GLOBAL DYNAMICS OF A PREDATOR-PREY MODEL WITH HASSELL-VARLEY TYPE FUNCTIONAL RESPONSE. Sze-Bi Hsu. Tzy-Wei Hwang. Yang Kuang

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SERIES B Volme, Nmber 4, November 28 pp GLOBAL DYNAMICS OF A PREDATOR-PREY MODEL WITH HASSELL-VARLEY TYPE FUNCTIONAL RESPONSE Se-Bi Hs Department of Mathematics National Tsing Ha University Hsinch, Taiwan, ROC Ty-Wei Hwang Department of Mathematics National Chng Cheng University Min-Hsing, Chia-Yi 62, Taiwan, ROC Yang Kang Department of Mathematics, Ariona state University Tempe, AZ , USA (Commnicated by Hal Smith Abstract. Predator-prey models with Hassell-Varley type fnctional response are appropriate for interactions where predators form grops have applications in biological control. Here we present a systematic global qalitative analysis to a general predator-prey model with Hassell-Varley type fnctional response. We show that the predator free eqilibrim is a global attractor only when the predator death rate is greater than its growth ability. The positive eqilibrim exists if the above relation reverses. In cases of practical interest, we show that the local stability of the positive steady state implies its global stability with respect to positive soltions. For terrestrial predators that form a fixed nmber of tight grops, we show that the existence of an nstable positive eqilibrim in the predator-prey model implies the existence of an niqe nontrivial positive limit cycle.. Introdction. Predator-prey models are argably the most fndamental bilding blocks of the any bio- ecosystems as all biomasses are grown ot of their resorce masses. Species compete, evolve disperse often simply for the prpose of seeking resorces to sstain their strggle for their very existence. Their extinctions are often the reslts of their failre in obtaining the minimm level of resorces needed for their sbsistence. Depending on their specific settings of applications, predator-prey models can take the forms of resorce-consmer, plantherbivore, parasite-host, tmor cells (virs-immne system, ssceptible-infectios interactions, etc. They deal with the general loss-win interactions hence may have applications otside of ecosystems. When seemingly competitive interactions 2 Mathematics Sbject Classification. 34D5, 34D2, 92D25. Key words phrases. Fnctional response, predator-prey model, global stability, limit cycles, extinction. The research of S.B. Hs T.W. Hwang are spported by NSC, R.O.C. The research of Yang Kang is spported in part by DMS DMS/NIGMS

2 858 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG are careflly examined, they are often in fact some forms of predator-prey interaction in disgise. The most poplar predator-prey model is the one with Michaelis-Menten type (or Holling type II fnctional response (Freedman 98 [2]: x (t ax( x/k cxy/(m x y (t y(fx/(m x D x( >, y( > where x, y st for prey predator density, respectively. The constants a, K, c, m, f, D are positive that st for prey intrinsic growth rate, carrying capacity, captring rate, half satration constant, maximal predator growth rate, predator death rate, respectively. This model exhibits the well-known bt highly controversial paradox of enrichment observed by Hairston et al (96 [4] by Rosenweig (969 [25] which is rarely reported in natre. To address this problem respond to the need of a simple deterministic model that prodcing the often observed extinction of prey species in isl ecosystems (Ebert 2 [9], Fan et al. 25 [], Arditi Ginbrg (989 [3] proposed the following predator-prey model with ratio-dependent type fnctional response: x (t ax( x/k cxy/(my x y (t y(fx/(my x D x( >, y( >. It is well known (Kang Beretta (998 [23], Jost et al. (999 [2], Hs et al. (2 [8], Xiao Ran (2 [26], Bereovskaya et al. (2 [4] that the system (2 can display richer more plasible dynamics than that of system (. It was known that the fnctional response can depend on predator density in other ways. One of the more widely known one is de to Hassell Varley (969 [7]. A general predator-prey model with Hassell-Varley type fnctional response may take the following form x (t ax( x/k cxy/(my γ x F(x, y, y (t y(d fx/(my γ x G(x, y, γ (,, x( x >, y( y >. In the following, we will call γ the Hassell-Varley constant. A nified mechanistic approach was provided by Cosner et al. (999 [8] where the fnctional response in system (3 was derived. In a typical predator-prey interaction where predators do not form grops, one can assme that γ, prodcing the so-called ratio-dependent predator-prey dynamics. For terrestrial predators that form a fixed nmber of tight grops, it is often reasonable to assme that γ /2. For aqatic predators that form a fixed nmber of tight grops, γ /3 maybe more appropriate. Since most predators do not form a fixed nmber of tight grops, it can be arged that for most realistic predator-prey interactions, γ [/2,. Or main reslts are applicable to these realistic cases. Mathematically, systems ( or (2 can be viewed as limiting cases of systems (3 if one chooses γ or in system (3. 2. Preliminary analysis. The main objective of this paper is to gain a detailed global ndersting of the dynamics of system (3. In this section, we present the basic reslts on the bondedness of positive soltions the local stabilities ( (2 (3

