Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales
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Theoretical Studies on the Effects of Dispersal Corridors on the Permanence of Discrete PredatorPrey Models in Patchy Environment
Abstract
We study two discrete predatorprey models in patchy environment, one without dispersal corridors and one with dispersal corridors. Dispersal corridors are passes that allow the migration of species from one patch to another and their existence may influence the permanence of the model. We will offer sufficient conditions to guarantee the permanence of the two predatorprey models. By comparing the two permanence criteria, we discuss the effects of dispersal corridors on the permanence of the predatorprey model. It is found that the dispersion of the prey from one patch to another is helpful to the permanence of the prey if the population growth of the prey is density dependent; however, this dispersion of the prey could be disadvantageous or advantageous to the permanence of the predator. Five numerical examples are presented to confirm the theoretical results obtained and to illustrate the effects of dispersal corridors on the permanence of the predatorprey model.
1. Introduction
The conception of permanence in ecological communities is first discussed in [1]. This notion of permanence is also known under different aliases, such as permanent coexistence or uniform persistence. Aptly termed, permanence of a population model reflects that all the species in the ecosystem will coexist for a relatively long time; that is, no extinction of any species will occur in the near future—this is very important in biological conservation. A permanent model may attain globally asymptotically stable positive equilibrium [2] or may have periodic solution [3, 4], and even a chaotic model can be permanent [5]. The sufficient and/or necessary conditions established to guarantee the permanence of various population models may offer helpful suggestions for environmental managers to take applicable measures for the sustainability of the ecosystems [6–8].
It is common that in the real world today the habitats of the species are separated into isolated patches due to spatial heterogeneity or the development of mankind, such as the construction of highways or railways [9–16]. This patchy environment may restrict the activities of the species and lead to adverse effects on the permanence of the ecological communities; therefore, some dispersal corridors may be built to allow movement between the patches. For example, the construction of QinghaiTibet railway in China separates the habitats of the Tibetan antelopes to pieces of small patches. In order to reduce the impact of patchy environment on the Tibetan antelopes, several underpasses (dispersal corridors) along the railway are constructed for the Tibetan antelopes to move from one side of the railway to the other side [17]. Indeed some researches [18] have verified that the Tibetan antelopes have really made use of these underpasses to cross the railway.
In the literature there are quite many papers that discuss the permanence of continuous time predatorprey or competitive systems in patchy environment: for example, one can refer to [19–23] and the many references cited therein. Though some continuous time models have their discrete time analogues, it is common knowledge that these discrete time analogues may present different dynamics, which are possibly more complicated [24–26]. To our best knowledge, there are few references on discrete time predatorprey or competitive models in patchy environment; see, for instance, [22, 27–30]. In [28, 30], the effects of dispersion on discrete models for single species in patchy environment have been discussed; two patches have been considered in [30] while patches have been tackled in [28]. In [27, 29], the effects of dispersion on discrete competitive models in patchy environment have been analyzed; in particular discrete LeslieGower type competitive model has been adopted in [27]. Motivated by the above papers as well as the importance of permanence in ecosystems, in this paper we will investigate the permanence of discrete time predatorprey models of Ricker type in patchy environment. To be specific, we will tackle two discrete predatorprey models in patchy environment; the first model has no dispersal corridors while the second model has dispersal corridors. Sufficient conditions to guarantee permanence will be established for both models. The effects of dispersal corridors on permanence are then analyzed by comparing the two sets of sufficient conditions. We remark that in the literature the numerous research papers on continuous/discrete predatorprey models in patchy environment have focused mainly on the permanence criteria and/or the existence of periodic solutions of the models; however, the effects of dispersal corridors on the permanence of the models are rarely studied until the present paper.
