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Uniform persistence and almost periodic solutions of a nonautonomous patch occupancy model
Advances in Difference Equations volume 2020, Article number: 143 (2020)
Abstract
In this paper, a nonlinear nonautonomous model in a rocky intertidal community is studied. The model is composed of two species in a rocky intertidal community and describes a patch occupancy with global dispersal of propagules and occupy each other by individual organisms. Firstly, we study the uniform persistence of the model via differential inequality techniques. Furthermore, a sharp threshold of global asymptotic stability and the existence of a unique almost periodic solution are derived. To prove the main results, we construct an appropriate Lyapunov function whose conditions are easily verified. The assumptions of the model are reasonable, and the results complement previously known ones. An example with specific values of parameters is included for demonstration of theoretical outcomes.
Introduction
Nonlocal problems concerning the conditions of the behavior of different classes of solutions play an important role in the qualitative theory of ordinary differential equations [1–16]. For more precision, we refer the readers to some specific problems such as boundedness, periodicity, almost periodicity, stability in the sense of Poisson and Ulam and to the problem of the existence of limit regimes of different types, convergence, dissipativity, and so on [17–30]. On the one hand, as pointed out in [31] in the real world, the delays in differential equations of populations, ecology, and dynamic problems are usually infinite time delays; for example, Zhou et al. [32] studied positive almost periodic solutions for a class of Lasota–Wazewska model with infinite time delays. On the other hand, assuming that a harvesting function is a function of the delayed estimate of the true ecological and dynamic models, Zhou et al. [33] presented an overview of the results on the classical Nicholson’s blowflies model with a linear harvesting term.
During the last few decades the question of the existence of oscillatory solutions for differential equations has been considered by many researchers. These systems arise while modeling several physical and natural phenomena in engineering, biology, chemistry, and ecology. Here we focus on the last application, which is ecology. The study of almost periodic models is relevant because they consider general seasonal oscillations of the weather that extend periodical variations of weather or habitat. The concept of almost periodicity was introduced by Danish mathematician Bohr around 1924–1926 [34–36] and later generalized by many other authors [24, 33, 37–40].
The almost periodic functions may capture the phenomena which may not be possible with periodic rates. For instance, the function \(x \mapsto \cos x + \cos5x \) is periodic, and this remains true when when 5 is replaced with any other rational number. However, the sum of the periodic functions \(x \mapsto\exp(ix)\) and \(x \mapsto\exp(i\sqrt{2} x)\) is not periodic. Hence, when such functions, obtained by using a combination of periodic functions, are not periodic, they are not without properties: they are almost periodic functions. More generally, we know that when all the solutions of an autonomous linear finite dimensional system are bounded, then all these solutions are almost periodic.
The notion of almost periodicity for certain functions was introduced by Fréchet [25]. In view of the literature, one can claim that the analysis of boundedness and stability is one of the central foci in the qualitative analysis. Classical ecological research of rocky intertidal communities has already been reported by some ecologists [27, 28]. Recently the authors studied a model of species interactions by a simple patch occupancy with global dispersal of propagules [18, 20]. In [20], Benicá et al. showed that natural ecosystems can sustain continued fluctuations, the model exposes the rocky habitat as small patches, there are three species which habit rocky intertidal community occupy by their individual organism.
This paper is concerned with an occupancy model consisting of two species in a rocky intertidal community, which describes a patch occupancy with global dispersal of propagules, and occupy each other by individual organisms. The model can be formulated by the following system:
and
where x and y are the coverage (i.e., the fraction of patches occupied) by barnacles and crustose algae, respectively, and z is the coverage by bare rock. Barnacles only can cover bare rock, and crustose algae can inhabit bare rock and barnacle. The variables a, b, c, \(m_{1}\), and \(m_{2}\) are all positive, where a is the colonization rate of barnacles on bare rock, b and c are colonization rates of crustose algae on barnacles and bare rock, and \(m_{1}\) and \(m_{2}\) are the mortality rates of barnacles and crustose algae, respectively.
Taking into account the practical significance of the species, it is always assumed that the initial conditions associated with system (1) are
In view of system (1) and initial conditions (3), it is easy to conclude that any solution of system (1) is positive.
Our paper is organized in four sections. In Sect. 2, we study Lyapunov function and general almost periodic functions. In Sect. 3, we provide the results on the uniform persistence and the global asymptotic stability of system (4). Furthermore, we give an example for the system. Finally, we compare main results in this work and the results in the literature in Sect. 4.
Essential preliminaries
For further consideration, we rewrite system (1) in the following form:
Let \(\mathscr{C}_{bd} \) be the collection of continuous and bounded functions from \(\mathbb{R} \times\mathbb{R}^{n}\) to \(\mathbb{R}^{n}\). Let \(\mathbb{R}_{+}=(0, +\infty)\), and define
where f is a continuous and bounded function.
For the relation between colonization and mortality rates, the following assumptions are assumed to hold true throughout the remaining part of the paper:
 \((A_{1})\):