3 HASSELL-VARLEY PREDATOR-PREY MODEL 859 of nonnegative eqilibria in (3. To this end, we nondimensionalie the system (3 with the following scaling t at, x x/k, y αy then the system (3 takes the form x (t x( x sxy/(x y γ F(x, y, y (t δy(d x/(x y γ G(x, y, x( x >, y( y >, where α ( m K c γ, s a K (K m f γ, δ a, d D f. (5 Observe that lim (x,y (, F(x, y lim (x,y (, G(x, y. We ths define that F(, G(,. Clearly, with this assmption, both F G are continos on the closre of R 2 where R2 {(x, y x >, y > }. The variational matrix of the system (4 is given by (4 A(x, y 2x sy x y γ sxy (x y γ 2, sx (x y γ 2 (x ( γyγ δy γ (x y γ 2, δ( x x y γ The following proposition shows that system (4 is dissipative. γxyγ (x y γ 2 d. (6 Proposition. Let (x(t, y(t be any soltion of (4 with (x(, y( R 2, then lim sp(x(t sy(t/δ t ( dδ2. 4dδ Proof. It follows immediately from the existence niqeness of soltions for ordinary differential eqations with initial conditions that the soltion is positive on its domain of definition. Let V (t x(t s δy(t differentiating V once yields V (t x(t( dδ x(t dδv (t ( dδ 2 /4 dδv (t. Hence, we have < V (t ( dδ 2 /4dδ (V ( ( dδ 2 /4dδe dδt. This gives the desired reslt. Notice that if d then (x(t, y(t (, as t for all (x(, y( R 2. In the following, we assme that d (,. For d (,, system (4 has three eqilibria. They are E (,, E (, E (x, y, where x >, y > x sy x y γ x x y γ, d. Since the vector field (F, G is not C at E, the stard local stability analysis method can not be applied to E. (7

4 86 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG At E, we have A(, [, s, δ( d ]. (8 This shows that E is a saddle point. At E, we have sy x ( (x y γ 2, sx (x y γ 2 (x ( γy γ A(x, y δy γ (x y γ 2, δ γx y γ. (9 (x y γ 2 A straightforward calclation shows that x 2 det A(x, y δγ y γ (x y γ 2 s( γδ x y 2γ (x y γ 4 sδ( γ x 2 y γ (x y γ 4 > ( sy tr A(x, y x (x y γ 2 δγyγ (x y γ 2. Hence, the stability of E is determined by the sign of tr A(x, y. This gives that E is locally asymptotically stable (or nstable if tr A(x, y < (or >. Smmariing these discssion, we arrive at the following proposition. Proposition 2. For system (4, the following statements hold. a. E is a saddle point. b. E is locally asymptotically stable if tra(x, y <. c. E is nstable if tra(x, y >. 3. Uniform persistence. The objective of this section is to present conditions ensring the system (4 is niformly persistent. To this end, we make the change of variables (x, y (, in system (4, where x/y γ, y σ σ will be chosen later. This redces it to the following system with (t g( ϕ ( σ ϕ 2 ( σ2 f (,, (t ψ( f 2 (,, ( >, ( > g( [ γδd ( γδd γδ]/(, ϕ ( 2, ϕ 2 ( s/(, ψ( σδ(/( d where σ γ/σ σ 2 ( γ/σ. Now let σ γ if γ (, 2 σ γ if γ [ 2, then { if γ (, σ 2, γ/( γ if γ [ 2, σ 2 { ( γ/γ if γ (, 2, if γ [ 2,. Hence, σ i, i, 2 the vector field (f, f 2 is C smooth on the closre of ( (

