THE EFFECTS OF HARVESTING AND TIME DELAY ON PREDATOR-PREY SYSTEMS WITH HOLLING TYPE II FUNCTIONAL RESPONSE

Transcription

1 SIAM J. APPL. MATH. Vol. 7 No. 4 pp. 78 c 9 Society for Industrial and Applied Mathematics THE EFFECTS OF HARVESTING AND TIME DELAY ON PREDATOR-PREY SYSTEMS WITH HOLLING TYPE II FUNCTIONAL RESPONSE JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN Abstract. In this paper the effects of harvesting and time delay on two different types of predator-prey systems with delayed predator specific growth and Holling type II functional response are studied by applying the normal form theory of retarded functional differential equations developed by Faria and Magalhães [J. Differential Equations 995 pp. 8 J. Differential Equations 995 pp. 4]. Hopf bifurcations are demonstrated in models with harvesting of the prey at a constant rate by taking the delay as a bifurcation parameter and numerical examples supporting our theoretical prediction are also given. Furthermore bifurcation analysis indicates that delayed predator-prey systems with predator harvesting exhibit Bogdanov Takens bifurcation. The versal unfoldings of the models at the Bogdanov Takens singularity are obtained and numerical simulations and bifurcation diagrams are given to illustrate the obtained results. Key words. predator-prey system harvesting time delay Hopf bifurcation Bogdanov Takens bifurcation AMS subject classifications. 34K8 37G5 37G 9D5 DOI..37/8785. Introduction. The Canadian Grand Banks off the coast of Newfoundland were prime fishing grounds for several centuries. The population of Atlantic cod Godus morhua used to be so abundant that in 968 about 4 people were employed in the fishing industry and more than 8 tons of Atlantic cod were harvested Schiermeier [34]. However the Atlantic cod stocks collapsed in about years and in 99 the Department of Fisheries and Oceans of Canada belatedly closed the fisheries Hutchings and Myers [] Myers et al. [7 8] Hutchings []. More than a decade later there is still no sign of recovery of Atlantic cod stocks in the Grand Banks. Today the fear of a rapid depletion of world fish stocks because of overexploitation is increasing Pauly et al. [3] Myers and Worm [9]. Depletions of marine fish stocks not only endanger the future of marine fisheries but also may lead to species extinction and ecosystem regime shifts Casey and Myers [6] Jackson et al. []. Thus understanding the dynamics of fishing becomes very important. There are several types of interactions within fisheries systems. Biological harvest technical and market interactions are the most important ones in bioeconomic modeling Flaaten [6]. The biological interactions are predator-prey relations and competition between species. In the case of predator-prey interactions it is well known that the reduction of the predator stock level may increase the surplus production of the prey. However harvesting becomes controversial when it comes to predators like whales and seals May et al. [6] Flaaten [5] Yodzis [38]. The purpose of this paper is to study the importance of harvesting on multispecies management within Received by the editors June 5 8; accepted for publication in revised form June 9; published electronically September School of Mathematical Sciences Beijing Normal University Beijing 875 People s Republic of China xiajing5@mail.bnu.edu.cn zhihualiu@bnu.edu.cn ryuan@bnu.edu.cn. The research of these authors was partially supported by the National Nature Science Foundation of China. Department of Mathematics University of Miami Coral Gables FL ruan@math. miami.edu. This author s research was partially supported by NSF grant DSM