The paper is organized as follows. In Section 2, we construct the two discrete predatorprey models in patchy environment with/without dispersal corridors; also in this section we give some preliminaries such as the definition of permanence. In Section 3, sufficient conditions for the permanence of the two discrete predatorprey models in patchy environment with/without dispersal corridors are offered. The effects of dispersal corridors on the permanence of the models are analyzed in Section 4. In Section 5, we present five examples where numerical simulations are performed to confirm the theoretical results and to illustrate the effects of dispersal corridors. Finally, we conclude in Section 6. The mathematical proofs of the theorems in Section 3 are presented in Appendix.
2. Models and Preliminaries
There are many types of discrete time mathematical models, which are governed by difference equations, established to reflect the fluctuation of the singlespecies population [31]. Among them, the Ricker type model, given as is first established for some kind of fishery population [32] and subsequently used to model other kinds of singlespecies population [31, 33, 34]. In model (1), is the population density of the species at the th time step, is the intrinsic growth rate of , is the strength of intraspecific competition of , and , are positive constants. Model (1) is logistic; that is, the fluctuation of the population of is undergoing the effect of density dependency. One can refer to [31] for more details about (1).
In the predatorprey model that we are considering in this paper, the prey is and the predator is , and the predator only feeds on the prey . Adopting model (1) to reflect the population growth of the prey , we have the following discrete predatorprey model of LotkaVolterra type: In model (2), is the intrinsic growth rate of the prey , is the intraspecific competition strength of the prey , is the predation strength, is the death rate of the predator if there is no prey for to feed on, reflects the predation efficiency or the rate of energy transferred from the prey to the predator through predation, and is the intraspecific competition strength of the predator . The parameters , , , and are all positive constants. The permanence and the global asymptotic stability of the positive equilibrium of model (2) have been discussed in [35, 36].
Now, suppose that the habitats of the predatorprey system (2) are separated into two patches, patch 1 and patch 2. We denote and as the respective population density of the prey and the predator in patch at the th time step, . Let be the intrinsic growth rate of the prey in patch and let be the death rate of the predator in patch , . Also, has the same biological meaning as in model (2) except that now it is the case in patch , , , . At this moment, there are no dispersal corridors between the two patches. Thus, we have the following discrete predatorprey model in patchy environment without dispersal corridors:
Next, we introduce dispersal corridors to the patchy model (3). It is assumed that the prey can migrate from one patch to another via these dispersal corridors, but the predator cannot use the dispersal corridors to migrate. We make some remarks to this assumption. Firstly, this assumption of only prey (but not predator) dispersal is usual in continuous/discrete time predatorprey models in patchy environment; see, for example, [22, 23, 37, 38]. Moreover, this assumption fits the common definition of prey refuge if the predator is fixed in one patch, and intensive studies have been done; refer to [39] and the references cited therein. Our assumption here is that the prey can move among patches while the predator is fixed in each patch—an example of such a system can be found in [40], where the prey (mayflies) is mobile among different patches, but the predator (charr) is confined to its patch since the river is fenced into patches. Indeed, systems of this type may exist naturally in swamp fields, where the prey (such as flying insects) can migrate to different ponds while the predator (such as fish) is fixed in each pond. Denote as the dispersal rate of the prey migrating from patch to patch , , , . We have the following discrete predatorprey model in patchy environment with dispersal corridors: Here, and represent the predation rates of the predator on the prey that migrates from another patch. Note that model (4) is reduced to model (3) if the dispersal rates and are both zero. Model (4) may be viewed as a discrete analogue of the continuous time predatorprey reactiondiffusion model for patchy environment proposed by Allen [41]. In [41] an extinction result, whereby some sufficient conditions are established to ensure the extinction of the species, has been developed. We remark that, from a practical standpoint, permanence results that guarantee species survival are more useful than extinction results; as such, we will focus on the permanence of the discrete model.
In view of the biological background, we let the initial values of models (3) and (4) be Also, throughout the paper we assume the following: (H_{1}), , , , , and , are all positive constants, and , are nonnegative constants.
It is obvious that all the solutions of either model (3) or model (4) with initial values (5) are positive, which corresponds well to the biological meanings of , , , noting that population density of any species is always nonnegative.
