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Existence and asymptotic behavior results of positive periodic solutions for discretetime logistic model
Advances in Difference Equations volumeÂ 2015, ArticleÂ number:Â 181 (2015)
Abstract
A discretetime logistic model with delay is studied. The existence of a positive periodic solution for a discretetime logistic model is obtained by a continuation theorem of coincidence degree theory, and a sufficient condition is given to guarantee the global exponential stability of a periodic solution. Finally, an example is given to show the effectiveness of the results in this paper.
1 Introduction
Examples of the discrete phenomena in nature abound and somehow the continuous version have commandeered all our attention  perhaps owing to that special mechanism in human nature that permits us to notice only what we have been conditioned to. The theory of difference equations has grown at an accelerated rate in the past decades; see [1â€“3].
In this paper, we discuss the existence and exponential stability of positive periodic solution for the following logistic model:
where \(\alpha(n)>1\), \(\beta(n)>0\) are Nperiodic sequences, Ï„ is a positive integer,
There have been many papers concerned with the properties of solution of difference equations. In [4], Zhang et al. studied the existence of periodic solutions of the equation without delay
under the assumptions that \(\mu\in(1,2)\), \(b(n)<\frac{(\mu1)^{2}}{4\mu}k\) hold for all \(n\in Z\). Parhi [5] considered the delay difference equation
and obtained the oscillatory and asymptotic behavior of solutions of (1.2). Liu and Ge [6] considered the difference equation
where \(p(n)\) is nonnegative, \(r(n)\) is a real numbers sequence. Suppose \(\lim_{x\rightarrow0}\frac{f(x)}{x}=b<0\) and \(xf(x)<0\), \(f(x)\leq\gammax\) for all \(x\neq0\), the authors proved that if
then every solution of (1.3) tends to zero as n tends to infinity. Li et al. [7] used the upper and lower solutions method to show that there exists a \(\lambda^{*}>0\) such that the nonlinear functional difference equation
has at least one positive Tperiodic solutions for \(\lambda\in(0,\lambda^{*})\) and does not have any positive Tperiodic solutions for \(\lambda>\lambda^{*}\), where \(a(n)\), \(h(n)\), and \(\tau(n)\) are Tperiodic solutions. Jiang et al.[8] presented the optimal existence theory for single and multiple positive periodic solutions to a class of functional difference equations based upon the fixed point theorem in cones.
For a complicated dynamical system, we note that discretetime neural networks have been studied by many authors; see e.g. Hu and Wang [9], Wang and Xu [10], Xiong and Cao [11], Yuan et al. [12], Zhao and Wang [13] and Zou and Zhou [14] for DNNs without time delays and Chen et al. [15], Liang et al. [16], Liang et al. [17] and Xiang et al. [18] for DNNs with discrete time delays. For more related results, see [19â€“28].
So far, to the best of the authors knowledge, there are few results for the existence and stability of positive periodic solutions to (1.1). The major challenges are as follows: (1)Â In order to obtain existence of positive periodic solutions, we must change (1.1) to the proper form by a variable transformation. How can we choose the above variable transformation, which is the key to the study of (1.1)? (2)Â Since it is very difficult to construct a Lypunov functional to (1.1), how can we choose a proper special function for obtaining the stability results, which is significant to our proof? (3)Â It is nontrivial to establish a unified framework.
It is, therefore, the main purpose of this paper to make the first attempt to handle the listed challenges.
Remark 1.1
Equation (1.1) was proposed by Pielou [29] in 1974, which is a discrete analog of the delay logistic equation
A classic logistic model has received great attention from theoretical and mathematical biologists and has been well studied; see e.g. [29â€“32].
The following sections are organized as follows: In SectionÂ 2, the existence of positive periodic solution to (1.1) is obtained. In SectionÂ 3, sufficient conditions are established for the global exponential stability of (1.1). In SectionÂ 4, an example is given to show the feasibility of our results.
2 Existence of positive periodic solution
Let X and Y be real Banach spaces and let \(L:D(L)\subset X\rightarrow Y\) be a Fredholm operator with index zero, here \(D(L)\) denotes the domain of L. This means that ImL is closed in Y and \(\operatorname{dim}\operatorname{Ker}L=\operatorname{codim}\operatorname{Im}L<+\infty\). If L is a Fredholm operator with index zero, then there exist continuous projectors \(P:X\rightarrow X\), \(Q:Y\rightarrow Y\) such that \(\operatorname{Im}P=\operatorname{Ker}L\), \(\operatorname{Im}L=\operatorname{Ker}Q=\operatorname{Im}(IQ)\). It follows that \(L_{D(L)\cap \operatorname{Ker}P}:(IP)X\rightarrow \operatorname{Im}L\) is invertible. Denote by \(K_{p}\) the inverse of \(L_{P}\).
Let Î© be an open bounded subset of X, a map \(N :\bar{\Omega}\rightarrow Y\) is said to be Lcompact in \(\bar{\Omega}\) if \(QN(\bar{\Omega})\) is bounded and the operator \(K_{p}(IQ)N(\bar{\Omega})\) is relatively compact. Because ImQ is isomorphic to KerL, there exists an isomorphism \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}L\). We first recall the famous Mawhin continuation theorem.
Lemma 2.1
[33]
Suppose that X and Y are two Banach spaces, and \(L:D(L)\subset X\rightarrow Y\), is a Fredholm operator with index zero. Furthermore, \(\Omega\subset X\) is an open bounded set and \(N:\bar{\Omega}\rightarrow Y\) is Lcompact on \(\bar{\Omega}\). If all the following conditions hold:

(1)
\(Lx\neq\lambda Nx\), \(\forall x\in\partial\Omega\cap D(L)\), \(\forall\lambda\in(0,1)\),

(2)
\(Nx\notin \operatorname{Im}L\), \(\forall x\in\partial\Omega\cap \operatorname{Ker}L\),

(3)
\(\operatorname{deg}\{JQN,\Omega\cap \operatorname{Ker}L,0\}\neq0\),
where \(J:\operatorname{Im}Q\rightarrow \operatorname{Ker}L\) is an isomorphism, then the equation \(Lx=Nx\) has a solution on \(\bar{\Omega}\cap D(L)\).
Lemma 2.2
[34]
Let \(g:Z\rightarrow R\) be Ï‰periodic. Then for any fixed \(k_{1},k_{2}\in I_{\omega}\), and any \(k\in Z\), one has
and
Now, we state the main results and give its proof.
Theorem 2.1
Suppose that assumptions (H_{1}) and (H_{2}) hold:
 (H_{1}):

there exists a constant \(C>0\) such that if \(x(n)\) is a Nperiodic sequences and satisfies
$$\sum_{n=0}^{N1} \bigl[\ln\bigl(1+ \beta(n)e^{y(n\tau)}\bigr)+\ln\alpha(n) \bigr]=0, $$then we have
$$\sum_{n=0}^{N1}\bigl\vert \ln\bigl(1+ \beta(n)e^{y(n\tau)}\bigr)+\ln\alpha(n)\bigr\vert \leq C; $$  (H_{2}):