2 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 79 the framework of predator-prey models Clark [8] Flaaten [6]. The study of predator-prey models with harvesting has attracted the attention of many researchers. Let xt andyt denote the density of the prey and predators at time t respectively. Consider constant-yield harvesting of a predator-prey system with Holling type II functional response:. ẋ = rx x K ẏ = y D + mx A + x m δ yx A + x H H r is the intrinsic growth rate of the prey; K is the carrying capacity of the prey; m is the maximum growth rate of predators; δ is the yield conversion factor for predators feedings on the prey; A is the half saturation constant for the predators which is the prey density at which the functional response is half maximal; D is the death rate of predators; H and H are constant harvesting rates for the prey and predators respectively. Note that we need to assume that ẋ andẏ for all time. Model. and its variants have been studied extensively; see for example Brauer and Soudack [3 4 5] Beddington and Cooke [] Dai and Tang [] Hogarth et al. [9] Myerscough et al. [3] Xiao and Jennings [35] and Xiao and Ruan [36]. Very rich and interesting dynamical behaviors such as the existence of multiple equilibria homoclinic loop and Hopf bifurcation have been observed. Let ȳ = y/δ t = mt r = r/m H = H /m D = D/m and H = H /mδ. Dropping the bars we obtain the dimensionless model. ẋ = rx x K ẏ = y D + x A + x yx A + x H H. When H = Xiao and Ruan [36] carried out a bifurcation analysis of model.. In particular they showed that codimension bifurcations occur in a two-dimensional parameter region. Under some conditions they proved that system. undergoes Bogdanov Takens bifurcation; i.e. it can exhibit qualitatively different dynamical behavior including Hopf bifurcation saddle-node bifurcation as well as homoclinic bifurcation. On the other hand population models with time delay are of current research interest in mathematical biology because of their realistic meaning; we refer to the monographs of Cushing [] Gopalsamy [7] and Kuang [3] for general delayed biological systems and a survey paper of Ruan [33] and the references cited therein for studies on delayed predator-prey systems. Brauer [] was the first to consider the combined effects of time delay and constant harvesting on predator-prey models. Further studies were performed by Martin and Ruan [5] who studied the combined effects of the prey harvesting and time delay on the dynamics of the generalized Gausetype predator-prey models and the Wangersky Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities while the prey harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular one of these models loses stability when the delay varies and then regains its stability when the prey harvesting rate is increased. In this paper following the work of Martin and Ruan [5] and Xiao and Ruan [36] we continue studying how time delay and harvesting affect the dynamics of the

3 8 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN predator-prey systems. We assume that a time delay τ> occurs in the predator response term. It represents a gestation time of the predators. The reproduction of predators after predating the prey is not instantaneous but will be mediated by some discrete time lag required for gestation of the predators. We consider two cases. Model. The prey population is harvested at a constant rate: ẋt =rxt xt xtyt.3 ẏt =yt D + K xt τ A + xt τ A + xt H. Model. The predator population is harvested at a constant rate: ẋt =rxt xt xtyt K A + xt.4 xt τ ẏt =yt D + H. A + xt τ The initial conditions for these systems are.5 xθ =φ θ yθ =φ θ for all θ [ τ] φ φ C[ τ] R + x = φ > and y = φ >. Following Xiao and Ruan [37] we first show that Hopf bifurcation occurs in Model and determine the direction of the Hopf bifurcation. We then show that Model exhibits Bogdanov Takens bifurcation. The paper is organized as follows. In section we show that the system.3 with prey harvesting undergoes Hopf bifurcation as the delays cross some critical values and determine the stability of the bifurcating periodic solutions by using the method of Faria and Magalhães [3]. Numerical simulations are performed to illustrate the obtained results. In section 3 following the technique of Faria and Magalhães [3 4] we show that the model.4 with predator harvesting exhibits Bogdanov Takens bifurcation and obtain versal unfoldings at the Bogdanov Takens singularity under some conditions. A brief discussion is given in section 4 to conclude the paper.. Prey harvesting. In this section we are concerned with Model i.e. system.3. We first study the interior equilibrium of system.3. An easy computation shows that when. radk KD AD HK D > and <D< system.3 has a unique positive equilibrium E =x y x = AD D y = r rx K H x A + x. Through transformation z t =xt x z t =yt y the equilibrium E =x y is translated to the origin O = and system.3 can be rewritten as the following equivalent system: ż t =α z t+α z t+ i!j! f ij zi tzj t i+j. ż t =β z t τ+ i!j! f ij zi t τz j t i+j

4 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 8 α = r x r K Ay A + x α = x ij = i+j f x i y j f ij = i+j f x i y j f f = rx x K xy We obtain the linearized system xy A + x H A + x β = f = y ij xy D + Ay A + x x. A + x.3 {ż t =α z t+α z t ż t =β z t τ. The characteristic equation of the linearized system.3 is a transcendental equation of the following form see Cooke and Grossman [9] and Ruan [3]:.4 λ λα α β e λτ =. In [5] Martin and Ruan considered the following system:.5 {ẋt =xt[fxt ythxt] H ẏt =yt[ d + cxt τhxt τ]. We note that when fx is the logistic growth function and xhx is the Holling type II response function in.5 then system.5 becomes system.3. Martin and Ruan studied the stability of the equilibria and existence of Hopf bifurcation for model.5. However the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions were not considered in [5]. In this section we shall determine the direction and stability of the bifurcating periodic solutions by employing the normal form theory due to Faria and Magalhães [3]. For convenience some useful lemmas and theorems from [5] are rewritten as follows. Lemma.. Assume that. holds and.6 rkad + D 3 HK ra D rkad 3 ra D 3 <. Then at.7 τ k = σ + arccos σ + +kπ k =... α β.4 has a simple pair of purely imaginary roots ±iσ +.8 σ + = α + α 4 +4α β. Furthermore if τ [τ allrootsof.4 have strictly negative real parts; if τ = τ allrootsof.4 except±iσ + have strictly negative real parts.