5 HASSELL-VARLEY PREDATOR-PREY MODEL 86 R 2. Observe that the nmbers of nontrivial positive eqilibria periodic orbits (if any of systems (4 ( are the same. Since < d <, we have ψ( where d/( d > ψ( σδ( d( /(. Moreover, g( > on R if γδ d g( has exactly one positive ero ( dγδ/[( dγδ ] if γδ > d. In last case, we have g(( < for. From system (, we see that the prey isocline, h(, is implicitly defined by f (,. Since f (, g(, lim f (, f (, < it follows from the implicit fnction theorem that h( is a C fnction defined on [, if γδ d or on [, ] if γδ > d. Moreover, h( (γδd s σ 2 h ( f f (, h( (, h( ( g( ϕ 2( ( ϕ( ϕ 2( [h(] σ ϕ ( ϕ σ 2( [h(] σ σ 2 [h(] σ2. (2 The qalitative behavior of h( is given in the following lemma (see Fig. (a-(d Fig.2 (a-(d. Lemma 3.. (a If γδ ( d, then h( > h ( for all [, ]. (b If γδ (, d ] (γδd s σ ( γδd γδ σ2 then h( > > h ( for all (,. (c If γδ (, d ] (γδd s σ < (γδdγδ σ2 then h( > for all [, h ( has exactly one positive ero. Moreover, h (( < for all. Proof. From (2, we have h ( < as long as h σ ( > ( g( ϕ 2 ( /( ϕ ( ϕ 2 ( γδd γδ. 2 Since γδd γδ < < h( for [,, so we have h σ ( > > γδd γδ 2 for all [,. Hence, the assertion (a follows immediately. Now let γδdγδ. It is sfficient to show that h has at most one positive ero in (,. To see this, notice that if h ( then ( 2 h ϕ ( ( 2h σ (/ ϕ 2 ( σ [h(] σ σ 2 [h(] σ2 <. This implies that h ( < (> if > (< near. Hence, h ( < for >. For otherwise, there is û > sch that h ( < on (, û h (û. ( 2 This implies that h (û 2h σ (û/ ϕ(û ϕ σ 2(û [h(û] σ σ 2 [h(û] σ2 <, a contradiction. Since h ( if h σ ( ( γδd s σ σ 2 γδd γδ. Hence, assertion (b follows.

