Mathematical ecological system has become one of the most important topics in the study of modern applied mathematics. During the last decade, Allee effects have received much attention from researchers, largely because of their potential role in extinctions of already endangered, rare or dramatically declining species. The Allee effect refers to a decrease in population growth rate at low population densities. There are several mechanisms that create Allee effects in populations; see, for example, [1-6].

Mathematical component of the available literature deals with differential equations or difference equations. Notice that, in the real world, there are many species whose developing processes are both continuous and discrete. Hence, using the only differential equation or difference equation can't accurately describe the law of their developments [7, 8]. Therefore, there is a need to establish correspondent dynamic models on new time scales.

A time scale is a nonempty arbitrary closed subset of reals. The theory of time scales was first introduced by Hilger in [9], in order to unify continuous and discrete analysis. The study of dynamic equations on time scales can combine the continuous and discrete situations, it unifies not only differential and difference equations, but also some other problems such as a mix of stop-start and continuous behaviors.

Although seasonality is known to have considerable impact on the species dynamics, to our knowledge there are few papers discussed the dynamics of a renewable resource subjected to Allee effects in a seasonally varying environment. Moreover, ecosystems are often disturbed by outside continuous forces in the real world, the assumption of almost periodicity of the parameters is a way of incorporating the almost periodicity of a temporally nonuniform environment with incommensurable periods (nonintegral multiples). In this paper, we introduce seasonality into the resource dynamic equation by assuming the involved coefficients to be almost periodic.

Motivated by the above statements, in the present paper, we shall study the following equation representing dynamics of a renewable resource $x$, that is subjected to Allee effects on time scales

$ x^\Delta(t)=a(t)x(t)(x(t)-b(t))(c(t)-x(t)), $

(1.1)

where $t\in\mathbb{T}$, $\mathbb{T}$ is an almost time scale; $a(t)$ represents time dependent intrinsic growth rate of the resource; the nonnegative functions $c(t)$ and $b(t)$ stand for seasonal dependent carrying capacity and threshold function of the species respectively satisfying $0 < b(t) < c(t)$. All the coefficients $a(t), b(t), c(t)$ are continuous, almost periodic functions. For the ecological justification of (1.1), one can refer to [5, 6].

For convenience, we introduce the notation

$ f^u=\sup\limits_{t\in \mathbb{T}} f(t), \ f^l=\inf\limits_{t\in \mathbb{T}} f(t), $

where $f$ is a positive and bounded function. Throughout this paper, we assume that the coefficients of equation (1.1) satisfy

$ \min\{a^l, b^l, c^l\}>0, \ \max\{a^u, b^u, c^u\} < +\infty. $

This is the first paper to study an almost equation representing dynamics of a renewable resource subjected to Allee effects on time scales. The aim of this paper is, by using exponential dichotomy of linear system and contraction mapping fixed point theorem, to obtain sufficient conditions for the existence of unique positive almost periodic solution of (1.1). We also investigate global exponential stability of the unique almost periodic solution by means of Lyapunov function.

Let us first recall some basic definitions which can be found in [10].

Let $\mathbb{T}$ be a nonempty closed subset (time scale) of $\mathbb{R}$. The forward and backward jump operators $\sigma, \rho : \mathbb{T}\rightarrow \mathbb{T}$ and the graininess $\mu$ : $\mathbb{T}\rightarrow \mathbb{R}^ +$ are defined, respectively, by

$ \sigma(t)=\inf\{s\in \mathbb{T}:s>t\}, \quad \rho(t)=\sup\{s\in \mathbb{T}:s < t\}, \quad \mu(t)=\sigma(t)-t. $

A point $t\in \mathbb{T}$ is called left-dense if $t>\inf{\mathbb{T}}$ and $\rho(t)=t$, left-scattered if $\rho(t) < t$, right-dense if $t < \sup{\mathbb{T}}$ and $\sigma(t)=t$, and right-scattered if $\sigma(t)>t$. If $\mathbb{T}$ has a left-scattered maximum $m$, then $\mathbb{T}^k=\mathbb{T}\backslash\{m\}$; otherwise $\mathbb{T}^k=\mathbb{T}$. If $\mathbb{T}$ has a right-scattered minimum $m$, then $\mathbb{T}_k=\mathbb{T}\backslash\{m\}$; otherwise $\mathbb{T}_k=\mathbb{T}$.