5 8 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN Lemma.. The following transversality condition { } dλ sign Re > dτ τ =τ k is satisfied λτ =μτ±iστ are the roots of.4 near τ = τ k k =... satisfying μτ k = στ k =σ + andk =... Lemma.3. Suppose that. and.6 hold then.4 has at least one eigenvalue with strictly positive real part for τ>τ. Theorem.4. Suppose that conditions. and.6 are satisfied. Then when τ [τ the zero solution of. is locally asymptotically stable; when τ>τ the zero solution of. is unstable; 3 τ k k =... are Hopf bifurcation values for system.. From the above theorem we know that system.3 undergoes Hopf bifurcations at the critical values τ k k =... In the following we shall determine the direction and stability of the bifurcating periodic solutions. Let z t =xτt x z t =yτt y. Then system.3 is transformed into functional differential equations in C[ ] R ż t =τ α z t+α z t+ i!j! f ij zi tz j t i+j.9 ż t =τ β z t + i!j f ij zi t zj t and the linearized system is i+j. {ż t =τ[α z t+α z t] ż t =τβ z t. System. has the characteristic equation. λ λτα τ α β e λ =. Letting λ = ξτ then. becomes. ξ α ξ α β e ξτ =. Obviously. is the same as.4. We know that for fixed k N. has a simple pair of conjugate complex roots ξτ =μτ ± iστ which satisfy μτ k = στ k =σ + μ τ k τ k is given in.7. Therefore the characteristic equation. has two complex roots λτ =τμτ ± iτστ which satisfy dreλτ dτ τ =τk = τ dreξτ τ =τk +Reξτ τ =τk = τ k μ τ k. dτ Let z =z z T. System.9 can further be written as.3 żt =Lτz t +F z t τ

6 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 83 α ϕ Lτϕ =τ + α ϕ β ϕ Σ i+j i!j! f ij ϕi ϕ j F ϕ τ =τ i!j f ij ϕi ϕ j i+j ϕ =ϕ ϕ T C[ ] R. By the Riesz representation theorem there exists an n n matrix ηθ τ whose elements are of bounded variation for θ [ ] such that for ϕ C[ ] R Lτϕ = dηθ τϕθ. Choose ηθ τ =τ α α Hθ+τ β Hθ + Hθ is the Heaviside function. expansion We expand F ϕ τ aboutϕ as the Taylor.4 F ϕ τ =! F ϕ τ+ 3! F 3ϕ τ+o ϕ 4 F l ϕ τ isgivenby.5 l! F lϕ τ =τ i+j=l i+j=l i!j! f ij ϕi ϕ j i!j! f ij ϕi ϕj. Setting a new parameter α = τ τ k system.3 is rewritten as.6 żt =Lτ k z t +F z t α F ϕ α =Lαϕ+F ϕ τ k + α. Let A be the infinitesimal generator corresponding to żt =Lτ k z t. Then A has a pair of purely imaginary characteristic roots ±iσ k σ k = τ k σ + which are simple and no other characteristic roots with zero real part. Consider Λ = { iσ k iσ k } and denote by P the invariant space of A corresponding to Λ dim P =. Let the phase space C = C[ ] R be decomposed by Λ as C = P Q by applying the formal adjoint theory for functional differential equations in Hale [8]. Consider complex coordinates and still write as C = C[ ] C. Suppose that Φ = Φ Φ is the basis for P and Φ θ =e iσkθ v T = e iσkθ v T Φ θ =Φ θ θ v =v T is a vector in C and v = τ kβ e iσ k iσ k which satisfies Lτ k Φ =iσ k v T.

