Almost periodic solutions of a discrete Lotka-Volterra model via exponential dichotomy theory

1.   Introduction

In recent years, using a wide variety of methods, a considerable amount of literature has been produced regarding the stability and existence of almost periodic solutions of differential equations ^{[2,3,4,5,6,7,8,9,10,11,12,13]}. Almost periodic solutions of most differential equations are vector-valued functions defined on the set of real numbers ${\mathbb{R}}$, but the concept of almost periodic solution, makes more sense on any additive group except ${\mathbb{R}}$, in mathematical modeling, discrete time models are preferred over the continuous time models. As it is directly applicable to various fields (economics, populations dynamics, species interactions, mathematical biology, ecology etc.), the theory of difference equation is much more substantiated than the corresponding theory of differential equation (see ^{[14,15,16,17,18,19,20]}).

Gopalsamy and Mohamad ^[21] established the following criteria for the existence of globally attractive positive almost periodic solutions for discrete Lotka-Volterra systems:

Xia et al. ^[22] studied the relationship between the global quasi-uniform asymptotic stability of a difference system and the existence of almost periodic solutions. Moreover, they applied their results to a discrete Lotka-Volterra system as follows:

Meng ^[23], Niu ^[24] and Xue ^[25] et al. obtained the results of global stability, global attraction and uniform stability of almost periodic solutions of discrete Lotka-Volterra systems by constructing Lyapunov functions. To the best of our knowledge, there is no paper employing exponential dichotomy theory (for difference system) to study the existence of almost periodic solutions of discrete Lotka- Volterra systems. Different from the above mentioned works ^{[21,22,23,24,25]}, in this paper, by using the exponential dichotomy theory for difference equations and Banach fixed point theorem, we prove the existence and uniqueness of solutions for the discrete almost periodic Lotka-Volterra model, formulated as follows:

where $r_{i}(n), a_{ij}(n), b_{ij}, \tau(n)$, $i, j = 1, \ldots, N$ are the almost periodic sequences.

The rest of this paper is organized as follows. In the second section, we introduce some preconditions for the existence of almost periodic solutions. In the third section, we prove the properties of the exponential dichotomy for the difference equationsand existence of exponential stability of almost periodic solutions for the Lotka-Volterra model. In the last section, the validity of the results is proved by numerical simulations.

2.   Preliminaries

For a bounded sequence $f$ defined on ${\mathbb{Z}}$, we define $f^{-}$ and $f^{+}$ as follows:

Taking $Z[a, b] = \{a, a+1, \ldots, b-1, b\}$, where $a, b\in {\mathbb{Z}}$. Similarly, we denote $Z[a, +\infty] = \{a, a+1, a+2, \ldots\}$.

Definition 2.1. ^[2,3,4] If for $\epsilon > 0$, there is a constant $l(\epsilon) > 0$ such that in any interval of length $l(\epsilon) > 0$ there exists a number $\bar{\tau}\in{\mathbb{Z}}$ such that the inequality

is satisfied for all $n\in{\mathbb{Z}}$. Then a sequence $x(n)$ is called to be almost periodic sequence.

Lemma 2.1. ^[2,3,4]Let $x: {\mathbb{Z}}\rightarrow {\mathbb{R}}$ and $y: {\mathbb{Z}}\rightarrow {\mathbb{R}}$. The following holds:

(a) If $x(n)$ is an almost periodic sequence, then it is bounded.

(b) If $x(n), y(n)$ are almost periodic sequences, $x(n)+y(n)$ and $x(n)\cdot y(n)$ arealmost periodic.

(c) If $x(n)$ is an almost periodic sequence, then $X(n)$ is almost periodicif and only if $X(n)$ is bounded on ${\mathbb{Z}}$ where

(d) If $x(n)$ is an almost periodic sequence and $X(\cdot)$ is defined on thevalue field of $x(n)$, then $X\circ x$ is almost periodic.

Consider the following almost periodic difference system:

where $A:{\mathbb{Z}}\rightarrow {\mathbb{Z}}^{m\times m}$ is an almost periodic matrix sequence and $f:{\mathbb{Z}}\rightarrow {\mathbb{Z}}^{m}$ is an almost periodic vector sequence. Its linear system is

Definition 2.2. ^[26] Suppose that $\Phi(n)$ is a fundamental matrix of the difference system (2.2). If there exists a projection $P$ such that $P^{2} = P$ and two positive constants $K, v,$ such that

then the difference system (2.2) is said to admit an exponential dichotomy.

