Stability and Bifurcation Analysis of a Nonlinear Discrete Logistic Model with Delay

Abstract

We consider a nonlinear discrete logistic model with delay. The characteristic equation of the linearized system at the positive equilibrium is a polynomial equation involving high order terms. We obtain the conditions ensuring the asymptotic stability of the positive equilibrium and the existence of Neimark-Sacker bifurcation, with respect to the parameter of the model. Based on the bifurcation theory, we discuss Neimark-Sacker bifurcation direction and the stability of bifurcated solutions. Finally, some numerical simulations are performed to illustrate the theoretical results.

1. Introduction

Logistic type models are often used to model a single species dynamics, for example, the underlying dynamics of tumour cells [1–4]. Moreover, time delay is sometimes necessary to better reflect the description of real processes [5, 6]. The continuous single population model described by the ordinary differential equations has been studied very deeply [7–9]. There are also very many results concerning the discrete logistic model (see [10–15] and references therein).

In [13], Zhou and Zou considered a discrete logistic equation: 𝑥()(𝑛)𝑟(𝑛)𝑥(𝑛)𝑛+1=𝑥exp1−𝐾(𝑛), (1) where {𝑟(𝑛)} and {𝐾(𝑛)} are positive 𝜔-periodic sequences. Sufficient conditions were obtained for the existence of a positive and globally asymptotically stable 𝜔-periodic solution.

In [14], Sun and Li considered the qualitative analysis of the following discrete logistic equation with several delays: 𝑥_𝑛+1=𝑥_𝑛exp^𝑚_𝑖=1𝑟_𝑖𝑥1−^{𝑛−𝑘𝑖}𝐾, (2) where 𝑟_𝑖∈(0,∞) for 𝑖=1,2,…,𝑚, 𝑘_𝑖  (𝑖=1,2,…,𝑚) are positive integers, and 𝐾∈(0,∞). They obtained sufficient conditions for the global attractivity of all positive solutions about the positive equilibrium of model (2). Moreover, the oscillation about the positive equilibrium of model (2) was also discussed.

Chen and Zhou [15] discussed the following nonlinear periodic delay difference equation: 𝑥_𝑛+1=𝑥_𝑛exp𝑎_𝑛+𝑏_𝑛𝑥^𝑝_{𝑛−𝑚𝜔}−𝑐_𝑛𝑥^𝑞_{𝑛−𝑚𝜔},𝑛=0,1,2,…, (3) where {𝑎_𝑛}, {𝑏_𝑛}, and {𝑐_𝑛} are periodic sequences with a common period 𝜔, 𝑎_𝑛>0, 𝑐_𝑛>0, 𝑚, 𝜔 are positive integers, and 𝑝, 𝑞 are positive constants with 𝑞>𝑝. Some sufficient conditions for the global attractivity and oscillation about the periodic solution of (3) were obtained.

Recently, Liu et al. [16] investigated the stability and bifurcation of a class of discrete-time Cohen-Grossberg neural networks with delays. Han and Liu [17] discussed the stability and bifurcation for a discrete-time model of Lotka-Volterra type with delay. He and Cao [18] studied the explicit formula for determining the direction of Neimark-Sacker bifurcation and the stability of periodic solution by using the normal form method and the center manifold theory.

To the best of our knowledge, no similar results have been reported regarding the nonlinear discrete logistic model with delay. In this paper, we are interested in the bifurcation analysis and the direction and the stability of the Neimark-Sacker bifurcations for the following nonlinear discrete logistic model: 𝑥_𝑛+1=𝑥_𝑛exp𝑟1−𝑎₁𝑥_𝑛−𝑚−𝑎₂𝑥²_𝑛−𝑚,𝑛=0,1,2,…;𝑚∈ℤ⁺, (4) where 𝑟, 𝑎₁, and 𝑎₂ are positive constants, together with the initial condition 𝑥_𝑛=𝜙_𝑛>0,𝑛∈{−𝑚,−𝑚+1,…,0}. (5)

Our works focus on the stability and bifurcation analysis and the direction analysis of the Neimark-Sacker bifurcations by applying the center manifold theorem and the normal form theory. The method of the paper is similar to the work of Yuri [19].