7 84 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN Also the two eigenvectors Ψ and Ψ of the formal adjoint operator A corresponding to iσ k and iσ k respectively form a basis Ψs =colψ s Ψ s of the adjoint space P of P Ψ s =e iσks u u Ψ s =Ψ s s. We know Ψ = Ψ θdηθ; therefore we can choose u = τ kα iσ k u. In order to have Ψ Φ = Ψ j Φ i ij = = I I is the second-order identical matrix.. is a bilinear inner product form θ ψ ϕ =ψϕ ψξ θdηθϕξdξ τ we have u = k βα τk βα+iσ k τ k α e iσ k +iσ k τ k α.notethat Φ τ =ΦB B is k αβ a diagonal matrix iσk B =. iσ k Take the enlarged phase space BC := {ϕ :[ ] C ϕ is continuous on [ lim θ ϕθ}. The elements of BC have the form ψ = ϕ + X αϕ C α C X = X θ isgivenby { I θ = X θ = θ<. The projection of C upon P ϕ ΦΨϕ associated with the decomposition C = P Q is replaced by π : BC P such that πϕ + X α=φ[ψϕ+ψα]. Thus we have the decomposition BC = P ker π. Use the decomposition z t = Φxt+y t xt C y t ker π C Q C = C [ ]; R denotes the space of continuously differentiable functions from [ ] to R. The equation.6 is equivalent to the system {ẋ = Bx +ΨF Φx + y α.7 d dt y = A Q y +I πx F Φx + y α. We have the Taylor expansions.8 ΨF Φx + y α = f x y α+ 3! f 3 x y α+h.o.t. I πx F Φx + y α = f x y α+ 3! f 3 x y α+h.o.t. fj x y α andf j x y α are homogeneous polynomials in x y α ofdegree j j = 3 with coefficients in C and ker π respectively and h.o.t. stands for higher order terms. The normal form method implies a normal form on the center manifold of the origin for.7 as.9 ẋ = Bx + g x α+ 3! g 3x α+h.o.t

8 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 85 g g 3 are the second- and third-order terms in x α respectively. Following [3] V m+p j X denotes the linear space of homogeneous polynomials of degree j in m + p real variables x =x...x m α =α...α p with coefficients in X; that is V m+p j X = c ql x q α l :q l N m+p c ql X. The operators M j ql =j are defined by M j px α =D x px αbx Bpx α j and M j act in V 3 j C V 3 j C =ImM j kerm j and kerm j =span{x q α l e k :q λ =λ k k = q N l N q l = j} {e e } is the canonical basis of C. Hence { } kerm x α =span x α { x kerm3 =span x x α x x } x α. For.8 it follows that. f x y α =Ψ[LαΦx + y+f Φx + y τ k ] F is given in.5. By computing we obtain. f A x α +A x α + a x +a x x + a x x α= Āx α +Āx α +ā x +ā x x +ā x. A = iσ k u + u v τ k A = iσ k u + u v τ k a = τ k [u f +u v f + u f e iσ k +u v f e iσ k ] a = τ k [u f + u f v + v +u f + u f e iσk v + e iσ k v ] a = τ k [u f +u v f + u f eiσ k +u v f eiσ k ]. Since the second-order term in α x of the normal form on the center manifold is given by it implies that.3 g x α= Proj kerm f x α A x g x α= α Ā x α

9 86 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN A = iσ k τ k u + u v. Note that { g3x α kerm3 x =span x x α x x } x α. However the terms O x α are irrelevant to determine the generic Hopf bifurcation. Hence we need only to compute the coefficients of x x and x x. Notice that 3! g 3 x α= 3! Proj kerm 3 f 3 x α= 3! Proj s f 3 x + O x α { x s := span x and the term f 3 x is given by x x } f 3 x = f 3 x + 3 [D xf U D xu g ] x + 3 [D yf h] x. Now we compute 3! g 3 x α step by step. Step. WecomputeProj s [D x f U ] x. Following [3] we take U x = M PI f x. We have from. that f a x +a x x + a x x = ā x +ā x x +ā x. According to the definition of M the equation M U x = f x can be written as the following differential equations: u u x x u = a x +a x x + a x x x iσ k u u x x + u = ā x x x iσ +ā x x +ā x. k A straightforward calculation shows that U x = iσ k iσ k a x a x x 3 a x 3āx +ā x x ā x Consequently Proj s [D x f U ] x = iσ k 3 a + a a a x x iσ k 3 a. + a ā ā x x.

10 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 87 Step. We compute Proj s [D x U g ] x. From.3 we know that g x =. Thus[D x U g] x =. Step 3. We compute Proj s [D y f h] x h = h h T is a second degree homogeneous polynomial in x x α with coefficients in Q which has the form h = hx x α=h x x + h x α + h x α + h x + h x + h α and h = hx x α is the unique solution in V 3 Q of the equation Since M hx α =I πx [LαΦx+F Φx τ k ]. M hx α =D xhx αbx A Q hx α = D x hx αbx ḣx α X [Lτ k hx α ḣx α] =I πx [LαΦx+F Φx τ k ]; thus h = hx θ can be evaluated by the system.4.5 ḣx D x hxbx =ΦΨF Φx τ k ḣx Lτ k hx = F Φx τ k ḣ denotes the derivative of hxθ with respect to θ. From. we obtain f x y = ΨF Φx + y τ k u u = τ ū ū k f p +f p p f l +f l p p = x + x + y p = v x + v x + y l = e iσ k x + e iσ k x + y. Hence we have [D y f h] x = τ k u T f p h + f p h + f p h f l h + f p h + f l h ū T f p h + f p h + f p h f l h + f p h + f l h.6 p = x + x p = v x + v x l = e iσ k x + e iσ k x. Thus Proj s [D y f c3 x h] x = x c 3 x x c 3 = u T ah τ + āh + bh + bh k ch + ch + dh + dh + eh + ēh

