Stability and bifurcation analysis of a discrete predator prey system with modified Holling Tanner functional response

Transcription

1 Zhao and Yan Advances in Difference Equations :40 R E S E A R C H Open Access Stability and bifurcation analysis of a discrete predator prey system with modified Holling Tanner functional response Jianglin Zhao 1* andyongyan 1 * Correspondence: ws @163.com 1 Faculty of Science and Technology Sichuan Minzu College Kangding China Abstract In this paper we study a discrete predator prey system with modified Holling Tanner functional response. We derive conditions of existence for flip bifurcations and Hopf bifurcations by using the center manifold theorem and bifurcation theory. Numerical simulations including bifurcation diagrams maximum Lyapunov exponents and phase portraits not only illustrate the correctness of theoretical analysis but also exhibit complex dynamical behaviors and biological phenomena. This suggests that the small integral step size can stabilize the system into the locally stable coexistence. However the large integral step size may destabilize the system producing far richer dynamics. This also implies that when the intrinsic growth rate of prey is high the model has bifurcation structures somewhat similar to the classic logistic one. Keywords: Discrete-time predator prey system; Flip bifurcation; Hopf bifurcation; Chaos 1 Introduction Predator prey interactions have long been studied and continue to be one of the dominant themes in both biology and mathematical biology due to their universal existence and importance 1. Recently a predator prey system with modified Holling Tanner functional response was given in 3 asfollows: du dt = ru1 u K dv dt kuv a+bu+cv = vs1 hv u 1 where r K k a b c s h are positive constants and u and v represent the population densities of prey and predator respectively. The prey grows logistically with carrying capacity K and intrinsic growth rate r in the absence of predator. The predator consumes the prey according to the functional response of Beddington DeAngelis type kuv/a + bu+ cvand grows logistically with intrinsic growth rate s and cv measures the mutual interference between predators. The parameters k a band c are the consumption rate the saturation constant the saturation constant for an alternative prey and the predator interference respectively. The carrying capacity u/h of predator is proportional to the population size of the prey. The parameter h is the number of preys required to support one predator at The Authors 018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made.

2 Zhao and Yan Advances in Difference Equations :40 Page of 18 equilibrium when v equals u/h.thetermv/u measures the loss in the predator population due to rarity of its favorite food. Applying the following scaling to 1 t t u u K v v k β bk a c a bk h m bk it becomes of the following form: du dt dv dt = ru1 u βuv a+u+mv h K = vs hv u. However for a mathematical biology model if the size of population is rarely small or the population has no overlapping generation or people study population changes within certain intervals of time the discrete-time model would indeed be more suitable and realistic than the continuous-time model 4 1. On the other hand numerical solutions or approximate solutions of discrete-time models can be obtained more easily and much work has shown that discrete-time prey predator models can produce a much richer set of patterns than those observed in continuous-time models Thus it is necessary to consider the corresponding discrete model of system. Using the forwardeuler scheme to system we therefore obtain the following discrete-time model: u u + δru1 u βuv a+u+mv v v + δvs hv u 3 where δ is the integral step size. In this paper we mainly focus on the dynamical behaviorofsystem3 in the interior of the first quadrant of R.Morepreciselywediscussthe stability of fixed points of system 3 and rigorously prove thatthe map 3undergoesthe flip bifurcation and Hopf bifurcation by using the center manifold theorem and bifurcation theory. Meanwhile resent numerical simulations not only to illustrate our results of theoretical analysis but also to explore complex dynamical behaviors. The outline of this paper is as follows. In Sect. we investigate in detail the existence and local stability of fixed points of model 3. In Sect. 3 we derive sufficient conditions for the existence of flip bifurcation and Hopf bifurcation. In Sect. 4 we present numerical simulations to check our results of theoretical analysis and exhibit some complex and new dynamical behaviors. In the end we give a brief conclusion in Sect. 5. Existence and stability of fixed points From a biological point of view system 3 must have positive values of u and v.wehave the following result. Theorem.1 Assume that = {u v u >0v >0rδu + βδv 1+rδ<0}. Then is an invariant set for system 3 if δ and s are enough small.

