On a Generalized Discrete Ratio-Dependent Predator-Prey System

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.


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 y y μϕ x y − D .

1.1
Here the predator-prey interactions are described by ϕ x/y ; this function replaces the functional response function ϕ x 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  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 ratiodependent predator-prey model.
For the sake of convenience, we introduce some notations and definitions.Denote Z, R, and R as the sets of all integers, real numbers, and nonnegative real numbers, respectively.Let C denote the set of all bounded sequences f : Z → R, C is the set of all f ∈ C such that f > 0, and We also define when f is a periodic continuous function with period ω.
In view of the periodicity of the actual environment, we begin with the following periodic continuous ratio-dependent predator-prey system: x t /y t denotes the ratio-dependent response function, which reflects the capture ability of the predator.Here we assume that g u satisfies the following monotonic condition, for short, we call it M : In 10 , we gave a sufficient condition for the permanence of the continuous model 1.9 In which a t , b t , c t , d t , and e t are all positive periodic continuous functions with period ω > 0; τ is a positive constant.In addition to condition M , the functional response function g also satisfies the following.
iv There exists a positive constant h such that u 2 g u ≤ h.
Without loss of generality, in this paper, we always assume that α 1 if α / 1, let c * t αc t and still denote c * t as c t .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 0 < m < ∞.
Through the above assumptions, we can see that, one of the main results in 10 can be given as follows.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 t denotes the integer part of t t > 0 and h N Correspondingly, the basic assumptions of 1.11 is the same as that in 1.8 , of cause, here b, d : Z → R and a, c, e : Z → R are periodic sequences with period ω > 0 and ω−1 k 0 b k > 0, ω−1 k 0 d k > 0, and g satisfies M .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 N 0 > 0, N k remains positive.In the remainder of this paper, for biological reasons, we only consider solutions N k with System 1.11 includes many biological models as its special cases, which have been studied by many authors; see 17-20 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: Then 1.13 has at least one positive ω-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 1.11only 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 τ 1 0 and obtained the following conclusion.
hold, where 1.17 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.
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 24-33 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.

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 λ 1 , λ 2 such that Lemma 2.2 20 .The problem has at least one periodic solution U if b ∈ C ω , a ∈ C and a is an ω-periodic sequence with a > 0, moreover, the following properties hold.
a U is positive ω-periodic.
b U has the following estimations for it's boundary: Lemma 2.3.For any positive constant K, the problem has at least one periodic solution U if d ∈ C and d is an ω-periodic sequence provided that (H2) holds.Moreover, the following properties hold.
a U is positive ω-periodic.
b U has the following estimations for its boundary:

2.10
We claim that there exist some n 1 and n 2 such that

2.11
If this is not true, then either 12 or for any n ∈ I ω , in any case, we can obtain ω−1 n 0 e n g K/x n / dω, this contradiction shows that our claim is true.
Note that for any n ∈ I ω , then by virtue of equality 2.10 , we complete the proof.

2.15
If b ∈ C ω , a ∈ C, and a is an ω-periodic sequence with a > 0, then any positive solutions x k of 2.15 satisfy where where

19
Proof.Consider the following auxiliary equation: by Lemma 2.2, 2.20 has at least one positive ω-periodic solution, denote it as z * k , then

2.23
Make the transformation u k u 1 k − u 2 k , we can obtain

2.24
Now we divide the proof into two cases according to the oscillating property of u k .First we assume that u k does not oscillate about zero, then u k will be either eventually positive or eventually negative.If the latter holds, that is, u 1 k < u 2 k , we have

2.25
Either if the former holds, then by 2.24 , we know u k 1 < u k , which means that u k is eventually decreasing, also in terms of its positivity, we know that lim k → ∞ u k exists.Then 2.24 implies lim k → ∞ u k 0, which leads to lim sup

2.26
Now we assume that u k oscillates about zero, by 2.24 , we know that u k > 0 implies u k 1 ≤ u k .Thus, if we let {u k l } be a subsequence of {u k }, where u k l is the first element of the lth positive semicycle of {u k }, then lim sup k → ∞ u k lim sup l → ∞ u k l .For the definition of semicycle and other related concepts, we refer to 34 .Notice that and u k l − 1 < 0, then we know