6 862 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG For part (c, we have h ( >. Hence, h ( > as close to. Since h is increasing as long as h σ is bonded by a decreasing fnction γδdγδ 2. There mst exist > sch that h (. Ths, h ( < for >. This proves the assertion (c. Remark. According to the Implicit Fnction Theorem, the fnction h is also dependent on s. The partial derivative of h with respect to s is given by h s (s, ( hσ2 (s, /(σ ϕ (h σ (s, σ 2 ϕ 2 (h σ2 (s, <. Remark 2. As a conseqence of Remark, the positive ero, is also a fnction of s, whenever it is defined. So, for ( γδd s σ < ( γδd γδ σ2 or eqivalently, s > ( γδd/( γδd γδ σ 2 σ, we have h σ (s, (s γδdγδ 2. Differentiating (s with respect to s yields 2( γδd γδ σ (s σ ( 2 (s σ h s (s, (s. Now from Remark, we obtain (s >. Moreover, is nbonded. Becase if it is not the case, then there are sitable positive constants c, c 2, c 3 sch that (s c c 2c 3s, which leads to a contradiction. Notice that ϕ ( [h( ] σ ϕ 2 ( [h( ] σ2 g( >. The system ( always has the trivial eqilibrim e (, the positive eqilibrim e (, where h(. Since g(, the system ( has a bondary eqilibrim e (, if only if γδ( d >. The variational matrix of the system ( is given by J(, [ g ( ϕ ϕ (σ 2 σ (σ2 ϕ ( σ σ 2 ϕ 2 ( σ2 σδ/( 2 σδ(/( d ].(3 The stability of eqilibria e, e e is determined by the eigenvales of the matrices J(e, J(e, J(e respectively is given in the following lemma. Lemma 3.2. For the system (, the following statements are tre. (a e is a saddle point with stable manifold {(, > }. (b If γδ ( d, then e is a saddle point with stable manifold {(, > }; e is locally asymptotically stable. (c If tr(j(e < then e is locally asymptotically stable. (d If tr(j(e > then e is an nstable focs or node. Proof. The variational matrix of the system ( at e is [ ] γδd J(e. σdδ Obviosly, the assertion (a hold. For part (b, the variational matrix at e is [ g J(e f ( (e σ/γ Since g ( ( γδd γδ /( <, so e is a saddle point. ].

7 HASSELL-VARLEY PREDATOR-PREY MODEL 863 To discss the stability of e, observe that the variational matrix at e is [ f J(e (e f (e ] σδ /( 2. Since f (e < so the determinant of J(e is positive the stability of e is determined by the sign of the trace of J(e. Ths e is an nstable focs or node if tr(j(e > e is locally asymptotically stable, if tr(j(e <. Moreover, if γδ ( d,, then from (2 Lemma 3. (a, one obtains tr(j(e f (e <. This proves the assertions (b, (c (d. Remark 3. Notice that, nder the change of variables, the bondary eqilibrim E is transformed to (, E splits into two eqilibria e e. Remark 4. Since (x, y ( γ σ, σ, we have sx y tra(e (x y γ 2 x δγx y γ (x y γ 2 γδ ( ( 2 σ s ( 2 σ2 tr J(e. So, the local stability of E e are the same. From the Proposition, we can prove (see below the system ( is niformly persistent dissipative. Lemma 3.3. The system ( is niformly persistent in R 2. Proof. Let ((t, (t be the soltion starting at A (, M where M [ (dδ2 4sd ] σ Γ be its orbit. Then since (x(t, y(t ((t γ σ (t, σ (t is a soltion of system (4 Proposition, we have limsp t (t M. Hence, Γ R (, M. The flow analysis gives that Γ mst intersect the prey isocline {(, h( < < }, let B be the first point that they intersect. Since e is a saddle point, there are two possibilities for Γ. Case. Γ {(, (, h( }. Let C (, be the first point of Γ {(, (, h( }, D (, be the intersection of {(, > } h(. Consider the bonded region Ω, enclosed by Γ, CD, DE EA where E (, M. Clearly, every trajectory will enter stay in Ω for all t sfficiently large. Case 2. Γ {(, (, h( }. This implies lim t ((t, (t e. Let Ω be the bonded region enclosed by Γ e A. Since e (if exists is a saddle point, ths every trajectory will either enter Ω or tend to e as t goes to. Hence, from the above discssion, we show that the system ( is permanent. Since every soltion of system (4 takes the form of ((t γ σ (t, σ (t, where ((t, (t is some soltion of system (. Ths, as a conseqence of Lemma 3.3, we have the following theorem for system (4.