3 Zhao and Yan Advances in Difference Equations :40 Page 3 of 18 Proof Suppose that u 0 v 0.Thenu 1 v 1 >0ifδ is small enough where u 1 = u 0 + δ ru 0 1 u 0 βu 0 v 0 a + u 0 + mv 0 Let L = rδu 1 + βδv 1 1+rδ. Then v 1 = v 0 + δ v 0 s hv 0. u 0 L = r δ u 0 + rδ1 + rδu 0 1+rδ+βδ 1+δ s hv 0 u 0 If δ is small enough then it follows that 1+δ s hv 0 rδu 0 hv0 =1+δs δ + u 0 a + u 0 + mv 0 u 0 Since v 0 < 1+rδ βδ wehave rδu 0 a + u 0 + mv 0 L < r δ u 0 + rδ1 + rδu 0 1+rδ+ 1+δ s hv 0 = r δ u 0 + rδ1 + rδu 0 + s hv 0 u 0 If s is small enough then we have ru 0 u 0 a + u 0 + mv 0 rδu 0 a + u 0 + mv 0 >0. rδu 0 a + u 0 + mv 0 δ1 + rδ. v rδ s < ru 0 a + u 0 + mv 0 + hv 0 u 0. Thus H =s hv 0 ru u 0 0 a+u 0 +mv 0 <0. Let Lu 0 = r δ u 0 + rδ1 + rδu 0 + δ1 + rδh.thenlu 0 =δ1 + rδh <0. Therefore L <0ifδ is small enough. Then u 1 v 1 if δ and s are small enough that is is an invariant set for system 3. It is clear that the fixed points of model 3 satisfy the following equations: u = u + δru1 u v = v + δvs hv u. βuv a+u+mv Lemma.1 Forallparametervalues system 3 has two fixed points the boundary fixed point A1 0 and the unique positive fixed point Bu v defined by u = βs + ahr hr msr+ βs + ahr hr mrs +4ahr h + ms rh + ms v = s h u. Now we perform the linear stability analysis of system 3 at each fixed point. The Jacobian matrix J of 3 evaluated at any point u visgivenby a 11 a 1 Ju v= 4 a 1 a

4 Zhao and Yan Advances in Difference Equations :40 Page 4 of 18 where βva + mv a 11 =1+δ r1 u a + u + mv δβua + u a 1 = a + u + mv a 1 = hδv u a =1+δ s hv u Moreover the characteristic equation of Ju v can be written as. λ + pu vλ + qu v=0 5 where pu v= a 11 + a qu v=a 11 a a 1 a 1. Lemma. 11 Let Fλ =λ + Pλ + Q where P and Q are constants. Suppose that F1 > 0 and λ 1 and λ are two roots of Fλ=0.Then i λ 1 <1and λ <1if and only if F 1 > 0 and Q <1; ii λ 1 <1and λ >1or λ 1 >1and λ <1 if and only if F 1 < 0; iii λ 1 >1and λ >1if and only if F 1 > 0 and Q >1; iv λ 1 = 1and λ 1if and only if F 1 = 0 and P 0 ; v λ 1 and λ aretheconjugatecomplexrootsand λ 1 = λ =1if and only if P 4Q <0and Q =1. Suppose that λ 1 and λ are two roots of 5 which are called eigenvalues of the fixed point u v. The point u v isasinkif λ 1 <1and λ < 1. A sink is locally asymptotic stable. The point u v isasourceif λ 1 >1and λ > 1. A source is locally unstable. The point u v is a saddle if λ 1 >1and λ <1or λ 1 <1and λ >1andu v is nonhyperbolic if either λ 1 =1or λ =1. Theorem. The eigenvalues of the fixed point A1 0 are λ 1 =1 δr andλ =1+δs. i A1 0 is a saddle if 0<δ < r ; ii A1 0 is a source if δ > r ; iii A1 0 is nonhyperbolic if δ = r. Let F A = {a h m r s δ β:δ = r } β a h m r s >0. It can be easily seen that one of the eigenvalues of A1 0 is 1 and the other is neither 1 nor 1 when all parameters of system 3 locateinf A. Then the center manifold of system 3atA1 0 is v = 0 when parameters are in F A.Themap3 restricted to this center manifold is the classic logistic model u ru1 u. It is well known that the predator population becomes extinct and the prey undergoes the period-doubling bifurcation to chaos by choosing the bifurcation parameter r. Therefore the fixed point A1 0 can undergo flip bifurcation when parameters vary in a small neighborhood of F A.

