consulting Ltd Edmonton 1987 ] R M Ma Time dela versus stabilit in population models with two three trophic levels Ecolog vol 4 pp ] F Brauer Absolute stabilit in dela equations Journal of Differential Equations vol 69 no pp ] Y Kuang Dela Differential Equations with Applications in Population Dnamics of Mathematics in Science Engineering Academic Press Boston Mass USA ] R V Culshaw S Ruan A Dela-differential equation model of HIV infection of CD4 + T-cells Mathematical Biosciences vol 165 pp ] J Dieudonne Foundations of Modern Analsis Academic Press 196 New Yor 5] S Ruan J Wei On the zeros of transcendental functions with applications to stabilit of dela differential equations Dnam Contin Discr Impuls Sst vol 1 pp ] B D Hassard N D Kazarinoff Y H Wan Theor Application of Hopf Bifurcation Cambridge Universit Cambridge ] K Gopalsam X He Dela-independent stabilit in bidirectional associative memor networs IEEE Trans Neural Netw vol 5 pp M r Debaldev Jana was born at Tamlu on 13 Januar 1984 He received his BSc MSc degree in Mathematics from Jadavpur Universit Kolata in 6 8 respectivel He has submitted his PhD Thesis in 1 under the guidance of Dr Nadulal Bairagi His research interests includes mathematical ecolog pattern formation fractional differential equation etc Revised online publication: 3 August 1

9 Engineering Letters :3 EL 3_5 Mr Swapan Charabort is woring as a teacher in Mathematics at Baruipara High School Halisahar 4 Parganas N West Bengal India He has submitted his PhD wor in Jadavpur Universit under the joint supervision of Dr Samares Pal Dr Nadulal Bairagi He has published five research articles in international journals of reputes His fields of interest are mathematical ecolog ecoepidemiolog bioeconomic harvesting Nadulal Bairagi 1967 did his MSc PhD from Jadavpur Universit Kolata India He joined as a facult of Mathematics Department of Jadavpur Universit in 1 He is now Associate Professor Coordinator of the Centre for Mathematical Biolog Ecolog He is the Founder Secretar of Biomathematical Societ of India He has published man papers in different fields of biomathematics His Research interests include ecolog epidemiolog bioeconomic harvesting phsiolog sstem biolog Revised online publication: 3 August 1