Sufficient and Necessary Conditions for the Permanence of a Discrete Model with Beddington-DeAngelis Functional Response
We give a sufficient and necessary condition for the permanence of a discrete model with Beddington-DeAngelis functional response with the form = / where , and are periodic sequences with the common period is nonnegative; , and are positive. It is because of the difference between the comparison theorem for discrete system and its corresponding continuous system that an additional condition should be considered. In addition, through some analysis on the limit case of this system, we find that the sequence has great influence on the permanence.
Many mathematical models have been established to describe the relationships between the species and the outer environment or among the different species in biomathematics. The dynamics of the growth of a population can be described if the functional behavior of the rate of growth is known. Of course, it is this functional behavior which is usually measured in the laboratory or in the field. Among the relationships between the species living in the same outer environment, the predator-prey theory plays an important and fundamental role. The dynamic relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. These problems may appear to be simple mathematically at first sight; they are, in fact, very challenging and complicated. Though most predator-prey theories are based on continuous models governed by differential equations, the discrete time models are more appropriate than the continuous ones when the size of the population is rarely small or the population has nonoverlapping generations. On the other hand, the concept of permanence has played an important role in mathematical ecology. Biologically, when a system of interacting species is persistent in a suitable sense, it means that all the species survive in the long term. For investigations on permanence of discrete predator-prey models, one can refer to [1–3] and references cited therein.
In 2006, Cui and Takeuchi studied the permanence, extinction, and periodic solutions for a predator-prey model with Beddington-DeAngelis functional response (see ); they gave a sufficient and necessary condition to guarantee the predator and prey to be permanent. As we all know, the continuous dynamic system and its corresponding discrete dynamic system, in some extent, have some similar properties, but also they have many differences. In this paper, we want to study the permanence for the following discrete predator-prey model with Beddington-DeAngelis functional response: where , and are periodic sequences with the common period is nonnegative; , and are positive. The reason and significance for the analysis on the properties of these biological models could be found in [5, 6]. The system (1) can be seen as the discrete form of the continuous situation which has been investigated in . And the discretization method could be found in .
As usual, we define the average value of periodic sequences with period as and we denote where .
In order to describe our main results, we need some lemmas below.
Lemma 1. If for all and , then has at least one nonnegative periodic solution . Moreover, if , then for all and if , then . Moreover, if , then the zero solution is globally asymptotically stable for any positive initial condition.
Proof. The existence conclusion could be found in . Now we only prove the globally asymptotical stability. That is, we consider the case . Notice that which implies that Therefore where represents the integer part of ; thus if , then . If , by (4), we have which implies that all the subsequences of are monotonically decreasing. Notice that they all have a lower bounded zero; thus, there exists some nonnegative constant such that . We claim that all . If all , then ; for sufficiently large, we have . While (7) implies that this contradiction shows that there exists at least one such that and then (4) implies that all . The proof is complete.
Remark 2. This lemma is different from the continuous one; here the condition can not support the globally asymptotical stability of (from the work of May  and Zhang and Zhou ). In addition, we can find that the continuous form of this lemma plays an important role in the proof of the permanence in .
Lemma 3. Assume that and are all -periodic sequences and is positive; if and the following inequality holds, then any solution for the periodic equation with positive initial condition has the property where is defined as that in Lemma 1.
Proof. By Lemma 1, we know that exists, and it is positive; for any positive solution of the equation denote and then satisfies Define , and then Notice that and here we use the inequality ; thus for any positive solution of (12), we have and by (10), we know that In particular, . Then equality (15) implies that ; that is, is nonincreasing; thus it converges, by (14), . The proof is complete.
Remark 4. In , Professor Zhou considered the existence and stability of the periodic solution of the equation . Here and are all positive -periodic sequences; under the condition the conclusion of Lemma 3 is satisfied. By Lemma 3, the condition (19) can be replaced by Notice that if , then the condition (19) can be simplified as In fact, in this case, by the work of Zhang and Zhou , the condition (21) could be generalized to It is worthy to say that, when , the conclusion of Lemma 3 is false. This is quite different from the corresponding continuous case. In particular, if , then (10) could be replaced by (22).
For the permanence of (1), we have the following.
Theorem 5. Assume that and are all -periodic sequences and is positive; if and hold true, then the system (1) is permanent and has at least one positive periodic solution provided that where is the unique periodic solution of given by Lemma 1.
2. Proof of the Main Results
Proof. Notice that, for any positive initial value, from mathematical induction, we can obtain that . Then we have Notice that If we let then by (27) and (28), we have If there exists some positive integer such that for , then . If there exists some positive integer such that for , then (30) implies that the sequence converges to zero, which shows that ; thus also hold true. If the sequence oscillates about zero, let be the first element of the th positive semicycle of the sequence ; then from (30), we have and therefore and from the above analysis, we can obtain From the second equation of (1), we have If does not oscillate about , then from (34) we have Otherwise, if we let be the first element of the th positive semicycle of the sequence , then from (34), we know that and by (35) and (36), we have The proof is complete.
