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Uniform ultimate boundedness of solutions of predatorprey system with MichaelisMenten functional response on time scales
Advances in Difference Equations volume 2016, Article number: 319 (2016)
Abstract
In this paper, a predatorprey system with MichaelisMenten functional response on time scales is investigated. First of all, we generalize the semicycle concept to time scales. Second, we obtain the uniformly ultimate boundedness of solutions of this system. Our results demonstrate that when the death rate of the predator is rather small without prey, whereas the intrinsic growth rate of the prey is relatively large, the species could coexist in the long run. In particular, if \(\mathbb{T}=\mathbb{R}\) or \(\mathbb{T}=\mathbb{Z}\), some wellknown results have been generalized. In addition, for the continuous case, we provide a new idea to prove its permanence. Finally, a numerical simulation is given to support our main results.
Introduction
The permanence is based on a global criterion for the coexistence of species, which describes a numerical technique for assembly of ecological communities of LotkaVolterra form [1]. During the last few years, the permanence of ecological models has been discussed by many authors [2–15]. In [4, 5, 10, 11], the comparison method was used, and some sufficient conditions of permanence of biological systems were established. In [6], by using the semicycle and related concepts, they discussed the permanence of a discrete biological system.
To the best of our knowledge, the permanence of biological system on time scales was first discussed by Zhang and Zhang [8]. Using the theory of differential inequality, they obtained the permanence of a cooperation system with feedback controls on time scales. But for the system itself, we could verify that some additional conditions in [8] may not be necessary. For further study, in [15], Li and Wang obtained a permanence result for a multispecies LotkaVolterra mutualism system by establishing some dynamic inequalities on time scales.
In the following we shall use the notation
where f is a periodic rdcontinuous function with period ω in \(\mathbb{T}\). We assume that \(s+\omega\in\mathbb{T}\) for any \(s\in \mathbb{T}\).
Remark 1.1
For any \(t>l\), \(t\in\mathbb{T}\), we have \(t\rho (t ) \leq\omega\).
Proof
For any \(t>l\), \(\rho ( t ) \in\mathbb{T}\), then \(\rho ( t ) +\omega\in\mathbb{T}\), if \(t\rho ( t ) >\omega\), then \(\rho ( t ) +\omega< t\), this implies \(\omega=0\), we arrive at a contradiction. Thus \(t\rho ( t ) \leq\omega\) holds. □
As is well known, predatorprey systems play an important role in ecosystems [2–7, 9–14, 16–27]. We find that there are few papers discussing the permanence of these systems on time scales.
In this paper, we are concerned with the following predatorprey system with MichaelisMenten functional response:
with the initial condition
on time scales \(\mathbb{T}\), where \(a,b,c,d,f,\beta,\gamma\in \mathrm{C}_{\mathrm{rd}} ( \mathbb{T},\mathbb{R} ) \) are positive ωperiodic functions. The Δ stands for the delta derivative.
Let \(x ( t ) =\exp \{ x_{1} ( t ) \} \), \(y ( t ) =\exp \{ x_{2} ( t ) \} \), \(\beta ( t ) =1\), if \(\mathbb{T}=\mathbb {R}\), then (1) reduces to the continuous predatorprey system
In [16], Fan et al. studied some basic problems of system (2), such as positive invariance, permanence, nonpersistence, extinction, dissipativity, and globally asymptotic stability. The methods they used were comparison theorems, coincidence degree theory, and the Lyapunov functional. Much attention has been paid to this predatorprey system or its analogs in [2–4, 17–21]. In [4], Fan and Li considered the permanence of the general delayed ratiodependent predatorprey model
where \(g ( u ) \) is monotonic increasing and there exists a constant k such that \(\lim_{u\rightarrow+\infty}g(u)=k\), by using comparison theorems. They obtained the following.
Theorem 1.1
Assume

(C1)
\(\overline{a}>m\overline{c}\);

