Uniqueness of limit cycles of the predator prey system with Beddington DeAngelis functional response

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1 J. Math. Anal. Appl Uniqueness of limit cycles of the preator prey system with Beington DeAngelis functional response Tzy-Wei Hwang 1 Department of Mathematics, Kaohsiung Normal University, 802 Kaohsiung, Taiwan, ROC Receive 4 December 2002 Submitte by H.R. Thieme Abstract The goal of this paper is to establish the uniqueness of limit cycles of the preator prey systems with Beington DeAngelis functional response. Through a change of variables, the preator prey system can be transforme into a better stuie Gause-type preator prey system. As a result, the uniqueness of limit cycles can be solve Elsevier Inc. All rights reserve. Keywors: Beington DeAngelis preator prey moel; Functional response; Global stability; Limit cycles 1. Introuction Generally, a preator-epenent preator prey moel takes the form [1,2] x t = xgx yp x, y, y t = εypx,y µy, 1.1 x0 = x 0 > 0, y0 = y 0 > 0, where xt,yt represent the population ensity of prey an preator at time t, respectively. g is the growth rate of prey an it is assume that g0 >0 an has exactly one positive zero, say K, which is calle the carrying capacity of prey. In most cases also in the following, the prey is always assume to grow logistically, i.e., gx = r1 x/k. The aress: t1445@nknucc.nknu.eu.tw. 1 Research supporte by National Council of Science, Republic of China X/$ see front matter 2003 Elsevier Inc. All rights reserve. oi: /j.jmaa

2 114 T.-W. Hwang / J. Math. Anal. Appl constant r is calle intrinsic growth rate of prey. The function P is the consumption rate of prey by a preator or the functional response of the preator. The constants ε, µ are the conversion rate an the eath rate of the preator, respectively. A functional response is calle of Beington DeAngelis type if it takes the form Px,y = mx/a + by + cx. This type of functional response was introuce by Beington [3] an DeAngelis et al. [6]. The term by measures the mutual interference between preators. A unifie mechanistic approach was provie by Cosner et al. [5]. Base on principle of mass action an the spatial grouping effect of preation, several types of functional response in preator prey moels are erive in their paper see [5] for more etails. The main purpose of this paper is to stuy the uniqueness of limit cycles of preator prey system with Beington DeAngelis type functional response [3 6] x t = rx1 x/k mxy/a + by + cx, y t = y µ + εmx/a + by + cx, 1.2 x0 = x 0 > 0, y0 = y 0 > 0, where r, K, m, a, b, c, µ, ε are positive constants. For simplicity, we nonimensionalize the system 1.2 with the following scaling: t rt, x x/k, y by/ck an then obtain the form x t = x1 x sxy/x + y + A, y t = δy + x/x + y + A, 1.3 x0 = x 0 > 0, y0 = y 0 > 0, where s = m br, mε δ= cr, = cµ mε, A= a ck. 1.4 It is known [4] that the solutions of system 1.3 are positive an boune for all t 0 an if 1 + A 1 then the equilibrium 1, 0 is globally asymptotically stable. This is the intuitive outcome of the extinction of the preator. So, in the following iscussion, we assume that A1 0 <<1 + A 1. Uner the assumption A1, there exist three equilibria 0, 0, 1, 0 an x,y,where x an y are positive an satisfy { 1 x sy x +y +A = 0, x x +y +A =. 1.5 From the first equation in 1.5, wehaves>sy /x + y + A = 1 x. Hence, y can be solve in terms of x. Substituting the expression into the secon equation in 1.5 yiels { 1 x x +A x +s 1 = y, x 2 + s 1 sx As =

3 T.-W. Hwang / J. Math. Anal. Appl a b c Fig. 1. Here = 1/4, δ = 1. a, b an c, show the global behaviors for systems 1.3 an 2.1, respectively. The positive equilibrium attracts all positive solutions in a an c, an there is an exactly one limit cycle in b an. Notice that from the secon equation in 1.6, we have the following inequality that will use several times later: x + s 1 >x + s 1 s = As > x The variational matrix of the system 1.3 is given by [ 1 2x sy sxy ] Jx,y = x+y+a + δyy+a x+y+a 2 sxy x+y+a 2 sx x+y+a + x+y+a 2 δx x+y+a δxy x+y+a 2 δ. 1.8 From the local stability analysis, we have that if trj x,y < 0thenx,y is locally asymptotically stable an if trj x,y > 0thenx,y is an unstable noe or focus. Hence, the existence of limit cycle follows from Poincaré Benixson theorem. Moreover, by using the ivergence criterion, the author [8] prove that for system 1.3, the local an global asymptotic stability of the positive equilibrium coincie. Thus, in the rest of the paper, we assume that A2 trj x,y > 0 or equivalently s δ x + y + A > x + y + A y.