5 Zhao and Yan Advances in Difference Equations :40 Page 5 of 18 The characteristic equation of the Jacobian matrix Ju v evaluated at the unique positive fixed point Bu v canbewrittenas λ +Gδλ + 1+Gδ + Hsδ =0 6 where Let G = s ru + H = ru + βu v a + u + mv βav a + u + mv. Fλ=λ +Gδλ + 1+Gδ + Hsδ. Then F1 = Hsδ >0 F 1 = 4 + Gδ + Hsδ. Theorem.3 i Bu v is a sink if one of the following conditions holds: i.1 Hs < G <0and 0<δ < G Hs ; i. G < Hs and 0<δ < G G 4Hs ; Hs ii Bu v is a source if one of the following conditions holds: ii.1 Hs < G <0and δ > G Hs ; ii. G < Hs and δ > G+ G 4Hs ; Hs ii.3 G 0; iii Bu v is a saddle if the following conditions hold: G < Hs and G G 4Hs Hs < δ < G + G 4Hs ; Hs iv Bu v is nonhyperbolic if one of the following conditions holds: iv.1 G < Hs and δ = G± G 4Hs Hs and δ G 4 G ; iv. Hs < G <0and δ = G Hs. From the Jury criterion and the preceding analysis it can be easily seen that one of the eigenvalues of the unique positive fixed point Bu v is 1 and the other is neither 1 nor 1ifiv.1ofTheorem.3 holds. When iv. of Theorem.3 is true the eigenvalues of the unique positive fixed point Bu v are a pair of conjugate complex numbers with modulus one. Let { F B1 = δ β a h m r s:δ = δ 1 = G G 4Hs G < } Hs β a h m r s >0 Hs

6 Zhao and Yan Advances in Difference Equations :40 Page 6 of 18 and { F B = δ β a h m r s:δ = δ 1 = G + G 4Hs G < } Hs β a h m r s >0. Hs ThentheuniquepositivefixedpointBu v may undergo the flip bifurcation when parameters vary in a small neighborhood of F B1 or F B. Let { H B = δ β a h m r s:δ = δ = G } Hs Hs < G <0β a h m r s >0. Then the unique positive fixed point Bu v may undergo the Hopf bifurcation when the parameters vary in a small neighborhood of H B. 3 Flip bifurcation and Hopf bifurcation Based on the previous analysis in this section we mainly focus on the flip bifurcation and Hopf bifurcation of the unique positive fixed point Bu v. Then we choose the integral step size δ as a bifurcation parameter for investigating the Flip bifurcation and Hopf bifurcation of Bu v by using the center manifold theorem and bifurcation theory. We first discuss the flip bifurcation of system 3atBu v when the parameters vary in a small neighborhood of F B1. Similar arguments can be applied to the other case of F B. Taking the parameters δ β a h m r s arbitrarily from F B1 we consider system 3with δ β a h m r s F B1 described by u u + δ 1 ru1 u βuv a+u+mv v v + δ 1 vs hv u. 7 Then the map 7 has a unique positive fixed point Bu v with eigenvalues λ 1 = 1and λ =3+Gδ 1 with λ 1byTheorem.3. Choosing δ as a bifurcation parameter we consider a perturbation of 7 as follows: u u +δ 1 + δ ru1 u βuv a+u+mv v v +δ 1 + δ vs hv u 8 where δ 1 is a small perturbation parameter. Assume that U = u u V = v v.thenwetransformthefixedpointbu v ofthe map 8 into the origin. For convenience we rewrite U and V as u and v respectively. Then we have u v a 11 u + a 1 v + a 13 u + a 14 uv + a 15 v + a 16 u 3 + a 17 u v + a 18 uv + a 19 v 3 + b 1 uδ + b vδ + b 3 u δ + b 4 uvδ + b 5 v δ + o u + v + δ 4 a 1 u + a v + a 3 u + a 4 uv + a 5 v + a 6 u 3 + a 7 u v + a 8 uv + a 9 v c 1 uδ + c vδ + c 3 u δ + c 4 uvδ + c 5 v δ + o u + v + δ 4