A function $f : \mathbb{T}\rightarrow \mathbb{R}$ is right-dense continuous provided it is continuous at right-dense point in $\mathbb{T}$ and its left-side limits exist at left-dense points in $\mathbb{T}$.

A function $p: \mathbb{T}\rightarrow \mathbb{R}$ is called regressive provided $1+\mu(t)p(t)\neq 0$ for all $t\in \mathbb{T}^k$. The set of all regressive and rd-continuous functions $p:\mathbb{T} \rightarrow \mathbb{R}$ will be denoted by $ \mathcal{R}=\mathcal{R}(\mathbb{T}, \mathbb{R})$. Define the set $\mathcal{R}^+=\mathcal{R}^+(\mathbb{T}, \mathbb{R})=\{p\in \mathcal{R}:1+\mu(t)p(t)>0, \forall \, t\in \mathbb{T}\}$.

If $r$ is a regressive function, then the generalized exponential function $e_r$ is defined by

$ e_r(t, s)=\exp\bigg\{\int_s^t \xi_{\mu(\tau)}(r(\tau))\Delta \tau\bigg\} $

for all $s, t\in \mathbb{T}$, with the cylinder transformation

$ \xi_h(z)=\left\{\begin{array}{ll} \frac{{\rm Log}(1+hz)}{h}, \ {\rm if}\ h\neq 0, \\ z, \ \ \ \ \ \ \ \ \ \ \ {\rm if}\ h=0. \end{array}\right. $

Lemma 2.1 (see [10]) If $p\in\mathcal{R}$ and $a, b, c \in \mathbb{T}$, then

$ \int_a^bp(t)e_p(c, \sigma(t))\Delta t=e_p(c, a)-e_p(c, b). $

Definition 2.1 [11] A time scale $\mathbb{T}$ is called an almost periodic time scale if

$ \Pi=\{\tau\in \mathbb{R}: t\pm \tau\in \mathbb{T}, \forall t\in \mathbb{T}\}\neq\{0\}. $

Definition 2.2 [12] Let $x\in \mathbb{R}^n$, and $A(t)$ be an $n\times n\ rd$-continuous matrix on $\mathbb{T}$, the linear system

$ x^\Delta (t)=A(t)x(t), \ t\in\mathbb{T} $

(2.1)

is said to admit an exponential dichotomy on $\mathbb{T}$ if there exist positive constant $k, \alpha$, projection $P$ and the fundamental solution matrix $X(t)$ of (2.1), satisfying

$ \begin{align} & \left| X\left( t \right) \right.P{{X}^{-1}}\left( \sigma \left( s \right) \right)\left| _{0} \right.\le k{{e}_{\theta \alpha }}\left( t,\sigma \left( s \right) \right),\ s,t\in \mathbb{T},t\ge \sigma \left( s \right), \\ & \left| X\left( t \right) \right.\left( 1-P \right){{X}^{-1}}\left( \sigma \left( s \right) \right)\left| _{0} \right.\le k{{e}_{\theta \alpha }}\left( \sigma \left( s \right),t \right),\ s,t\in \mathbb{T},t\le \sigma \left( s \right), \\ \end{align} $

where $|\cdot|_0$ is a matrix norm on $\mathbb{T}$$\big(A=(a_{ij})_{n\times m}$, then$|A|_0=\big(\sum\limits_{i=1}^n\sum\limits_{j=1}^m|a_{ij}|^2\big)^{\frac{1}{2}}\big)$.

Considering the following almost periodic system

$ x^\Delta (t)=A(t)x(t)+f(t), \ t\in\mathbb{T}, $

(2.2)

where $A(t)$ is an almost periodic matrix function, $f(t)$ is an almost periodic vector function.

Lemma 2.2 (see [12]) If the linear system (2.1) admits exponential dichotomy, then system (2.2) has a unique almost periodic solutions $x(t)$ as follows

$ x(t)=\int^t_{-\infty}X(t)PX^{-1}(\sigma(s))f(s)\Delta s-\int_t^{+\infty}X(t)(I-P)X^{-1}(\sigma(s))f(s)\Delta s, $

(2.3)

where $X(t)$ is the fundamental solution matrix of (2.1).