The paper is organized as follows. In Section 2, we analyze the distribution of the characteristic equation associated with the model (4) and obtain the existence of the Neimark-Sacker bifurcation. In Section 3, the direction and stability of closed invariant curve from the Neimark-Sacker bifurcation of the model (4) are determined. In Section 4, some numerical simulations are performed to illustrate the theoretical results. A brief discussion is given in Section 5.

2. Stability Analysis

In this section, we will employ the techniques of Guo et al. [20] to study the distribution of the characteristic roots of model (4). Then we will obtain the stability of the positive equilibrium and the existence of local Neimark-Sacker bifurcations. Clearly, model (4) has a unique positive equilibrium 𝑥^∗=2/(𝑎²₁+4𝑎₂+𝑎₁).

Set 𝑦_𝑛=𝑥_𝑛−𝑥^∗; then there follows that 𝑦_𝑛+1=𝑦_𝑛+𝑥^∗×exp𝑟1−𝑎₁𝑦_𝑛−𝑚+𝑥^∗−𝑎₂𝑦_𝑛−𝑚+𝑥^∗2−𝑥^∗. (6) By introducing a new variable 𝑌_𝑛=(𝑦_𝑛,𝑦_𝑛−1,…,𝑦_𝑛−𝑚)^𝑇, we can rewrite (6) in the form 𝑌_𝑛+1=𝐹𝑌_𝑛, (7) where 𝐹=(𝐹₀,𝐹₁,…,𝐹_𝑚)^𝑇 and 𝐹_𝑘=𝑦_𝑛+𝑥^∗exp𝑟1−𝑎₁𝑦_𝑛−𝑚+𝑥^∗−𝑎₂𝑦_𝑛−𝑚+𝑥^∗2−𝑥^∗𝑦,𝑘=0,_{𝑛−𝑘+1},1≤𝑘≤𝑚. (8) Clearly, the origin is a fixed point of (7) and the linear part of (7) is 𝑌_𝑛+1=𝐴𝑌_𝑛, (9) where 𝑎𝐴=10⋯0−𝑟₁+2𝑎₂𝑥^∗𝑥^∗10⋯0001⋯00⋮⋮⋱⋮⋮00⋯10. (10)

The characteristic equation of 𝐴 is given by 𝑃(𝜆)=𝜆^𝑚+1−𝜆^𝑚+𝑟𝑎₁+2𝑎₂𝑥^∗𝑥^∗=0. (11)

Lemma 1. There exists 𝑟>0 such that for 0<𝑟<𝑟 all roots of (11) have modulus less than one.

Proof. When 𝑟=0, (11) becomes 𝜆^𝑚+1−𝜆^𝑚=0. (12) The equation has, at 𝑟=0, an 𝑚-fold root 𝜆=0 and a simple root 𝜆=1.
Consider the root 𝜆(𝑟) such that |𝜆(0)|=1. This root depends continuously on 𝑟 and is a differential function of 𝑟. From (11), we have 𝑑|𝜆|²𝑑𝑟_{𝑟=0,𝜆=1}=𝜆𝑑𝜆+𝑑𝑟𝜆𝑑𝜆𝑑𝑟_{𝑟=0,𝜆=1}=−2𝑎₁+2𝑎₂𝑥^∗𝑥^∗<0. (13) Consequently, |𝜆|<1 for all sufficient small 𝑟>0. Thus, all roots of (11) are inside the unit circle for sufficient small positive 𝑟, and the existence of the maximal 𝑟 follows.
In the sequel, we define a parametric curve Σ with 𝑢(𝑡)()𝑣(𝑡)()=cos𝑚+1𝑡−cos𝑚𝑡,=sin𝑚+1𝑡−sin𝑚𝑡. (14) Let 𝜃(𝑡)=𝑣(𝑡)/𝑢(𝑡). It is easy to see that 𝜃^′(𝑡)>0 for all 𝑡∈ℝ such that 𝑢(𝑡)≠0. Therefore, as 𝑡 increases from 0 to 𝜋, the corresponding point (𝑢(𝑡),𝑣(𝑡)) on the curve Σ moves from origin and anticlockwise around origin. Moreover, it follows from 𝑢(𝑡)²+𝑣(𝑡)²=2(1−cos𝑡) that the part of the curve Σ with parameter 𝑡∈[0,𝜋] is simple; that is, it cannot intersect itself. Let 0≤𝜔₀≤𝜔₁<𝜔₂<⋯<𝜔_𝑚<𝜔_𝑚+1≤𝜋 be the 𝑚+2 zeros of 𝑣(𝑡) in the interval [0,𝜋] and 𝑐_𝑗=𝑢(𝜔_𝑗)  (𝑗∈{0,1,2,…,𝑚+1}). Obviously, we have 𝜔₀=0,𝜔_𝑗=2𝑗−1𝑐2𝑚+1𝜋,𝑗=1,2,…,𝑚+1,_𝑗=2(−1)^𝑗𝜔sin^𝑗2,𝑗=0,1,2,…,𝑚+1. (15) Moreover, from the anticlockwise property of the curve Σ, we further have that (−1)^𝑗𝑣^′𝜔_𝑗>0,𝑗∈{0,1,2,…,𝑚+1}. (16) From 𝑢(𝑡)²+𝑣(𝑡)²=2(1−cos𝑡), it follows that 0≤𝑢(𝑡)²+𝑣(𝑡)²≤2. This means that the curve is contained in the region (𝑢,𝑣)∈ℝ²∣0≤𝑢(𝑡)²+𝑣(𝑡)²≤2. (17)
Accordingly, we set 𝑟_𝑘=2𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝜔sin^2𝑘−12𝑚,𝑘=1,2,…,2+1, (18) where [⋅] is the greatest integer function.
By Lemma 1 and the idea of Guo et al. (see Lemma  1 in [20]), we can obtain the distribution of the roots of the characteristic equation (11).