8 864 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG Theorem 3.4. The system (4 is niformly persistent in R 2. Remark 5. From Proposition 2, Theorem 3.4 the Poincaré-Bendixson Theorem [5], the system (4 has at least one limit cycle in R 2, provided tra(x, y >. 4. Global stability reslts. As we have mentioned at the end of section, the most biologically interesting cases for the system (4 are when γ /2 or 2/3. We ths will focs on the cases when γ /2 in this next sections. To stdy the global behavior of soltions for system (4, we need following lemma. Lemma 4.. Let γ [ 2, Γ(t ((t, (t be any periodic soltion of system ( with period T >. Then tr(j(γ(tdt tr(j(e T P(, dd Ω where Ω is the bonded region enclosed by Γ. The fnction P is given as follow P(, ( q( 2s σ σδ( q( s( d sd( dq ( ( q( s( d 2 { σ σ where q( if σ σ if Proof. First, let s consider the following fnction: q(, θ { θ θ θ( θ if if where θ >. Clearly, q(, θ is a positive, C fnction on [, q(, for. Moreover, q (, θ > (< for > if θ > (<. Since γ [ 2,, we have σ γ γ σ 2. Hence, q ( q (, σ for >. Let A γδ(d B dγδ. From (, we have (t (t (t (t A(t B (t σ (t (t σδ( d (t (t, s (t (t A(t B ( s (t σ (t (t (t (t A B σ s (B A( (t ( ( (t ((t s( σ (t (t ( ( (t (t ( ( σ (t σ s( d((t ( σ σδ( d σ s σδ (t ( ( q( s( d. This gives (t (t (t σδ( d (t (4

9 HASSELL-VARLEY PREDATOR-PREY MODEL 865 Observe that (t (t q( s( d (t tr(j(γ(t ϕ 2 ( ϕ 2 ( (t ϕ 2 ( So, from (, we have σδσ ( d s( d ( σ σδ( d( q( s( d ( ( g( ϕ 2 ( ϕ 2((t ϕ 2 ((t (tdt, tr(j(γ(tdt ( A(t (t (t (t. (5 ( ϕ ( σ (t ϕ 2 ( (t Γ(t (t dt. (t ( ( ( g( ϕ ( ϕ 2 ( σ dt (6 ϕ 2 ( ϕ 2 ( dt. (7 (t( 2(t σ (t (t Using the fact that d /(, we have tr(j(γ(tdt tr(j(e T ( A(t (t( 2(t (t (t ( A ( 2 σ A (t σδ (t dt ( d q((t((t dt. Now from (, (4 (7, we obtain Γ tr(j(γ(tdt tr(j(e T d 2(t σ (t (tdt σδ( d σ (t dt dt ( d 2(t((t σ (tdt (t q( (t ( d q( s( d (t dt q((σδσ ( d s( d ( σ ( d σδ( d( q( s( d ( d q( q( s( d (t (t dt d 2 σδ( d σ (t (t dt ( d q((σδσ ( d s( d ( σ σδ( d( q( s( d M(, d N(, d, (t (t dt (t (t dt