\(m_{1}^{} < a^{+}\),
 \((A_{2})\):

\(m_{2}^{} < c^{+}\).
Definition 1
([24])
A Lyapunov function is a scalar function \(V(x)\) defined on a region D that is continuous, positive definite, i.e., \(V(x)>0\) for all \(x \neq 0\), and has continuous firstorder partial derivatives at every point of D. The derivative of V with respect to the system \(x'= f(x)\), written as \(V^{*} (x)\), is defined as a dot product \(V^{*} (x)= \nabla V(x) . f(x)\).
The existence of a Lyapunov function, for which \(V^{*}(y) \leq0\) on some region D containing the origin, guarantees the stability of the zero solution of \(y' = f(y)\), while the existence of a Lyapunov function, for which \(V^{*}(y)\) is negative definite on some region D containing the origin, guarantees the asymptotic stability of the zero solution of \(y' = f(y)\). For example, given the system
and the Lyapunov function \(V(y, z) = \frac{y^{2} + z^{2} }{2}\), we obtain
which is nonincreasing on every region containing the origin, and thus the zero solution is stable.
Definition 2
([24])
Let \(f: \mathbb{R}\to\mathbb{R}^{n}\) be a continuous function. Given \(\varepsilon>0\), we call \(\tau>0\) and εtranslation for f if and only if, for all \(t \in\mathbb{R}\), \(f(t + \tau)  f(t) < \varepsilon\). The set consisting of all εtranslations for f is denoted by
The function f is an almost periodic function if and only if, for each \(\varepsilon>0\), the set \(E(f, \varepsilon)\) is relatively dense in \(\mathbb{R}\), which means that there exists a constant \(l>0\) such that any subinterval of \(\mathbb{R}\) of length l meets \(E(f, \varepsilon)\).
Definition 3
Let
be one bounded positive solution of (4). Then u is said to be globally asymptotically stable if, for any positive bounded solution
of (4), the following is satisfied:
We state the following essential lemma.
Lemma 4
([23])
The following hold:

(i)
If\(a>0\), \(b>0\), and\(\dot{x}(t) \leq x (t)( b  ax(t))\), whenever\(t\geq0\)and\(x(0)>0\), then\(\limsup_{t \to +\infty} x(t)\leq\frac{b}{a}\).

(ii)
If\(a>0\), \(b>0\), and\(\dot{x} (t)\geq x(t)(bax(t))\), whenever\(t\geq0\)and\(x(0)>0\), then\(\liminf_{t\to+\infty} x(t) \geq\frac{b}{a}\).
Lemma 5
For any solution\((x(t), y(t))\)of system (4) with initial conditions (3), there exists\(T_{1} >0\)such that
for all\(t \geq T_{1}\), where\(M_{1} = \frac{a^{+}  m_{1}^{}}{ a^{} }\)and\(M_{2} = \frac{c^{+} + b^{+}M_{1}  m_{2}^{}}{c^{}}\).
Proof
It is clear that \(x(t)>0\) and \(y(t)>0\) for all \(t\in\mathbb{R}\). Since
and from (i) of Lemma 4, one has
By (7), for any \(\varepsilon>0\) small enough, there exists \(T_{1}> 0\) such that \(x(t) < M_{1} + \varepsilon\) for all \(t\geq T_{1}\). For \(t\geq T_{1}\), we have
From (i) of Lemma 4, one has
Setting \(\varepsilon\to0\) with the righthand side of (8), then
□
Remark 1
From Theorem 8.10 in [29] and Lemma 5 above, it is clear that the boundedness of solution of system (4) is independent of the initial conditions, which implies that any solution of (4) is ultimately bounded.
Main results
This section is devoted to the main results of this paper. We provide results on the uniform persistence and the global asymptotic stability of system (4).
In what follows, we set
Theorem 6
Assume that
 \((A_{3})\):