7 Zhao and Yan Advances in Difference Equations :40 Page 7 of 18 where a 11 =1+δ ru + βu v a + u + mv a 13 = δ r + βv a + mv a + u + mv 3 δβ a 14 = a + u + mv a + a 1 = δβu a + u a + u + mv mu v a + u + mv a 15 = δβmu a + u a + u + mv a 3 16 = δβv a + mv a + u + mv 4 δβ a 17 = a + u + mv 3 a 18 = δβm a + u + mv 3 b 1 = ru + a mv + a u 3mu v a + u + mv 3mu v a + u + mv βu v a + u + mv b = βu a + u a + u + mv b 3 = r + βv a + mv a + u + mv 3 β b 4 = a + u + mv a + mu v a + u + mv a 1 = δs h a =1 δs a 3 = δs hu a 6 = δs hu a 7 = δs u c 1 = s h c = s c 3 = s hu a 19 = δβm u a + u a + u + mv 4 ; b 5 = βmu a + u a + u + mv 3 a 4 = δs u a 8 = δh a u 9 =0 c 4 = s u c 5 = h u a 5 = δh u 10 and δ = δ 1. We construct the invertible matrix a 1 a 1 T = 1 a 11 λ a 11 and apply the translation u v T = Tũ ṽ T. Then the map 9 can be changed into where ũ 1 0 ũ f u v δ + 11 ṽ 0 λ ṽ gu v δ f u v δ = λ a 11 a 13 a 1 a λ a 1 u + λ a 11 a 14 a 1 a λ a 1 uv + λ a 11 a 15 a 1 a λ a 1 v + λ a 11 a 16 a 1 a λ a 1 u 3 + λ a 11 a 17 a 1 a λ a 1 u v

8 Zhao and Yan Advances in Difference Equations :40 Page 8 of 18 + λ a 11 a 18 a 1 a λ a 1 uv + λ a 11 a 19 a 1 a λ a 1 v 3 + λ a 11 b 1 a 1 c λ a 1 uδ + λ a 11 b a 1 c 1 + λ a 1 vδ + λ a 11 b 3 a 1 c λ a 1 u δ + λ a 11 b 4 a 1 c λ a 1 uvδ + λ a 11 b 5 a 1 c λ a 1 v δ + o u + v + δ 4 gu v δ = 1 + a 11a 13 + a 1 a λ a 1 u a 11a 14 + a 1 a λ a 1 uv a 11a 15 + a 1 a λ a 1 v a 11a 16 + a 1 a λ a 1 u a 11a 17 + a 1 a λ a 1 u v a 11a 18 + a 1 a λ a 1 uv a 11a 19 + a 1 a λ a 1 v a 11b 1 + a 1 c λ a 1 uδ a 11b + a 1 c 1 + λ a 1 vδ a 11b 3 + a 1 c λ a 1 u δ a 11b 4 + a 1 c λ a 1 uvδ a 11b 5 + a 1 c λ a 1 v δ + o u + v + δ 4 and u = a 1 ũ + a 1 ṽ v = 1+a 11 ũ +λ a 11 ṽ. Next we apply the center manifold theorem 19todeterminethedynamicsofthefixed point ũ ṽ=00atδ = 0. Then there exists a center manifold of the map 11 which can be represented as follows: W c 0 0 = { ũ ṽ ṽ = hũ δ h0 0 = 0 Dh0 0 = 0 }. Assume that hũ δ =a 1 ũ + a ũδ + a 3 δ + O ũ + δ 3 1 where O ũ + δ 3 is a function of order at least three in their variables ũ δ. Then the center manifold must satisfy N hũ δ = h ũ + f ũ hũ δ δ δ λ hũ δ g ũ hũ δ δ δ = 0. 13