10 866 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG where M(, ( d q( ( q( s( d N(, ( d 2 q((σδσ ( d s( d ( σ σδ( d σ ( d. σδ( d( q( s( d The Green s Theorem implies that tr(j(γ(tdt tr(j(e T sd( dq ( Ω ( q( s( d 2 dd ( 2 Ω σδ( d σ ( Ω P(, dd, Ω Ω ( N M dd ( d q( σ σδ( d( q( s( d sd( dq ( ( q( s( d 2 ( q( 2s σ σδ( q( s( d where Ω is the bonded region enclosed by Γ. This proves the lemma. dd dd Lemma 4.2. Let γ [ 2,. If e is locally asymptotically stable, then the system ( has no nontrivial periodic orbit in R 2. Proof. Let Γ(t (x(t, y(t be any one nontrivial periodic orbit of system ( with period T >. It is sfficient to show that tr(j(x(t, y(tdt <. (8 Bt (4.4 follows immediately form Lemma 4.. Hence, the lemma holds. Since the systems (4 ( have same nmbers of periodic soltions in R 2, so we have the following theorem for system (4. Theorem 4.3. For system (4, the local global asymptotic stability of e coincide, provided γ [ 2,. Notice that the fnction q (, θ < if θ (,. So, the Lemma 4. can not be applied to the case γ (, 2. In sch case, we may constrct a Lyapnov fnction for system (, if γδd γδ. A global stability reslt for system ( its conseqence are given as follows. Lemma 4.4. Let γδdγδ. Then the eqilibrim e is globally asymptotically stable for system ( in R 2. Proof. To show that e is globally asymptotically stable in R 2. Consider the following Lyapnov fnction V (, g( ϕ 2 ( exp ( ϕ ( σ σ2 e ϕ 2 ( σ σ 2 ψ(ξ ϕ 2 (ξ dξ

11 HASSELL-VARLEY PREDATOR-PREY MODEL 867 for (, R 2. The derivative of V along the soltion of system ( is V (, V (, ( g( ϕ 2 ( g( ( ϕ ( ψ( ϕ 2 ( ϕ 2 ( ϕ ( ψ( σ ϕ 2 ( s ψ(( ( γδd γδ ( σ. Clearly, γδd γδ implies V (, for (, R 2. Hence, the lemma follows from Lyapnov-LaSalle s invariance principle (Hale (98 [5]. Theorem 4.5. Let γδd γδ. Then the eqilibrim E is globally asymptotically stable for system (4 in R Uniqeness of limit cycle for the case γ /2.. The most interesting case for system (4 is when γ /2, which corresponds to the scenario of a terrestrial predator-prey interaction where predators form grops (Cosner et al. 999 [8]. In this case, σ γ 2, σ σ 2 the system ( is eqivalent to the following Gase type predator-prey system: (t g( (ϕ ( ϕ 2 ( g( ϕ(, (t ψ(, ( >, ( > (9 h( g( ϕ( A B 2 s, (2 where A γδ(d, B γδd. A straightforward comptation yields ( 2 s 2 h ( A 2 2B As B l(. (2 Lemma 5.. Let A. Then ϕ(h γδ( d ( σδ( d 2 A ψ(h( l( ( (s 2 C A A 2 s 2. where C ( γδ( db. ( da Proof. From (9, (2 (2 we have ϕ(h ( l( ( (s 2, ψ(h( σδ( d A B s 2.

12 868 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG Since l( l( (l( l( (l( l( The lemma follows immediately. ( (A( 2B ( (A 2 (A 2B ( (A 2 ( γδd d ( (A 2 A ( d γδb A ( ( d γδ( B A. Lemma 5.2. If h ( > then system (9 has at most one limit cycle in R 2. Moreover, if it exists, then it is a stable limit cycle. Proof. As a conseqence of h ( > Lemma 3., we have A > C >. Now, it is sfficient to show d d ( ϕ(h ( ψ(h( < for R { }. From Lemma 5., we obtain ϕ(h ( ( γδ( d ψ(h( σδ( d σδ( d 2 A l( A B C A( 2 A B ( γδ( d σδ( d σδ( d 2 A l( q ( q 2 (. Since l( >, ( 2 (A B 2 q ( ( (A B (A B A( A 2 B <, (A B 2 q 2 ( A( 2(A B (AC A 2 ( 2 A 2 2 2AB AB AC <, ths we have d d ( ϕ(h ( ψ(h( < for R { }. Now according to Theorem 2.2 in Hwang (999 [9], the system (9 has at most one limit cycle if it exists then it is stable. A parallel reslt for system (4 can be obtained easily from the fact that both systems ( (4 has the same nmber of periodic soltions in R 2. This is given in the following theorem. Theorem 5.3. Let γ 2. The system (4 has at most one limit cycle in R2, provided tr A(x, y >. Moreover, if a limit cycle exists, then it is orbitally asymptotically stable.