Definition 2.3 The almost periodic solution $x^*$ of equation(1.1) is said to be exponentially stable, if there exist positive constants $\alpha>0$, $\alpha\in\mathcal{R}^+$ and $N=N(t_0)\geq1$ such that for any solution $x$ of equation (1.1) satisfying

$ \|x-x^*\|\leq N |x(t_0)-x^*(t_0)| e_{-\alpha}(t, t_0), t\in [t_0, +\infty)_\mathbb{T}. $

Clearly, the trivial solution $x(t)\equiv 0$ is an almost periodic solution of (1.1). Since the study deals with resource dynamics, we are interested in the existence of positive almost periodic solutions of the considered equation.

First, we make the following assumptions:

$(\text{H}1)$$-abc\in\mathcal{R}^+$;

$(\text{H}2)$ there exist two positive constants $L_1>L_2>0$, such that

$ \frac{4a^u(b^u+c^u)^3}{27\inf\limits_{t\in\mathbb{T}}\{abc\}}\leq L_1\leq b^l+c^l;\ \frac{2}{3}(b^l+c^l)\leq L_2\leq \frac{a^lL_1^2(b^l+c^l-L_1)}{\sup\limits_{t\in\mathbb{T}}\{abc\}}; $

$(\text{H}3)$$\frac{\lambda a^u}{\inf\limits_{t\in\mathbb{T}}\{abc\}} < 1$, where $\lambda=\max\{|2(b^l+c^l)L_2-3L_1^2|, |2(b^u+c^u)L_1-3L_2^2|\}.$

Theorem 3.1 Assume that $(\text{H}1)-(\text{H}3)$ hold, then equation (1.1) has a unique almost periodic solution.

Proof Let $Z=\{z|z \in C(\mathbb{T}, \mathbb{R}), z\;\text{is}\;\text{an}\;\text{almost}\;\text{periodic}\;\text{function}\}$ with the norm $\|z\|=\sup\limits_{t\in \mathbb{T}} |z(t)|$, then $Z$ is a Banach space.

Equation (1.1) can be written as

$ x^\Delta(t)=-a(t)b(t)c(t)x(t)+a(t)x^2(t)[b(t)+c(t)-x(t)], t\in\mathbb{T}. $

For $z\in Z$, we consider the almost periodic solution $x_z(t)$ of the nonlinear almost periodic differential equation

$ x^\Delta(t)=-a(t)b(t)c(t)x(t)+a(t)z^2(t)[b(t)+c(t)-z(t)], t\in\mathbb{T}. $

(3.1)

Since $\inf\limits_{t\in \mathbb{T}} [a(t)b(t)c(t)]\geq a^lb^lc^l>0$, from Lemma 2.15 [12] and $(\text{H}1)$, the linear equation

$ x^\Delta(t)=-a(t)b(t)c(t)x(t) $

admits exponential dichotomy on $\mathbb{T}$.

Hence, by Lemma 2.2, equation (3.1) has exactly one almost periodic solution

$ x_z(t)=\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)z^2(s)[b(s)+c(s)-z(s)]\Delta s. $

Define an operator $\Phi:Z\rightarrow Z$,

$ (\Phi z)(t)=\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)z^2(s)[b(s)+c(s)-z(s)]\Delta s. $

(3.2)

Obviously, $z$ is an almost periodic solution of equation (1.1) if and only if $z$ is the fixed point of operator $\Phi$.

Let $\Omega=\{z|z\in Z, L_2\leq z(t)\leq L_1, t\in \mathbb{T}\}$.

Now, we prove that $\Phi\Omega\subset\Omega$. In fact, $\forall z\in\Omega$, we have

$ \begin{align} & \left( \Phi z \right)\left( t \right)=\int_{-\infty }^{t}{{{e}_{-abc}}}\left( t,\sigma \left( s \right) \right)a\left( s \right){{z}^{2}}\left( s \right)\left[ b\left( s \right)+c\left( s \right)-z\left( s \right) \right]\Delta s \\ & \ \ \ \ \ \ \ \ \ \ \ \le \frac{{{a}^{u}}}{2}\int_{-\infty }^{t}{{{e}_{-abc}}}\left( t,\sigma \left( s \right) \right)z\left( s \right)z\left( s \right)\left[ 2\left( {{b}^{u}}+{{c}^{u}} \right)-2z\left( s \right) \right]\Delta s \\ & \ \ \ \ \ \ \ \ \ \ \le \frac{4{{a}^{u}}{{\left( {{b}^{u}}+{{c}^{u}} \right)}^{3}}}{27}\int_{-\infty }^{t}{{{e}_{-abc}}}\left( t,\sigma \left( s \right) \right)\Delta s \\ & \ \ \ \ \ \ \ \ \ \ \ \le \frac{4{{a}^{u}}{{\left( {{b}^{u}}+{{c}^{u}} \right)}^{3}}}{27{\inf\limits_{t\in \mathbb{T}}}\,\left\{ abc \right\}}\le {{L}_{1}}. \\ \end{align} $