11 88 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN and a = f + f v b = f c = f e iσ k d = f e iσ k e = f v. In the following we compute h θ andh θ. Combining.4 with.5 we know that h =h h T and h =h h T are respectively the solution of the following two systems: h a =Φ Φ ā.7 h a Lτ k h =τ k b and.8 a h iσ k h =Φ Φ ā h a Lτ k h =τ k b a =f +v + v f b =f +f v e iσk + e iσ k v a = f Solving.7 and.8 we obtain +f v b = f e iσ k +f v e iσ k..9 h θ = iσ k a e iσ kθ v ā e iσ kθ v+c and.3 h θ = iσ k c =c c T c =c c T and a e iσkθ v + 3 āe iσ kθ v + e iσkθ c c = b β c = α b β a α β c = c iσ k τ k a + τk α b iσ k iσ k τ k α τk α β e iσ k = τ k a β e iσ k +iσ k τ k b τk α b iσ k iσ k τ k α τk α β e iσ. k Step 4. WecomputeProj s f3 x. From.8 f 3 x is given by f 3 x = ΨF 3Φx τ k F 3 Φx τ k is defined in.5. Thus F 3 can be computed as follows: f 3 F 3 Φx τ k =τ 3 p +3f p p k f 3 3 l +3f l p

12 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 89 p p l are given in.6. Therefore we obtain Proj s f3 3a x x = x 3ā x x a = τ k u f 3 + f v +f v +τ k u f 3 e iσ k + f e iσk v +f v. Summarizing Steps 4 we obtain 3! g 3x = A3 x x Ā 3 x x.3 A 3 = iσ k 3 a + a a a + a + c 3. In consequence the normal form of.9 has the form ẋ = Bx + g x α+ 3! g 3 x α+h.o.t. A x = Bx + α A3 x + x Ā x α Ā 3 x x + O x α + x 4. The normal form.9 relative to P can be written in real coordinates w w through the change of variables x = w iw x = w + iw. Followed by the use of polar coordinates ρ ξ w = ρ cos ξ w = ρ sin ξ thisformalformbecomes.3 { ρ = k αρ + k ρ 3 + Oα ρ + ρ α 4 ξ = σ k + O ρ α k =ReA k =ReA 3. Following [7] we know that the sign of k k determines the direction of the bifurcation and the sign of k determines the stability of the nontrivial periodic solution bifurcating from Hopf bifurcation. Therefore summarizing the above discussion we have the following theorem. Theorem.5. The flow of.6 on the center manifold of the origin at τ = τ k is given by.3. Hopf bifurcation is supercritical if k k < and subcritical if k k >. Moreover the nontrivial periodic solution is stable if k < and unstable if k >. As an example we consider system.3 with A =D = 3 H = K =4 and r = ;thatis.33 ẋ = x x 4 ẏ = y xy +x. xt τ + 3 +xt τ In this case system.33 has a unique positive equilibrium x y = 7 4 and conditions. and.6 are satisfied. Thus the equilibrium x y is asymptotically stable when τ [τ τ =.579. By means of the software Maple we

13 y y y y x x 9 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN a b t t a.9.8 b t t a3.9.8 b x x Fig.. The solution trajectories and phase portraits of the system.33 before a a3 and after b b3 Hopf bifurcation. In a a3 τ =.45; inb b3 τ =.6. can compute the following values: v v = = v.5.36i and u i u= = u i a =.6.399i a = i c 3 = i. a =.58.5i a = i Therefore from. and.3 we can compute A = i A 3 = i. Thus k =.34 k =.74. By Theorem.5 we know that there is a supercritical Hopf bifurcation for.33 at τ = τ and the nontrivial periodic solution associated with Hopf bifurcation is stable. The numerical simulations are depicted in Figure.