Lemma 2. Consider the following.(i)𝑃(𝜆) has zeros on the unit circle if and only if 𝑟=𝑟_𝑘 for some 𝑘∈{1,2,…,[𝑚/2]+1}. Moreover, if 𝑟=𝑟_𝑘 for some 𝑘∈{1,2,…,[𝑚/2]+1}, then all the zeros of 𝑃(𝜆) of modulus 1 are 𝑒^{±𝑖𝜔2𝑘−1}, which are simple.(ii)𝑃(𝜆) has a simple zero −1 if and only if 𝑚 is even and 𝑟=2/(𝑎₁+2𝑎₂𝑥^∗)𝑥^∗.(iii)For a fixed 𝑘∈{1,2,…,[𝑚/2]+1}, there exist 𝛿>0 and a 𝐶¹-mapping 𝜆∶(𝑟_𝑘−𝛿,𝑟_𝑘+𝛿)→ℂ such that 𝜆(𝑟_𝑘)=𝑒^{𝑖𝜔2𝑘−1} and 𝜆(𝑟) is a zero of 𝑃(𝜆) for all 𝑟∈(𝑟_𝑘−𝛿,𝑟_𝑘+𝛿).(iv)All zeros of 𝑃(𝜆) are inside the unit circle if and only if 0<𝑟<𝑟₁=2𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝜋sin2().2𝑚+1 (19)Now we verify that the transversality condition is satisfied.

Lemma 3. Along (11), one obtains 𝑑|𝜆|²𝑑𝑟_{𝑟=𝑟𝑘}>0, (20) where 𝑘=1,2,…,[𝑚/2]+1.

Proof. By Lemma 2(i), from (14) and (16), it follows that 𝑑|𝜆|²𝑑𝑟_{𝑟=𝑟𝑘}=𝜆𝑑𝜆+𝑑𝑟𝜆𝑑𝜆𝑑𝑟_{𝑟=𝑟𝑘}=−2ℜ𝜆𝑎₁+2𝑎₂𝑥^∗𝑥^∗(𝑚+1)𝜆^𝑚−𝑚𝜆^𝑚−1_{𝑟=𝑟𝑘}=−2𝑎₁+2𝑎₂𝑥^∗𝑥^∗×(𝑚+1)cos(𝑚+1)𝜔_2𝑘−1−𝑚cos𝑚𝜔_2𝑘−1(𝑚+1)²+𝑚²−2𝑚(𝑚+1)cos𝜔_2𝑘−1=−2𝑎₁+2𝑎₂𝑥^∗𝑥^∗×𝑣^′𝜔_2𝑘−1(𝑚+1)²+𝑚²−2𝑚(𝑚+1)cos𝜔_2𝑘−1>0. (21)

Remark 4. Since 𝜔_2𝑘−1≠𝜋/2,2𝜋/3, 𝑒^{𝑖𝑠𝜔2𝑘−1}≠1 for 𝑠=1,2,3,4  (𝑘=1,2,…,[(𝑚−1)/2]+1); that is, the nondegeneracy condition is satisfied.