4 116 T.-W. Hwang / J. Math. Anal. Appl Uner assumption A2, it follows that s>δ. Moreover,combining 1.6 with A2, we have s δx + s 1 > 1. sx + A Hence, we get the restrictions on the parameters s>max{δ,1 + A}. 1.9 The main result is to show that uner the assumptions A1 an A2, the system 1.3 has exactly one limit cycle in R 2 + see Fig. 1. The rest of the paper is organize as follows. In Section 2 a transformation will be introuce to engener system 1.3 into a generalize Gause type preator prey system, to which a wealth of existing methos an results are applicable [7,9]. Taking avantage of this, we can prove the uniqueness of limit cycles for the system 1.3. Finally, some biological implications will be iscusse in Section Main result Uner the assumptions A1 an A2 an through the change of variables t = x + y + A τ, u = σxy, system 1.3 becomes x = ϕx[hx u], u = ψxu, 2.1 x0 = x 0 > 0, u0 = u 0 > 0, where θ σx=, θ = δ x s 1 > 0, x ϕx =, σx 1 xx + A hx = σx, ψx = δ x2 + s 1 sx As [ = δ 1 x A ] 1 xx + A. 2.2 Conversely, the change of variables y = u/σ x, t = x + y + A τ converts system 2.1 to 1.3. Hence, system 1.3 has a unique limit cycle in R 2 + if an only if 2.1 oes. Notice that the set Ω = 0, 1 R + R+ 2 is positively invariant an any trajectory must intersect it from the exterior to the interior provie x0 1. So, the limit cycles must lie in Ω. From 1.6 an 1.7, we have ψx = 0 an, consequently, ψ can be written as ψx = δ x x x + x + s 1 s. 2.3

5 T.-W. Hwang / J. Math. Anal. Appl We see that the prey isocline of 2.1 is given by u = hx. Clearly, h1 = 0an h x = x 1 x x + A x + δ σx x 1 xx + A. 2.4 Since h1 = 0, system 2.1 has equilibria E 1 = 1, 0 an E = x,hx. The local stability of E 1 an E are etermine by the eigenvalues of the matrix JE 1 an JE, respectively, where Jx,uis the variational matrix of the system 2.1 an is given by [ Jx,u= ϕ x hx u + ϕxh ] x ϕx ψ. 2.5 xu ψx At E 1, we have [ ] JE 1 + A s 1 θ 1 =. 0 ψ1 This gives E 1 is a sale point. At E, we have [ JE ϕx h = ] x ϕx ψ. x hx 0 Since the eterminant of JE is positive, the trace of JE is ϕx h x an from assumption A2, an 2.4, we have h x = σx x + y + A tr Jx,y > x x + s 1 So, E is unstable. Now we are in a position to state the main theorem. Theorem 2.1. Uner the assumptions A1 an A2, the system 2.1 has at most one limit cycle in Ω. Moreover,ifitexiststhenitisstable. Proof. Accoring to Hwang [7], it suffices to show that ϕxh x/ψx < 0 x for x 0, 1 {x }. From 2.2, one obtains ϕxh 1 xx + A x = x1 A 2x x + δ 1 = x1 A 2x x + δ x A + 1 x + δψx δ = 1 + x 2 + [ 1 δ1 ] x + Aδ + 1 x + δψx 2.7 δ px + 1 x + δψx. 2.8 δ Note that px is efine by 2.7 an 2.8. So, from 1.7, 2.6 an 2.3, we have px = ϕx h x >0 2.9

6 118 T.-W. Hwang / J. Math. Anal. Appl an px px = x x = x x 1 + x + x δ 1 + x + x + s 1 s s 1 s+ 1 1 δ = 1 + ψx δ s 1 s+ 1 1 δx x. Division of ϕxh x by ψx/δ yiels δ ϕxh x = δ px px + px + ψx x + δ ψx ψx δ = δ 1 + ψx ψx δ [ ] s 1 s+ 1 1 δ x x + px x x x + x + s 1 s + x + δ where = x + δ 1 + s 1 + px x x x + x + s 1 s s 1 s+ δ1 x + x + s 1 s = x 1 + s δ x s[1 + s 1 s+ δ1 ] x + x + s 1 s x + s 1 + px 2x + s 1 s 1 x x x s + 2x + s 1 s 1 x + x + s 1 s = x 1 + s δ + px x + s 1 1 2x + s 1 s x x x s + 2x + s 1 s D x + x + s 1 s,

7 T.-W. Hwang / J. Math. Anal. Appl D = px 2x + s 1 s 1 + s 1 s+ δ1 = 1 + x s 1 sx 1 1 δ x + 2 Aδ 1 + s 1 s 2 s 1 s 1 δ = 1 + x + s 1 s δ x + s 1 s+ 2 Aδ = x + s 1 s 2 x x + s 1 s s 1 sx + s 1 s 1 1 δ x + s 1 s+ 2 Aδ = x + s 1 s 2 2 As δ 1 s δx + s 1 s. Now from A1, 1.7 an 1.9, we have D<0. Since s + s 1ψs = sδs 1 A > 0, hence from 1.7, 1.9 an 2.3, we obtain x s < 0. This implies δ ϕxh x x + s 1 = x ψx 2x + s 1 s px x x 2 x s 2x + s 1 s D x + x + s 1 s 2 < 0 for x 0, 1 {x }. This proves the theorem. 3. Discussion For completeness, we summarize global results Theorem 3.1 in [4], Corollary 2.1 in [8] an Theorem 2.1 for system 1.3 in Table 1. To facilitate the iscussion, we nee a lemma. Lemma 3.1. Assume that 0 <<1 + A 1. a If s max{δ,δ/ /1 2 } then trj x,y 0. b If s>max{δ,δ/ /1 2 } then there exists 0 <A <1 / A 1 such that trj x,y < >0 if an only if A > <A. Proof. From 1.8 an 1.5, we have tr Jx,y = x + s δx y x + y + A Table 1 Complete global results of system 1.2 Conitions Results A 1 1, 0 is globally asymptotically stable 2 <1 + A 1,trJ x,y 0 x,y is globally asymptotically stable 3 <1 + A 1,trJ x,y > 0 There is an exactly one limit cycle

