STABILITY ANALYSIS AND HOPF BIFURCATION OF DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE

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1 Electronic Journal of Differential Equations Vol. 6 (6 No. 55 pp.. ISSN: URL: or STABILITY ANALYSIS AND HOPF BIFURCATION OF DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS WITH BEDDINGTON-DEANGELIS FUNCTIONAL RESPONSE XIN JIANG ZHIKUN SHE ZHAOSHENG FENG Abstract. In this article we study a density-dependent predator-prey system with the Beddington-DeAngelis functional response for stability and Hopf bifurcation under certain parametric conditions. We start with the condition of the eistence of the unique positive equilibrium and provide two sufficient conditions for its local stability by the Lyapunov function method and the Routh-Hurwitz criterion respectively. Then we establish sufficient conditions for the global stability of the positive equilibrium by proving the non-eistence of closed orbits in the first quadrant R. Afterwards we analyze the Hopf bifurcation geometrically by eploring the monotonic property of the trace of the Jacobean matri with respect to r and analytically verifying that there is a unique r such that the trace is equal to. We also introduce an auiliary map by restricting all the five parameters to a special one-dimensional geometrical structure and analyze the Hopf bifurcation with respect to all these five parameters. Finally some numerical simulations are illustrated which are in agreement with our analytical results.. Introduction The dynamic relationship between predators and their preys is one of the dominant themes in both mathematical biology and theoretical ecology. Better understanding eact trends of population dynamics can contribute to the environmental protection and resource utilization. In the pt decades considerable attention h been dedicated to various predator-prey models of which the following predator-prey system with the Beddington-DeAngelis functional response originally proposed by Beddington 4 and DeAngelis 9 independently h been etensively studied by applied mathematicians and biologists theoretically and eperimentally: d(t dy(t ( sy(t (t c b(t m m (t m 3 y(t ( y(t d f(t m m (t m 3 y(t (. Mathematics Subject Clsification. 34D 34E5 37G5. Key words and phres. Density-dependent; local and global stability; Hopf bifurcation; monotonicity; geometrical restriction. c 6 Te State University. Submitted March 6 6. Published September 6.

2 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 where c b s m m m 3 d and f are positive constants (t and y(t represent the population density of the prey and the predator at time t respectively c is the intrinsic growth rate of the prey d is the death rate of the predator and b stands for the mutual interference between preys. The predator consumes the prey s(ty(t with the functional response of Beddington-DeAngelis type f(ty(t m m (tm 3y(t m m (tm 3y(t and contributes to its growth with the rate. This is similar to the well-known Holling type II model with an etra term m 3 y(t in the denominator. System (. is a population model which h received much attention in biology and ecology. Cantrell and Cosner 6 presented qualitative analysis of system (. on permanence which implies the eistence of a locally ymptotically stable positive equilibrium or periodic orbits. Hwang 5 considered the local and global ymptotic stability of the positive equilibrium (ˆ ŷ by the divergence criterion. That is for system (. under the conditions (f dm c b > dm and tr(j(ˆ ŷ (ˆ ŷ is globally ymptotically stable. Recently Hwang 6 presented the condition (f dm c b > dm and tr(j(ˆ ŷ > to ensure the uniqueness of the limit cycle of system (.. For more details about biological background of system (. we refer the reader to However abundant evidence suggests that predators do interfere with each other s activities so to result in competition efforts and that the predator may be of density dependence because of the environmental factors 3. Kratina et al have demonstrated a fact that the predator density dependence is significant at both high predator densities and low predator densities 7. Hence the following more realistic predator and prey density-dependent model with the Beddington- DeAngelis functional response w proposed 8 : d(t ( sy(t (t c b(t m m (t m 3 y(t dy(t ( f(t y(t d ry(t m m (t m 3 y(t (. where r represents the rate of predator density dependence. Compared with system (. system (. contains not only b (t which stands for intrpecific action of prey species but also ry (t which stands for intrpecific action of predator species. Li and She 8 considered dynamics of system (. by showing (f dm c b > dm the sufficient and necessary condition for the permanence and eistence of the unique positive equilibrium and (f dm c b dm the sufficient and necessary condition for the global ymptotical stability of boundary solution. Furthermore the dynamics of the stage-structured model of system (. w studied in 3 and the non-autonomous ce of system (. w tackled in. From the point of view of ecological managers it may be desirable to have a unique positive equilibrium which is globally ymptotically stable. Although considerable attention h been undertaken on system (. it seems that sufficient and necessary conditions for local stability and even global stability of the positive equilibrium have not been comprehensively presented yet. In addition the eistence of (stable or unstable limit cycles and Hopf bifurcation with respect to the parameters are rarely discussed. In this article we will discuss the local and global stability of the positive equilibrium and analyze the Hopf bifurcation of system (. in the first quadrant. For simplicity by the re-scaling t ct b c y bm3 cm y s s cm 3 a bm cm b f cm

