Complex dynamic behaviors of a discrete-time predator prey system

Transcription

1 Chaos, Solitons and Fractals 3 (7) Comple dnamic behaviors of a discrete-time predator pre sstem Xiaoli Liu *, Dongmei Xiao Department of Mathematics, Shanghai Jiao Tong Universit, Shanghai 4, PR China Accepted October 5 Abstract The dnamics of a discrete-time predator pre sstem is investigated in the closed first quadrant R þ. It is shown that the sstem undergoes flip bifurcation and Hopf bifurcation in the interior of R þ b using center manifold theorem and bifurcation theor. Numerical simulations are presented not onl to illustrate our results with the theoretical analsis, but also to ehibit the comple dnamical behaviors, such as the period-5, 6, 9,, 4, 8,, 5 orbits, cascade of period-doubling bifurcation in period-, 4, 8, quasi-periodic orbits and the chaotic sets. These results reveal far richer dnamics of the discrete model compared with the continuous model. The Lapunov eponents are numericall computed to confirm further the compleit of the dnamical behaviors. Ó 5 Elsevier Ltd. All rights reserved.. Introduction In population dnamics, there are two kinds of mathematical models: the continuous-time models described b differential equations or dnamical sstems, and the discrete-time models described b difference equations, discrete dnamical sstems or iterative maps. The simplest continuous-time population model is the logistic differential equation of a single species, first introduced b Verhulst [] and later studied further b Pearl and Reed []: _ ¼ r ð:þ k where (t) denotes the population of a single species at time t, k is the carring capacit of the population, and r is the intrinsic growth rate. Eq. (.) describes the growth rate of the population size of a single species. However, the population size of a single species ma has a fied interval between generations or possibl a fied interval between measurements. For eample, man species of insect have no overlap between successive generations, and thus their population evolves in discrete-time steps. Such a population dnamics is described b a sequence { n } that can be modeled b the logistic difference equation nþ ¼ n þ r n n k. ð:þ We can see that (.) is time discretization of Eq. (.) b the forward Euler scheme with step one. * Corresponding author. Tel.: addresses: iaoliliu@sjtu.edu.cn (X. Liu), iaodm@sjtu.edu.cn (D. Xiao) /$ - see front matter Ó 5 Elsevier Ltd. All rights reserved. doi:6/j.chaos.5..8