The permanence results of models (3) and (4) will be presented in Section 3. To be precise, we state the definition of permanence as follows.
Definition 1. Model (3) (or (4)) with initial values (5) is said to be permanent if there exist positive constants , , , and , , such that hold for each solution of model (3) (or (4)) with initial values (5).
3. Permanence of the Models
In this section, we will establish the permanence of models (3) and (4) with initial values (5). The mathematical proofs of Theorems 2, 4, and 5 will be presented in Appendix.
3.1. Permanence without Dispersal Corridors
We consider model (3) together with initial values (5). Notice that in (3) the first two equations are independent of the last two equations; therefore, we only need to consider the permanence of the model comprising just the first two equations of (3); namely, with initial values
Theorem 2. Suppose that are satisfied, where Then, model (7) with initial values (8) is permanent.
Remark 3. (a) The permanence or global asymptotical stability for the positive equilibrium of model (7) has been studied in [35, 36, 42]. In [35] it is shown that, for sufficiently small and , model (7) has a positive equilibrium which is globally attractive, and as a consequence model (7) is permanent. So the conditions for the permanence of (7) are not directly derived in [35]. On the other hand, based on the study of a nonautonomous version of (7), it is proved in [36] (Corollary 4) that the positive equilibrium of (7) is globally asymptotically stable if together with other conditions. Further, in [42] it is proved that the positive equilibrium of (7) can also be globally asymptotically stable if . It is known that a system is permanent if its positive equilibrium is globally asymptotically stable [8]. Hence, in [36, 42] once again permanence is a consequence but the conditions to ensure it are indirectly obtained.
(b) Our method to obtain the permanence of (7) (or (3)) is different from that in [35, 36, 42]. In particular, the inequalities and in Definition 1 can be verified here (refer to , , , and in Appendix). In most cases if it is proved that each solution of a model is bounded both below and above by some positive constants, then the model is considered to be permanent. However, it is not easy to show the inequalities such as and .
Using Theorem 2, we give the following theorem on the permanence of model (3).
Theorem 4. Suppose that conditions (9) and (10) are satisfied, and hold, where Then, model (3) with initial values (5) is permanent.
3.2. Permanence with Dispersal Corridors
The following theorem gives the sufficient conditions to guarantee the permanence of predatorprey model (4) in patchy environment with dispersal corridors.
Theorem 5. Suppose that are satisfied, where Then, model (4) with initial values (5) is permanent.
Remark 6. Unlike the proof of Theorem 4, in the proof of Theorem 5 (refer to Appendix) the inequalities , , , and in Definition 1 cannot be verified. This also reinforces Remark 3(b) where we indicate that such inequalities are not easy to prove in most cases.
4. Effects of Dispersal Corridors
In Section 3, sufficient conditions for the permanence of models (3) and (4) with initial values (5) are offered in Theorems 4 and 5, respectively. In this section, we will analyze the effects of dispersal corridors on the permanence of discrete predatorprey system by comparing the two sets of sufficient conditions obtained. Recall that the sufficient conditions to guarantee the permanence of predatorprey system without dispersal corridors in patchy environment are (9)–(13), while (15)–(18) are the sufficient conditions to guarantee the permanence of the similar system with dispersal corridors. From the proofs of Theorems 4 and 5 (see Appendix), these conditions can be classified in Table 1.
4.1. Effects of Dispersal Corridors on the Permanence of the Prey
We begin with conditions (9) and (15), which are mainly concerned with the permanence of the prey in patch 1 under the situation of without or with dispersal corridors. Rewrite (9) as Also, rewrite (15) as
On the left sides of (20) and (21), is the intrinsic growth rate of the prey in patch 1, which makes positive contribution to the population growth of the prey since it is greater than 0. The right sides of (20) and (21) can be understood as the negative contribution, caused by the intracompetition of species in patch 1, the predation by the predator in patch 1, and also the dispersion to patch 2 if dispersal corridors exist, to the population growth of the prey. Thus, and can be interpreted as the net reproduction rates of the prey in patch 1 without or with dispersal corridors, respectively. Consequently, the biological interpretation of (20) or (21) is that the prey in patch 1 is permanent if its net reproduction rate is positive.