there exists a constant \(D>0\) such that when \(y>D\),
$$\ln\bigl(1+\beta(n)e^{y(n)}\bigr)>\ln\alpha(n) $$and
$$\ln\bigl(1+\beta(n)e^{y(n)}\bigr)< \ln\alpha(n) $$uniformly hold for \(n\in Z\).
Then (1.1) has at least one positive Nperiodic solution.
Proof
In order to obtain the positive periodic solution of (1.1), let \(x(n)=e^{y(n)}\), then from (1.1) we have
with the initial condition
where \(\alpha(n)>1\), \(\beta(n)>0\) are Nperiodic sequences, Ï„ is a positive integer. Define
Let \(l_{N}\subset l\) denote the subspace of all Nperiodic sequences equipped with the norm
where \(I_{N}=\{0,1,2,\ldots,N1\}\). Then \(l_{N}\) is a Banach space. Let
Then \(l_{N}^{0}\) and \(l_{N}^{c}\) are both closed linear subspaces of \(l_{N}\), and \(l_{N}=l_{N}^{0}\oplus l_{N}^{c}\), \(\operatorname{dim}l_{N}^{c}=1\). Take \(X=Y=l_{N}\). Now, for \(y\in X\), \(n\in Z\), we define a linear operator
and a nonlinear operator
Then L is a bounded linear operator with \(\operatorname{Ker}L=l_{N}^{c}\) and \(\operatorname{Im}L=l_{N}^{0}\). So it follows that L is a Fredholm mapping of index zero. Define the continuous projectors P, Q,
and
Let
then
Since \(\operatorname{Im}L\subset Y\) and \(D(L)\cap\operatorname{Ker}P\subset X\), \(K_{p}\) is an embedding operator. Hence \(K_{p}\) is a completely operator in ImL. By the definitions of Q and N, one knows that \(QN(\bar{\Omega})\) is bounded on \(\bar{\Omega}\). Hence the nonlinear operator N is Lcompact on \(\bar{\Omega}\). We complete the proof in three steps.
Step 1. Let \(\Omega_{1}=\{y\in D(L)\subset X:Ly=\lambda Ny, \lambda\in(0,1)\}\). We show that \(\Omega_{1}\) is a bounded set. If \(\forall y\in\Omega_{1}\), then \(Ly=\lambda Ny\), i.e.,
\(\forall x\in\Omega_{1}\), summing on both sides of (2.2) from 0 to \(N1\) with respect to n, we have
Thus, from (2.3) and condition (H_{1}) we obtain
We claim that there exist a point \(k\in Z\) and a constant \(M_{1}>0\) such that
Otherwise, for any \(M_{1}>0\) and each \(n\in I_{N}\), one has
In view of assumption (H_{2}), we see that this contradicts (2.3). Hence (2.4) holds. Denote
Then
In a similar way, from (2.3) and assumption (H_{2}), there exist a point \(\xi_{2}\in I_{N}\) and a constant \(M_{2}>0\) such that
Therefore, it follows from LemmaÂ 2.2, (2.1), (2.5), and (2.6) that
and
Thus
Step 2. We will show that condition (2) in LemmaÂ 2.1 satisfies. Let
where \(A=\max\{M,D\}\). Obviously, condition (1) in LemmaÂ 2.1 satisfies. When \(\forall y\in\partial\Omega\cap \operatorname{Ker}L\), y is a constant with \(\y\= A\). Then we claim \(QNy\neq0\). In fact, if \(QNy=0\), then
which contradicts assumption (H_{2}) when \(\y\= A\).
Step 3. We will show that condition (3) in LemmaÂ 2.1 holds. Take the homotopy
We claim \(H(y,\mu)\neq0\) for all \(y\in\partial\Omega\cap \operatorname{Ker}L\). If this is not true, then
Since \(y\in\partial\Omega\cap \operatorname{Ker}L\), \(\mu\in[0,1]\), \(yH(y,\mu)>0\), one has \(H(y,\mu)\neq0\). By the degree theory,
From LemmaÂ 2.1, we know that \(Lx(n)=Nx(n)\) has at least one periodic solution in \(\bar{\Omega}\). That is, (1.1) has at least one positive Nperiodic solution. The proof is completed.â€ƒâ–¡
Corollary 2.1
Let \(\mathcal {F}(n,y)=\ln(1+\beta(n)e^{y(n\tau)})+\ln\alpha(n)\), \(n\in Z\), \(y\in R\). There exist constants \(D_{1}\) and \(D_{2}\) such that

(i)
\(y\mathcal{F}(n,y)>0\) for \(y>D_{1}\), \(n\in I_{N}\),

(ii)
one of the following two conditions holds:

(a)
\(y\mathcal{F}(n,y)\leq D_{2}\) for \(y\geq D_{1}\), \(n\in I_{N}\),

(b)
\(y\mathcal{F}(n,y)\geq D_{2}\) for \(y\leqD_{1}\), \(n\in I_{N}\).

