Abstract

Verifiable criteria are established for the permanence and existence of positive periodic solutions of a delayed discrete predator-prey model with monotonic functional response. It is shown that the conditions that ensure the permanence of this system are similar to those of its corresponding continuous system. And the investigations generalize some well-known results. In particular, a more acceptant method is given to study the bounded discrete systems rather than the comparison theorem.

1. Introduction

Since the end of the 19th century, many biological models have been established to illustrate the evolutionary of species, among them, predator-prey models attracted more and more attention of biologists and mathematicians. There are many different kinds of predator-prey models in the literature. And since 1990s, the so-called ratio-dependent predator-prey models play an important role in the investigations on predator-prey models, which can be roughly stated as that the per capita predator growth rate should be a function of the ratio of prey to predator abundance. Under some simple assumptions, a general form of a ratio-dependent model is

Here the predator-prey interactions are described by ; this function replaces the functional response function in the traditional prey-dependent model. For the study of ratio-dependent predator-prey models, most works have been done on the Michaelis-Menten type model

or its periodic type

see [17] and references therein. It is easy to see that here the functional response function is , as we know, this functional response function was first used by Holling [8], and later biologists call it Holling type II functional response function, it usually describes the uptake of substrate by the microorganisms in microbial dynamics or chemical kinetics [9]. And in the present paper, we will concentrate on the general form of the ratio-dependent predator-prey model.

For the sake of convenience, we introduce some notations and definitions. Denote and as the sets of all integers, real numbers, and nonnegative real numbers, respectively. Let denote the set of all bounded sequences , is the set of all such that , and We define

for any Obviously, if is an -periodic sequence, then

We also define

if is an -periodic sequence. And denote

when is a periodic continuous function with period

In view of the periodicity of the actual environment, we begin with the following periodic continuous ratio-dependent predator-prey system:

where and represent the densities of the prey population and predator population at time respectively; are real constants; and are continuous periodic functions with period and , is not always zero; and (here ) denotes the ratio-dependent response function, which reflects the capture ability of the predator. Here we assume that satisfies the following monotonic condition, for short, we call it ():

(i)(ii) for (iii)lim.

In [10], we gave a sufficient condition for the permanence of the continuous model

In which and are all positive periodic continuous functions with period ; is a positive constant. In addition to condition (), the functional response function also satisfies the following.

(iv)There exists a positive constant such that .

Without loss of generality, in this paper, we always assume that (if let and still denote as ).

Some special cases of system (1.8) have been studied, see [11, 12] and so forth. In those papers, the authors mainly concentrated on the existence of periodic solutions and permanence for systems they considered.

Set

from (ii) and (iii), we can easily obtain

Through the above assumptions, we can see that, one of the main results in [10] can be given as follows.

Theorem 1.1. If(H1)and(H2)hold, then system (1.9) is permanent.

Remark 1.2. By similar methods proposed in [10], we can show that under conditions (H1) and (H2), system (1.8) is also permanent.

We also need to mention that conditions (H1) and (H2) are sufficient to assure the existence of positive periodic solutions of (1.8); this problem has been solved in [13].

However, when the size of the population is rarely small or the population has nonoverlapping generations [14, 15], a more realistic model should be considered, that is, the discrete time model. Just as pointed out in [16], even if the coefficients are constants, the asymptotic behavior of the discrete system is rather complex and “chaotic” than the continuous one, see [16] for more details. Similar to the arguments of [17], we can obtain a discrete time analogue of (1.8):

where denotes the integer part of () and . Correspondingly, the basic assumptions of (1.11) is the same as that in (1.8), of cause, here and are periodic sequences with period and , , and satisfies (). To the best of our knowledge, a few investigations have been carried out for the permanence on delayed discrete ecological systems, since the dynamics of these systems are usually more complicated than the continuous ones.