Theorem 5. Consider the following.(i)If 0<𝑟<𝑟₁=2𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝜋sin2(),2𝑚+1 (22)then the fixed point of model (4) 𝑥^∗ is asymptotically stable.(ii)If 𝑟>𝑟₁=2𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝜋sin2(),2𝑚+1 (23)then the fixed point of model (4) 𝑥^∗ is unstable.(iii)If 𝑟=𝑟_𝑘=2𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝜔sin^2𝑘−12,𝑘=1,2,…,𝑚−12+1, (24)then the model (4) undergoes a Neimark-Sacker bifurcation at the positive fixed point 𝑥^∗; that is, a unique closed invariant curve bifurcates form 𝑥^∗. Moreover, if 𝑚 is even, the model (4) undergoes a flip bifurcation when 2𝑟=𝑎₁+2𝑎₂𝑥^∗𝑥^∗. (25)

3. Direction and Stability of the Neimark-Sacker Bifurcation

In this section, we will study the direction and the stability of the Neimark-Sacker bifurcation in model (4). The method we used is based on the techniques developed by Yuri [19].

Without loss of generality, denote 𝜔_2𝑘−1 by 𝜔^∗ and the critical value 𝑟_𝑘  (𝑘=1,2,…,[(𝑚−1)/2]+1) by 𝑟^∗ at which map (7) undergoes a Neimark-Sacker bifurcation at the origin.

For map (7), we obtain 𝑌_𝑛+1=𝐴𝑌_𝑛+12𝐵𝑌_𝑛,𝑌_𝑛+16𝐶𝑌_𝑛,𝑌_𝑛,𝑌_𝑛+𝑂𝑌_𝑛⁴, (26) where 𝐵𝑌_𝑛,𝑌_𝑛=𝐵₀𝑌_𝑛,𝑌_𝑛,0,…,0^𝑇,𝐶𝑌_𝑛,𝑌_𝑛,𝑌_𝑛=𝐶₀𝑌_𝑛,𝑌_𝑛,𝑌_𝑛,0,…,0^𝑇,𝐵₀𝑌_𝑛,𝑌_𝑛=−2𝑟𝑎₁+2𝑎₂𝑥^∗𝑦_𝑛−𝑚𝑦_𝑛+𝑟𝑎₁+2𝑎₂𝑥^∗2−2𝑟𝑎₂𝑥^∗𝑦²_𝑛−𝑚,𝐶₀𝑌_𝑛,𝑌_𝑛,𝑌_𝑛=3𝑟²𝑎₁+2𝑎₂𝑥^∗2−2𝑟𝑎₂𝑦_𝑛𝑦²_𝑛−𝑚+6𝑎₂𝑟²𝑎₁+2𝑎₂𝑥^∗−𝑟𝑎₁+2𝑎₂𝑥^∗3𝑥^∗𝑦³_𝑛−𝑚. (27) Let 𝑞∈ℂ^𝑚+1 be a complex eigenvector corresponding to 𝑒^𝑖𝜔∗, 𝐴𝑞=𝑒^𝑖𝜔∗𝑞,𝐴𝑞=𝑒^{−𝑖𝜔∗}𝑞. (28) We also introduce an adjoint eigenvector 𝑞^∗∈ℂ^𝑚+1 having the properties 𝐴^𝑇𝑞^∗=𝑒^{−𝑖𝜔∗}𝑞^∗,𝐴^𝑇𝑞^∗=𝑒^𝑖𝜔∗𝑞^∗ (29) and satisfying the normalization ⟨𝑞^∗,𝑞⟩=1, where ⟨𝑞^∗,𝑞⟩=∑^𝑚_𝑗=0𝑞^∗_𝑗𝑞_𝑗.