Proof . We prove it by contradiction. If (38) is false, then for every , there exists a solution with initial condition such that
Choose sufficiently small positive constants and such that
From (39), for any given , there exists a positive integer such that
Equation (42) shows that there exists a sufficiently large such that
Then the second equation of (1) now yields
and by (40), we have
This implies that the subsequence of is monotonically decreasing; thus it is convergent; by (45),
Notice that is bounded; thus by (44), using mathematical induction, we can easily obtain
and thus there exists a sufficiently large such that
therefore the first equation of (1) yields
By (41), utilizing Lemma 1, we know that the equation has at least one positive -periodic solution called . It is obvious that is independent of . Let
If does not oscillate about zero, then
which implies that when is large enough,
If oscillates about zero, let be the first element of the th negative semicycle of the sequence ; then from (51), we know that Notice that and thus when is sufficiently large, Inequalities (53) and (56) imply that when is sufficiently large, . Since is independent of , then and this contradicts (42). The proof is complete.
Proof. If (58) is not true, then, for any , there exists a positive initial value which may be dependent on such that
where is the solution of (1) with positive initial values .
By (25) and (24), we can choose the constant and sufficiently small such that From (59), we can choose sufficiently large such that ; then the first equation of (1) implies that By (61) and Lemmas 1 and 3, the following equation has a unique positive -periodic solution for any sufficiently small positive number and We claim that, for any , there exists a sufficiently large such that In fact, by (62) and (63), if we set where the sequence is the solution of (63) with initial condition , then Thus Denote and then (68) implies that Notice that and by (23), we have Define a function Then From (70), we can obtain . By (73), (74), and (72), we know that which implies that Under condition (23), by Lemma 3, we have and then (64), (77), and (76) imply that (65) holds. Now the second equation of (1) yields and thus (60) shows that This contradicts (59). The proof is complete.
Proof. If (80) is false, then there exists an initial value such that for any , where represents the solution of (1) with initial value . Thus, there is a subsequence of such that On the other hand, by Lemma 9, there exists a constant (which is independent of the initial value ) such that Notice that, for any , there exists a which satisfies , (82), and (83). Choose sufficiently large such that Obviously, such exists. Fixing it, and by the first equation of (1), we know and by mathematical induction, we can easily obtain and this contradicts (83). The proof is complete.
Proof. If (87) is false, then for any , there exist an initial value and a positive integer sequence such that
where represents the solution of (1) with initial value .
From the proof of Lemma 10, we can find that, for any , when is sufficiently large, , here is sufficiently large. On the other hand, by Lemma 10, there also exist a constant (which is independent to ) and a subsequence of such that
The rest of the proof is similar to that of Lemma 11; we omit it here.
Proof of Theorem 6. Assume that (1) is permanent; then there exist two constants and such that
for simplicity, the inequality holds true only for sufficiently large ; we omit the explanation of the domain for in what follows. Choose sufficiently small such that
Consider the following equation:
by (93) and Lemma 3, (94) has a unique positive solution which is globally asymptotically stable. Obviously,
In addition, . Notice that
then (98) implies that
this is a contradiction.
If then the second equation of (1) implies that and by (101), then (102) also implies (100), which also contradicts the permanence. The proof is complete.
3. Examples and Discussion
In this section, we give some examples to illustrate our main results and also give some discussions.
Example 1. Let ,. Then condition (10) holds true; by Lemma 3, we know that the periodic equation has only one positive 3-periodic solution. By numerical computation, we know ,,. See Figure 1 for more assistance.
Example 2. Let and are defined as in Example 1, ,,,,, and .
By the above definition, we can easily obtain and by Theorem 5, we know that system (1) is permanent and has at least one positive periodic solution. In fact, from the numerical results, we can get one of the periodic solutions: ,, ,,. See Figure 2.
Now let us go back to conditions (10) and (23). Obviously, condition (23) can be included by (10), and condition (10) assures the globally asymptotical stability of the positive periodic solution of (12). But under condition (10), for the solution of the inequality , we can not obtain for sufficiently large; for the solution of the inequality , we can not obtain for sufficiently large. One crucial reason is that the comparison theorem for discrete system is quite different from its corresponding continuous system. In , we gave a qualitative analysis for the permanence of ratio-dependent prey-predator model; there we used the comparison theorem of difference equations; unfortunately, an additional condition should be considered, while in  we deleted this additional condition (the method we used is not the comparison theorem of difference equations but is the semicycle theory). In the present paper, can condition (23) be deleted? We leave it for further investigation.
In addition, we consider an extreme situation: . In this extreme case, we have already obtained the permanence for system (1) (see ). Recall that the condition which assures the permanence is and . Let us see the limit condition under this extreme situation. Now condition (24) was deduced to , while the condition remains unchanged. Notice that the deduced condition is independent of the sequences and