2 One epects the deterministic models ma provide a useful wa of gaining sufficient understanding about the dnamics of the population of a single species. As it is well known, the dnamics of (.) is trivial, i.e. ever non-negative solution of (.) ecept the constant solution tends to the other constant solution k as t!for all permissible values of r and k. Hence, the population (t) approaches the limit k as time evolves (cf. [3 5] and reference therein). On the other hand, the dnamics of (.) is more comple. It is remarkable that for (.), period-doubling phenomenon and the onset of chaos in the sense of Li Yorke [6] occur for some values of r [7,8]. Now Eq. (.) becomes a prototpe for chaotic behavior of discrete dnamical sstems well beond the discipline of mathematical biolog. It is also remarkable from a biological point of view that such a simple discrete model leads to unpredictable dnamic behaviors. This suggests the possibilit that the governing laws of ecological sstems ma be relativel simple and therefore discoverable. Ma [9,] have clearl documented the rich arra of dnamic behavior possible in simple discrete-time models. Hence, the discrete version has been an important subject of stud in diverse phenomenolog or as an object interesting to analze b itself from the mathematical point of view [ 4]. The purpose of this paper is to analze qualitativel the dnamical compleit of a discrete-time predator pre model, which can be regarded as a coupling perturbation of (.) in R or time discretization of a Lotka Volterra tpe predator pre sstem [5] b Euler method. About the discrete-time predator pre models, an earl work was done b Beddington et al. [6]. From then, there has been a considerable amount of literatures on discrete-time predator pre models (e.g. see [7 ] and references therein). The basic forms of dnamics observed in their models are stable fied points, periodic orbits and some random motions. We consider a Lotka Volterra tpe predator pre sstem [3,5] _ ¼ r ð Þb k ð:3þ _ ¼ðd þ cþ where (t) and (t) denote pre and predator densities respectivel, b is the predator functional response, which represents the number of pre individuals consumed per unit area per unit time b an individual predator, c is the conversion efficienc of pre into predators, c is the predator numerical response, and d is the predator mortalit rate. In the absence of predator (i.e. ), this model reduces to (.). Let us introduce scaled variables, X ¼, Y ¼ b, and s ¼ t, sstem (.3) is reduced to k ck k 8 dx >< ds ¼ r kx ð X Þk cxy dy >: ds ¼ðd k þ k cx ÞY. For the sake of simplicit, we rewrite the sstem above as _ ¼ rð Þb _ ¼ðd þ bþ X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) where r, b and d are positive parameters, r = r k, b = k c and d = d k. Appling the forward Euler scheme to sstem (.4), we obtain the discrete-time predator pre sstem as follows:! þ d½rð ÞbŠ ð:5þ! þ dðd þ bþ where d is the step size. It is clear that sstem (.5) can be regarded as a coupling perturbation of (.) in R. From the point of view of biolog, we will focus on the dnamical behaviors of (.5) in the closed first quadrant R þ, and show that the dnamics of the discrete time model (.5) can produce a much richer set of patterns than those discovered in continuous-time model (.3). More precisel, in this paper, we rigorousl prove that (.5) undergoes the flip bifurcation and the Hopf bifurcation b using center manifold theorem and bifurcation theor. Meanwhile, numerical simulations are presented not onl to illustrate our results with the theoretical analsis, but also to ehibit the comple dnamical behaviors. These results reveal far richer dnamics of the discrete model compared with the continuous model. This paper is organized as follows. In Section, we discuss the eistence and stabilit of fied points for sstem (.5) in the closed first quadrant R þ. In Section 3, we show that there eist some values of parameters such that (.5) undergoes the flip bifurcation and the Hopf bifurcation in the interior of R þ. In Section 4, we present the numerical simulations, which not onl illustrate our results with the theoretical analsis, but also ehibit the comple dnamical behaviors such as the period-5, 6, 9,, 4, 8,, 5 orbits, cascade of period-doubling bifurcation in period-, 4, ð:4þ

3 8 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) , quasi-periodic orbits and the chaotic sets. The Lapunov eponents are computed numericall to confirm further the dnamical behaviors. A brief discussion is given in Section 5.. The eistence and stabilit of fied points In this section and through out the paper, from the point of view of biolog, we consider the discrete-time model (.5) in the closed first quadrant R þ of the (,) plane. We first discuss the eistence of fied points for (.5), then stud the stabilit of the fied point b the eigenvalues for the variational matri of (.5) at the fied point. It is clear that the fied points of (.5) satisf the following equations: ¼ þ d½rð ÞbŠ ¼ þ dðd þ bþ. B a simple computation, it is straightforward to obtain the following results: Lemma. (i) For all parameter values, (.5) has two fied points, O(,) and A(,) (ii) ifb> d, then (.5) has, additionall, a unique positive fied point, B( *, * ), where ¼ d b ¼ rðbdþ b. Now we stud the stabilit of these fied points. Note that the local stabilit of a fied point (,) is determined b the modules of eigenvalues of the characteristic equation at the fied point. Let the vector function ( + d(r( ) b), + d(d + b)) T be denoted b V(,). Then the variational matri of (.5) at a fied point (,) is þ rd rd bd bd DV ð Þ ¼. bd dd þ bd The characteristic equation of the variational matri can be written as k þ pð Þk þ qð Þ ¼ ð:þ which is a quadratic equation with one variable, p(,)= +dd rd +(rd bd) + bd, and q(,) =b d - +(+rd rd bd)( dd + bd). In order to stud the modulus of eigenvalues of the characteristic equation (.) at the positive fied point B( *, * ), we first give the following lemma, which can be easil proved b the relations between roots and coefficients of the quadratic equation. Lemma.. Let F(k) =k +Bk + C. Suppose that F() >, k and k are two roots of F(k) =. Then (i) jk j < and jk j < if and onl if F() > and C < (ii) jk j < and jk j > (or jk j > and jk j < ) if and onl if F() < (iii) jk j > and jk j > if and onl if F() > and C > (iv) k = and jk j 5 if and onl if F() = and B 5, (v) k and k are comple and jk j = jk j = if and onl if B 4C < and C =. Let k and k be two roots of (.), which called eigenvalues of the fied point (,). We recall some definitions of topological tpes for a fied point (,). (,) is called a sink if jk j < and jk j <. A sink is locall asmptotic stable. (,) is called a source if jk j > and jk j >. A source is locall unstable. (,) is called a saddle if jk j > and jk j < (or jk j < and jk j > ). And (,) is called non-hperbolic if either jk j =orjk j =. Substituting the coordinates of the fied point O(, ) for (, ) of(.) and computing the eigenvalues of the fied point O(, ) straightforward, we can obtain the following proposition. Proposition.3. The fied point O(,) is a saddle if < d < d, O(,) is a source if d > d, and O(,) is non-hperbolic if d ¼ d. We can see that when d ¼, one of the eigenvalues of the fied point O(,) is and the other is not one with d module. Thus, the flip (or period-doubling) bifurcation ma occur when parameters var in the neighborhood of d ¼. However, one can see the flip bifurcation can not occur for the original parameters of (.5) b computation, d and O(, ) is degenerate with higher codimension.