Lemma 6. Let 𝑞=(𝑞₀,𝑞₁,…,𝑞_𝑚)^𝑇 be the eigenvector of 𝐴 corresponding to eigenvalue 𝑒^𝑖𝜔∗ and let 𝑞^∗=(𝑞^∗₀,𝑞^∗₁,…,𝑞^∗_𝑚)^𝑇 be the eigenvector of 𝐴^𝑇 corresponding to eigenvalue 𝑒^{−𝑖𝜔∗}. Then 𝑞=1,𝑒^{−𝑖𝜔∗},𝑒^{−2𝑖𝜔∗},…,𝑒^{−𝑚𝑖𝜔∗𝑇},𝑞^∗=𝑞^∗₀−𝑒^{𝑖𝑚𝜔∗}𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞^∗₀−𝑒^{𝑖(𝑚−1)𝜔∗}𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞^∗₀⋮−𝑒^𝑖𝜔∗𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞^∗₀, (30) where 𝑞^∗₀=11−𝑒^{−𝑖(𝑚+1)𝜔∗}𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑚. (31)

Proof. Let 𝑞=(𝑞₀,𝑞₁,…,𝑞_𝑚)^𝑇 be an eigenvector of 𝐴 corresponding to eigenvalue 𝑒^𝑖𝜔∗. Then 𝑞₀−𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞_𝑚=𝑒^𝑖𝜔∗𝑞₀,𝑞_𝑗=𝑒^𝑖𝜔∗𝑞_𝑗+1,𝑗=0,1,2,…,𝑚−1. (32) Similarly, 𝑞^∗=𝑞^∗₀−𝑒^{𝑖𝑚𝜔∗}𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞^∗₀−𝑒^{𝑖(𝑚−1)𝜔∗}𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞^∗₀⋮−𝑒^𝑖𝜔∗𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑞^∗₀. (33) For satisfying the normalization ⟨𝑞^∗,𝑞⟩=1, we choose 𝑞^∗₀=11−𝑒^{−𝑖(𝑚+1)𝜔∗}𝑟^∗𝑎₁+2𝑎₂𝑥^∗𝑥^∗𝑚. (34)
Let 𝑇^𝑐 denote a real eigenspace corresponding to 𝑒^{±𝑖𝜔∗}, which is two-dimensional and is spanned by {ℜ(𝑞),ℑ(𝑞)}, and let 𝑇^𝑠 be a real eigenspace corresponding to all eigenvalues of 𝐴^𝑇 other than 𝑒^{±𝑖𝜔∗} being (𝑚−1) dimensional.
For any 𝑢∈ℝ^𝑚+1, we have its decomposition 𝑢=𝑤𝑞+𝑤𝑞+𝑣, (35) where 𝑤∈ℂ, 𝑤𝑞+𝑤𝑞∈𝑇^𝑐, and 𝑣∈𝑇^𝑠. The complex variable 𝑤 can be viewed as a new coordinate on 𝑇^𝑐, and 𝑤=𝑞^∗,𝑞,𝑢𝑣=𝑢−^∗,𝑢𝑞−𝑞^∗,𝑢𝑞. (36) In this coordinate, the map 𝐹 at 𝑟=𝑟^∗ has the form 𝑤↦𝑒^𝑖𝜔∗𝑤+𝑞^∗,𝐹𝑤𝑞+𝑤𝑞+𝑣,𝑣↦𝐴𝑣+𝐹𝑤𝑞+𝑤𝑞+𝑣−𝑞^∗,𝐹𝑤𝑞+𝑤𝑞−𝑞+𝑣𝑞^∗,𝐹𝑤𝑞+𝑤𝑞+𝑣𝑞. (37)