(3.3)

On the other hand, we have

$ \begin{align} & \left( \Phi z \right)\left( t \right)=\int_{-\infty }^{t}{{{e}_{-abc}}}\left( t,\sigma \left( s \right) \right)a\left( s \right){{z}^{2}}\left( s \right)\left[ b\left( s \right)+c\left( s \right)-z\left( s \right) \right]\Delta s \\ & \ \ \ \ \ \ \ \ \ \ \ \ge {{a}^{l}}\int_{-\infty }^{t}{{{e}_{-abc}}}\left( t,\sigma \left( s \right) \right){{z}^{2}}\left( s \right)\left[ {{b}^{l}}+{{c}^{l}}-z\left( s \right) \right]\Delta s. \\ \end{align} $

(3.4)

Note that

$ \frac{2}{3}(b^l+c^l)\leq L_2\leq z(t)\leq L_1, t\in\mathbb{T}. $

Since the function $g(u)=u^2[b^l+c^l-u]$ is increasing on $u\in[0, \frac{2}{3}(b^l+c^l)]$ and decreasing on $u\in[\frac{2}{3}(b^l+c^l), +\infty]$, then we have $g(z(t))\geq g(L_1)$ for $t\in\mathbb{T}$, that is

$ z^2(t)[b^l+c^l-z(t)]\geq L_1^2(b^l+c^l-L_1), t\in\mathbb{T}. $

Thus by (3.4), we obtain

$ \begin{align} & \left( \Phi z \right)\left( t \right)\ge {{a}^{l}}L_{1}^{2}\left( {{b}^{l}}+{{c}^{l}}-{{L}_{1}} \right)\int_{-\infty }^{t}{{{e}_{-abc}}}\left( t,\sigma \left( s \right) \right)\Delta s \\ & \ \ \ \ \ \ \ \ \ \ \ \ge \frac{{{a}^{l}}L_{1}^{2}\left( {{b}^{l}}+{{c}^{l}}-{{L}_{1}} \right)}{\sup \limits_{t\in \mathbb{T}}\,\left\{ abc \right\}}\ge {{L}_{2}}. \\ \end{align} $

(3.5)

It follows from (3.3) and (3.5) that

$ L_2\leq (\Phi z)(t)\leq L_1. $

(3.6)

In addition, for $z\in\Omega$, equation (3.1) has exactly one almost periodic solution

$ x_z(t)=\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)z^2(s)[b(s)+c(s)-z(s)]\Delta s. $

Since $x_z(t)$ is almost periodic, then $(\Phi z)(t)$ is almost periodic. This, together with (3.6), implies $\Phi z\in \Omega$. So we have $\Phi\Omega\subset\Omega$.

Next, we prove that $\Phi$ is a contraction mapping on $\Omega$. In fact, in view of $(\text{H}1)-(\text{H}3)$, for any $z_1, z_2\in \Omega$,

$ \|\Phi z_1-\Phi z_2\| \\ =\sup\limits_{t\in\mathbb{T}}|(\Phi z_1)(t)-(\Phi z_2)(t)|\\ =\sup\limits_{t\in\mathbb{T}} \bigg|\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)z_1^2(s)[b(s)+c(s)-z_1(s)]\Delta s\\ -\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)z_2^2(s)[b(s)+c(s)-z_2(s)]\Delta s\bigg|\\ =\sup\limits_{t\in\mathbb{T}} \bigg\{\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)|(b(s)+c(s))(z_1^2(s)-z_2^2(s))-(z_1^3(s)-z_2^3(s))|\Delta s\bigg\}\\ =\sup\limits_{t\in\mathbb{T}} \bigg\{\int_{-\infty}^te_{-abc}(t, \sigma(s))a(s)|z_1(s)-z_2(s)|\\ |(b(s)+c(s))(z_1(s)+z_2(s))-(z_1^2(s)+z_1(s)z_2(s)+z_2^2(s))|\Delta s\bigg\}\\ \leq\lambda a^u\sup\limits_{t\in\mathbb{T}} \bigg\{\int_{-\infty}^te_{-abc}(t, \sigma(s))\Delta s\bigg\}\|z_1-z_2\|\\ \leq\frac{\lambda a^u}{\inf\limits_{t\in\mathbb{T}}\{abc\}}\|z_1-z_2\|, $