The only difference between (20) and (21) is whether the dispersal rate is involved or not. Recall that is the dispersion rate of the prey from patch 1 to patch 2. It is obvious that or equivalently This inequality shows that the net reproduction rate of the prey in patch 1 is larger when dispersal corridors are introduced. Hence, the existence of dispersal corridors for the prey to migrate from patch 1 to patch 2 is helpful to the permanence of the prey in patch 1. Moreover, if , then always holds; but if , then we are not certain whether . In other words, with the intrinsic growth rate of the prey in patch 1 keeping the same value regardless of the existence or nonexistence of dispersal corridors, if the prey in patch 1 can be permanent without dispersal corridors, then it is also permanent in patch 1 with dispersal corridors; but if the prey in patch 1 is permanent with dispersal corridors, it may or may not be permanent without the dispersal corridors. In fact, the population growth of the prey in patch 1 is density dependent; hence, an appropriate dispersion from patch 1 to patch 2 may counterbalance the effect of density dependency. Certainly only the dispersion rate in an appropriate range is helpful to the permanence of the prey in patch 1. If the dispersion rate is too large, then the residence of the prey in patch 1 will decrease to a very small number, which will lead to a decrease in the intrinsic growth rate and will consequently increase the value of , the intraspecific competition strength of the predator in patch 1, due to limited food sources. Hence, in the case of large dispersion rate , the value of will decrease. If the decrease in is larger than the decrease in , then the net reproduction rate of the prey in patch 1 may decrease and (21) may not hold; that is, the permanence of the prey in patch 1 may not be guaranteed.
In summary, an appropriate degree of dispersion of the prey from patch 1 to patch 2, measured by the dispersion rate , is helpful to the permanence of the prey in patch 1. Similar conclusion can be drawn in patch 2 by comparing the condition (12) with (17). An appropriate dispersion rate of the prey from patch 2 to patch 1 is helpful to the permanence of the prey in patch 2.
4.2. Effects of Dispersal Corridors on the Permanence of the Predator
We begin with conditions (10) and (16), which are mainly concerned with the permanence of the predator in patch 1 under the situation of without or with dispersal corridors. Rewrite (10) as and rewrite (16) as For convenience, we also list the expressions of and (refer to Theorems 2 and 5) as follows:
On the left sides of (24) and (25), is the death rate of the predator in patch 1. The right sides of (24) and (25) can be understood as the positive contribution to the population growth of the predator in patch 1 due to the predation. Hence, and can be interpreted as the net reproduction rates of the predator in patch 1 without or with dispersal corridors, respectively. Consequently, the biological interpretation of (24) or (25) is that the predator in patch 1 is permanent if its net reproduction rate is positive.
There are two “places” in (25) where the dispersal rate appears; one is in the denominator of , while another is in the term . We will discuss the biological implications of them. First, it is clear that the that is explicit in the denominator of (25) has a diminishing effect on . Therefore, this dispersal rate in the denominator is disadvantageous to the net reproduction rate of the population growth, further the permanence, of the predator in patch 1. In fact, the migration of the prey from patch 1 to patch 2, while the predator remains fixed to patch 1, causes the decrease in the food sources of the predator, which in turn is disadvantageous to the permanence of the predator in patch 1.
Next, we consider the effect of the dispersal rate in the term . It is obvious from (26) that ; thus or equivalently (see (20) and (21)) which is exactly (23). We have observed earlier that the net reproduction rate of the prey in patch 1 with dispersal corridors is larger than that without dispersal corridors. In the following, we continue to analyze how this larger net reproduction rate of the prey in patch 1 impacts the net reproduction rate of the predator in patch 1.