Using Taylor expansions, we obtain 𝑤↦𝑒^𝑖𝜔∗1𝑤+2𝑔₂₀𝑤²+𝑔₁₁𝑤1𝑤+2𝑔₀₂𝑤²+12𝐺₂₁𝑤²𝑤+𝐺₁₀,𝑣𝑤+𝐺₀₁,𝑣1𝑤,𝑣↦𝐴𝑣+2𝐻₂₀𝑤²+𝐻₁₁𝑤1𝑤+2𝐻₀₂𝑤²+𝑂|𝑤|³, (38) where 𝑔_𝑖𝑗∈ℂ, 𝐺₁₀, 𝐺₀₁∈ℂ^𝑚+1, and 𝑔₂₀=𝑞^∗,𝐵𝑞,𝑞,𝑔₁₁=𝑞^∗,𝐵𝑞,𝑞,𝑔₀₂=𝑞^∗,𝐵𝑞,𝑞,𝐺₂₁=𝑞^∗,𝐶𝑞,𝑞,𝑞,𝐺₁₀,𝑣=𝑞^∗,𝐺,𝐵𝑞,𝑣₀₁,𝑣=𝑞^∗,𝐵,𝐻𝑞,𝑣₂₀=𝐵𝑞,𝑞−𝑞^∗,𝐵𝑞,𝑞𝑞−𝑞^∗,𝐵𝑞,𝑞𝐻𝑞,₁₁=𝐵𝑞,𝑞−𝑞^∗,𝐵𝑞,𝑞𝑞−𝑞^∗,𝐵𝑞,𝑞𝐻𝑞,₀₂=𝐵𝑞,𝑞−𝑞^∗,𝐵𝑞,𝑞𝑞−𝑞^∗,𝐵𝑞,𝑞𝑞. (39)
Now, we seek the center manifold which has the representation 𝑣=𝑉(𝑤,𝑤)=12𝑧₂₀𝑤²+𝑧₁₁𝑤1𝑤+2𝑧₀₂𝑤²+𝑂|𝑤|³, (40) where ⟨𝑞^∗,𝑧_𝑖𝑗⟩=0. Substituting (40) into (38), we have 𝑧₂₀=𝑒^2𝑖𝜔∗𝐼−𝐴⁻¹𝐻₂₀,𝑧₁₁=(𝐼−𝐴)⁻¹𝐻₁₁,𝑧₀₂=𝑒^{−2𝑖𝜔∗}𝐼−𝐴⁻¹𝐻₀₂. (41) When restricting (38) to the center manifold, up to cubic term, then we have 𝑔₂₀=𝑞^∗,𝐵𝑞,𝑞,𝑔₁₁=𝑞^∗,𝐵𝑞,𝑞,𝑔₀₂=𝑞^∗,𝐵𝑞,𝑞,𝑔₂₁=𝑞^∗,𝐵𝑞,𝑧₂₀+2𝑞^∗,𝐵𝑞,𝑧₁₁+𝑞^∗,𝐶𝑞,𝑞,𝑞=𝑞^∗,𝐶𝑞,𝑞,𝑞−2𝑞^∗,𝐵𝑞,𝐼−𝐴⁻¹𝐵𝑞,𝑞+𝑞^∗,𝐵𝑞,𝑒^2𝑖𝜔∗𝐼−𝐴⁻¹𝐵+𝑒𝑞,𝑞_{−𝑖𝜔∗}1−2𝑒^2𝑖𝜔∗1−𝑒^{−𝑖𝜔∗}𝑞^∗,𝐵𝑞,𝑞𝑞^∗,𝐵𝑞,𝑞−21−𝑒^{−𝑖𝜔∗}𝑞^∗,𝐵𝑞,𝑞²−𝑒^𝑖𝜔∗𝑒^3𝑖𝜔∗𝑞−1^∗,𝐵𝑞,𝑞². (42)
Define 𝑐(𝑟)=𝑔₂₀𝑔₁₁(𝑤−3+2𝑤)2𝑤²−𝑤()+𝑔𝑤−11−𝑤¹¹²+𝑔₀₂²2𝑤²−𝑤+𝑔₂₁2. (43) Substituting 𝑤=𝑒^{−𝑖𝜔∗} into (43), we can obtain 𝑐(𝑟^∗).

From the above argument, we have the following result.

Theorem 7. The direction and stability of Neimark-Sacker bifurcation of model (4) can be determined by the sign of ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)]; that is, the Neimark-Sacker bifurcation of model (4) at 𝑟=𝑟_𝑗 is supercritical (resp., subcritical) and unique closed invariant curve bifurcation from 𝑥^∗ is asymptotically stable (resp., unstable) when ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)]<0 (resp., ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)]>0).