where $\lambda=\max\{|2(b^l+c^l)L_2-3L_1^2|, |2(b^u+c^u)L_1-3L_2^2|\}.$

Since $\frac{\lambda a^u}{\inf\limits_{t\in\mathbb{T}}\{abc\}} < 1$, this implies that $\Phi$ is a contraction mapping. Thus, $\Phi$ has exactly one fixed point $z^*$ in $\Omega$ such that $\Phi (z^*)=z^*$. Otherwise, it is easy to verify that $z^*$ satisfies equation (1.1). This means that equation (1.1) has a unique almost periodic solution in $z^*(t)$, and $L_2\leq z^*(t)\leq L_1$. This completes the proof.

Next, we shall construct a suitable Lyapunov functional to study the global exponential stability of the almost periodic solution of (1.1).

Theorem 3.2

Assume that $(\text{H}1)-(\text{H}3)$ hold. Suppose further that

$(\text{H}4)$ $\gamma=4(b^lc^l+L_2^2)-(b^u+c^u+L_1)^2>0$;

$(\text{H}5)$ Let $0 < \alpha < \frac{a^l\gamma}{4}$, and $-\alpha\in\mathcal{R}^+$;

then equation (1.1) has a unique globally exponentially stable almost periodic solution.

Proof According to Theorem 3.1, we know that (1.1) has an almost periodic solution $x^*(t)$, and $L_2\leq x^*(t)\leq L_1$. Suppose that $x(t)$ is arbitrary solution of (1.1) with initial condition $x(t_0)>0, t_0\in\mathbb{T}$. Now we prove $x^*(t)$ is globally exponentially stable.

Let $V(t)=|x(t)-x^*(t)|$. Calculating the upper right derivatives of $V(t)$ along the solution of equation (1.1), from $(\text{H}4)$ and $(\text{H}5)$, then

$ D^+V^\Delta(t) ={\rm sgn}(x(t)-x^*(t))(x^\Delta(t)-x^{*\Delta}(t))\nonumber\\ ={\rm sgn}(x(t)-x^*(t))[-a(t)b(t)c(t)+a(t)(b(t)+c(t))(x(t)+x^*(t))\nonumber\\ -a(t)(x^2(t)+x(t)x^*(t)+x^{*2}(t))](x(t)-x^*(t))\nonumber\\ =-a(t)[b(t)c(t)-(b(t)+c(t))(x(t)+x^*(t))\nonumber\\ +(x^2(t)+x(t)x^*(t)+x^{*2}(t))]|x(t)-x^*(t)|\nonumber\\ =-a(t)\bigg[\bigg(x(t)-\frac{b(t)+c(t)-x^*(t)}{2}\bigg)^2-\frac{(b(t)+c(t)+x^*(t))^2}{4}\nonumber\\ +b(t)c(t)+x^{*2}(t)\bigg]|x(t)-x^*(t)|\nonumber\\ \leq-a^l\bigg[b^lc^l+L_2^2-\frac{(b^u+c^u+L_1)^2}{4}\bigg]|x(t)-x^*(t)|\nonumber\\ \leq-\alpha V(t). $

(3.7)

Integrating (3.7) from $t_0$ to $t$, we get $V(t)\leq e_{-\alpha}(t, t_0)V(t_0), $ that is

$ |x(t)-x^*(t)|\leq e_{-\alpha}(t, t_0)|x(t_0)-x^*(t_0)|, $

then $\|x-x^*\|\leq|x(t_0)-x^*(t_0)|e_{-\alpha}(t, t_0). $

From Definition 2.3, the almost periodic solution $x^*$ of (1.1) is globally exponentially stable. This completes the proof.