The exponential form of system (1.11) assures that, for any initial condition , remains positive. In the remainder of this paper, for biological reasons, we only consider solutions with

System (1.11) includes many biological models as its special cases, which have been studied by many authors; see [1720] and so forth. Among them, Fan and Wang (see [17]) considered the existence of positive periodic solutions for delayed periodic Michaelis-Menten type ratio-dependent predator-prey system

and obtained the following theorem.

Theorem 1.3. Assume that the following conditions hold:(A1)(A2)Then (1.13) has at least one positive -periodic solution.

Later in [20], we proved that under conditions (A1) and (A2), system (1.13) is also permanent, so by the main result in [21], we can also obtain Theorem 1.3, which gives another method to prove the existence of periodic solutions.

From the works above, it is not difficult to find: that for the continuous time model (1.3) and the discrete time model (1.13), conditions that assure the existence of positive periodic solutions are exactly the same. In addition, when we comparing the work in [11] with that in [20], we found amazedly that conditions that assure the permanence of the discrete models are also the same as those of the continuous models. This motivated us to consider the permanence of system (1.11) only under conditions (H1) and (H2), since we have already obtained the permanence of system (1.9).

Until very recently, Yang [22] studied the permanence of system (1.11) when and obtained the following conclusion.

Theorem 1.4. Assume that(B1)(B2)(B3) hold, where Then (1.11) is permanent.

Remark 1.5. In Theorem 1.4, condition (B3) implies condition (B1); and (B3) is an equality, it is too strong to satisfy.

As pointed out in [23], if we use the method of comparison theorem, then the additional condition (like (B3)), to some extent, is necessary. But for system itself, this condition may be not necessary. In this paper, our aim is to improve the above results. One of the main results in this paper is given below, furthermore, we can conclude Corollary 3.5 similarly, from which we could show that condition (B3) can be deleted. Now we list the main result in the following.

Theorem 1.6. Assume that (H1) and (H2)hold. Then system (1.11) is permanent.

Remark 1.7. Condition (H2) is a necessary condition.

Corollary 1.8. Assume that (H1) and (H2) hold, then system (1.11) has at least one positive -periodic solution.

Clearly, Theorem 1.6 extends and improves [19, Theorem  3.1], [20, Theorem  1.4]; Theorem 1.6 also extends and improves Theorem 1.4 by weaker conditions (H1) and (H2) instead of (B1–B3) when the coefficients are all periodic. In particular, our investigation gives a more acceptant method to study the bounded discrete systems, which is better than the comparison theorem.

For the permanence of biology systems, one can refer to [2433] and the references cited therein.

The tree of this paper is arranged as follows. In the next section, we give some useful lemmas which are essential to prove our conclusions. And in the third section, we give a proof to the main result.

2. Preliminary

In this section, we list the definition of permanence and establish some useful lemmas.

Definition 2.1. System (1.11) is said to be permanent if there exist two positive constants such that for any solution of (1.11).

Lemma 2.2 ([20]). The problem has at least one periodic solution if and is an -periodic sequence with , moreover, the following properties hold.(a) is positive -periodic.(b) has the following estimations for it's boundary: especially, if

Lemma 2.3. For any positive constant , the problem has at least one periodic solution if and is an -periodic sequence provided that (H2) holds. Moreover, the following properties hold.(a) is positive -periodic.(b) has the following estimations for its boundary: especially, if where represents the inverse of

Proof. We only prove that (2.6) holds, for the rest of the proof, one can refer to [17]. Let be any possible -periodic positive solution of (2.5), then therefore this leads to We claim that there exist some and such that If this is not true, then either or for any in any case, we can obtain this contradiction shows that our claim is true.
Note that for any
then by virtue of equality (2.10), we complete the proof.