9 Zhao and Yan Advances in Difference Equations :40 Page 9 of 18 Substituting 11 and1 into13 and comparing the coefficients of 1 we obtain a 1 = a = a 3 =0. 1 { a a 1 1 λ a11 a 13 + a 1 a 3 a a a 11 a 14 + a 1 a 4 +1+a a 11 a 15 + a 1 a 5 } 1 { a1 1 + a11 b a λ 1 + a 1 c 1 +1+a a 11 b + a 1 c } Therefore the map 10 restricted to the center manifold W c 0 0 is given by F : ũ ũ + h 1 ũ + h ũδ + h 3 ũ δ + h 4 ũδ + h 5ũ 3 + O ũ + δ 4 14 where h 1 = 1 { a 1 + λ a 1 λ a 11 a 13 a 1 a 3 1 a a 11 λ a 11 a 14 a 1 a 4 +1+a 11 } λ a 11 a 15 a 1 a 5 1 { h = a1 λ a 11 b 1 a 1 c 1 1+a11 } λ a 11 b a 1 c 1 + λ a 1 a { h 3 = a 1 + λ a 1 λ a 11 a 13 a 1 a a 1 λ a 11 1 λ a 11 a 14 a 1 a 4 1+a 11 λ a 11 λ a 11 a 15 a 1 a 5 } + + a 1 { a1 λ a 11 b 1 a 1 c 1 +λ a 11 } λ a 11 b a 1 c 1 + λ a 1 1 { a 1 + λ a 1 λ a 11 b 3 a 1 c 3 a1 1 + a 11 λ a 11 b 4 a 1 c a 11 λ a 11 b 5 a 1 c 5 } a { h 4 = a1 λ a 11 b 1 a 1 c 1 +λ a 11 } λ a 11 b a 1 c 1 + λ a 1 a 1 { h 5 = a 1 + λ a 1 λ a 11 a 13 a 1 a a 1 λ a 11 1 λ a 11 a 14 a 1 a 4 λ a a 11 λ a 11 a 15 a 1 a 5 } + 1 { a λ a 1 λ a 11 a 16 a 1 a a 11 3 λ a 11 a 19 a 1 a 9 }.

10 Zhao and Yan Advances in Difference Equations :40 Page 10 of 18 Let α 1 = F ũ δ + 1 F F δ ũ 00 = h and α = 1 3 F + 1 F 6 ũ 3 ũ 00 = h 5 + h 1. Then we have the following results. Theorem 3.1 If α 1 0and α 0then the map 3 undergoes a flip bifurcation at the unique positive fixed point Bu v when the parameter δ varies in a small neighborhood of F B1. Moreover if α >0resp. α <0then the period- orbits that bifurcate from Bu v are stable resp. unstable. Next we discuss the Hopf bifurcation of Bu v when the parameters δ β a h m r s vary in a small neighborhood of H B. Taking the parameters δ β a h m r s arbitrary from H B we consider system 3withδ β a h m r s H B represented by u u + δ ru1 u βuv a+u+mv v v + δ vs hv u. 15 Then the map 15 has a unique positive fixed point Bu v. Then we choose δ as a bifurcation parameter and consider a perturbation of 15 as follows: u u +δ + δ ru1 u βuv a+u+mv v v +δ + δ vs hv u 16 where δ 1 is a small perturbation parameter. Assume that U = u u and V = v v.thenwetransformthefixedpointbu v of the map 16 into the origin. For convenience we rewrite U and V as u and vrespectively. Then we have a 11 u + a 1 v + a 13 u + a 14 uv + a 15 v + a 16 u 3 + a 17 u v u + a 18 uv + a 19 v 3 + o u + v 4 v a 1 u + a v + a 3 u + a 4 uv + a 5 v + a 6 u 3 + a 7 u v + a 8 uv + a 9 v 3 + o u + v 4 17 where a 11 a 1 a 13 a 14 a 15 a 16 a 17 a 18 a 19 a 1 a a 3 a 4 a 5 a 6 a 7 a 8 a 9 are given in 10 by substituting δ for δ + δ. Then the characteristic equation associated with the linearization of model 17 at u v=00isgivenby λ + p δ λ + q δ =0 where p δ = Gδ + δ q δ =1+Gδ + δ +Hsδ + δ.