(C2)
\(k\overline{e}>\overline{d}\)
hold, here \(m=\sup_{z\in{}[0,+\infty)}\{g(z)/z\}\). Then system (3) is permanent.
Let \(N_{1} ( k ) =\exp \{ x_{1} ( k ) \} \), \(N_{2} ( k ) =\exp \{ x_{2} ( k ) \} \), \(\beta ( k ) =1\), if \(\mathbb{T}=\mathbb{Z}\), then (1) is reformulated as
this predatorprey system or its other forms has attracted the attention of many authors [5–7]. Fan and Li [5] considered the permanence of a delayed discrete predatorprey model with Hollingtype III functional response
the comparison theorem, and the theory of population equation. They obtained the following.
Theorem 1.2
Assume

(B1)
\(2m\overline{b_{1}}>\overline{\alpha_{1}}\);

(B2)
\(\overline{\alpha_{2}}>\overline{b_{2}}\);

(B3)
\((b_{2}(k))^{M}<1\)
hold. Then system (5) is permanent.
In fact, (B3) is not necessary. In [6], by using the semicycle and related concepts, Fan and Li considered permanence of the system (4), and obtained the following.
Theorem 1.3
Assume

(H1)
\(\overline{a}>\overline{c/\gamma}\);

(H2)
\(\overline{f}>\overline{d}\)
hold. Then system (4) is permanent.
In recent years, the existence of periodic solutions of predatorprey systems on time scales has been obtained by coincidence degree theory in many articles [22–26], since the existence result could be obtained by coincidence degree theory both in the continuous case and the discrete case. Fazly and Hesaaraki [22] obtained the existence of periodic solutions of nonautonomous predatorprey dynamical system with BeddingtonDeAngelis functional response by coincidence degree theory. Tong et al. [23] investigated the existence of periodic solutions of a predatorprey system with sparse effect and BeddingtonDeAngelis or a Holling III functional response. By using a continuation theorem based on coincidence degree theory, they obtained sufficient criteria for the existence of periodic solutions for the system.
Since the permanence of this system has been obtained by the comparison theorem in the continuous case, while it has been obtained by semicycle concept in discrete case. Also, in order to delete the additional condition (B3) and the additional conditions of Theorem 3.1 in [8], we need to extend the semicycle concept of the discrete case to that of time scales.
Noticing that system (2) and (4) is derived from (1) by exponential transformations, \(x ( t ) =\exp \{ x_{1} ( t ) \} \), \(y ( t ) =\exp \{ x_{2} ( t ) \} \), and \(N_{1} ( k ) =\exp \{ x_{1} ( k ) \} \), \(N_{2} ( k ) =\exp \{ x_{2} ( k ) \} \), respectively, when \(\mathbb{T}=\mathbb{R}\) and \(\mathbb{T}=\mathbb{Z}\). Obviously, when solutions of system (1) are uniformly ultimate bounded, systems (2) and (4) are both permanent, and the contrary also holds true. Thus, our aim is to prove the uniform ultimate boundedness of solutions of system (1) by using the semicycle concept on time scales instead of comparison theorems. Our result is an unification and extension of a continuous and discrete analysis.
The rest of the paper is organized as follows. In Section 2, we state some basic properties about time scales, and generalize the semicycle concept to time scales. Section 3 is devoted to the uniform ultimate boundedness of solutions of system (1). A discussion is presented in Section 4. The final section of the paper contains a numerical example supporting the result.
Preliminary
First we will give some definitions about time scales before presenting our main result (see [28, 29]).
Definition 2.1
A time scale is an arbitrary nonempty closed subset \(\mathbb{T}\) of the real number \(\mathbb{R}\).
Definition 2.2
For \(t\in\mathbb{T}\) we define the forward jump operator \(\sigma:\mathbb{T}\rightarrow\mathbb{T}\), and the backward jump operator \(\rho:\mathbb{T}\rightarrow\mathbb{T}\), by
respectively.
Throughout this paper we often assume \(a\leq b\), \(a,b\in\mathbb{T}\), and define the interval \([ a,b ] \) in \(\mathbb{T}\) by
Definition 2.3
Assume \(f:\mathbb{T}\rightarrow\mathbb{R}\) is a function and let \(t\in\mathbb{T}^{k}\), where throughout the paper
Then we define \(f^{\Delta} ( t ) \) to be the number (provided it exists) with the property that, given any \(\varepsilon>0\), there exists a neighborhood U of t such that
We call \(f^{\Delta} ( t ) \) the delta (or Hilger) derivative of f at t.
The following lemmas will be useful to prove our main result. Their proofs are similar to [6], we omit them here.
Lemma 2.1
The equation
has at least one periodic solution \(x ( t ) \) if \(a ( t )\), \(b ( t ) \) are both ωperiodic rdcontinuous functions and \(\overline {a}>0\), \(b(t)>0\); moreover, the inequalities
hold.
Lemma 2.2
Assume that d, f, β, γ are all positive ωperiodic rdcontinuous functions, then, for any positive constant M, the equation
has at least one periodic solution \(x ( t ) \) provided with \(\overline{d}<\overline{f/\beta}\); moreover, the inequalities
hold.
Similar to the definition of semicycle in discrete case (see [30]), we give the definition of a semicycle on time scales.
Definition 2.4
A positive semicycle of a rdcontinuous function \(f: \mathbb{T}\rightarrow\mathbb{R}\) consists of a ‘string’ of terms \(\{ f ( t ) ,t\in [ s,t ] ,s,t\in\mathbb{T} \} \), all greater than or equal to 0. A negative semicycle of a rdcontinuous function \(f: \mathbb{T}\rightarrow \mathbb{R}\) consists of a ‘string’ of terms \(\{ f ( t ) ,t\in [ p,q ] ,p,q\in\mathbb{T} \} \), all less than or equal to 0.
The uniform ultimate boundedness
Before giving our main result, we list the definition of uniform ultimate boundedness.
Definition 3.1
Solutions of (1) are said to be uniformly ultimate bounded if there exist two constants \(\lambda_{1}\) and \(\lambda_{2}\) such that, for any initial condition \(( x_{1} ( 0 ) ,x_{2} ( 0 ) ) ^{\mathbb{T}}\in\mathbb{R}^{2}\),
Here we say ‘uniformly’, because \(\lambda_{1}\) and \(\lambda_{2}\) are independent on \(( x_{1} ( 0 ) ,x_{2} (0 ) ) ^{\mathbb{T}}\).
We now state the main result as follows.
Theorem 3.1
Assume