Lemma 2.2. ^[26]Let $r(n) > 0$ be an almost periodic sequence on ${\mathbb{Z}}$ and $\underset{n\in{\mathbb{Z}}}\inf r(n) > 0$, then the linear system

admits an exponential dichotomy on ${\mathbb{Z}}.$

Lemma 2.3. ^[27] [Chapter 7, Exercise 4]Suppose that system (2.2) admits an exponential dichotomy satisfying definition 2.Then system (2.2) does not admit a non-trivial bounded solution. i.eIf there is a bounded solution, it's only a zero solution.

Lemma 2.4. ^[27] [Chapter 7, Theorem 7.6.5]Suppose that $\Phi(n)$ is the normal fundamental matrix of the linear system (2.2).If the system (2.2) admits exponential dichotomy with the projection $P$ and the constants$K$, $v$, $M$ with $\left\|f(n)\right\|\leq M$. Then the system (2.1) has a unique bounded solution which canbe uniquely expressed by

3.   Main results

We consider the discrete almost periodic Lotka-Volterra system as follows:

where $r_{i}(n), a_{ij}(n)$, $b_{ij}(n)$ and $\tau(n)$ are positive almost periodic sequences defined on ${\mathbb{Z}}$. Throughout this paper, we use the notations $\frac{e^{r_{i}^{+}}}{a_{ii}^{-}} = B_{i}$ and $B = \max\{B_{i}\}$. Further, we always assume that system (3.1) satisfies the following conditions:

$(H_{1})\quad r_{i}^{-} > \sum^{N}_{j\neq i}a_{ij}^{+}B_{j}+\sum^{N}_{j = 1}b_{ij}^{+}B_{j}$, $\frac{1}{a_{ii}^{+}}(r_{i}^{-}-\sum^{N}_{j\neq i}a_{ij}^{+}B_{j} -\sum^{N}_{j = 1}b_{ij}^{+}B_{j}) = D_{i}.$

Set $D = \min\limits_{1\leq n\leq n}\{D_{i}\}$. We further suppose that Eq (3.3) satisfies :

$(H_{2}) \quad\frac{(D+B)\sum^{N}_{j = 1}(a_{ij}^{+}+b_{ij}^{+})}{D^{2}r_{i}^{-}} < 1.$

$(H_{3}) \quad r_{i}^{-} > 1+\frac{(D+B)\sum^{N}_{j = 1}(a_{ij}^{+}+b_{ij}^{+})}{D^{2}}.$

Let $u_{i}(t) = \frac{1}{x_{i}(t)}$, $\dot{u_{i}}(t) = \frac{-1}{x_{i}^{2}(t)}\dot{x_{i}}(t)$, $\dot{x_{i}}(t) = \dot{u_{i}}(t)\frac{-1}{u_{i}^{2}(t)}$, then we have,

then we can get the difference equation as follow:

Theorem 3.1. If $(H_1)-(H_3)$ hold, then there exists a unique positive almost periodic solution $u_{i}(n)$ of (3.3), i.e, system (3.3) has a unique almost periodic solution $x_{i}(n)$.

Theorem 3.2. Let $x^{*}(n) = \{x^{*}_{1}(n), \ldots, x^{*}_{N}(n)\}$be a positive almost periodic solution of (3.1)in the $\mathcal{B}$. If $(H_1)-(H_3)$ hold, then the solution $x(n) = \{x_{1}(n), \ldots, x_{N}(n)\}$of (3.1) converges exponentially to $x^{*}(n)$ as $n\rightarrow \infty$.

4.   Proof of main results

4.1.   Existence of bounded solution

We consider the almost periodic Lotka-Volterra system as follows:

where $x_{i}(t)$ denotes the density of prey species $x_{i}$ at time $t$; $r_{i}(t), a_{ij}(t)$, $b_{ij}(t)$ and $\tau(t)$ are positive continuous almost periodic functions defined on $t\in (-\infty, +\infty)$.

For the purpose of convenience, we should use piecewise constant variable differential equations to obtain the discrete model. Assume that the average growth rate changes over regular time intervals in (4.1), and then we can incorporate this aspect in (4.1) and obtain the following semi-discretization modified system (4.1):

where $t\neq\ldots-2, -1, 0, 1, 2, \ldots$ and $[t]$ denotes the integer part $t$, $t\in(-\infty, +\infty).$

Integrating over $n\leq t < n+1, n = \ldots, -2, -1, 0, 1, 2, \ldots,$ we have

Let $t\rightarrow n+1$, then lead us to

where $r_{i}(n), a_{ij}(n)$ and $b_{ij}(n)$ are positive almost periodic sequences defined on ${\mathbb{Z}}$.