4 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) The following propositions show the local dnamics of fied point A(,) and positive fied point B( *, * ) from Lemma.. Proposition.4. There eist at least four different topological tpes of A(, ) for all permissible values of parameters. (i) A(,) is a sink if b < d and < d < min r db (ii) A(,) is a source if b < d and d > ma r db (or b > d and d > ) r (iii) A(,) is not hperbolic if either b = d, or d ¼ or d ¼ r bd (iv) A(,) is a saddle for the other values of parameters ecept those values in (i) (iii). We can easil see that one of the eigenvalues of fied point A(,) is and the other is neither nor if the term (iii) of Proposition.4 holds. And the conditions in the term (iii) of Proposition.4 impl all parameters locate in the following set: F A ¼ ðb d r dþ : r ¼ b 6¼ d d 6¼ b > d > d >. d b d The fied point A(,) can undergo flip bifurcation when parameters var in the small neighborhood of F A, since when parameters are in F A a center manifold of (.5) at A(,) is = and (.5) restricted to this center manifold is the logistic model (.). Hence, in this case the predator becomes etinction and the pre undergoes the period-doubling bifurcation to chaos in the sense of Li-Yorke b choosing bifurcation parameter r. Proposition.5. When b > d, sstem (.5) has a unique positive fied point B( *, * ) and (i) it is a sink if one of the following conditions pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi holds: (i.) rd 4b(b d) P and < d < rd rdðrd4bðbdþþ rdðbdþ (i.) rd 4b(b d) < and < d <. bd (ii) it is a source if one of the following conditions holds: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (ii.) rd 4b(b d) P and d > rdþ rdðrd4bðbdþþ rdðbdþ (ii.) rd 4b(b d) < and d >. bd (iii) it is not hperbolic if one of the following conditions holds: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (iii.) rd 4b(b d) P and d ¼ rd rdðrd4bðbdþþ rdðbdþ (iii.) rd 4b(b d) < and d ¼. bd From Lemma., we can easil see that one of the eigenvalues of the positive fied point B( *, * )isand the other is neither nor if the term (iii.) of Proposition.5 holds. We rewrite the conditions in the term (iii.) of Proposition.5 as the following sets: ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) rd rdðrd 4bðb dþþ F B ¼ ðb d r dþ : d ¼ b > d > rd > 4bðb dþ r > rdðb dþ or ( pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) rd þ rdðrd 4bðb dþþ F B ¼ ðb d r dþ : d ¼ b > d > rd > 4bðb dþ r >. rdðb dþ When the term (iii.) of Proposition.5 holds, we can obtain that the eigenvalues of the positive fied point B( *, * ) are a pair of conjugate comple numbers with module one. The conditions in the term (iii.) of Proposition.5 can be written as the following set: H B ¼ ðb d r dþ : d ¼ b > d 4bðb dþ > rd b > d > r >. b d In the following section, we will stud the flip bifurcation of the positive fied point B( *, * ) if parameters var in the small neighborhood of F B (or F B ), and the Hopf bifurcation of B( *, * ) if parameters var in the small neighborhood of H B.