Lemma 2.4. Consider the inequality problem If and is an -periodic sequence with , then any positive solutions of (2.15) satisfy where Moreover, if then where

Proof. Consider the following auxiliary equation: by Lemma 2.2, (2.20) has at least one positive -periodic solution, denote it as , then Let then Make the transformation we can obtain Now we divide the proof into two cases according to the oscillating property of . First we assume that does not oscillate about zero, then will be either eventually positive or eventually negative. If the latter holds, that is, we have Either if the former holds, then by (2.24), we know which means that is eventually decreasing, also in terms of its positivity, we know that exists. Then (2.24) implies which leads to
Now we assume that oscillates about zero, by (2.24), we know that implies Thus, if we let be a subsequence of where is the first element of the th positive semicycle of , then For the definition of semicycle and other related concepts, we refer to [34]. Notice that
and then we know Therefore By the medium of (2.22), (2.25), and (2.26), we have

Corollary 2.5. Any positive solution of the inequality problem (2.15) satisfies where , and

Proof. Define the function it is easy to see which immediately leads to From (2.15), we have By Lemma 2.4, for any let we can obtain by (2.34) and (2.37), we complete the proof.

Remark 2.6. Note that when
Similarly, we can obtain the following result.

Lemma 2.7. If any positive solution of the inequality problem satisfies here is a positive constant. Then if and is an -periodic sequence with , one has Moreover, if then where

The proof is similar to that of Lemma 2.4.

Corollary 2.8. If any positive solution of the inequality problem satisfies here is a positive constant, then where and

3. Proof of the Main Result

For the rest of this paper, we only consider the solution of (1.11) with initial conditions (1.12). To prove Theorem 1.6, we need the following several propositions.

Proposition 3.1. There exists a positive constant such that

Proof. Given any positive solution of (1.11), from the first equation of (1.11), we have Set then thus which is equivalent to hence Therefore By Lemma 2.4, we have where

Proposition 3.2. Under condition (H1), there exists a positive constant such that

Proof. Given any positive solution of (1.11), from the first equation of (1.11), we have Set then which yields that is, thus Therefore Since (H1) holds, then by Lemma 2.7 and Proposition 3.1, we have

Proposition 3.3. If (H2) holds, then there exists a positive constant such that

Proof. Given any positive solution of (1.11). Set from the second equation of (1.11) and notice that conditions (ii) and (iii) on imply that then thus which is equivalent to hence Therefore for any given we have for sufficiently large . Here we use the monotonicity of the function
Consider the following auxiliary equation:
By Lemma 2.3 and condition (H2), we can obtain that (3.23) has at least one positive -periodic solution, denote it as and where Let then Denote we have First we assume that does not oscillate about zero, then will be either eventually positive or eventually negative. If the latter holds, that is, we have Either if the former holds, then by (3.28), we have which means that is eventually decreasing, also in terms of its positivity, we obtain that exists. Then (3.28) leads to this implies
Now we assume that oscillates about zero, in view of (3.28), we know that implies Thus, if we let be a subsequence of where is the first element of the th positive semicycle of , then Also, from
and we know Therefore Thus we have where

Proposition 3.4. Under conditions (H1) and (H2), there exists a positive constant such that

Proof. Given any positive solution of (1.11), from the second equation of (1.11), we have then hence Therefore from the second equation of (1.11), we have Consider the auxiliary equation by Lemma 2.3 and (H2), (3.40) has at least one positive -periodic solution, denoted it as then Where If we set then And let we have If does not oscillate about zero, then by a similar analysis as that in Proposition 3.1, we have Otherwise, if oscillates about zero, by (3.45), we know that implies Thus, if we denote as a subsequence of where is the first element of the th negative semicycle of , then On the other hand, from and we can obtain Therefore By the medium of (3.43), we have Hence where

Proof of Theorem 1.6. From the Propositions 3.13.4, we can easily know that system (1.11) is permanent. The proof is complete.

By a similar process as above, we can obtain the following result.

Corollary 3.5. Assume that and . If(C1)(C2) then system (1.11) is permanent.

Obviously, (B2) includes (C2), (B1), and (B3) include (C1). Thus, Corollary 3.5 generalizes and improves Theorem 1.4.

Acknowledgments

This work was supported by NNSF of China (10771032), the Natural Science Foundation of Ludong University (24070301, 24070302, 24200301), Program for Innovative Research Team in Ludong University, and China Postdoctoral Science Foundation funded project.