3 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 3 d md f and r crm bfm 3 system (. is nondimensionalized to d(t dy(t ( Q((t y(t : by(t P ((t y(t : (t( (t d ry(t s(ty(t (t y(t a (t (t y(t a. (.3 We start with the eistence of the unique positive equilibrium of system (.3 and show that this unique positive equilibrium cannot be a saddle under the given conditions. To ensure the local stability of this positive equilibrium we first provide a sufficient condition by constructing a Lyapunov function and then give a concrete condition depending only on the parameters by the Routh-Hurwitz criterion. By virtue of two clsical criteria we discuss the global ymptotic stability of the positive equilibrium and present several nonequivalent sufficient conditions. That is under the permanence condition we firstly present a condition by Dulac s criterion for the global attractiveness of the positive equilibrium. Secondly by the divergency criterion we provide a sufficient condition for the stability of all possibly eisting closed orbits under which we prove that the positive equilibrium is locally and globally ymptotic stable because of the non-eistence of stable closed orbits. Thirdly under a stronger permanence condition for providing concrete bounds of all possible closed orbits we apply Grammer s rule and Green s theorem to present the divergency integral and obtain another sufficient condition on global ymptotical stability of the positive equilibrium. Afterwards we eplore the Hopf bifurcation with respect to the parameter r. We first geometrically eplore the monotonic property of the trace of the Jacobian matri with respect to r and then analytically verify that there is a unique r such that the trace is equal to. Bed on these arguments we analyze the Hopf bifurcation with respect to the parameter r. Moreover in order to generalize our conclusion on the Hopf bifurcation we consider the Hopf bifurcation with respect to all five parameters by introducing an auiliary map and restrict the five parameters to a special one-dimensional geometrical structure. Some numerical simulations are performed to illustrate our analytical results. Note that Hwang 5 verified that for system (. the local stability and global stability of the positive equilibrium coincide. However from the Hopf bifurcation analysis we find that for system (.3 the coincidence between local stability and global stability does not hold due to the eistence of the parameter r. Consequently the analysis results show that the rate r of predator density dependence h a significant effort on the global dynamics of system (.3. The rest of this paper is organized follows. In Section we consider the local stability of the positive equilibrium. In Section 3 we establish several sufficient conditions for the global stability of the positive equilibrium by two clsical criteria. In Section 4 we analyze Hopf bifurcations with respect to the parameter r and all five parameters respectively. Section 5 presents some numerical simulations and Section 6 is a brief conclusion.. Local stability of positive equilibrium We know from 8 that system (.3 h a unique positive equilibrium if and only if the condition ad < d (.

4 4 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 holds which is the sufficient and necessary condition for permanence of the system. This unique positive equilibrium cannot be a saddle point. According to the definition of permanence 8 and the Poincáre-Bendison theorem 3 for any trajectory starting in R : {( y R : > y > } there are three ces for its ω-limit set in R : the positive equilibrium which is not a saddle a closed orbit or a saddle together with possible homoclinic orbits. For the third ce in addition to this saddle there eists at let another positive equilibrium inside the region enclosed by the homoclinic orbit simultaneously. Since the unique positive equilibrium is not a saddle we now eplore the conditions to guarantee the local ymptotic stability of this positive equilibrium. Let ( y be the unique positive equilibrium which is short for the epression ( (a b d r s y (a b d r s and satisfies sy y a d ry y a. (. Clearly < < and < y < d r. By 7 Theorem. ( y smoothly depends on the parameters a s r and d. Let (t (t and y(t y y(t. Then system (.3 is equivalent to d F F y y g ( y dy G y G yy y g ( y where F sy ( y a F y s( a ( y a G b(y a ( y a b G y ( y a br and both g ( y and g ( y are of o( y given by: ( g ( y F Fy y sy y a ( g ( y G y G yy y bdy bry by y a. (.3 (.4 Applying the Lyapunov stability theorem 6 we can find a condition to ensure the local ymptotic stability of ( for system (.3 follows: Theorem.. If condition (. holds and F < then the positive equilibrium ( y of system (.3 is locally ymptotically stable. Proof. Let V ( y F y y G y. For system (.3 we have dv V d ( y : V dy y F Fy G y G y o( y y. Obviously V ( y and V ( y if and only if ( y (. Under the condition F < there eists a neighborhood N of the origin such that dv ( y

