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

1 Department of Mathematics, Anqing Normal University, Anqing, Anhui 246133, China 2Department of Mathematics, Research Center for Complex Systems and Network Sciences, Southeast University, Nanjing, Jiangsu 210096, China 3Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 4Department of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan


Introduction
Logistic type models are often used to model a single species dynamics, for example, the underlying dynamics of tumour cells [1][2][3][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][8][9].There are also very many results concerning the discrete logistic model (see [10][11][12][13][14][15] and references therein).
In [13], Zhou and Zou considered a discrete logistic equation: where {()} and {()} are positive -periodic sequences.Sufficient conditions were obtained for the existence of a positive and globally asymptotically stable -periodic solution.
In [14], Sun and Li considered the qualitative analysis of the following discrete logistic equation with several delays: where   ∈ (0, ∞) for  = 1, 2, . . ., ,   ( = 1, 2, . . ., ) are positive integers, and  ∈ (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: where {  }, {  }, and {  } are periodic sequences with a common period ,   > 0,   > 0, ,  are positive integers, and ,  are positive constants with  > .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: where ,  1 , and  2 are positive constants, together with the initial condition 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.

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  * = 2/(√ 2  1 + 4 2 +  1 ).Set   =   −  * ; then there follows that By introducing a new variable   = (  ,  −1 , . . .,  − )  , we can rewrite (6) in the form where  = ( 0 ,  1 , . . .,   )  and Clearly, the origin is a fixed point of (7) and the linear part of (7) is where The characteristic equation of  is given by Lemma 1.There exists  > 0 such that for 0 <  <  all roots of (11) have modulus less than one.
Consider the root () such that |(0)| = 1.This root depends continuously on  and is a differential function of .From (11), we have Consequently, || < 1 for all sufficient small  > 0. Thus, all roots of (11) are inside the unit circle for sufficient small positive , and the existence of the maximal  follows.
In the sequel, we define a parametric curve Σ with Let () = V()/().It is easy to see that   () > 0 for all  ∈ R such that () ̸ = 0. Therefore, as  increases from 0 to , the corresponding point ((), V()) on the curve Σ moves from origin and anticlockwise around origin.Moreover, it follows from () Moreover, from the anticlockwise property of the curve Σ, we further have that From () This means that the curve is contained in the region Accordingly, we set where (iv) All zeros of () are inside the unit circle if and only if Now we verify that the transversality condition is satisfied.
then the fixed point of model (4)  * is asymptotically stable.
then the fixed point of model (4)  * is unstable.
then the model (4) undergoes a Neimark-Sacker bifurcation at the positive fixed point  * ; that is, a unique closed invariant curve bifurcates form  * .Moreover, if  is even, the model (4) undergoes a flip bifurcation when (25)

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  2−1 by  * and the critical value   ( = 1, 2, . . ., [( − 1)/2] + 1) by  * at which map (7) undergoes a Neimark-Sacker bifurcation at the origin.
From the above argument, we have the following result.

Numerical Simulations
In this section we will give an example to illustrate the analytic results.Here we take model ( 4 4) at  =  1 is supercritical and unique closed invariant curve bifurcating from  * is asymptotically stable.These are shown by Figures 1 and 2.

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  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  passes the critical  * .
(11)s 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).