5 84 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) Flip bifurcation and Hopf bifurcation Based on the analsis in Section, we discuss the flip bifurcation and Hopf bifurcation of the positive fied point B( *, * ) in this section. We choose parameter d as a bifurcation parameter to stud the flip bifurcation and Hopf bifurcation of B( *, * ) b using center manifold theorem and bifurcation theor in [,3]. We first discuss the flip bifurcation of (.5) at B( *, * ) when parameters var in the small neighborhood of F B. The similar arguments can be applied to the other case F B. Taking parameters (b,d,r,d ) arbitraril from F B, we consider sstem (.5) with (b,d,r,d ), which is described b! þ d ½r ð Þb Š ð3:þ! þ d ðd þ b Þ. Then map (3.) has a unique positive fied point B( *, * ), whose eigenvalues are k =, k =3r * d with jk j 5 b Proposition.5, where ¼ d b ¼ r ðb d Þ. b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Since (b,d,r,d ) F B, d ¼ r d r d ðr d 4b ðb d ÞÞ r d ðb d. Choosing d as a bifurcation parameter, we consider a perturbation of (3.) as follows: Þ! þðd þ d Þ½r ð Þb Š! þðd þ d ð3:þ Þðd þ b Þ where jd * j, which is a small perturbation parameter. Let u = * and v = *. Then we transform the fied point B( *, * ) of map (3.) into the origin. We have a u þ a v þ a 3 uv þ a 4 u þ b ud u B! þb vd þ b 3 uvd þ b 4 u d A ð3:3þ v a u þ v þ a 3 uv þ c ud þ c uvd where a ¼ þ d ðr r b Þ a ¼b d a 3 ¼b d a 4 ¼r d b ¼ r r b b ¼b b 3 ¼b b 4 ¼r a ¼ b d a 3 ¼ b d c ¼ b c ¼ b. We construct an invertible matri a a T ¼ a k a and use the translation u ~ ¼ T v ~ for (3.3), then the map (3.3) becomes ~! ~ þ f ðu v d Þ ~ k ~ gðu v d ð3:4þ Þ where f ðu v d Þ¼ ½a 3ðk a Þa a 3 Š uv þ a 4ðk a Þ a ðk þ Þ þ b ðk a Þ a ðk þ Þ vd þ ½b 3ðk a Þa c Š a ðk þ Þ gðu v d Þ¼ ½a 3ð þ a Þþa a 3 Š uv þ a 4ð þ a Þ a ðk þ Þ þ b ð þ a Þ a ðk þ Þ vd þ ½b 3ð þ a Þþa c Š a ðk þ Þ u þ ½b ðk a Þa c Š ud a ðk þ Þ uvd þ b 4ðk a Þ a ðk þ Þ u d a ðk þ Þ u þ ½b ð þ a Þþa c Š ud a ðk þ Þ uvd þ b 4ð þ a Þ a ðk þ Þ u d a ðk þ Þ uv ¼a ð þ a Þ~ þ½a ðk a Þa ð þ a ÞŠ~~ þ a ðk a Þ~ u ¼ a ð~ þ ~~ þ ~ Þ.