5 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 5 holds in N and dv ( y if and only if (t y(t. This implies that V ( y is a Lyapunov function for system (.3 and thus ( is locally ymptotical stable. That is the positive equilibrium ( y of system (.3 is locally ymptotically stable. Remark.. Since F y y G < min{ y G y F } implies that F < we here obtain a weaker local ymptotic stability condition than the one in 8. Note that the condition of local stability in Theorem.3 depends on and y and the positive equilibrium ( y usually needs to be solved numerically at first. It is evident that the condition F < can be directly derived by s a. So in the following we attempt to seek a weaker condition than s a by applying the Routh-Hurwitz criterion for the linerization of system (.3 instead of constructing a Lyapunov function. Clearly the characteristic equation of the linerization of system (.3 is where a r λ a λ a (.5 a bry (b s y ( y a rsy ( y a ( y a 3 b y. (.6 Since a is equivalent to bry (b s y ( y a it follows from the first equation in (. that a s (s b ( s s( bry ( a. sy y a Because of the relation s > a implies that s > b and s > a. Thus if s b or s a then a > i.e. the trace of the Jacobian matri at ( y is negative. From (. we have rsy sd s( y y a > sd s. According to then r a > r rsy ( y a ( y a 3 b y > sd s ( y a ( y a > implies a 3 >. Since r sd s ( y a ( y a 3 > can be derived by s sd ra we see that if s sd ra then a >. Especially when s d then a > i.e. the determinant of the Jacobian matri at ( y is positive. Let S ma{a b a} and S ma{a ra d }. By the Routh-Hurwitz criterion it is straightforward to reach a condition depending only on parameters for the local stability of ( y follows: Theorem.3. If condition (. and s min{s S } (.7 hold then the unique positive equilibrium ( y of system (.3 is locally ymptotically stable.

6 6 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 Remark.4. In it shows that for system (.3 the positive equilibrium is locally ymptotically stable if d < ad < ( d ( s d r or s d r < ad < ( d ( s d r. Clearly this condition can also generate condition (.7 but (.7 looks succinct. Remark.5. Notice that the unique positive equilibrium is not a saddle under condition (. and we have derived that when s d (and even s S then a > which will be used for analyzing Hopf bifurcation in Section 4. Here it is still of our interest whether a > always holds or not while we analyze Hopf bifurcation. 3. Global stability of the positive equilibrium In the previous section we have presented conditions for local ymptotic stability of the positive equilibrium. In this section we try to establish sufficient conditions for its global ymptotic stability. For this by 8 Theorem 4.3 we need to provide conditions to ensure that there is no closed orbit ecept for the positive equilibrium ( y in the first quadrant R. We will use Dulac s criterion and the divergency criterion for analyzing the global attractiveness of ( y respectively. Theorem 3.. For system (.3 if condition (. and min{bd a b} (3. hold then the positive equilibrium is globally ymptotically stable. Proof. For system (.3 by choosing B( y y a there holds BP BQ y ( y a( bd bry ( b ( bry sy bdy < in the simply connected region R due to conditions (. and (3.. Thus by Dulac s criterion 3 there is no periodic solution in R and this implies that the positive equilibrium ( y is globally ymptotically stable. Lemma 3. (divergency criterion 3. Assume that L is a closed orbit with period T. If the condition div( y < (> holds then L is a single stable (unstable limit cycle. Bed on Lemma 3. we obtain the following result. Theorem 3.3. For system (.3 if condition (. and s ma{b 4abr b 4a} (3. hold then the local and global ymptotic stability of the positive equilibrium ( y coincide. Proof. For system (.3 the Jacobian matri is ( J J J((t y(t J 3 J 4

7 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 7 where sy(t J (t (t y(t a s(ty(t ((t y(t a J s(t((t a ((t y(t a by(t(y(t a J 3 ((t y(t a J (t 4 b d ry (t y(t a (ty(t ((t y(t a. Assume that l(t ((t y(t is an arbitrary but fied nontrivial periodic orbit of system (.3 with period T > then there holds If s b then If s > b then (t (t y (t y(t trj((t y(t trj((t y(t sy(t (t (t y(t a b (t d ry(t. (t y(t a (s b(ty(t (t bry(t <. ((t y(t a (t (s by(t 4a (s by(t < 4a because of the condition s b 4abr and trj((t y(t (t (s b(t 4a bry(t < by the condition s b 4a. Consequently by the divergency criterion the closed orbit l(t is stable which yields a contradiction with the local ymptotic stability of the positive equilibrium. So if ( y is locally ymptotically stable system (.3 h no nontrivial periodic orbit in R. This indicates that the positive equilibrium must be also globally ymptotically stable. Then from Theorems.3 and 3.3 we can directly obtain the following corollary. Corollary 3.4. For system (.3 if conditions (. (.7 and (3. hold then the unique positive equilibrium is globally ymptotically stable. Remark 3.5. Provided that conditions (. and (3. hold we can additionally obtain that system (.3 h the unique (stable limit cycle in the first quadrant if the positive equilibrium is unstable which will be described Corollary 4.5. Note that in the proof of Theorem 3.3 we simply apply the mean inequality to sure the negativeness of the divergency integral. In fact the divergency integral can be further epanded by applying Grammer s rule and Green s theorem. However this requires boundary values of periodic orbits of system (.3. For this similar to 9 Theorem. by defining a set where s y r(a s d r Γ : {( y R : y y y} d r(a s r ( d( s da 4 r and y d(a dr(a we have the following stronger permanent condition for providing concrete boundary values.