(A1)
\(\overline{a}>\overline{c/\gamma}\);

(A2)
\(\overline{d}<\overline{f/\beta}\)
hold. Then the solutions of system (1) are uniformly ultimate bounded.
Proof
We will prove \(\lambda_{1}\leq\lim\inf_{t\rightarrow \infty}x_{i} ( t ) \leq\lim\sup_{t\rightarrow\infty }x_{i} ( t ) \leq\lambda_{2}\), \(\lambda_{1}\) and \(\lambda_{2}\) are constants, \(i=1,2\). Thus, we divide the proof into four parts.
Part 1. \(\lim\sup_{t\rightarrow\infty}x_{1} ( t ) \leq K_{1}\), that is to say, \(x_{1} ( t ) \) is uniformly ultimate bounded above.
From (1), when t is sufficiently large, we know
and construct the following auxiliary equation:
Note that \(a(t)\), \(b(t)\) are positive ωperiodic functions, we can see \(\overline{a}>0\). By Lemma 2.1, (7) has at least one ωperiodic solution, denote it as \(y_{1}^{\ast} ( t ) \), we have
let \(u_{1} ( t ) =x_{1} ( t ) y_{1}^{\ast} ( t ) \), then
Now, the proof has two cases according to the oscillating property of \(u_{1} ( t ) \). First we assume that \(u_{1} ( t ) \) does not oscillate about zero, then \(u_{1} ( t ) \) will be either eventually positive or eventually negative. If the latter holds, then
If the former holds, then \(( u_{1} ( t ) ) ^{\Delta}<0\), which means that \(u_{1} ( t ) \) is eventually decreasing, also in terms of its positivity, we know that \(\lim_{t\rightarrow\infty}u_{1} ( t ) \) exists. Then (9) yields \(\lim_{t\rightarrow\infty}u_{1} ( t ) =0\), which leads to
Now we assume \(u_{1} ( t ) \) oscillates about zero, by (7), we know that \(u_{1} ( t ) \geq0\) implies \(( u_{1} ( t ) ) ^{\Delta}\leq0\). Thus, by the semicycle concept, we let \(u_{1} ( t ) \geq0\), for \(t\in [ s_{\alpha},t_{\alpha } ] \), \(s_{\alpha},t_{\alpha}\in\mathbb{T}\), \(\alpha\in\mathcal{L}\), where \(\mathcal{L}\) is an index set, and the interval \([ s_{\alpha },t_{\alpha} ] \) satisfies:

(a1)
For any \(\alpha,\beta\in\mathcal{L}\), if \(\alpha\neq \beta\), \([ s_{\alpha},t_{\alpha} ] \cap [ s_{\beta },t_{\beta} ] =\varnothing\).

(b1)
If \(s_{\alpha}\) is leftscattered, then \(u_{1} ( \rho ( s_{\alpha} ) ) <0\).

(c1)
If \(s_{\alpha}\) is leftdense, then there exists a hollow left neighborhood \(\mathring{U}_{} ( s_{\alpha} ) \) of \(s_{\alpha}\) such that \(u_{1} ( t ) <0\), for \(t\in\mathring{U}_{} ( s_{\alpha} ) \).

(d1)
If \(t_{\alpha}\) is rightscattered, then \(u_{1} ( \sigma ( t_{\alpha} ) ) <0\).

(e1)
If \(t_{\alpha}\) is rightdense, then there exists a hollow right neighborhood \(\mathring{U}_{+} ( t_{\alpha} ) \) of \(t_{\alpha}\) such that \(u_{1} ( t ) <0\), for \(t\in\mathring{U}_{+} ( t_{\alpha} ) \).
Notice that \(\lim\sup_{t\rightarrow\infty}u_{1} ( t ) =\lim \sup_{\alpha\rightarrow\infty}u_{1} ( s_{\alpha} ) \). If \(s_{\alpha}\) is leftscattered, by integrating (9) over the set \([ \rho ( s_{\alpha} ) ,s_{\alpha} ] \), we have
by (b1), it follows that
if and only if
it is easy to see (13) holds from Remark 1.1.
If \(s_{\alpha}\) is leftdense, we choose \(t_{1}\in\mathring{U}_{} ( s_{\alpha} ) \), such that \(s_{\alpha}t_{1}\leq\omega\). By integrating (9) over the set \([ t_{1},s_{\alpha} ] \), we have
notice that \(u_{1} ( t_{1} ) <0\), it follows that
Then from (12) and (14), \(u_{1} ( t ) = x_{1} ( t ) x_{1}^{\ast} ( t ) \) is uniformly ultimate bounded above, thus
Therefore from (10), (11), and (15), \(x_{1} ( t ) \) is uniformly ultimate bounded above.
Part 2. \(k _{1}\leq\lim\inf_{t\rightarrow\infty}x_{1} ( t ) \), that is to say, \(x_{1} ( t ) \) is uniformly ultimate bounded below.
Also from (1), when t is sufficiently large, we know
and construct the following auxiliary equation:
From (A1) and Lemma 2.1, (17) has at least one ωperiodic solution, denote it as \(z_{1}^{\ast} ( t ) \), we have
let \(u_{2} ( t ) =x_{1} ( t ) z_{1}^{\ast} ( t ) \), then
from (15) and (18), we know when t is sufficiently large,
Now, the proof has two cases according to the oscillating property of \(u_{2} ( t ) \). First we assume that \(u_{2} ( t ) \) does not oscillate about zero, then \(u_{2} ( t ) \) will be either eventually positive or eventually negative. If the former holds, then
If the latter holds, then \(( u_{2} ( t ) ) ^{\Delta}>0\), which means that \(u_{2} ( t ) \) is eventually increasing, also in terms of its negativity, we know that \(\lim_{t\rightarrow\infty}u_{2} ( t ) \) exists. Then (26) yields \(\lim_{t\rightarrow\infty}u_{2} ( t ) =0\), which leads to
Now we assume \(u_{2} ( t ) \) oscillates about zero, by (19), we know that \(u_{2} ( t ) \leq0\) implies \(( u_{2} ( t ) ) ^{\Delta}\geq0\). Thus, by the semicycle concept, we let \(u_{2} ( t ) \leq0\), for \(t\in [ p_{\alpha},q_{\alpha } ]\), \(p_{\alpha},q_{\alpha}\in\mathbb{T}\), \(\alpha\in\mathcal{L}\), where \(\mathcal{L}\) is an index set, and the interval \([ p_{\alpha },q_{\alpha} ] \) satisfies:

(a2)
For any \(\alpha,\beta\in\mathcal{L}\), if \(\alpha\neq \beta\), \([ p_{\alpha},q_{\alpha} ] \cap [ p_{\alpha },q_{\alpha} ] =\varnothing\).

(b2)
If \(p_{\alpha}\) is leftscattered, then \(u_{2} ( \rho ( p_{\alpha} ) ) >0\).

(c2)
If \(p_{\alpha}\) is leftdense, then there exists a hollow left neighborhood \(\mathring{U}_{} ( p_{\alpha} ) \) of \(p_{\alpha}\) such that \(u_{2} ( t ) >0\), for \(t\in\mathring{U}_{} ( p_{\alpha} ) \).

(d2)
If \(q_{\alpha}\) is rightscattered, then \(u_{2} ( \sigma ( q_{\alpha} ) ) >0\).

(e2)
If \(q_{\alpha}\) is rightdense, then there exists a hollow right neighborhood \(\mathring{U}_{+} ( q_{\alpha} ) \) of \(q_{\alpha}\) such that \(u_{2} ( t ) >0\), for \(t\in\mathring{U}_{+} ( q_{\alpha} ) \).
Notice that \(\lim\inf_{t\rightarrow\infty}u_{2} ( t ) =\lim \inf_{\alpha\rightarrow\infty}u_{2} ( p_{\alpha} ) \). If \(p_{\alpha}\) is leftscattered, by integrating (19) over the set \([ \rho ( p_{\alpha} ) ,p_{\alpha} ] \), we have
by (b2), from (20), it follows that
from Remark 1.1, we can see
then \(u_{2} ( t_{n} ) \) is lower bounded.
If \(p_{\alpha}\) is leftdense, we choose \(t_{2}\in\mathring{U}_{} ( p_{\alpha} ) \), such that \(p_{\alpha}t_{2}\leq\omega\). Notice that \(u_{2} ( t_{2} ) >0\), by integrating (19) over the set \([t_{2},p_{\alpha} ] \), we have
from (20), we have
in this case, \(u_{2} ( p_{\alpha} ) \) is also lower bounded.
Then \(u_{2} ( t ) =x_{1} ( t ) z_{1}^{\ast} ( t ) \) is uniformly ultimate bounded below, hence
Thus from (21), (22), and (23), \(x_{1} ( t ) \) is uniformly ultimate bounded below. Then Part 2 holds.
Therefore, from Part 1 and Part 2, \(x_{1} ( t ) \) is uniformly ultimate bounded, we can assume \(m\leq\exp \{ x_{1} ( t ) \}\leq M\) for any \(t\in{}[ T_{0},\infty)\cap\mathbb{T}\), where \(T_{0}\) is sufficiently large.
Part 3. \(k_{1}\leq\lim\inf_{t\rightarrow\infty}x_{2} ( t ) \), that is to say, \(x_{2} ( t ) \) is uniformly ultimate bounded below.
From (1), when t is sufficiently large, we know
and we construct the following auxiliary equation:
from Lemma 2.2 and (A2), we find that (25) has at least one ωperiodic solution, denote it as \(z_{2}^{\ast} ( t ) \), we have
Similarly, let \(v_{2} ( t ) =x_{2} ( t ) z_{2}^{\ast } ( t ) \), then
Now, the proof has two cases according to the oscillating property of \(v_{2} ( t ) \). First we assume that \(v_{2} ( t ) \) does not oscillate about zero, similar to \(u_{2}(t)\) in Part 2, we obtain
Now we assume \(v_{2} ( t ) \) oscillates about zero, by (26), we know that \(v_{2} ( t ) \leq0\) implies \(( v_{2} ( t ) ) ^{\Delta}\geq0\). Thus, by the semicycle concept, we let \(v_{2} ( t ) \leq0\), for \(t\in [ p_{\alpha},q_{\alpha } ] \), \(p_{\alpha},q_{\alpha}\in\mathbb{T}\), \(\alpha\in\mathcal{L}\), where \(\mathcal{L}\) is an index set, the interval \([ p_{\alpha },q_{\alpha} ]\) satisfies (a2)(e2) by replacing \(u_{2}(t)\) in Part 2 with \(v_{2}(t)\).