14 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 9 Remark.6. Martin and Ruan [5] studied the Hopf bifurcation in the following predator-prey system with prey harvesting and delayed prey specific growth: xt τ ẋt =rxt K xtyt A + xt H.34 ẏt =yt D + xt A + xt. Similarly as in Theorem.5 we can determine the direction of the Hopf bifurcation in system Predator harvesting. Following Xiao and Ruan [36] we know that system.4 has a unique interior equilibrium E =x y provided that 3. D + AD K hold and x y isgivenbyx = K D+AD D DADr + H 4 = and K> AD Kr D and y = r x K A+x. We assume throughout this section that <D<. We translate the equilibrium x y of system.4 to the origin. Setting z t = xt x z t =yt y system.4 can be written as the following system: ż t =α z t+α z t+ i!j! f ij zi tz j t i+j 3. ż t =β z t τ+β z t+ i!j! f ij zi t τzj t i+j 3.3 α = r x r K Ay A + x α = x A + x Ay β = A + x β = D + x A + x f ij = i+j f x i y j f ij = i+j f xy x i y j ij xy f = rx x xy K A + x f = y D + x H. A + x Consider the linearized system of 3. at the zero equilibrium 3.4 {ż t =α z t+α z t ż t =β z t τ+β z t. The characteristic equation for system 3.4 takes the form 3.5 λ λα + β +α β α β e λτ =. In fact we have α β α β = the proof can be found in the Appendix. We can easily see that 3.5 has two zero eigenvalues and no other eigenvalues on the imaginary axis if and only if τ = τ = α+β α β and τ α β. Because τ > and α β < we know that τ α β.

15 9 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN Normalizing the delay τ in system.4 by the time-scaling t t/τ system.4 is transformed into [ ẋt =τ rxt xt xtyt ] K A + xt 3.6 [ ] xt ẏt =τyt D + τh. A + xt Setting z t =xt x z t =yt y system 3.6 can be rewritten as functional differential equations in C := C[ ] R ż t =τ α z t+α z t+ i!j! f ij zi tzj t i+j 3.7 ż t =τ β z t + β z t+ i!j! f ij zi t z j t. Let z =z z T. System 3.7 is written as i+j 3.8 żt =τlz t +τfz t L : C R F : C R are given by 3.9 Lϕ = α ϕ + α ϕ β ϕ + β ϕ Fϕ = i+j i+j i!j! f ij ϕi ϕj i!j! f ij ϕi ϕ j and ϕ =colϕ ϕ. We consider the formal Taylor expansion of F 3. F ϕ = j j! F jϕ ϕ C and define L = τ L. L ϕ is a continuous linear function. Therefore there exists a matrix function ηθ θ whose elements are of bounded variation such that Choose ηθ =τ α α β L ϕ = dηθϕθ. Hθ+τ β Hθ +. Let A be the infinitesimal generator corresponding to żt =L z t. Then A has two zero characteristic roots. Define Λ = {} and denote by P the invariant space of A corresponding to Λ the dimension of P equals to. Let Φ = Φ Φ and Ψ=colΨ Ψ be the bases for P and P the adjoint space of P respectively. We note that Φ =ΦB thusforφ=φ Φ and the matrix B is given by B =.

16 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 93 We know that Φ i Ψ i i = satisfy Φ i = dηθφ iθ Ψ i = Ψ i θdηθ and Ψ Φ = Ψ j Φ i ij = = I. Accordingly we can obtain α α Φθ = θ α α α θ + α τ es + g fs+ h Ψs = θ s e f e f g h satisfy the following equations: α e + β f = τ α g + τ β h = f 3. hα τ α β α g + τ α β f = h α τ τ β α α g + 3 τ β α f = eα + f α τ τ β α =. According to [4] we obtain that the normal form for 3.8 is as follows {ẋ = x + O x x 3 ẋ = B x + B x x + O x x 3 B = τα ef + ff τα α ef + ff B = τα gf + hf + ef τα α gf +ef +hf + ff +ττ α ef + ff and α i f ij f ij i j = are given in 3.3. In fact the Bogdanov Takens bifurcation for a planar system with two discrete delays have been studied in Faria []. Applying the formula in [] we can also derive the above normal form. Therefore we have the following result. Theorem 3.. Suppose that 3. holds. Then the equilibrium E of.4 is a Bogdanov Takens singularity for τ = τ = α+β α β. We know that system.4 undergoes Bogdanov Takens bifurcation as τ crosses the critical value τ. Next we are interested in determining a versal unfolding for system.4 with a Bogdanov Takens singularity. Note that α β α β =see the Appendix. τ = τ + μ β = αβ α We introduce two bifurcation parameters μ =μ μ by setting + μ. System 3.8 is rewritten as 3. żt =L z t +L μz t + F z t μ 3.3 L μϕ = μ Lϕ+τ μ ϕ