11 Zhao and Yan Advances in Difference Equations :40 Page 11 of 18 Since δ β a h m r s H B there exists a pair of complex conjugate eigenvalues λ λ with modulus 1 at u v=00bytheorem.3where λ λ = p δ ± i 4q δ p δ =1+ Gδ + δ ± iδ + δ 4Hs G. 18 Then we have λ = q δ l = d λ d δ δ =0 = G >0. In addition we require that when δ =0λ n λ n 1n =134whichisequivalentto p Note that δ β a h m r s H B sop0. Thus we only need to satisfy p0 01whichleadsto G Hs3Hs. 19 In the following we investigate the normal form of the map 17at δ =0.Putμ =1+ Gδ and ω = δ 4Hs G. Using the translation u a 1 0 ũ = v μ a 11 ω ṽ the model 17becomes where ũ μ ω ũ f ũ ṽ + 0 ṽ ω μ ṽ gũ ṽ f ũ ṽ= 1 a 1 a13 u + a 14 uv + a 15 v + a 16 u 3 + a 17 u v + a 18 uv + a 19 v 3 + o u + v 4 gũ ṽ= μ a 11a 13 a 1 a 3 u + μ a 11a 14 a 1 a 4 uv ωa 1 ωa 1 + μ a 11a 15 a 1 a 5 ωa 1 v + μ a 11a 16 a 1 a 6 ωa 1 u 3 + μ a 11a 17 a 1 a 7 ωa 1 u v + μ a 11a 18 a 1 a 8 ωa 1 uv + μ a 11a 19 a 1 a 9 ωa 1 v 3 + o u + v 4 and u = a 1 ũ v =μ a 11 ũ ωṽ. Let fũũ = a 1 a 1 a 13 + a 14 a 1 μ a 11 +a 15 μ a 11 fũṽ = 1 a 1 ωa1 a 14 +ωμ a 11 a 15 fṽṽ = ω a 15 a 1

12 Zhao and Yan Advances in Difference Equations :40 Page 1 of 18 fũũũ = 6 a 1 a16 a a 17a 1 μ a 11+a 18 a 1 μ a 11 + a 19 μ a 11 3 fũũṽ = a 1 ωa17 a 1 + ωa 18a 1 μ a 11 +3ωμ a 11 fũṽṽ = ω a 18 +3a 19 fṽṽṽ = 6ω3 a 19 a 1 gũũ = ωa 1 { a 1 a 13 μ a11 a 13 a 1 a 3 + a1 μ a 11 a 14 μ a11 a 14 a 1 a 4 + a 15 μ a 11 μ a 11 a 15 a 1 a 5 } gũṽ = 1 { a1 μ a11 a 14 a 1 a 4 +μ a11 } μ a 11 a 15 a 1 a 5 a 1 gṽṽ = ω μ a11 a 15 a 1 a 5 a 1 gũũũ = 6 ωa 1 { a 3 1 μ a11 a 16 a 1 a 6 + a 1 μ a 11 μ a 11 a 17 a 1 a 7 + a 1 μ a 11 μ a 11 a 18 a 1 a 8 +μ a11 3 } μ a 11 a 19 a 1 a 9 gũũṽ = { a a 1 μ a11 a 17 a 1 a 7 +a1 μ a 11 μ a 11 a 18 a 1 a μ a 11 μ a 11 a 19 a 1 a 9 } gũṽṽ =ω { μ a 11 a 18 a 1 a 8 +3μ a11 a 19 a 1 a 9 } gṽṽṽ = 6ω a 1 μ a11 a 19 a 1 a 9. Then the map 0 can undergo the Hopf bifurcation when the following discriminatory quantity is not zero: 1 λ λ γ = Re ξ 0 ξ λ ξ 11 ξ 0 + Re λξ 1 1 δ =0 where ξ 0 = 1 8 fũũ fṽṽ + gũṽ +i gũũ gṽṽ fũṽ ξ 11 = 1 4 fũũ + fṽṽ +i gũũ + gṽṽ ξ 0 = 8 1 fũũ fṽṽ gũṽ +i gũũ gṽṽ + fũṽ ξ 1 = 1 fũũũ + fũṽṽ + gũũṽ + gṽṽṽ +i gũũũ + gũṽṽ fũũṽ fṽṽṽ. 16 From the preceding analysis and theorem in 0 we have the following result. Theorem 3. If condition 19 holds and γ 0then the map 3 undergoes a Hopf bifurcation at the unique positive fixed point Bu v when the parameter δ varies in a small neighborhood of H B. Moreover if γ <0resp. γ >0then an attracting resp. repelling invariant closed curve bifurcates from the fixed point for δ > δ resp. δ > δ.