Proposition 4.1. For every solution $\{(x_{1}(n), \ldots, x_{N}(n))^{T}\}$ of (4.3), we have

${\it Proof}$: Obviously, when $n\geq n_{0}$ $x_{i}(n) > 0$. Suppose that there exists a $l_{0}\in Z[n_{0}, +\infty)$, such that $x_{i}(l_{0}+1)\geq x_{i}(l_{0})$, then

then we get,

It follows that

By the fact that $\underset{x\in{\mathbb{R}}}\max (x\exp(a-bx)) = \frac{\exp(a-1)}{b}$, for $a, b > 0.$ Hence,

We claim that $x_{i}(n)\leq B_{i}$, for $n\geq l_{0}$. By way of contradiction, we assume that there exists a $q_{0}\geq l_{0}$, such that $x_{i}(q_{0}) > B_{i}$, then, $q_{0}\geq l_{0}+2$. Let $x_{i}(\tilde{q}_{0}) > B_{i}$, where $\tilde{q}_{0}\geq l_{0}+2$ is the smallest integer, then, $x_{i}(\tilde{q}_{0}-1) < x_{i}(\tilde{q}_{0})$. The above argument produces that $x_{i}(\tilde{q}_{0})\leq B_{i}$, which is contradictory. The proof of our claim is complete.

Now, considering that $x_{i}(n+1) < x_{i}(n)$ for all $n\in Z[n_{0}, +\infty)$. Then $\lim\limits_{n\rightarrow +\infty} x_{i}(n)$ exists, we denote $\lim\limits_{n\rightarrow +\infty} x_{i}(n) = \bar{x}_{i}$. We claim that $\bar{x}_{i}\leq \frac{e^{r_{i}^{+}}}{a_{ii}^{-}}$. Let us assume that $\bar{x}_{i} > \frac{e^{r_{i}^{+}}}{a_{ii}^{-}}$. Taking the limit of both sides in system (3.1), gives us

Hence,

However,

which is a contradiction. The proof is complete.

Proposition 4.2. Assume that $(H_1), (H_2)$ and $(H_3)$ hold. For every solution $\{(x_{1}(n), \ldots, x_{N}(n))^{T}\}$ of (3.1), we have

${\it Proof}$: By proposition 1, there exists a $n^{*}\in{\mathbb{N}}$, such that

Suppose that there exists a $k_{0}\geq n^{*}$, such that $x_{i}(k_{0}+1)\leq x_{i}(k_{0})$. Note that

When $n = k_{0}$, we have

Then we obtain,

Thus,

Then,

We claim that $x_{i}(n)\geq\tilde{x}_{i}$, for $n\geq k_{0}$. Suppose that there exists a $p_{0}\geq k_{0}$, such that $x_{i}(p_{0}) < \tilde{x}_{i}$, then $p_{0}\geq k_{0}+2$. Let $x_{i}(\tilde{p}_{0}) < \tilde{x}_{i}$, where $\tilde{p}_{0}\geq k_{0}+2$ is the smallest integer, then, $x_{i}(\tilde{p}_{0}-1)\geq x_{i}(\tilde{p}_{0})$. The above argument produces that $x_{i}(\tilde{p}_{0})\geq \tilde{x}_{i}$, which is a contradiction. The proof of our claim is complete.

Next, considering that $x_{i}(n+1) > x_{i}(n)$ for all $n\in Z[n_{0}, +\infty)$. Then $\lim\limits_{n\rightarrow +\infty} x_{i}(n)$ exists, we denoted $\lim\limits_{n\rightarrow +\infty} x_{i}(n) = \underline{x}_{i}$. we state that $\underline{x}_{i}\geq\frac{1}{a_{ii}^{+}}(r_{i}^{-}-\sum^{N}_{j\neq i}a_{ij}^{+}B_{j} -\sum^{N}_{j = 1}b_{ij}^{+}B_{j})\geq D_{i}$. We write $\underline{x}_{i} < \frac{1}{a_{ii}^{+}}(r_{i}^{-}-\sum^{N}_{j\neq i}a_{ij}^{+}B_{j} -\sum^{N}_{j = 1}b_{ij}^{+}B_{j})$. Taking the limit of both sides in system (3.1), we obtain

Hence, we get

which is contradictory. Since

then for $n\geq n_{0}$, we have $x_{i}(n)\geq D_{i}$.

Proof of Theorem 2.1: Let $\mathcal{B} = \{\varphi(n) = (\varphi_{1}(n), \varphi_{2}(n), \ldots, \varphi_{N}(n)) |\varphi_{i}(n): {\mathbb{Z}}\rightarrow {\mathbb{R}}$ is almost periodic sequence$\}.$ For any $\varphi\in\mathcal{B}$, we consider equation

Let $u^{\varphi}(n) = (u^{\varphi}_{1}(n), u^{\varphi}_{2}(n), \ldots, u^{\varphi}_{N}(n))^{T}$ be a solution of (4.6).