6 Net, we determine the center manifold W c (,) of (3.4) at the fied point (,) in a small neighborhood of d * =. From the center manifold theorem, we know there eists a center manifold W c (,), which can be approimatel represented as follows: W c ð Þ ¼fð~ ~Þ : ~ ¼ a d þ a ~ þ a ~d þ a 3 d þ Oððj~jþjd jþ 3 Þg where Oððj~jþjd jþ 3 Þ is a function with order at least three in their variables ð~ d Þ, and a ¼ a ¼ ð þ a Þ½a a 4 a 3 ð þ a Þa a 3 Š k a ¼ b ð þ a Þc a ð þ k Þ a 3 ¼. þ b ð þ a Þ a ð þ k Þ Therefore, we consider the map which is map (3.4) restricted to the center manifold W c (,) f : ~!~ þ h ~ þ h ~d þ h 3 ~ d þ h 4 ~d þ h 5 ~ 3 þ Oððj~jþjd jþ 4 Þ where h ¼ k þ fa a 4 ðk a Þðþa Þ½a 3 ðk a Þa a 3 Šg h ¼ a ðk þ Þ ½b a ðk a Þa c b ðk a Þð þ a ÞŠ h 3 ¼ k þ fa ðk a Þ½a 3 ðk a Þa a 3 Šþa a a 4 ðk a Þþa ½b ðk a Þc a Š ðþa Þ½b 3 ðk c a ÞŠ þ a b 4 ðk a Þg þ a ðk þ Þ a b ðk a Þ h 4 ¼ k þ fa 3ðk a Þ½a 3 ðk a Þa a 3 Šþa 3 a a 4 ðk a Þ a b þ a ½b ðk a Þc a Šg þ a ðk þ Þ ðk a Þ h 5 ¼ a k þ fðk a Þ½a 3 ðk a Þa a 3 Šþa a 4 ðk a Þg. In order for map (3.5) to undergo a flip bifurcation, we require that two discriminator quantities a and a are not zero, where and a ¼ a ¼ 6 o f o~d þ o 3 f o~ þ 3 of o f od o~ o f o~ ðþ! ðþ. From a simple calculation, we obtain X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) ð3:5þ and a ¼ r d ðb d Þðd Þ 4b b d ð4 r d Þ 6¼ ( a ¼ h 5 þ h ¼ ðb d Þ 4 d ð þ k Þ d d ) r d d r d ð þ d d Þ. b b From the above analsis and the theorem in [], we have the following theorem. Theorem 3.. If a 5, then map (3.) undergoes a flip bifurcation at the fied point B( *, * ) when the parameter d * varies in the small neighborhood of the origin. Moreover, if a > (resp., a < ), then the period- points that bifurcate from B( *, * ) are stable (resp., unstable).

7 86 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) 8 94 Fig. 4.. (a) Bifurcation diagram of map (.5) with d covering [.6,.4], r =, b =.6, d =, the initial value is (.83, 5). (b) Maimum Lapunov eponents corresponding to (a). In Section 4 we will give some values of parameters such that a 5, thus the flip bifurcation occurs as d varies (see Fig. 4.). Finall we discuss the Hopf bifurcation of B( *, * ) if parameters var in the small neighborhood of H B. Taking parameters (b,d,r,d ) arbitraril from H B, we consider sstem (.5) with parameters (b,d,r,d ), which is described b! þ d ½r ð Þb Š ð3:6þ! þ d ðd þ b Þ. Then map (3.6) has a unique positive fied point B( *, * ), where ¼ d b, ¼ r ðb d Þ. b Note that the characteristic equation associated with the linearization of the map (3.6) at B( *, * ) is given b k þ pk þ q ¼ where p ¼ þ d r d b q ¼ d r d þ r d d ðb d Þ. b b Since parameters (b,d,r,d ) H B, the eigenvalues of B( *, * ) are a pair of comple conjugate numbers k, and k with modulus b Proposition.5, where k ¼ p þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p p 4q ¼ þ r d þ i r d ð4b 4b d r d Þ. ð3:7þ b ðb d Þ Now we consider a small perturbation of (3.6) b choosing the bifurcation parameter d as follows! þðd þ dþ½r ð Þb Š ð3:8þ! þðd þ dþðd þ b Þ where jdj, which is a small parameter. Moving B( *, * ) to the origin, let u = * and let v = * we have u! u þðd þ dþ½r uð uþ r u b v b uðv þ ÞŠ v v þðd þ dþb uðv þ Þ The characteristic equation associated with the linearization of the map (3.9) at (u, v) = (, ) is given b k þ pðdþk þ qðdþ ¼. ð3:9þ