13 Zhao and Yan Advances in Difference Equations :40 Page 13 of 18 4 Numerical simulations In this section we present the bifurcation diagrams phase portraits and maximum Lyapunov exponents for system 3 to illustrate our theoretical analysis and show the complex dynamical behaviors by using numerical simulations. The bifurcation parameters are considered for the following three cases: i Varying δ in the range 1 δ 1.4 and fixing r = β =0.5 a =0. m =0. s =0.6 h =. ii Varying δ in the range δ 3 and fixing r =1 β =0.5 a =0. m =0. s =0.6 h =. iii Varying r in the range r.99 and fixing δ =1 β =0.5 a =0. m =0. s =0.6 h =. Case i. On the basis of Lemma.1 weknowthatthemap3 hasoneuniquepositive fixed point. By calculation the flip bifurcation of system 3 emergesfromthefixed point u v = at δ = with α 1 = α = and δ β a h m r s F B1 which illustrates Theorem 3.1.FromFigs.1a b we observe that the fixed point Bu v isstablefor0<δ < and loses its stability at the flip bifurcation parameter value δ = Also there is a cascadeofperiod 46816orbitsemerging.ThemaximumLyapunovexponentscorresponding to Figs. 1a b are shown in Fig. 1c. The phase portraits associated with Figs. 1a b are displayed in Fig.. It can be seen from Fig. that there are chaotic sets when δ = Further Fig. 1d shows that if δ = then the max- Figure 1 a Flip bifurcation diagram of map 3 in theδ u plane with initial value b Flip bifurcationdiagram of map 3 in theδ v plane.c Maximum Lyapunov exponents corresponding to aandb. d Local amplification of cforδ

14 Zhao and Yan Advances in Difference Equations :40 Page 14 of 18 Figure Phase portraits for various values of δ corresponding to Figs. 1a b imum Lyapunov exponents are larger than 0 which confirms the existence of chaotic sets. Case ii. According to Lemma.1 we know that the map 3 hasoneuniquepos- itive fixed point. After calculation the Hopf bifurcation emerges from the fixed point u v = at δ = with γ = and δ β a h m r s H B. This shows the correctness of Theorem 3.. FromFigs.3a b we observe that the fixed point Bu v isstablefor0<δ < and loses its stability at the Hopf bifurcation parameter value δ = Then an attracting invariant cycle bifurcates from the fixed point since γ = < 0 by Theorem 3.. The maximum Lyapunov exponents corresponding to Figs. 3a b are calculated and shown in Fig. 3c. Figure 3d is a local amplifications for δ It can be easily