13 HASSELL-VARLEY PREDATOR-PREY MODEL 869 Figre (a.2 Figre (b 2.5 d/2 γ/3 δ3 s d/2 γ/3 δ s Figre (c Figre (d d/2 γ/3 δ s d/2.35 γ/3.3 δ.25 s Figre. Typical isoclines dynamics of system (3 for d 2 γ Figre 2(a d/2 γ2/3 δ3 s Figre 2(b d/2 γ2/3 δ s Figre 2(c d/2 γ2/3 δ s Figre 2(d d/2 γ2/3 δ s Figre 2. Typical isoclines dynamics of system (3 for d 2 γ 2 3. In Figres 2, panels (a-(c show that the local global stability for the positive eqilibrim of (3 coincide. When the positive eqilibrim of (3 is nstable, then a niqe limit cycle is observed (Figres (d, 2(d.

14 87 SZE-BI HSU AND TZY-WEI HWANG AND YANG KUANG Conditions Reslts. d, γ (, E (, is G. A. S. 2. d <, γ (,, tr(a(x, y > At least one limit cycle. 3. d <, γ [ 2,, tr(a(x, y E (x, y is G. A. S. 4. d <, γ (,, tr(a(x, y, E (x, y is G. A. S. δd δ 5. d <, γ 2, tr(a(x, y > There is an niqe limit cycle. Table. Qalitative Behavior of Soltions of System (4. The G. A. S. sts for globally asymptotically stable. 6. Discssion. To facilitate the discssion section, we smmarie or findings into the following table (Table 6.. Recall that s c a K ( K m γ, δ f a, d D f. Since d is eqivalent to D f, from the first assertion in Table 6., we conclde that, if the growth ability of predator (f is no larger than its death rate (D, then the predators are doomed. In the following, we assme that D < f, i.e. < d <. From Theorem 3.4, the system (4 or eqivalently (3, is niformly persistent. This means neither predator nor prey can die ot. Moreover, there is only one positive eqilibrim the existence of limit cycles is garanteed by Poincaré-Bendixson Theorem when the system (3 possesses an nstable positive eqilibrim. From (2, Lemmas d 3., 3.2, Remark 4, we have, if < (> d D fd then E is locally asymptotically stable (nstable. Since Remark 2 shows that is an increasing, nbonded fnction with respect to s (or eqivalently, K. So, the stability of E changes from stable to nstable as K increases. Notice that the eqilibrim density of both species are increasing if K increases. The above discssion strongly spports that phenomena exhibited by systems ( (3 are similar, althogh the smoothness of their vector fields are different. (The vector field of (3 is not smooth at (,. It is qite natre to make the following conjectres:. The local global stability of the positive eqilibrim of (3 coincide. 2. There is a niqe limit cycle if the positive eqilibrim of (3 is nstable. Or findings (assertions 2 5 in Table 6., partially answer these conjectres. However, significant improvements appear to be difficlt. Acknowledgements. The athors wold like to thank the referees very mch for their valable comments sggestions. REFERENCES [] P. A. Abrams L. R. Ginbrg, The natre of predation: prey dependent, ratio-dependent or neither?, Trends in Ecology Evoltion, 5 (2, [2] R. Arditi A. A. Berryman, The biological paradox, Trends in Ecology Evoltion, 6 (99, 32. [3] R. Arditi L. R. Ginbrg, Copling in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 39 (989, [4] F. S. Bereovskaya, G. Karev R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, J. Math. Biol. 43 (2, [5] F. S. Bereovsky, G. Karev, B. Song C. Castillo-Chave, A simple epidemic model with srprising dynamics, Math. Biosci. Eng., (24, 2.

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