8 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) where pðdþ ¼ þ r d ðd þ dþ b qðdþ ¼ r d ðd þ dþ þ r d ðb d Þðd þ dþ. b b Correspondingl, when d varies in a small neighborhood of d = the roots of the characteristic equation are qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k ¼ pðdþ pðdþ 4qðdÞ ¼ þ r d ðd þ dþþiðd þ dþ r d ð4b 4b d r d Þ b and there have jk j¼ðqðdþþ = l ¼ djk j dd ¼ r d >. d¼ b In addition, it is required that when d =,k m 6¼ m ¼ 3 4 which is equivalent to p() 5,,,. Note that (b,d,r,d ) H B.So 4b r d > b d >. Thus, p() 5,. We onl need to require that p() 5,, which leads to 6¼ jb j ¼ 3. ð3:þ b d r d Therefore, the eigenvalues k, of fied point (,) of (3.9) do not la in the intersection of the unit circle with the coordinate aes when d = and (3.) holds. Net we stud the normal form of (3.9) when d =. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Let a ¼ r d b ¼ r d ð4b 4b d r d Þ b ðb d Þ b ðb d Þ T ¼ b a and use the translation u ¼ T ~ v ~ the map (3.9) becomes ~! a b ~ þ ~ b a ~ f ~! ð~ ~Þ ð3:þ ~gð~ ~Þ where ~f ð~ ~Þ ¼ ða þ Þd b ~ðb~ þ a~þþ ad b b r ~ ~gð~ ~Þ ¼d ½b ~ðb~ þ a~þþr ~ Š and ~f ~~ ¼ f ~ ~~ ¼ ad ðb a þ b þ r Þ f ~ b ~~ ¼ b ða þ Þd f ~ ~~~ ¼ f ~ ~~~ ¼ f ~ ~~~ ¼ f ~ ~~~ ¼ ~g ~~ ¼ ~g ~~ ¼d ðr þ b aþ ~g ~~ ¼b d b ~g ~~~ ¼ ~g ~~~ ¼ ~g ~~~ ¼ ~g ~~~ ¼. In order for map (3.) to undergo Hopf bifurcation, we require that the following discriminator quantit a is not zero [3]: " a ¼Re ð kþ # k n k n kn k kn k þ Reð kn Þ ð3:þ

9 88 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) 8 94 where n ¼ 8 ½f ~~ f ~~ þ g ~~ þ iðg ~~ g ~~ f ~~ ÞŠ n ¼ 4 ½f ~~ þ f ~~ þ iðg ~~ þ g ~~ ÞŠ n ¼ 8 ½f ~~ f ~~ þ g ~~ þ iðg ~~ g ~~ þ f ~~ ÞŠ n ¼ 6 ½f ~~~ þ f ~~~ þ g ~~~ þ g ~~~ þ iðg ~~~ þ g ~~~ f ~~~ f ~~~ ÞŠ δ =.8 δ =.83 δ = δ =.36 δ =.348 δ = δ =.36 δ =.38 Fig. 4.. Phase portraits for various values of d corresponding to Fig. 4.(a).

10 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) Fig (a) Bifurcation diagram of map (.5) with d covering [,.95], r =3,b = 3.5, d =, the initial value is (743,6735). (b) Local amplification corresponding to (a). Maimum Lapunov Eponent (a) Mamum Lpunov Eponent (b) Fig (a) Maimum Lapunov eponent with d covering [,.95], r = 3, b = 3.5, d =, the initial value is (743, 6735). (b) Local amplification corresponding to (a). After some manipulation, we obtain a ¼ 3ð aþðb d Þ 8 h i < ð 9a þ 4a þ a 3 6a 4 Þ ab þ 4a b þ a3 b þ ð aþ 3 b ðb r Þþab r þ 4a b r þ a 3 b r þ a r : a h i ) þ 3 6a a þ 6a 3 a þ a a b ðb þ ab þ r Þaðb r Þðb þ b a þ r Þ 3 b 3a ðb þ ab þ r Þ ð aþ b 6ð a Þðb d Þ 3ðb d Þ ðb þ ab þ r Þ. 6ðb d Þ From above analsis and the theorem in [3], we have the following theorem. Theorem 3.. If the condition (3.) holds and a 5, then map (3.8) undergoes Hopf bifurcation at the fied point B( *, * ) when the parameter d varies in the small neighborhood of the origin. Moreover, if a < (resp., a > ), then an attracting (resp., repelling) invariant closed curve bifurcates from the fied point for d > (resp., d < ).

11 9 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) 8 94 Fig Phase portraits for various values of d corresponding to Fig. 4.3(a). In Section 4 we will choose some values of parameters to show the process of Hopf bifurcation for map (3.8) in Fig. 4.5 b numerical simulation.