Notice that \(\lim\sup_{t\rightarrow\infty}v_{2} ( t ) =\lim \sup_{\alpha\rightarrow\infty}v_{2} ( p_{\alpha} ) \). By a similar analysis to Part 2, if \(p_{\alpha}\) is leftscattered, integrating inequality (26) from \(\rho ( p_{\alpha} ) \) to \(p_{\alpha}\), we have
by (b2), it follows that
by Remark 1.1, \(v_{2} ( p_{\alpha} ) \) is lower bounded.
If \(p_{\alpha}\) is leftdense, we choose \(t_{3}\in\mathring{U}_{} ( p_{\alpha} ) \), such that \(p_{\alpha}t_{3}\leq\omega\). By integrating (26) over the set \([ t_{3},p_{\alpha} ] \), we have
notice that \(v_{2} ( t_{3} ) >0\), hence
in this case, \(v_{2} ( p_{\alpha} ) \) is also lower bounded.
Then \(v_{2} ( t ) =x_{2} ( t ) z_{2}^{\ast} ( t ) \) is uniformly ultimate bounded below, hence
Thus from (27) and (28), \(x_{2} ( t ) \) is uniformly ultimate bounded below.
Part 4. \(\lim\sup_{t\rightarrow\infty}x_{2} ( t ) \leq K_{2}\), that is to say, \(x_{2} ( t ) \) is uniformly ultimate bounded above.
Also from (1), when t is sufficiently large, we know
and we construct the following auxiliary equation:
from Lemma 2.2 and (A2), we find that (30) has at least an ωperiodic solution, denote it as \(y_{2}^{\ast} ( t ) \), we have
Similarly, let \(v_{1} ( t ) =x_{2} ( t ) y_{2}^{\ast } ( t ) \), then
from (28) and (31), we know, when t is sufficiently large,
Now, the proof has two cases according to the oscillating property of \(v_{1} ( t ) \). First we assume that \(v_{1} ( t ) \) does not oscillate about zero, similar to \(u_{1} ( t ) \) in Part 1, we obtain
Now we assume \(v_{1} ( t ) \) oscillates about zero, by (32), we know that \(v_{1} ( t ) \geq0\) implies \(( v_{1} ( t ) ) ^{\Delta}\leq0\). Thus, by the semicycle concept, we let \(v_{1} ( t ) \geq0\), for \(t\in [ s_{\alpha},t_{\alpha } ] \), \(s_{\alpha},t_{\alpha}\in\mathbb{T}\), \(\alpha\in\mathcal{L}\), where \(\mathcal{L}\) is an index set, the interval \([ s_{\alpha },t_{\alpha} ]\) satisfies (a1)(e1) by replacing \(u_{1}(t)\) in Part 1 with \(v_{1}(t)\).
Notice that \(\lim\sup_{t\rightarrow\infty}v_{1} ( t ) =\lim \sup_{\alpha\rightarrow\infty}v_{1} ( s_{\alpha} ) \). By a similar analysis to Part 1, if \(s_{\alpha}\) is leftscattered, integrating inequality (32) from \(\rho ( s_{\alpha} ) \) to \(s_{\alpha}\), we have
by (b1), from (33), it follows that
If \(s_{\alpha}\) is leftdense, we choose \(t_{4}\in\mathring{U}_{} ( s_{\alpha} ) \), such that \(s_{\alpha}t_{4}\leq\omega\). By integrating (32) over the set \([ t_{4},s_{\alpha} ] \), we have
notice that \(v_{1} ( t_{4} ) <0\), hence
Then from (35) and (36), \(v_{1} ( t ) = x_{2} ( t ) y_{2}^{\ast} ( t ) \) is uniformly ultimate bounded above, thus
Thus from (34) and (37), \(x_{2} ( t ) \) is uniformly ultimate bounded above.
Therefore, from Part 3 and Part 4, \(x_{2} ( t ) \) is uniformly ultimate bounded.
Finally, we choose \(\lambda_{1}=\min\{k_{1},k_{2}\}\), \(\lambda_{2}=\max \{K_{1},K_{2}\}\). This completes the proof of Theorem 3.1. □
Discussion