Consider the function where in the interval . The derivative of is , so is monotonically increasing in . Thus, noting (27) we have Noting the definitions of and , inequality (29) shows that the dispersal rate which appeared in is advantageous to the increase of the net reproduction rate of the predator in patch 1, which in turn is helpful to the permanence of the predator in patch 1. In fact, from (27) (or (23)) this dispersion rate in increases the net reproduction rate of the prey in patch 1, which consequently increases the food supply for the predator in patch 1. As a result, this contributes positively to the net reproduction rate of the predator in patch 1, which further helps to sustain the permanence of the predator in patch 1.
In summary, the effects of the dispersion of the prey from patch 1 to patch 2, measured by the dispersal rate , on the permanence of the predator in patch 1 are twofold: one is disadvantageous—contributed by the presence of in the denominator of , while the other is advantageous—contributed by the presence of in . Similar conclusion can be drawn in patch 2 by comparing the condition (13) with (18). The dispersion of the prey from patch 2 to patch 1, measured by the dispersal rate , has multiple effects, both advantageous and disadvantageous, on the permanence of the predator in patch 2.
Remark 7. From the two sets of permanence conditions in Theorems 4 and 5, no conclusion can be drawn on the effects of on the permanence of the prey and the predator in patch 1; also the effects of on the permanence of the prey and the predator in patch 2 are unknown.
5. Numerical Simulation
In this section, we present five numerical examples to confirm the theoretical results obtained earlier as well as to illustrate the effects of the dispersal corridors. All the examples are processed by MATLAB. The first example below confirms the correctness of Theorem 4.
Example 8. Consider the following predatorprey model (3) in patchy environment without dispersal corridors: Direct computation gives Hence, the conditions of Theorem 4 are all satisfied. By Theorem 4, model (30) is permanent and this is illustrated in Figure 1.
The next example illustrates Theorem 5.
Example 9. Consider the following predatorprey model (4) in patchy environment with dispersal corridors: In model (32), the dispersal rates are and . Also, we set and ; that is, we assume that the predation rate of the predator on the prey which migrated from another patch is the same as the predation rate on the prey originally in the patch. By direct computation, we have all the conditions of Theorem 5 satisfied as follows: Hence, by Theorem 5 model (32) is permanent. Figure 2 illustrates the permanence of model (32).
Next, we present an example to show that dispersal corridors may be disadvantageous to the permanence of the ecosystem.
Example 10. Consider the following two models: The difference between (34) and (35) is only whether dispersal corridors exist or not; all other coefficients are the same. Model (34) is without dispersal corridors, and a direct computation shows that all the conditions of Theorem 4 are satisfied as follows: Hence, by Theorem 4 model (34) is permanent. Figure 3(a) illustrates the permanence of model (34).
(a)
(b)
Model (35) has dispersal corridors with , , and . Checking the conditions of Theorem 5, we find Hence, not all the conditions of Theorem 5 are satisfied and we cannot ensure the permanence of model (35). In fact, by numerical simulation we find that the predator in patch 1 is driven to extinction and so model (35) is nonpermanent. This is depicted in Figure 3(b).
This example shows that the dispersion of the prey from patch 1 to patch 2 by the dispersal corridors may be disadvantageous to the predator in patch 1, which has been discussed in Section 4.2. Further, it is observed from the above calculations that the introduction of increases the net reproduction rate () of the prey in patch 1 from 0.1230 to 0.1635, which has been noted in Section 4.1.
Next, we give an example to show that dispersal corridors may be helpful to the permanence of the ecosystem.
Example 11. Consider the following two models: The difference between (38) and (39) is only whether dispersal corridors exist or not; all other coefficients are the same. Model (38) is without dispersal corridors, and a direct computation yields Hence, not all the conditions of Theorem 4 are satisfied and we cannot guarantee the permanence of (38). Indeed, by numerical simulation it is found that the predator in patch 1 will be driven to extinction. So model (38) is nonpermanent and this is illustrated in Figure 4(a).