8 8 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 Theorem 3.6. Suppose that system (.3 satisfies the condition ad < ( d( s and s <. (3.3 Then for any solution ((t y(t of system (.3 with the positive initial condition (i.e. ( > and y( > there is a T > such that for all t > T ((t y(t Γ holds. Obviously since ad < ( d( s implies that condition (. holds and s < implies that condition (.7 holds by combining Theorem.3 and Corollary 3.4 we directly have the following corollary. Corollary 3.7. If condition (3.3 holds then the unique positive equilibrium is locally ymptotically stable. If conditions (3. and (3.3 hold then the unique positive equilibrium is globally ymptotically stable. From Theorem 3.3 and Corollary 3.7 we know that if conditions (3.3 holds and s b the unique positive equilibrium of system (.3 is globally ymptotically stable. Further we can derive from Theorem 3.6 that if condition (3.3 holds all possible periodic orbits must lie in Γ. Thus instead of (. we attempt to employ condition (3.3 for the ce of s > b to derive the divergency integral and obtain the following theorem. Theorem 3.8. For system (.3 if the condition (s b( sb r < min{ ( y a ( y a s b( y a } or (s b( sb > r < min { ( y a ( y a s } b( y a ( y a > by( d ry (3.4 holds then the unique positive equilibrium ( y is globally ymptotically stable provided that condition (3.3 holds. Proof. Assume that l(t ((t y(t is an arbitrary but fied nontrivial periodic orbit of system (.3 with period T >. Similar to the proof of Theorem 3.3 it suffices to show that under conditions (3.3 and (3.4 trj((t y(t <. Since ((t y(t is an orbit of system (.3 and (s b(ty(t trj((t y(t (t bry(t ((t y(t a we have trj((t y(t y(t (t bry(t (s bd (t y(t a y(t (s b (t y(t a ( (t (t y(t a d

9 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 9 Further since (t bry(t (s b d s ( (t (t (t y(t (s b (t y(t a (ry(t y (t by(t (t bry(t (s b d s ( (t ry(t (s b (t y(t a s b y (t b (t y(t a (t bry(t (s b s ( (t(d ry(t trj( y it is ey to show that Clearly (s b ry(t (t s (t s b b trj((t y(t trj( y y (t (t y(t a. bry s (s b( (d ry s b y (t b (t y(t a s b s ( ((t br(y(t y s b s trj( y ry(t (t (t ((d ry(t( (t (d ry ( ry(t (t (t s b y (t b (t y(t a s b s ( s (s b(d ry(t ((t ( r s (s b( b (y(t y. s b y (t b (t y(t a s b s ( bry d s b bs l s y a dy. ry(t (t (t (3.5

10 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 Moreover since ((t y(t satisfies system (.3 and ( y satisfies system (. we can obtain that and (t (t (t sy(t (t y(t a sy y a (t sy(t (t y(t a sy ( y a((t y(t a ( as ( y a((t y(t a (y(t y ((t y (t (t d ry(t by(t (t y(t a ry y a ry(t (t (t y(t a y a ( y a((t y(t a ((t r Thus by Crammer s Rule we have ( y a((t y(t a y a r((t y(t a (t (t ( as r((t y(t a rsy y a y y y a y a (t rsy y a sy (y(t y. y a y (t by(t ((ty(ta( y a (t y a ((t y(t a y (t by(t r((t y(t a ((ty(ta( y a Let D denote the region encloused by the periodic orbit l(t and M ( y s (s b(d ry(t r((t y(t a rsy y a r s M ( y (s b( b r((t y(t a rsy M 3 ( y r y a ((ty(ta ( y a by ( r((t y(t a rsy y a Then from (3.5 and (3.6 we obtain s (s b(d ry(t ((t r s (s b( b (y(t y ( M ( y ( y a((ty(ta ( y a((ty(ta ( y a((ty(ta y a r((t y(t a (t (t.. (3.6

11 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS ( as M ( y y a ( M ( y y (t by(t sy y y a ((t y(t a (t by(t y a (t M ( y y a (t ( d M ( y l y a r( y a y a d M ( y l y a ( sy dy M ( y ( y a l y a by l M ( y ( as y a In the following we denote that dy by. K (s b(d ry s K r( s (s b( b. Then by applying Green s theorem we deduce that trj((t y(t trj( y D D D D D D D It is ey to see that s b D bs ( s ( y a br d dy s( a K M 3 ( y y d dy a K M 3 ( y by( y a r( y a d dy by(y a K M 3 ( y y d dy a sy K M 3 ( y ( y a ( y a d dy r by s (s b( b r( y a rsy y a r s (s b y a r( y a rsy y a ( y a(ya r( y a r s (s b(d ry r( y a rsy y a ( y a(ya ( y a(ya d dy d dy d dy. trj( y < since ( y is locally ymptotically stable under conditions (3.3. D ( s (ya br d dy < because of the condition r < s b(ya. M 3 ( y > because of r < (ya ( y a.