In Theorem 3.1, if we let \(\mathbb{T}=\mathbb {R}\) and \(\mathbb{T}=\mathbb {Z}\), respectively, then the result is exactly changed into Theorem 1.1 (by the comparison theorem) and Theorem 1.3 (by the semicycle theory). That is, we provide a unified method to study the permanence for the continuous system and discrete system. In addition, we give a new method to investigate the permanence for the continuous system.
From the proof of Theorem 3.1, we can easily see that our methods can also be used to study the following generalized predatorprey system with functional response:
here \(h(u)=g(u)/u\), where \(g ( u ) \) is monotonic increasing and there exists a constant k such that \(\lim_{u\rightarrow+\infty }g(u)=k \). The coefficient functions are all bounded.
As is well known, the permanence of the periodic biological system is closely associated with the existence for the periodic solutions of the system, in general, when the periodic system is permanent, then there must exist at least one positive periodic solution. By a similar analysis to that in Fan and Wang [27], we can obtain the following remark.
Remark 4.1
Assume (A1), (A2) hold, then system (1) has at least one ωperiodic solution.
This shows that the conditions for uniform ultimate boundedness of solutions are the same as that for the existence of periodic solutions of the system.
Numerical example
In this section, we give a numerical example to support our main result. Assume that \(a(t)=1.2+0.1\sin(\pi t)\), \(b(t)=0.006+0.005\sin(\pi t)\), \(c(t)=0.4+0.1\cos(\pi t)\), \(d(t)=0.5+0.3\sin(\pi t)\), \(f(t)=0.5+0.05\cos (\pi t)\), \(\gamma(t)=0.5+0.1\cos(\pi t)\), \(\beta(t)=0.45+0.05\cos (\pi t)\), and \(\mathbb{T}=\bigcup_{k=0}^{\infty}[2k,2k+1]\). In this case, we can see
here \(k=1,2,3,\ldots\) . In applying a numerical analysis using Matlab, we assume that \(x_{1}(0)=4.7\), \(x_{2}(0)=4.8\), and then obtain Figure 1. It is easy to see \(x_{1}(t)\), \(x_{2}(t)\) are uniformly ultimate bounded. We also obtain the relationship between \(x_{1}(t)\) and \(x_{2}(t)\) (see (B) of Figure 1). Our numerical simulation supports our theoretical findings (see the figures). We conclude that it is valid for any initial condition \(( x_{1} ( 0 ) ,x_{2} ( 0 ) ) ^{\mathbb{T}}\in\mathbb{R}^{2}\).
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Acknowledgements
This work is supported by NSF of China (11201213, 11371183), NSF of Shandong Province (ZR2015AM026, ZR2013AM004) and the Project of Shandong Provincial Higher Educational Science and Technology (J15LI07).
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Yu, YY., Wang, LL. & Fan, YH. Uniform ultimate boundedness of solutions of predatorprey system with MichaelisMenten functional response on time scales. Adv Differ Equ 2016, 319 (2016). https://doi.org/10.1186/s1366201610376
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DOI: https://doi.org/10.1186/s1366201610376
MSC
 34N05
 37N25
Keywords
 ωperiodic solutions
 uniform ultimate boundedness
 semicycle
 time scales
 predatorprey system