(a)
(b)
Model (39) has dispersal corridors with , , , and . Suppose By numerical simulation we find that the predator in patch 1 is now permanent, and model (39), as a whole, is permanent. This is depicted in Figure 4(b). Note, however, that model (39) does not satisfy some of the conditions of Theorem 5, as This simply reinforces the fact that the conditions in Theorem 5 are sufficient conditions.
We observe that the introduction of dispersal corridors leads to (i) an increase in the net reproduction rates of both the prey and the predator in patch 1 as well as the prey in patch 2 and (ii) a decrease in the net reproduction rate of the predator in patch 2. The overall effect on the permanence of model (39) is positive. This example shows that the introduction of dispersal corridors to a patchy ecosystem may be advantageous to the permanence of the ecosystem.
We will further investigate the effects of dispersal corridors on model (39). For this, in Table 2, for different values of and we list the net reproduction rate of the prey in patch , the net reproduction rate of the predator in patch , , and the permanence status of model (39). Recall that, for a model with dispersal corridors,

From Table 2, we observe the following. (a)The effect of dispersal corridors on the net reproduction rate of prey is positive; specifically, increases as increases and increases as increases. This coincides with the discussion in Section 4.1. (b)The effects of dispersal corridors on the net reproduction rate of predator are twofold: advantageous and disadvantageous. For a fixed , first increases and then decreases as the value of increases from 0 to 0.28. (c)The entire ecosystem cannot be permanent by the sole effect of (i.e., ). If the value of increases to sufficiently large, even the prey in patch 1 will undergo extinction due to unpredicted environmental changes, as the population density of the prey in patch 1 will become very small. If the prey in patch 1 becomes extinct, so will the predator in patch 1, since there are no food sources. (d)Noting (c) above, we further observe that the ecosystem is also nonpermanent if is too small; an appropriate value of is needed to induce a permanent (39).
So far equilibrium dynamics have been observed in Examples 8–11. Since it is well known that Ricker type model has complex dynamics, we give an example with more complex dynamics to illustrate our results.
Example 12. Consider the following two models:
The difference between (44) and (45) is only whether dispersal corridors exist or not; all other coefficients are the same. Model (44) is without dispersal corridors, and a direct computation yields
Though not all the conditions of Theorem 4 are satisfied, model (44) is permanent which is illustrated in Figure 5(a). It is observed from Figure 5(a) that the subsystem of patch 2 has a stable positive equilibrium while the subsystem of patch 1 does not have such stable equilibrium though it is permanent. Further illustration of this subsystem of patch 1 is given in Figure 5(b). Figure 5(b) is the phase diagram of and it is observed that the subsystem of patch 1 of model (44) has a periodic solution of period 3. In fact, the periodic solution is , , .
Model (45) has dispersal corridors with , , and . The net reproduction rates are computed as follows:
The dynamics of (45) are presented in Figure 6(a). It is observed that the predator in patch 2 will become extinct and consequently model (45) is not permanent. Figure 6(b) illustrates that the periodic solution observed in the subsystem of patch 1 of model (44) now vanishes due to the dispersion.
Comparing (46) and (47), we observe that when , (i) the net reproduction rate of the prey in patch 2 increases, which has been noted in Section 4.1 and (ii) the net reproduction rate of the predator in patch 2 decreases its value from positive to negative; this shows that the dispersion of the prey from patch 2 to patch 1 has an adverse effect on the permanence of the predator in patch 2, which has been discussed in Section 4.2. This example shows that large dispersal rates of the prey may destroy the permanence of a predatorprey system.
(a)
(b)
(a)
(b)
6. Conclusion
The paper is mainly involved with the effects of dispersal corridors on the permanence of discrete predatorprey models in patchy environment. We have introduced two models of Ricker type undergoing the effect of density dependency—model (3) is in patchy environment without dispersal corridors and model (4) is with dispersal corridors. Two sets of sufficient conditions to guarantee the permanence of the two models are then obtained through theoretical studies, respectively. In Theorem 4, conditions (9)–(13) guarantee the permanence of model (3); this result is illustrated by Example 8. In Theorem 5, conditions (15)–(18) ensure the permanence of model (4); this result is also numerically illustrated by Example 9.