(a)
Then (2.2) has at least one periodic solution.
Remark 2.1
The initial condition \(x(n)=\phi(n)>0\), \(n\in[\tau,0]_{Z}\), of (1.1) assures that the initial condition \(y(n)=\ln x(n,)\), \(n\in[\tau,0]_{Z}\), of (2.1) is meaningful.
Remark 2.2
In [35], Li and Huo studied a class of abstract delay difference equation and obtained the existence of positive periodic solutions. In the present paper, based on the work of [35], we investigate a concrete model and obtain the population dynamics of the model.
3 Global exponential stability of periodic solution
In this section, we establish some results for exponential stability of the Nperiodic solution of (1.1).
Definition 3.1
The periodic solution of (2.1), \(y^{*}(n)\) is globally exponentially stable if there exist constants \(\mu>1\) and \(L>0\) such that
where \(y(n)\) is a solution of (1.1) with the initial value condition \(y(n)=\psi(n)\), \(\psi^{*}\) is the initial value of \(y^{*}(n)\), and
Theorem 3.1
Under the conditions of TheoremÂ 2.1, assume further that:

(i)
$$\alpha^{+}+\lambda_{0}^{\tau+1}L\beta^{+}< 1, $$
where \(\alpha^{+}=\max\{\alpha(n),n\in Z\}\), \(\beta^{+}=\max\{\beta(n),n\in Z\}\), \(\lambda_{0}>1\) with
$$\lambda_{0}\bigl(\alpha^{+}+\lambda_{0}^{\tau+1}L\beta^{+} \bigr)< 1. $$ 
(ii)
If \(f(x,y)=xy\), \(x,y\in R\), then \(f(x,y)f(x^{*},y*)\leq Lyy^{*}\), where L is a positive constant.
Then system (1.1) has a Nperiodic solution \(x^{*}(n)\), and there exists \(\lambda_{0}>1\) such that
Proof
By (1.1), we have
For \(\lambda\in R\), define the function
From condition (i), we have \(F(1)>0\). So, there exists some constant \(\lambda_{0}>1\) such that \(F(\lambda_{0})>0\). Then by (3.1), we have
Define \(u(n)=\lambda_{0}^{n}x(n)x^{*}(n)\), \(n\in[\tau,+\infty)_{Z}\), then by (3.2) and condition (ii), we have
Assume that \(K=\max_{s\in[\tau,0]_{Z}}\psi(s)\psi^{*}(s)\). Then we claim that
Otherwise, there exists \(n_{0}\in Z^{+}\) such that
By (3.3) and condition (i) we have
which is a contradiction. So \(u(n)\leq K\), \(n\in Z^{+}\). Therefore,
The proof is completed.â€ƒâ–¡
Remark 3.1
Because (3.2) contains the nonlinear term \(x^{*}(n+1)x^{*}(n\tau)x(n+1)x(n\tau)\), which results in great difficulty in obtaining exponential stability, we add condition (ii).
Remark 3.2
In general, the Lyapunov functional method is crucial for studying stability problems. In the present paper, due to the stronger nonlinearity of (1.1), the Lyapunov functional method is not valid. We overcome these difficulties by constructing a novel functional, which is different for the corresponding ones of past work.
4 Numerical simulations
This section presents an example to demonstrate the validity of our theoretical results:
where \(\alpha(n)=\frac{5\cos n\pi}{3}>1\), \(\beta(n)=0.2\). We can choose a proper parameter Ï„ and L such that all conditions of TheoremÂ 3.1 hold. So there exists a periodic solution for (4.1) which is globally exponentially stable. The corresponding numerical simulations are presented in FigureÂ 1 with different initial conditions.
In this paper, we discussed the existence and stability of positive periodic solutions for (1.1). First, the sufficient conditions that ensure the existence of a positive periodic solution were obtained by using the continuation theorem and some inequality techniques. Then a nonLyapunov method was used to establish the criteria for the global exponential stability of the periodic solution. Finally, a numerical example was presented to demonstrate the effectiveness of our theoretical results. The proposed criteria in this paper are easy to verify. The proposed analysis method is also easy to extend to the case of other differential equations.
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Acknowledgements
This paper is supported by the Postdoctoral Foundation of Jiangsu (1402113C) and the Postdoctoral Foundation of China (2014M561716).
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Du, B., Zhu, S. Existence and asymptotic behavior results of positive periodic solutions for discretetime logistic model. Adv Differ Equ 2015, 181 (2015). https://doi.org/10.1186/s1366201505272
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DOI: https://doi.org/10.1186/s1366201505272