12 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 the denominator in M is negative since condition (3.3 implies that rsy > sd s >. Thus if (s b( sb then s > b and we have trj((t y(t <. And if (s b( sb > then s > b and we have by(y a ( y a sy y a ( y a < by the inequality ( y a > by( d ry which also implies that trj((t y(t <. Consequently under conditions (3.3 and (3.4 the positive equilibrium ( y is globally ymptotically stable. Remark 3.9. From the proof of Theorem 3.8 we can find that when r trj((t y(t < always holds. Thus for system (.3 with r the local and global ymptotic stability of the positive equilibrium coincide which w also proved in 5. Remark 3.. Similarly it is also interesting to find conditions to ensure that trj((t y(t > ; that is (s b(ty(t (t bry(t ((t y(t a > (3.7 such that the local and global ymptotic stability of the positive equilibrium ( y coincide. In Theorem 3.8 we obtained a sufficient condition for global stability of the positive equilibrium bed on the permanence condition (3.3 which is also the local ymptotic stability condition. But this condition contains and y which we need to numerically solve the positive equilibrium first. Intuitively condition (3.4 can be simplified by replacing and y with boundary values defined in Γ. However for the ce of s > b the sub-condition (s b( sb > in condition (3.4 can not be simplified further since. 4. Hopf bifurcations with respect to one parameter r and all five parameters Compared to the systems studied in system (.3 involves the density dependence of the predators and thus h the etra term ry with r >. In this section we will first focus on the Hopf bifurcation with respect to the parameter r and then etend our discussions to all other parameters. From 6 we know that system (.3 with r h a unique positive equilibrium if and only if condition (. holds. Thus system (.3 with r h a unique positive equilibrium if and only if condition (. holds. Following 7 one can see that this positive equilibrium ( y must smoothly depend on the parameters a > d > s > b > and r. For the Hopf bifurcation we perform two different choices of parameters to analyze the change of Sign(a by following the ide in 6.

13 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 3 Lemma 4.. For system (.3 with r if (. holds and s > ma{b bd d } then we have a < if and only if a < s b (s bd s s (s bd s (s bd s sd. d Proof. Let ρ (s bd s(s bd and â ρ sd (ρ s sd. Then we see â > if and only if s > bd d d. As r in system (. it gives ads ( s sd. Namely when a â it h ρ. As r in system (.3 if s > b we have a ( b s ( ρ. Because sd s ( sd s 4ads increes a increes. So we see that a < if and only if a < â. Lemma 4.. Under condition (. if s > S and b b s then a r >. Proof. According to the equation rsy dry sd s ry increes increes. Moreover with the condition b b s a increes increes since a d( b s (b b s ry ( b s r y d( b s. Hence it suffices to prove that increes r increes. Since ( y satisfies system (. under the condition s > S ( y can be determined by the two hyperbolic equations whose corresponding locations can be roughly shown in Figure (left and Figure (middle by the clsifications provided in 8. Notice that the hyperbolic curves in Figure (left do not depend on the parameter r and ( ad d lies on the right hyperbolic curve in Figure (middle. Further for the second equation in system (. dy d d ry r(yad > holds for the right curve in the first quadrant especially for r < r ( y ( ad d y r > (yad d ry r (yad d r and. Moreover in the first quadrant when r < r the curve corresponding to r lies above the curve corresponding to r which is shown in Figure (right. Otherwise the curve corresponding to r must intersect with the curve corresponding to r at a point ( y in the first quadrant. d r y r ( y ad < d r y r ( y ad However since it arrives at a contradiction. As shown in Figure (left and Figure (right r increes we see that increes. Thus we complete the proof. Bed on the above two lemm let us first focus on the Hopf bifurcation with respect to the parameter r regarding other parameters fied constants. Under condition (. if s > S and b b da(r s then dr > holds for system (.3 with r. Furthermore by Lemm 4. and 4. we have the following result: Lemma 4.3. For system (.3 under the same conditions shown in Lemm 4. and 4. there eists a unique r such that a (r. Moreover a (r < if r < r and a (r > if r > r. Proof. Let ( y be the point in Figure (left with d( b s d( b s. Under the conditions shown in Lemm 4. and 4. we can obtain that for the sufficiently large r holds and hence a (r d( b s (b b s ry ( b s r y d( b s > holds. Otherwise for all r and < due to the