12 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) Fig (a) Bifurcation diagram of map (.5) with d covering [5,.8], r =3,b =4,d =, the initial value is (,74). (b) Local amplification corresponding to (a). 4. Numerical simulations In this section, we present the bifurcation diagrams, phase portraits and Maimum Lapunov eponents for sstem (.5) to confirm the above theoretical analsis and show the new interesting comple dnamical behaviors b using numerical simulations. The bifurcation parameters are considered in the following three cases: () Varing d in range.6 < d 6.4, and fiing b =.6, d =, r =. () Varing d in range < d 6.95, and fiing b = 3.5, d =,r =3. (3) Varing d in range < d 6.8, and fiing b =4,d =,r =3. For case (). b =.6, d =, r =, based on Lemma., we know the map (.5) has onl one positive fied point. 5 After calculation for the positive fied point of map (.5), the flip bifurcation emerges from the fied point p at d ¼ ffiffiffiffiffi p 76 with a =.5598, a = 76 and ðb d r dþ ¼ð:6 :5 ffiffiffiffiffi 76 ÞFB. It shows the correctness of Theorem 3.. p From Fig. 4.(a), we see that the fied point is stable for d < ffiffiffiffiffi p 76 and loses its stabilit at the flip bifurcation parameter value d ¼ ffiffiffiffiffi 76, we also observe that there is a cascade of period-doubling. The maimum Lapunov eponents corresponding to Fig. 4.(a) are computed in Fig. 4.(b). The phase portraits which are associated with Fig. 4.(a) are disposed in Fig. 4.. For d (.8,.35), there are period-, 4, 8, 6 orbits. When d =.36,.38, we can see the chaotic sets. The maimum Lapunov eponents corresponding to d =.36,.38 are larger than that confirm the eistence of the chaotic sets. For case (). b = 3.5, d =,r = 3, according to Lemma., we know the map (.5) has onl one positive fied point. 4 After calculation for the positive fied point of map (.5), the Hopf bifurcation emerges from the fied point at d ¼ with a =.5344 and ðb d r dþ ¼ H B. It shows the correctness of Theorem From Fig. 4.3(a), we observe that the fied point of map (.5) is stable for d < loses its stabilit at d ¼, and 3 3 an invariant circle appears when the parameter d eceeds. From Fig. 4.3, we see that there are period-doubling phenomenons. Fig. 4.3(b) is the local amplification for d [.78,.873]. 3 The maimum Lapunov eponents corresponding to Fig. 4.3(a) are calculated and plotted in Fig. 4.4 where we can easil see that that the maimum Lapunov eponents are negative for the parameter d (,.87), that is to sa, the non-chaotic region is bigger than the chaotic region (.87,.95). For d (.87,.855) some Lapunov eponents are bigger than, some are smaller than, so there eist stable fied point or stable period windows in the chaotic region. In general the positive Lapunov eponent is considered to be one of the characteristics impling the eistence of chaos.

13 9 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) 8 94 The phase portraits which are associated with Fig. 4.3(a) are disposed in Fig. 4.5, which clearl depicts the process of 4 how a smooth invariant circle bifurcates from the stable fied point When d eceeds there appears a circle curve 3 4 enclosing the fied point , and its radius becomes larger with respect to the growth of d. When d increases at certain values, for eample, at d =.785, the circle disappears and a period-5 orbit appears, and some cascades of perioddoubling bifurcations lead to chaos. From Fig. 4.5 we observe that there are period-5, period-, period-, period-9 and quasi-periodic orbits. For case (3). b =4,d =,r = 3 and d is varing. We draw the bifurcation diagram in Fig. 4.6(a) with local amplification in Fig. 4.6(b). The phase portraits of various d corresponding to Fig. 4.6(a) are plotted in Fig From Fig. 4.7, we see that the fied point 3 8 of map (.5) is stable for d < and loses its stabilit at d ¼, an invariant δ = δ=.6 δ = δ =.68 δ =.7 δ = δ =.7 δ =.73 δ =.73 Fig Phase portaits for various values of d corresponding to Fig. 4.6(a).