17 94 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN 3.4 F ϕ μ =μ μ ϕ ϕ +τ + μ F ϕ for ϕ =. ϕ In the phase space BC = P ker π we decompose z t in the form z t =Φx + y. Thus the system 3. is equivalent to ẋ = Bx + Ψ[L μφx + y+ F Φx + y μ] 3.5 d dt y = A Q y +I πx [L μφx + y+ F Φx + y μ] L μϕ F ϕ μ are given by 3.3 and 3.4 and ϕ =colϕ ϕ. Define 3.6 Ψ[L μφx + y+ F Φx + y μ] = f x y μ+ 3! f 3 x y μ+h.o.t I πx [L μφx + y+ F Φx + y μ] = f x y μ+ 3! f 3 x y μ+h.o.t.. Therefore system 3.5 becomes 3.7 ẋ = Bx + f x y μ+h.o.t. d dt y = A Q y + f x y μ+h.o.t. f x y μ andf x y μ are homogeneous polynomials in x y μ ofdegree with coefficients in R and ker π respectively. Thus the normal form for 3. on the center manifold of the origin has the form 3.8 ẋ = Bx + g x μ+h.o.t. g is the second-order term in x μ and determined by g x μ= Proj ImM f c x μ ImM c is a complementary space of ImM inv 4 R andv 4 R isthe space of homogeneous polynomials of degree in the variables x x μ μ. According to and 3.6 we know that f x μ=ψ[l [ μφx+τ F Φx] α =Ψ τ μ x β α τ μ x τ α μ x ] + τ F Φx F is given in 3.9 and 3.. We consider the canonical basis of V 4 R x x x x x μ i x μ i μ μ μ μ x x x x x μ i x μ i μ μ μ μ i =.

18 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY 95 The operators M j are defined by M j px =D xpxbx Bpx j ; thus M acting in V 4 R has the following form M px x = x p x p x p x p V 4R. The images of the basis under M x x x x μ i μ x x x μ μ x x x μ x i =. are respectively x μ i x μ i x μ i A complementary space of ImM inv 4R is { ImM c =span x x x x μ μ x μ μ μ μ }. x μ x μ Following [] we know that the normal form of 3. on the center manifold is given by { ẋ = x + h.o.t. 3.9 ẋ = λ x + λ x + B x + B x x + h.o.t. 3. λ = τ α fμ λ = β α fμ τ α hμ and 3. B = τ [ α ef + ff α α ef + ff ] B = τ { α gf [ α α τ f + hf α α gf + hf +α ef e +α τ f f]}. The above arguments imply the following theorem. Theorem 3.. Let μ μ be defined by τ = τ + μ β = αβ α + μ τ = α+β α β.fore =x y and μ = μ =system.4 exhibits a Bogdanov Takens bifurcation. The normal form on the center manifold for.4 of E is given by 3.9 withλ λ B B defined by 3. and 3..

19 96 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN Fig.. The Bogdanov Takens bifurcation diagram and phase portraits for system 3.. As an example we consider system.4 with r =K =A =D = 3 ;thatis ẋt =xt xt xtyt +xt 3. ẏt =yt xt τ + H. 3 +xt τ According to Theorem 3. we know that system 3. has a Bogdanov Takens singularity point x y = whenh τ = 3 6. We will show that system 3. undergoes Bogdanov Takens bifurcation when τ and H vary in a small neighborhood of τ and H. Now we introduce two bifurcation parameters μ μ by setting τ = + μ β = αβ α + μ ; i.e. H = H + μ. Following the analysis in this section we obtain the versal unfolding for system 3. as follows: {ẋ = x + h.o.t. 3.3 ẋ = λ x + λ x + B x + B x x + h.o.t. λ =.34676μ λ = μ μ B = B = Therefore we know that system 3. exhibits Bogdanov Takens bifurcation when the parameters μ μ vary in a small neighborhood of the origin. On the lines H ± there exists stable Hopf bifurcation while there exists curves HL ± corresponding to homoclinic bifurcation. The bifurcation diagram is depicted in Figure. Remark 3.3. Similarly we can study Bogdanov Takens bifurcation in the following predator-prey model with predator harvesting and delayed prey specific growth xt τ ẋt =rxt xtyt K 3.4 A + xt ẏt =yt D + xt H A + xt and obtain results similar to Theorems 3. and 3..