Through mathematical analysis of the two sets of sufficient conditions, combined with the actual biological background of models (3) and (4), we have discussed the effects of dispersal corridors on the permanence of discrete predatorprey model. It is found that the existence of dispersal corridors for the prey to migrate, such as from patch to patch , while the predator remains fixed to the original patch, is helpful to the permanence of the prey in patch under density dependent circumstances due to an increased net reproduction rate of the prey in patch . Moreover, it is also found that the migration of prey, such as from patch to patch , is both advantageous and disadvantageous to the predator in patch because the dispersal rate has both positive and negative effects on the net reproduction rate of the predator in patch . Examples 10–12 are presented to illustrate the positive/negative effects of the dispersal corridors.
Appendix
In this section, we present the proofs of Theorems 2, 4, and 5. First we state a lemma which is useful in proving the theorems; this lemma is the autonomous case of Lemma 2 in [43].
Lemma 13. Suppose satisfies for , where and are positive constants, is a positive integer, and . Further, assume that and , where A is a positive constant. Then,
Proof of Theorem 2. We will first establish the first inequality in Definition 1. To begin, from the first equation of (7) we have
It is straightforward that the maximum of the function where in the interval is . Therefore, it follows from (A.3) that
Let be arbitrarily given. From (A.4), there exists a positive integer such that
Using (A.5) in the second equation of (7) gives
Hence, using the maximum of as earlier, we obtain
Since is arbitrary, letting in (A.7) yields
Now, in view of inequality (9), we can choose such that
Further, noting (A.8) we have from the first equation of (7)
for sufficiently large . Using the inequality which holds for , together with the definition of (see (A.4)), we find
or
Inequalities (A.4) and (A.12) imply that Lemma 13 can be applied to (A.10). Thus, we have
Again let in (A.13) and it follows that
It is clear from (9) that .
We will further show that . From the derivation of (A.12), we have
that is, , and so
Therefore, from the definition of (see (A.14)) we find
In summary, combining (A.4), (A.14), and (A.17) we get
In the following, we will establish the second inequality in Definition 1. In fact, we will prove that there exists such that
Recalling that and noting inequality (10), we can choose sufficiently small such that and . From the second equation of (7) and (A.14), we have
for sufficiently large . Using the inequality which holds, for , and the fact that , we find
Hence,
which, together with (A.8), indicates that we can apply Lemma 13 to (A.20) to give
Letting , we get
It is obvious from (10) that .
We will further show that . From (A.21), we have
and so
Combining (A.8), (A.24), and (A.26), we get
In view of (A.18) and (A.27), we have obtained both inequalities in Definition 1 and so the permanence of model (7) with initial values (8) is established. This completes the proof of Theorem 2.
Proof of Theorem 4. Using a similar argument as in the proof of Theorem 2, we see that, under the assumptions (12) and (13), the following model
with initial values , is permanent. In fact, analogous to (A.18) and (A.27), we have
where , , and are given in Theorem 4 and
Coupling with Theorem 2, the permanence of model (3) with initial values (5) follows immediately. This completes the proof of Theorem 4.
Proof of Theorem 5. Consider the following auxiliary system:
with initial values , .
Comparing (7) and (A.31), it follows from Theorem 2 that model (A.31) is permanent under conditions (15) and (16). Therefore, there exist positive constants and such that
From the first two equations of (4), it is clear that, for any , where represents the set of all positive integers, we have and . Hence,
Next, we consider the following auxiliary system:
with initial values , . By a similar argument as above, we see that (A.34) is permanent under conditions (17) and (18). Thus, there exist positive constants and such that
In view of the last two equations of (4), we have and