14 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) δ =.735 δ =.74 δ = δ =.765 δ =.77 δ =.8 Fig. 4.7 (continued) circle appears when the parameter d eceeds. There is an invariant circle for more large regions of d (,.66). when d increases, there are: period-6 orbits, period-5 orbits and attracting chaotic sets. 5. Discussion It is well known that the dnamics of sstem (.3) is trivial in the first quadrant for all parameters. More precisel, for some parameter values the sstem has no positive equilibria and the one of boundar equilibria, (k, ), attracts all orbits of the sstem in the interior of the first quadrant otherwise, the sstem has a unique positive equilibrium and the positive equilibrium attracts all orbits of the sstem in the interior of the first quadrant. Thus, sstem (.3) has no limit ccles for all parameter values. However, the discrete-time predator pre model (.5) has comple dnamics. In this paper, we show that the unique positive fied point of (.5) can undergo flip bifurcation and Hopf bifurcation. Moreover, sstem (.5) displas much interesting dnamical behaviors, including period-5, 6, 9,, 4, 8,, 5 orbits, invariant ccle, cascade of period-doubling, quasi-periodic orbits and the chaotic sets, which implies that the predator and pre can coeist in the stable period-n orbits and invariant ccle. These results reveal far richer dnamics of the discrete model compared to the continuous model. Acknowledgments This work was supported b the National Natural Science Foundations of China (No. 3) and Program for New Centur Ecellent Talents in Universities of China.

15 94 X. Liu, D. Xiao / Chaos, Solitons and Fractals 3 (7) 8 94 References [] Verhulst PF. Notice sur la loi que la population suit dans son accroissement. Corr Math Phs 838:3. [] Pearl R, Reed LJ. On the rate of growth of the population of the United States since 79 and its mathematical representation. Proc Natl Acad Sci 96: [3] Brauer F, Castillo-Chavez C. Mathematical models in population biolog and epidemiolog. New York: Springer-Verlag. [4] Freedman HI. Deterministic mathematical models in population ecolog. Marcel Dekker: New York 98. [5] Murra JD. Mathematical biolog. nd ed. Berlin: Springer-Verlag 993. [6] Li TY, Yorke JA. Period three implies chaos. Amer Math Monthl 9758: [7] Strogatz S. Nonlinear dnamics and chaos. Reading, Mass: Addison-Wesle 994. [8] Saha P, Strogatz S. The birth of period 3. Math Mag 99568:4 7. [9] Ma RM. Biological populations with nonoverlapping generations: stable points, stable ccles, and chaos. Science 97486: [] Ma RM. Simple mathematical models with ver complicated dnamics. Nature 9766: [] Summers D, Cranford JG, Heale BP. Chaos in periodicall forced discrete-time ecosstem models. Chaos, Solitons & Fractals :33 4. [] Gao YH. Chaos and bifurcation in the space-clamped FitzHugh Nagumo sstem. Chaos, Solitons & Fractals 4: [3] Jing ZJ, Jia Z, Wang R. Chaos behavior in the discrete BVP oscillator. Int J Bifurcat & Chaos :69 7. [4] Jing ZJ, Chang Y, Guo B. Bifurcation and chaos in discrete FitzHugh Nagumo sstem. Chaos, Solitons & Fractals 4: 7. [5] Rosenzweig ML, MacArthur RH. Graphical representation and stabilit conditions of predator pre interactions. Amer Naturalist 96397:9 3. [6] Beddington JR, Free CA, Lawton JH. Dnamic compleit in predator pre models framed in difference equations. Nature :58 6. [7] Lopez-Ruiz R, Fournier-Prunaret R. Indirect Allee effect, bistabilit and chaotic oscillations in a predator pre discrete model of logistic tpe. Chaos, Solitons & Fractals 54:85. [8] Xiao YN, Cheng DZ, Tang SY. Dnamic compleities in predator pre ecosstem models with age-structure for predator. Chaos, Solitons & Fractals 4:43. [9] Fan M, Wang K. Periodic solutions of a discrete time non-autonomous ratio-dependent predator pre sstem. Math Comput Model 35:95 6. [] Wang WD, Lu ZY. Global stabilit of discrete models of Lotka Volterra tpe. Nonlinear Anal 99935:9 3. [] Huo HF, Li WT. Stable periodic solution of the discrete periodic Leslie Gower predator pre model. Math Comput Model 4 4:6 9. [] Robinson C. Dnamical sstems, stabilit, smbolic dnamics and chaos. nd ed. London, New York, Washington (DC): Boca Raton 999. [3] Guckenheimer J, Holmes P. Nonlinear oscillations, dnamical sstem and bifurcation of vector fields. New York: Springer- Verlag 983. p.6 5.