8 120 T.-W. Hwang / J. Math. Anal. Appl If δ<sthen substituting 1.5 into 3.1, one obtains tr Jx,y s δ s δ = 1 + x + s s s δ s δ = 1 + s s + s δ x 1 + s δ s 3.2 C x. 3.3 Note that C is efine by 3.2, 3.3 an C 0, 1. Moreover, C + s 1 s 0ifan only if s δ/ /1 2. Now for part a, if s δ then from 3.1, trj x,y < 0. If δ<s δ/ /1 2 then CC + s 1 s 0 <As= x x + s 1 s. From 1.7, we have C x for all A 0,A 1. Consequently, trj x,y 0. For part b, let A = CC + s 1 s/s.hence, 0 <A < s 1 s/s = A 1 an x A x A +s 1 s = A s = CC+s 1 s. This gives x A = C. From the secon equation in 1.6, we can solve x in terms of A an give x A = s1 + s As. Differentiation x with respect to A an using 1.7, yiels x A = s 2x + s 1 s > 0. Thus, from 3.3, we have A < >A if an only if trj x,y > <0. This completes the proof. Recall that s = m/br, δ = mε/cr, = cµ/mε, A = a/kc. Since µ Kmε/Kc + a is equivalent to 1 + A 1 an the first assertion in Table 1, we conclue that while the eath rate of preator is larger than its maximum growth rate then the preator will go extinction. Next, assume <1 + A 1. This is equivalent to 0 <A<A 1. From Lemma 3.1, we know that the global behavior of solutions of system 1.3 are etermine by the following two cases. Case 1. s max{δ,δ/ /1 2 } or equivalently { c b min ε, m 2 ε 2 c 2 µ 2 } µεmε cµ + mrε 2. It follows from the secon assertion in Table 1 that x,y is globally asymptotically stable. In other wors, while the eath rate of preator is less than its maximum growth rate an the prouct of b is large enough then the prey an preator coexist in the form of equilibrium. Case 2. s>max{δ,δ/ /1 2 } or equivalently { c b<min ε, m 2 ε 2 c 2 µ 2 } µεmε cµ + mrε 2.

9 T.-W. Hwang / J. Math. Anal. Appl a b c Fig. 2. In a, the parameter region for the existence of A given by Lemma 3.1, when = 1/4. The unboune region, enote by, is boune by the s-axe an the two lines efine by s = δ an s = s/ /1 2. The graphs of A = A s are given in b with = 1/4 an ifferent δ. Let K = a/ca an K 1 = a/ca 1. Then from the assertions 2 an 3 in Table 1, we obtain that the prey an preator will coexist in equilibrium form if K 1 <K<K an the prey an preator populations exhibit perioic oscillation if K>K. From the above iscussion, we know that when the interference between preators is too weak then the scenario of system 1.2 is similar to the corresponing system with b = 0. On the other han, if the interference between preators is strong enough then the system 1.2 is stable i.e., the stability of the positive equilibrium will not change no matter how large the carrying capacity K is. Acknowlegment The author woul like to thank the referees for their helpful suggestions that improve the presentations in this paper. References [1] P.A. Abrams, L.R. Ginzburg, The nature of preation: prey epenent, ratio-epenent or neither?, Trens Ecol. Evol

10 122 T.-W. Hwang / J. Math. Anal. Appl [2] R. Ariti, L.R. Ginzburg, Coupling in preator prey ynamics: ratio-epenence, J. Theor. Biol [3] J.R. Beington, Mutual interference between parasites or preators an its effect on searching efficiency, J. Animal Ecol [4] R.S. Cantrell, C. Cosner, On the ynamics of preator prey moels with the Beington DeAngelis functional response, J. Math. Anal. Appl [5] C. Cosner, D.L. DeAngelis, J.S. Ault, D.B. Olson, Effects of spatial grouping on the functional response of preators, Theor. Pop. Biol [6] D.L. DeAngelis, R.A. Golstein, R.V. O Neill, A moel for trophic interaction, Ecology [7] T.W. Hwang, Uniqueness of limit cycle for Gause-type preator prey systems, J. Math. Anal. Appl [8] T.W. Hwang, Global analysis of the preator prey system with Beington DeAngelis functional response, J. Math. Anal. Appl [9] Y. Kuang, H.I. Freeman, Uniqueness of limit cycles in Gause-type preator prey systems, Math. Biosci