Since $r_{i}^{-} > 0$, then $\Delta u_{i}(n) = -r_{i}(n)u_{i}(n)$ admits an exponential dichotomy on ${\mathbb{Z}}.$ By Lemma 2.4, we have a bounded solution of the form

In view of $\Delta u_{i}(n) = -r_{i}(n)u_{i}(n)\Longrightarrow u_{i}(n+1) = (1-r_{i}(n))u_{i}(n),$ we have

We can get

According to Lemma 2.1, and using the almost periodicity of $\prod\limits^{n-1}_{s = m+1}(1-r_{i}(s))$, we deduce that $u_{i}^{\varphi}$ is also almost periodic.

We define a mapping $\mathfrak{T}: {\bf{B}}\rightarrow {\bf{B}}$ by setting

with the norm $\left\|\mathfrak{T}\right\| = \sup_{n\in{\mathbb{Z}}}\left|\mathfrak{T}\right|$.

Let $\varphi(n) = \left(\varphi_{1}(n), \varphi_{2}(n), \ldots, \varphi_{N}(n)\right), \psi(n) = \left(\psi_{1}(n), \psi_{2}(n), \ldots, \psi_{N}(n)\right)\in {\bf{B}}$, we have

We observe that

Similarly,

where $j = 1, \ldots, N$. Therefore,

Hence, by $(H_2)$, the mapping $\mathfrak{T}$ is contractive on ${\bf{B}}$. It follows that the mapping $\mathfrak{T}$ possesses a unique fixed point $\varphi^{*}\in{\bf{B}},$ such that $\mathfrak{T}\varphi^{*} = \varphi^{*}$. thus $\varphi^{*}$ is an almost periodic solution.

Proof of Theorem 2.2: Let $y(n) = u(n)-u^{*}(n)$, where $y_{i}(n) = \{y_{1}(n), \ldots, y_{N}(n)\}$. Then, for $i = 1, \ldots, N$,

Define a function $\Phi(\mu)$ by setting

Obviously, $\Phi$ is continuous on $[0, 1]$. Then, by $(H_3)$, we have

which implies that there exist a constant $\lambda\in(0, 1]$, such that

We consider the discrete Lyapunov functional $V(n) = |y(n)|e^{\lambda n}$. We calculate the difference of $V_{i}(n)$, for all $n\geq n_{0}$, $i = 1, \ldots, N$, it yields

Let $M_{i} = e^{\lambda n_{0}}(\max_{n\in{\mathbb{Z}}[n_{0}, \infty)}|x_{i}(n)-x_{i}^{*}(n)|)$, for all $n\geq n_{0}$. We claim that

where $M = \max{M_{i}}$. We prove our claim by contradiction. Let be a $n_{*} > n_{0}$, such that $V_{i}(n_{*}+1)\geq M_{i}$, and $V_{i}(n) < M$, $\forall n\in{\mathbb{Z}}[n_{0}, n_{*})$, which implies that $V_{i}(n_{*})-M_{i}\geq 0$ and $V_{i}(n)-M_{i} < 0$, $\forall n\in{\mathbb{Z}}[n_{0}, n_{*})$. However,

Thus,

which is contradictory. Hence (4.7) holds, then $|y(n)| < Me^{-\lambda n}$ for all $n\geq n_{0}$. This complete the proof.

5.   Examples

We consider two specific examples.

$Example$ 1: Consider the following system, when $N = 1$,

Discretization of the differential equations, we get

It is easy to verity that system (5.1) satisfies all the assumptions. Thus, system (5.1) admits an almost periodic solution that is quasi-uniformly asymptotically stable. Figure 1 shows the dynamics of system (5.1).

Therefore, the system satisfies all the assumptions. It follows that this system has an almost periodic solution which is exponentially stable.

$Example$ 2: Consider the following two dimensional predator-prey system, that is $N = 2$,

Discretization of the above differential equations, we obtain

It is easy to verity that system (5.2) satisfies all the assumptions. Thus, system (5.2) admits an almost periodic solution that is exponential convergence. Figure 2 shows the dynamics of system (5.2).

Therefore, the system satisfies all the assumptions. It follows that this system has an almost periodic solution which is exponentially stable.

6.   Conclusions

In this paper, by applying exponential dichotomy, we obtain the existence and exponential convergence of positive non-autonomous discrete Lotka-Volterra systems. The method is different from the previous works ^{[21,22,23,24,25]}. I believe that this result has potential applications in population dynamics.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant (No. 11931016).

Conflict of interest

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