\(\frac{a^{}  m_{1}^{+}}{ \gamma_{1}^{+}} > M_{2}\),
 \((A_{4})\):

\(\frac{ c^{}m_{2}^{+}}{ \gamma_{2}^{+}} > M_{1}\).
which implies that system (4) is uniformly persistent, where
and
Proof
From Eq. (6) and \((A_{3})\), for any \(\epsilon> 0\) small enough, there exist \(T_{2} > T_{1}\) such that \(y(t) < M_{2} + \epsilon\) and
for all \(t\geq T_{2}\). Also, for \(t\geq T_{2}\), we get
From (ii) of Lemma 4, it follows that
Setting \(\varepsilon\to0\) with the righthand side of (11), then
From equations (7), (12) and \((A_{4})\), for any \(\varepsilon_{*} > 0\) small enough, there exist \(T_{3} > T_{2}\) such that \(x(t) < M_{1} + \varepsilon_{*}\) and
for all \(t\geq T_{3}\). By the same arguments,
From (ii) of Lemma 4, one has
Setting \(\varepsilon\to0\) with the righthand side of (13), then
Let \(T_{*}=T_{3}\), from equations (13) and (14), system (4) is uniformly persistent. The proof is completed. □
Lemma 7
([19])
Letrbe a real number andfbe a nonnegative function defined on\([r, + \infty)\)such thatfis integrable and is uniformly continuous on\([r, + \infty)\). Then\(\lim_{t\to+ \infty} f(t)=0\).
The following theorem shows that relation (5) holds. That is, system (4) with initial condition (3) is globally asymptotically stable. First, we shall propose the following assumption:
Theorem 8
Assume that\((A_{1})\)–\((A_{4})\)hold and\(\varLambda> 1\), then system (4) with initial condition (3) is globally asymptotically stable.
Proof
Suppose that
are two positive solutions of system (4). Since \(\varLambda >1\), one has \(\frac{c^{}}{\gamma_{1}^{+}} > \frac{\gamma _{2}^{+}}{a^{}}\), then there exist positive constants α and β such that
Define a Lyapunov function by
for \(t > T\). Calculating directly the upper right derivative of V along the solutions of model (4), we obtain that
From (15), it follows that there exist constant \(\rho>0\) and large enough \(T>0\) such that
for \(t > T\). Integrating both sides of (17) from T to t, and by the integral comparison theorem, one has
for each \(t>T\), which yields
for \(t > T\). Then \(x(t)  x^{*}(t) + y(t)y^{*}(t) \) is integrable on \([T, + \infty)\). On the other hand, from Remark 1, x, \(x^{*}\), y, and \(y^{*}\) are ultimately bounded. It is clear that they have bounded derivatives. It follows that \(x(t)x^{*}( t) + y(t)  y^{*}(t)\) is uniformly continuous on \([T, +\infty)\). The conditions of Lemma 7 are satisfied, thus
The proof is complete. □
Before discussing the almost periodic system, we need to recall the following Lemma 9, which can be referred to in the monographs [24, 26]. Consider the following system:
where \(f(t, x) \in\mathscr{C}_{bd}\).
Lemma 9
(Theorem 6.6 in [26])
Assume that\(f(t, x)\in\mathscr{C}_{bd}\)is almost periodic intuniformly for\(x\in\mathbb{R}^{n} \)and the solution of (18) is ultimately bounded. And suppose that there exists a Lyapunov function\(V(t, x, y)\)defined on\(\mathbb{R}_{+} \times\mathbb{R}^{n} \times\mathbb{R}^{n}\)which has the following conditions:

(1)
For continuous nondecreasing functionsuandv,
$$u\bigl( \vert xy \vert \bigr)\leq V(t, x, y)\leq v\bigl( \vert xy \vert \bigr). $$ 
(2)
There exists a positive constantLsuch that
$$\bigl\vert V(t, x_{1}, y_{1}) V(t, x_{2}, y_{2}) \bigr\vert \leq L \bigl[ \vert x_{1}  x_{2} \vert + \vert y_{1}y_{2} \vert \bigr]. $$ 
(3)
Ifλis a continuous and positive definite function, then\(\dot{V}\leq \lambda(xy)\).
Then system (4) has a unique almost periodic solution, which is globally uniformly asymptotically stable.
Assume \((A_{1})\)–\((A_{4})\) hold and \(\varLambda>1\), from Lemma 5 and Theorem 6, for any bounded solution \((x(t), y(t))\) of system (4) with initial conditions (3). Then there exist positive constants \(n_{1}^{*}\), \(n_{2}^{*}\), \(M_{1}^{*}\), and \(M_{2}^{*}\) such that \(0< n_{1}^{*} \leq x(t) \leq M_{1}^{*}\) and \(0< n_{2}^{*} \leq y(t)\leq M_{2}^{*}\). For almost periodic system (4), we consider the associated product system of (4) as follows:
Define a Lyapunov function
for \(t \geq0\), where α and β satisfy (15). We still assume that almost periodic system (4) satisfies \((A_{1})\)–\((A_{4})\) and \(\varLambda>1\). From Remark 1, almost periodic solution \((x(t), y(t))\) and \((u(t), v(t))\) of system (19) is ultimately bounded. It is not difficult to verify that conditions (1) and (2) of Lemma 9 are satisfied. Moreover, \(y(t)u(t)\leq K\) and \(x(t)v(t)\leq K\), where
By the same arguments followed in the proof of Theorem 8, one can have that
where ρ is defined as in (17). Taking \(\lambda(r)=2\rho r+K\), \(r\geq0\), condition (3) of Lemma 9 is satisfied.
By virtue of Lemma 9, the following theorem is an immediate consequence.
Theorem 10
Let\((A_{1})\)–\((A_{4})\)and\(\varLambda>1 \)be satisfied and system (4) is an almost periodic. Then there exists a unique almost periodic solution of system (4). Moreover, the unique solution is globally asymptotically stable.
Remark 2
Theorem 10 demonstrates that \(\varLambda> 1\) is the sharp threshold of existence and global asymptotic stability of almost periodic solution of system (4). Based on the ultimate boundedness of almost periodic solution of system (4), the Lyapunov function in Theorem 8 can be applied to prove Theorem 10. Condition (3) of Lemma 9 is easier to be checked than the usual condition \(\dot{V}(t, x,y)\leq\rho V(t,x, y)\). If system (4) is a periodic system, then the following corollary can be concluded.
Corollary 1
If\((A_{1})\)–\((A_{4})\)and\(\varLambda>1 \)hold, then periodic system (4) has a unique positive periodic solution, which is globally asymptotically stable.
Here, we provide an example to illustrate our main results.
Example 1
Consider system (4) with specific values for its parameters of the form
with initial conditions (3). Clearly,
and \(b(t)=\frac{1}{3t^{2}}\). By direct computations, one can conclude that all the assumptions of Theorem 8 hold and \(\varLambda> 1\). Therefore, system (21) with initial conditions (3) is globally asymptotically stable.
Conclusion
Dynamic properties of nonlinear nonautonomous models are rarely considered in the literature. In this paper, we try to fill in this gap by studying a nonautonomous model in a rocky intertidal community. The uniform persistence of the model is examined prior to proceeding to the main results. We establish the threshold condition \(\varLambda>1\) to guarantee the global asymptotic stability and the existence of a unique positive almost periodic solution of the addressed system. Furthermore, an appropriate Lyapunov function is constructed to prove the main results. A particular example with specific parameters that are consistent with the theoretical assumptions is constructed for the sake of demonstrating the validity of outcomes. The authors claim that the results of this paper are new and present different approach, the main theorem of the paper improved the known results of some models [6, 13, 27]. Indeed, it is a challenging problem to search for a suitable Lyapunov function to discuss a threedimensional model in a rocky intertidal community. Precisely saying, we described the essence of the species interactions by a simple patch occupancy model with global dispersal of propagules. The model can be used in the rocky habitat as a large set of very small patches, each of which can be empty (bare rock) or occupied by an individual organism.
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Acknowledgements
JA would like to thank Prince Sultan University for supporting this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM), group number RGDES20170117.
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The research is supported by Higher Education Foundation of Anhui (KJ2019A0712, KJ2019A0713, and gxyq2019067).
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Zhou, H., Alzabut, J., Rezapour, S. et al. Uniform persistence and almost periodic solutions of a nonautonomous patch occupancy model. Adv Differ Equ 2020, 143 (2020). https://doi.org/10.1186/s13662020026032
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MSC
 34K13
 34C25
 92D25
 34D40
Keywords
 Nonautonomous dynamical species
 Uniform persistence
 Almost periodic solution
 Global asymptotic stability