4. Numerical Simulations

In this section we will give an example to illustrate the analytic results. Here we take model (4) with 𝑎₁=1.6, 𝑎₂=0.8, 𝑚=2, and 𝑚=5, respectively. From 𝑎₁=1.6, 𝑎₂=0.8, and 𝑚=2, we have 𝑥^∗=0.5 and 𝑟₁=0.515. By the simple calculation, we obtain ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)]=−0.1452. Similarly, when 𝑚=5, we have 𝑥^∗=0.5, 𝑟₁=0.2372, and ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)] = −0.0948.

From Theorem 5, we conclude that, with 𝑎₁=1.6, 𝑎₂=0.8, and 𝑚=2, the positive equilibrium 𝑥^∗=0.5 of model (4) is asymptotically stable when 𝑟<𝑟₁=0.515, and model (4) undergoes the Neimark-Sacker bifurcation at 𝑥^∗ when 𝑟=𝑟₁=0.515. Furthermore, from Theorem 7 and ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)]=−0.1452<0 the Neimark-Sacker bifurcation of model (4) at 𝑟=𝑟₁ is supercritical and unique closed invariant curve bifurcating from 𝑥^∗ is asymptotically stable. These are shown by Figures 1 and 2.

(a)

(b)

Figure 1 

The positive equilibrium is asymptotically stable, where 𝑎1=1.6, 𝑎2=0.8, 𝑚=2, and 𝑟=0.49<𝑟1=0.515.

(a)

(b)

Figure 2 

A stable invariant closed circle bifurcating from 𝑥∗, where 𝑎1=1.6, 𝑎2=0.8, 𝑚=2, and 𝑟=0.5156>𝑟1=0.515.

Similarly, with 𝑎₁=1.6, 𝑎₂=0.8, and 𝑚=5, the positive equilibrium 𝑥^∗=0.5 of model (4) is asymptotically stable when 𝑟<𝑟₁=0.2372, and, from Theorem 7 and ℜ[𝑒^{−𝑖𝜔∗}𝑐(𝑟^∗)]=−0.0948<0, the Neimark-Sacker bifurcation of model (4) at 𝑟=𝑟₁ is supercritical and unique closed invariant curve bifurcating from 𝑥^∗ is asymptotically stable. These are shown by Figures 3 and 4.

(a)

(b)

Figure 3 

The positive equilibrium is asymptotically stable, where 𝑎1=1.6, 𝑎2=0.8, 𝑚=5, and 𝑟=0.23<𝑟1=0.2372.

(a)

(b)

Figure 4 

A stable invariant closed circle bifurcating from 𝑥∗, where 𝑎1=1.6, 𝑎2=0.8, 𝑚=5, and 𝑟=0.238>𝑟1=0.2372.

5. Conclusion and Discussion

There has been a large body of work discussing the stability and bifurcation in logistic model, but most of them dealt with only the continuous logistic models, or only the discrete logistic model without delays. In this paper, we discuss the dynamical behaviors of nonlinear discrete logistic model with delay. The characteristic equation (11) is a polynomial equation involving high order terms, which make it difficult to find all parameters such that the characteristic roots have modulus 1 or less. By analyzing the characteristic equation, some sufficient and necessary conditions are derived to ensure that all the characteristic roots have modulus less than 1. Moreover, we obtain that when the parameter 𝑟 varies, the positive equilibrium of the model (4) exchanges its stability and Neimark-Sacker bifurcation occurs. Furthermore, the direction and stability of closed invariant curve from the Neimark-Sacker bifurcation of the model (4) are determined. Numerical simulations also show the occurrence of the stable bifurcate periodic solutions when 𝑟 passes the critical 𝑟^∗.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are very grateful to the Associate Editor, Professor Zhan Zhou, and the anonymous reviewers for their valuable time and professional comments. This paper is jointly supported by the National Natural Science Foundation of China under Grant nos. 11072059 and 61272530, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant no. 20110092110017, the Natural Science Foundation of Jiangsu Province of China under Grant no. BK2012741, and the Programs of Educational Commission of Anhui Province of China under Grant nos. KJ2011A197 and KJ2013Z186. This work is also funded by the Deanship of Scientific Research, King Abdulaziz University (KAU), under Grant no. 3-130/1434/HiCi. Therefore, the authors acknowledge technical and financial support of KAU.