14 y y y 4 X. JIANG Z. K. SHE Z. FENG EJDE-6/ r r r r r r Figure. Left: Curves for the hyperbolic equation sy ( ( y a. Middle: Curves for the hyperbolic equation (d ry( y a. Right: Curves for (d ry( y a with r > r > r. hyperbolic curves in Figure (left and Figure (middle y > y r and so ry r. This contradicts the fact that ry < d. From Lemm 4. and 4. there eists one unique r such that a (r and a (r < if and only if r < r. From Lemma 4.3 a (r under the conditions shown in Lemm 4. and 4.. Moreover recalling from Section that when s d we have a (r > and thus a (r >. Since a and a smoothly depend on the parameters a > b > d > s > and r there eists a neighborhood W of r such that for all r W there holds a (r < 4a (r. This implies that the characteristic equation (.5 h conjugate comple roots for all r W denoted λ : α(r ± iω(r a 4a a ± i. Notice that α(r and ω(r >. Hence the characteristic equation (.5 h one pair of conjugated pure imaginary roots λ ±iω(r. By the center manifold theorem 6 the orbit structure near ( y r ( y r can be determined by the vector field (.3 restricted to the center manifold which h the following form ( ( ( α(r ω(r f ω(r α(r y ( y r f ( y r ( d dy where f and f are nonlinear in and y. Then by the Hopf bifurcation theory 6 we can directly obtain the following result on Hopf bifurcation for system (.3. Theorem 4.4. For system (.3 if the following conditions ad < d s d b b s s > ma{b a a bd d d } and a < s s b s (s bd (s bd s (s bd s sd hold then the positive equilibrium ( y is an unstable focus for < r r and an ymptotically stable focus for < r r. Moreover when u(r > there eists a neighborhood U of the positive equilibrium ( y such that the system h a unique unstable periodic orbit in U for < r r (and hence a stable periodic orbit eists outside this unstable periodic orbit. When u(r < there eists a neighborhood U of the positive equilibrium ( y such that the system h a unique stable periodic orbit in U for < r r where u(r is given in 6

15 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 5 : u(r 6 f fyy fy fyyy 6ω(r f y(f fyy fy(f fyy ff fyyf yy. In Theorem 4.4 we have discussed the eistence of limit cycles for sufficiently small r r. In fact combining Lemm 4.3 and 3. with the proof of Theorem 3.3 it is straightforward to obtain the following corollary. Corollary 4.5. For system (.3 if the conditions shown in Theorem 4.4 hold then there is at let one stable limit cycle when r < r. Moreover the limit cycle is unique if condition (3. is also satisfied. Additionally we have the following stability results on the positive equilibrium. Corollary 4.6. For system (.3 if the conditions shown in Theorem 4.4 hold then the positive equilibrium is unstable if r < r and locally ymptotically stable if r > r. Moreover the positive equilibrium is globally ymptotically stable when r > r and condition (3. holds. Up to now we have discussed the Hopf bifurcation only with respect to the parameter r. It is also interesting to discuss the Hopf bifurcation with respect to all parameters. In the rest of this section we would like to set a special geometrical structure such that the analysis process can be simplified by applying the clsical Hopf bifurcation theory on systems with one single parameter. For this let µ (b s y ( y a bry be an auiliary map which can be also regarded an auiliary parameter and will be used to analyze the Hopf bifurcation. Intuitively by this parameter µ we restrict the five parameters of system (.3 to a special geometrical structure such that the five-dimensional parameters can be simply mapped to one-dimensional parameter. Thus it enables us to consider the Hopf bifurcation along with this special geometrical structure where the five parameters to some etent affect together one parameter µ. Note that from Lemm 4. and 4.3 there certainly eist conditions on the five parameters a b d r s to sure the possibilities of µ > µ < and µ respectively. Moreover since a and a smoothly depend on the five parameters a > if s d and µ can be regarded a continuous map of the five parameters. For the parameters a b d r s satisfying µ(a b d r s there eists a neighborhood K R 5 of (a b d r s (and thus a neighborhood Ω R of µ such that for all (a b d r s K (and thus µ Ω a (a b d r s < 4a (a b d r s holds. This implies that the characteristic equation (.5 h conjugate comple roots in K 4a a denoted λ : β(µ ± iϕ(µ a ± i. Clearly when µ β( β ( < and ϕ( > the characteristic equation (.5 h one pair of conjugated pure imaginary roots λ ±iϕ(. By the center manifold theorem 6 we can see that the orbit structure near ( y a b d r s with µ(a b d r s (i.e. near ( y µ with µ is determined by the vector field (.3 restricted to the center manifold which h the form ( d ( ( ( β(µ ϕ(µ f dy ( y µ ϕ(µ β(µ y f ( y µ where f and f are nonlinear in and y.

16 6 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 Since β ( < by the clsical Hopf bifurcation theory on dynamical systems with one single parameter 6 we can obtain the following theorem on the Hopf bifurcation with respect to all the five parameters along with the special geometrical structure µ (b s y ( y a bry. Theorem 4.7. For system (.3 if s d then the positive equilibrium ( y is an unstable focus for µ < or an locally ymptotically stable focus for < µ. Moreover when ψ( > there eists a neighborhood Ū of the positive equilibrium ( y such that the system h a unique unstable periodic orbit in Ū for < µ (and hence a stable periodic orbit eists outside this unstable periodic orbit. When ψ( < there eists a neighborhood Ū of the positive equilibrium ( y such that the system h a unique stable periodic orbit in Ū for µ < where the coefficient ψ( is given in 6: ψ( 6 f f yy f y f yyy f y(f f yy f f f yyf yy 6ϕ(. 5. Numerical simulations In this section we illustrate some numerical eamples. f y(f f yy Eample 5.. Let a 3 b 5 s d. and r then system (.3 becomes (ty(t (t (t( (t (t y(t 3 y (t 5y(t ( (5. (ty(t. y(t. (t y(t 3 Since.6 ad < d.8 and s < a 6 according to Theorem.3 the positive equilibrium ( y ( of system (5. is locally ymptotically stable. In fact the positive equilibrium is globally ymptotically stable too since bd and a b and condition (3. in Theorem 3. holds. The global ymptotic stability can be seen from Figure (left. Note that in Figure (left all si orbits go to ( t tends to starting from initial points (.. (.5.3 (.7.6 (.9. (..5 and (.5.35 respectively. Eample 5.. Let a b s.5 d. and r then system (.3 becomes (t (t( (t.5(ty(t (t y(t ( y (ty(t (5. (t y(t. y(t. (t y(t Since. ad < ( d( s.45 and.5 s < according to Theorem 3.6 the positive equilibrium ( y ( of system (5. is locally ymptotically stable. In fact by Theorem 3.3 the positive equilibrium is globally ymptotically stable too since.5 s < b. The global ymptotic stability can be seen from Figure (right. Note that in Figure (right all si orbits all go to ( t tends to starting from initial points (.. (.5.3 (.7.6 (.9. (..5 and (.5.35 respectively.