2.28
Therefore By the medium of 2.22 , 2.25 , and 2.26 , we have lim sup

2.30
Corollary 2.5.Any positive solution of the inequality problem 2.15 satisfies where a ∈ C, a M > 0 and b ∈ C .
Proof.Define the function it is easy to see which immediately leads to lim sup

2.34
From 2.15 , we have By Lemma 2.4, for any ω > 0, lim sup 37 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 x k of the inequality problem
here H is a positive constant.Then if b ∈ C ω , a ∈ C, and a is an ω-periodic sequence with a > 0, one has

2.40
Moreover, if a ∈ C ω , then where The proof is similar to that of Lemma 2.4.

Corollary 2.8. If any positive solution x k of the inequality problem
here H is a positive constant, then where a ∈ C and b ∈ C .

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
Proof.Given any positive solution N 1 k , N 2 k of 1.11 , from the first equation of 1.11 , we have which is equivalent to

3.6
Therefore By Lemma 2.4, we have lim sup where 3.9 Proposition 3.2.Under condition (H1), there exists a positive constant Proof.Given any positive solution N 1 k , N 2 k of 1.11 , from the first equation of 1.11 , we have 12 that is,

3.15
Since H1 holds, then by Lemma 2.7 and Proposition 3.1, we have lim inf

3.16
Proposition 3.3.If (H2) holds, then there exists a positive constant K 2 such that lim sup 3.17 Proof.Given any positive solution N 1 k , N 2 k of 1.11 .Set N 2 k exp{u 1 k }, from the second equation of 1.11 and notice that conditions ii and iii on g imply that g k, u k ≤ 1, then which is equivalent to

3.21
Therefore for any given ε > 0, we have for sufficiently large k.Here we use the monotonicity of the function g u .
Consider the following auxiliary equation: By Lemma 2.3 and condition H2 , we can obtain that 3.23 has at least one positive ωperiodic solution, denote it as z * 1 k and where First we assume that u k does not oscillate about zero, then u k will be either eventually positive or eventually negative.If the latter holds, that is, u 1 k < u 2 k , we have

3.29
Either if the former holds, then by 3.28 , we have u k 1 < u k , which means that u k is eventually decreasing, also in terms of its positivity, we obtain that lim k → ∞ u k exists.Then 3.28 leads to lim k → ∞ u k 0, this implies lim sup

3.30
Now we assume that u k oscillates about zero, in view of 3.28 , we know that u k > 0 implies u k 1 ≤ u k .Thus, if we let {u k l } be a subsequence of {u k } where u k l is the first element of the lth positive semicycle of {u k }, then lim sup Proof.Given any positive solution N 1 k , N 2 k of 1.11 , from the second equation of 1.11 , we have

3.38
Therefore from the second equation of 1.11 , we have Consider the auxiliary equation

3.44
And let u k u 1 k − u 2 k , we have If u k does not oscillate about zero, then by a similar analysis as that in Proposition 3.1, we have lim inf

3.46
Otherwise, if u k oscillates about zero, by 3.45 , we know that u k < 0 implies u k 1 ≥ u k .Thus, if we denote {u k l } as a subsequence of {u k } where u k l is the first element of the lth negative semicycle of {u k }, then lim inf k → ∞ u k lim inf l → ∞ u k l .On the other hand, from

3.51
Proof of Theorem 1.6.From the Propositions 3.1-3.4,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.Obviously, B2 includes C2 , B1 , and B3 include C1 .Thus, Corollary 3.5 generalizes and improves Theorem 1.4.
and y t represent the densities of the prey population and predator population at time t respectively; τ 1 ≥ 0, τ 2 ≥ 0 are real constants; b, d : R → R and a, c, e : R → R are continuous periodic functions with period ω > 0 and eProof.We only prove that 2.6 holds, for the rest of the proof, one can refer to 17 .Let x n be any possible ω-periodic positive solution of 2.5 , then if d ∈ C ω , where g −1 represents the inverse of g.