20 PREDATOR-PREY SYSTEMS WITH HARVESTING AND DELAY Discussion. Predator-prey models play a crucial role in studying the management of renewable resources Clark [8]. The effect of constant-rate harvesting on the dynamics of predator-prey systems has been investigated by many authors; see for example Brauer and Soudack [3 4 5] Beddington and Cooke [] Dai and Tang [] Hogarth et al. [9] Myerscough et al. [3] Xiao and Jennings [35] and Xiao and Ruan [36]. Very rich and interesting dynamical behaviors such as the existence of multiple equilibria existence of Hopf bifurcation limit cycles homoclinic loops and Bogdanov Takens bifurcations have been observed. It is also observed that in some cases before a catastrophic harvest rate is reached the effect of harvesting is to stabilize the equilibrium of the population system. Martin and Ruan [5] studied the combined effects of prey harvesting and delay on the dynamics of predator-prey systems and focused on three very well-studied delayed predator-prey models. Namely they considered a generalized Gause-type predator-prey model with prey harvesting and a time delay in the prey specific growth term; a generalized Gause-type predator-prey model with prey harvesting and a time delay in the predator response function is analyzed; and the Wangersky Cunningham predator-prey model with prey harvesting. It was shown that in the first and third models the time delay could cause not only instability and oscillations but also the switching of stabilities while the prey harvesting changes only the equilibrium values but not the properties of solutions. In the second model the time delay induces instability and bifurcation but there is no switching of stabilities; however increasing the prey harvesting level will help the system to regain its stability. This indicates that the prey harvesting has a stabilizing effect on the dynamics of the model. In this paper following the work of Martin and Ruan [5] and Xiao and Ruan [36] we continued studying the combined effects of time delay and constant harvesting on the dynamics of predator-prey systems with Holling type II functional response. Two different types of models have been analyzed; namely predator-prey systems with delayed predator response and i prey harvesting or ii predator harvesting. There are two types of bifurcation phenomena: Hopf bifurcation and Bogdanov Takens bifurcation. More precisely in the model with prey harvesting there is no bifurcation on the number of positive equilibria; time delay can induce oscillations of both species via Hopf bifurcation. While in the model with predator harvesting multiple positive equilibria and degenerate equilibria can exist Bogdanov Takens bifurcation can occur. In predator-prey interactions within fisheries systems it is well known that the reduction of the predator stock level may increase the surplus production of the prey. Harvesting predators becomes controversial May et al. [6] Flaaten [5] Yodzis [38]. The Bogdanov Takens bifurcation diagram in the predator-prey models with predator harvesting in section 3 indicates that there are some parameter regions in which both predator and prey species can be driven to extinction. This may provide some explanations for the collapse of the Atlantic cod stocks in the Canadian Grand Banks Hutchings and Myers [] Myers et al. [7 8] Hutchings []. Our study demonstrates that appropriate harvesting of predator population is crucial in the long-term survival of both predator and prey species and in turn the fisheries systems. This is significant and useful in designing fishing policies for the fishery industry Pauly et al. [3] Myers and Worm [9]. Fishing is a seasonal activity. It will be very interesting to study how seasonal harvesting affects the existing Hopf and Bogdanov Takens bifurcations in predator-prey systems with Holling type II functional response. We leave this for future consideration.

21 98 JING XIA ZHIHUA LIU RONG YUAN AND SHIGUI RUAN Appendix A. In this section we verify that α β α β =. Since [ α β α β = r x ] r K Ay A + x D + x + Ax y A + x A + x 3 = r x D + x + DAy K A + x A + x = r x H x DAr K + K y A + x = x H K x K A + x + DAr x K A + x [ x K = H + DAr x ] A + x x K K [ ] K x K x = H + DAr A + x K x K KK x H + DArK x = A + x KK x in order to prove α β α β = we need only to prove KK x H + DArK x =. In fact KK x H + DArK x [ K D+AD = K K D = ADKH D + DAr [ K AD D ] [ H + DAr K ] ] K D+AD D = 4KADH D+DAr[ K D AD ] 4 D = DArK[ D AD K 4H D ] Kr 4 D [ = DArK 4 D D + AD ] 4ADr D+4H D K Kr [ = DArK 4 D D + AD ] 4 DADr + H =. K Kr Acknowledgments. We would like to thank Baochuan Tian for his help in plotting Figure. We are grateful to the referees for their careful reading valuable comments and helpful suggestions. Finally we thank Teresa Faria for sending a copy of []. REFERENCES [] J. R. Beddington and J. G. Cooke Harvesting from a prey-predator complex Ecol. Modelling 4 98 pp [] F. Brauer Stability of some population models with delay Math. Biosci pp [3] F. Brauer and A. C. Soudack Stability regions and transition phenomena for harvested predator-prey systems J. Math. Biol pp

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