17 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS y y Figure. Left: Evolutions of si orbits for system (5.. Right: Evolutions of si orbits for system (5.. Eample 5.3. Let a b / s / d /6 and r /75 then system (.3 becomes (t (t( (t (ty(t (t y(t y (t y(t ( 6 75 y(t (ty(t (t y(t. (5.3 The positive equilibrium is ( y ( A straightforward calculation gives that s < 3 ad < ( d( s 5 (s b( sb.664 > 75 r < (a d r < s( s (a d 34 and b < d(a ( s. This implies that conditions (3.3 and (3.4 in Theorem 3.8 are satisfied. Thus ( y is globally attractive which can be seen from Figure 3. Note that in Figure 3 the si orbits starting from initial points (..7 ( (.5.3 (.7 (..5 (.5 3 respectively go to ( t tends to y Figure 3. Evolutions of si orbits for system (5.3.

18 8 X. JIANG Z. K. SHE Z. FENG Eample 5.4. Let a EJDE-6/55 b s 34 and d /4 then system (.3 becomes 4 3 (ty(t (t (t( (t (t y(t (t y (t y(t ry(t. 4 (t y(t A direct calculation shows that 9 bd d a a d } 5 4 ad < d (5.4 s > ma{b s b (s bd 3 a< s sd s s (s bd s (s bd 89 and 34 b sb. This indicates that the conditions in Lemm 4. and 4. hold and the condition 43 s d 43 also holds. Hence there is one unique r which can be calculated numerically r.5395 and then u(r >. We can choose r.8 > r and find.45 a >. > 4 a. By Theorem 4.4 the positive equilibrium ( y (.4. is locally ymptotically stable and there is at let one unstable closed orbit and one stable closed orbit in the first quadrant. Note that in Figure 4 (left the orbit starting from initial points (.46.9 and (.43.8 tends to a stable limit cycle t approaches ; and in Figure 4 (middle and 4 (right ((t y(t starting from initial points (.46.9 tends to a periodic form t approaches. Note that the unstable limit cycle in the region enclosed by the stable limit cycle is very close to the positive equilibrium and thus h not been shown in Figure 4 (left y.45 y t t Figure 4. Phe diagram (left for system (5.4 with evolution of (t (middle and y(t (right t. Conclusions and future works. In this article we considered the stability of the unique positive equilibrium and Hopf bifurcation with respect to parameters in a density-dependent predator-prey system with the Beddington-DeAngelis functional response. We started with the eistence and uniqueness of the positive equilibrium which can not be a saddle and provided first a weaker sufficient condition by the Lyapunov function method and then a concrete condition only depending on parameters by the Routh-Hurwitz criterion for local stability. Moreover we presented several sufficient conditions for global stability of the positive equilibrium by two clsical criteria. That is by Dulac s criterion we directly obtained (. and (3. for global stability. By the divergency criterion we established (. (.7 and (3. the sufficient conditions of global attractiveness.