15 Zhao and Yan Advances in Difference Equations :40 Page 15 of 18 Figure 3 a Hopf bifurcation diagram of map 3 intheδ u plane with initial value b Hopf bifurcation diagram of map 3 intheδ v plane.c Maximum Lyapunov exponents corresponding to Figs. 3a b. d Local amplification diagram of a for δ.75.9 seen from Figs. 3c d that the maximum Lyapunov exponents are nonnegative for the parameter δ which implies the existence of chaos. Also we observe from Fig. 4 that there are period-10 period-0 period-30 and period-73 orbits and attracting chaotic sets. Case iii. The bifurcation diagrams of system 3 inther u andr v planesfor r.86 are disposed in Figs. 5a b. The maximum Lyapunov exponent corresponding to Fig. 5a is computed and plotted in Fig. 5c and the local amplification diagram corresponding to c for δ.8.6 is shown in Fig. 5d. From Figs. 5a b we can see that there is a cascade of period-doubling emerging. From Fig. 5d we see that the maximum Lyapunov exponents corresponding to r =.85 are greater than 0 which implies the existence of chaotic sets. 5 Conclusion In this paper we have mainly considered the complex behaviors of a predator prey system with modified Holling Tanner functional response in R. By using the center manifold theorem and bifurcation theory we prove that the unique positive fixed point of system 3 can undergo flip bifurcation and Hopf bifurcation. Most importantly when the integral step size δ is chosen as a bifurcation parameter numerical simulations show that system 3 shows very rich nonlinear dynamical behaviors including stable coexistence period-doubling bifurcation leading to chaos attracting invariant circles and even stranger chaotic attractors. According to Figs. 1 and we can observe that the small integral step size δ can stabilize the dynamical system 3 but the large integral step size

16 Zhao and Yan Advances in Difference Equations :40 Page 16 of 18 Figure 4 Phase portraits for various values of δ corresponding to Figs. 3a c may destabilize the system producing more complex dynamical behaviors. Then it reminds of us that the integral step size may play a key role in exploring the dynamical behaviors. In addition from Fig. 5 we can see that the appropriately intrinsic growth rate r of prey can stabilize the dynamical system 3. However the high intrinsic growth rate may destabilize system 3. From a biological point of view when the prey population is submitted to the high intrinsic growth rate the number of preys are abundant and the prey consumption by predator may have a marginal effect on the dynamics of prey. Hence the dynamical behavior of prey mainly depends on the population itself. Then system 3 becomes the classic logistic model and exhibits the period-doubling leading to chaos.

17 Zhao and Yan Advances in Difference Equations :40 Page 17 of 18 Figure 5 a Bifurcation diagram of map 3 inther u plane with initial value b Bifurcation diagram ofmap 3 in ther v plane.c Maximum Lyapunov exponents corresponding to Figs. 3a b. d Local amplification diagram of aforδ.8.6 Funding This work was supported by the National Natural Science Foundation of China no and by Sichuan Minzu College nos. XYZB17001 and sfkc Competing interests Both authors declare that they have no competing interests. Authors contributions JZ is responsible for the model formulation and study planning. JZ and YY have done the calculation the proof and the simulation. Both authors have read and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: June 018 Accepted: 6 September 018 References 1. Berryman A.A.: The origins and evolution of predator prey theory. Ecology Shi H. Li W. Lin G.: Positive steady states of a diffusive predator prey system with modified Holling Tanner functional response. Nonlinear Anal. Real World Appl Yang W.: Global asymptotical stability and persistent property for a diffusive predator prey system with modified Leslie Gower functional response. Nonlinear Anal. Real World Appl Zhang L. Zhang C. Zhao M.: Dynamic complexities in a discrete predator prey system with lower critical point for the prey. Math. Comput. Simul Choo S.: Global stability in n-dimensional discrete Lotka Volterra predator prey models. Adv. Differ. Equ Cao H. Zhou Y. Ma Z.: Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection. Math. Biosci. Eng Ren J. Yu L. Siegmund S.: Bifurcations and chaos in a discrete predator prey model with Crowley Martin functional response. Nonlinear Dyn Huang T. Zhang H. Yang H. Wang N. Zhang F.: Complex patterns in a space- and time-discrete predator-prey model with Beddington-DeAngelis functional response. Commun. Nonlinear Sci Huang J. Liu S. Ruan S. Xiao D.: Bifurcations in a discrete predator prey model with nonmonotonic functional response. J. Math. Anal. Appl

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