19 EJDE-6/55 DENSITY-DEPENDENT PREDATOR-PREY SYSTEMS 9 By Grammer s rule and Green s theorem we derived the divergency integral and further obtained (3.3 and (3.4 the sufficient conditions of global attractiveness. Afterwards we analyzed the Hopf bifurcation with respect to the parameter r by eploring the monotonicity of a (r and successively obtaining a unique r such that a (r. Furthermore we introduced an auiliary map µ (b s y ( y a bry to restrict the five parameters to a special one-dimensional geometrical structure and analyzed the Hopf bifurcation with respect to all five parameters along with this geometrical restriction. Note that Hwang verified that for system (. the local stability and global stability of the positive equilibrium coincide in 5. However from analysis in Section 4 we find that for system (.3 the coincidence between local stability and global stability does not hold because of the occurrence of the parameter r. Consequently the analysis results show that the predator density dependence rate r h a significant effort on the global dynamics of system (.3. Finally some numerical simulations have been performed to illustrate our analytical results. Notice that from the analysis in Section 4 system (.3 h limit cycles under certain conditions. However the problem on the number of limit cycles is not involved at this stage. So it is interesting to further eplore the number of the limit cycles with their location estimate in our future work well the necessary and sufficient conditions for the uniqueness of limit cycles. Acknowledgments. This work is supported by National Science Foundation of China under and 4. References P. A. Abrams L. R. Ginzburg; The nature of predation: prey-dependent ratio-dependent or neither? Trends Ecol. Evol (. D. D. Bainov P. S. Simeonov; Systems with impulse effect: stability theory and applications. Ellis Horwood Limited Chichester ( D. D. Bainov P. S. Simeonov; Impulsive differential equations: periodic solutions and applications. Longman Scientific and Technical New York (993 4 J. R. Beddington; Mutual interference between parites or predators and its effect on searching efficiency J. Animal. Ecol ( E. Beretta Y. Kuang; Geometric stability switch criteria in delay differential systems with delay dependent parameters SIAM J. Math. Anal (. 6 R. S. Cantrell C. Cosner; On the dynamics of predator-prey models with the Beddington- DeAngelis functional response J. Math. Anal. Appl (. 7 C. Cosner D. L. DeAngelis J. S. Ault D. B. Olson; Effects of spatial grouping on the functional response of predators Theor. Pop. Biol ( J. M. Cushing; Periodic time-dependent predator-prey systems SIAM J. Appl. Math ( D. L. DeAngelis R. A. Goldstein R. V. O Neil; A model for trophic interaction Ecology (975. Z. J. Du X. Chen Z. Feng; Multiple positive periodic solutions to a predator-prey model with Leslie-Gower Holling-type II functional response and harvesting terms Discrete Contin. Dyn. Syst. Ser. S 7(6 3-4 (4. Z. J. Du Z. Feng; Periodic solutions of a neutral impulsive predator-prey model with Beddington-DeAngelis functional response with delays J. Comput. Appl. Math (4. W. Hahn; Stability of Motion. Springer ( J. Hale; Ordinary Differential Equations. Krieger Malabar (98.

20 X. JIANG Z. K. SHE Z. FENG EJDE-6/55 4 J. K. Hale P. Waltman; Persistence in infinite-dimensional systems SIAM J. Math. Anal ( T. W. Hwang; Global analysis of the predator-prey system with Beddington-DeAngelis functional response J. Math. Anal. Appl (3. 6 T. W. Hwang; Uniqueness of limit cycles of the predator-prey system with Beddington- DeAngelis functional responsee J. Math. Anal. Appl. 9 3 (4. 7 P. Kratina M. Vos A. Bateman B. R. Anholt; Functional response modified by predator density Oecologia (9. 8 H. Li Z. She; A Density-Dependent Predator-Prey Model with Beddington-DeAngelis Type Electron. J. Differ. Eq. 9-5 (4. 9 H. Li Z. She; Uniqueness of periodic solutions of a nonautonomous density-dependent predator-prey system J. Math. Anal. Appl (5. H. Li Z. She; Dynamics of a nonautonomous density-dependent predator-prey model with Beddington-DeAngelis functional response Int. J. Biomath. 9(4 Article ID pages (6. doi:.4/s H. Li Y. Takeuchi; Dynamics of the density dependent predator-prey system with Beddington- DeAngelis functional response J. Math. Anal. Appl (. S. Liu E. Beretta; A stage-structured predator-prey model of Beddington-DeAngelis type SIAM J. Appl. Math (6. 3 Z. She H. Li; Dynamics of a desity-dependent stage-structured predator-prey system with Beddington-DeAngelis functional response J. Math. Anal. Appl (3. 4 H. R. Thieme; Persistence under relaed point-dissipativity (with application to an endemic model SIAM J. Math. Anal ( Z. Wang J. Wu; Qualitative analysis for a ratio-dependent predator-prey model with stagestructure and diffusion Nonlinear Anal. RWA (8. 6 S. Wiggins; Introduction to applied nonlinear dynamical systems and chaos. Spring-Verlag New York (99. 7 Z. Zhang C. Li Z. Zheng W. Li; Bifurcation theory of vector fields (in Chinese. Higher Education Press Beijing ( J. Zhao J. Jiang; Permanence in nonautonomous Lotka-Volterra system with predator-prey Appl. Math. Comput (4. 9 H. Zhu S. Campbell G. Wolkowicz; Bifurcation analysis of a predator-prey system with nonmonotonic functional response SIAM J. Appl. Math (. Xin Jiang School of Mathematics and Systems Science Beihang University Beijing 9 China address: jiangin99@6.com Zhikun She (corresponding author School of Mathematics and Systems Science Beihang University Beijing 9 China address: zhikun.she@buaa.edu.cn Zhaosheng Feng School of Mathematical and Statistical Sciences University of Te-Rio Grande Valley Edinburg TX USA address: zhaosheng.feng@utrgv.edu