Abstract
In this paper, we propose and analyze a commensalism model with nonmonotonic functional response and density-dependent birth rates. The model can have at most four nonnegative equilibria. By applying the differential inequality theory, we show that each equilibrium can be globally attractive under suitable conditions. However, commensalism can be established only when resources for both species are large enough.
1. Introduction
Commensalism is a long-term biological interaction in which members of one species gain benefits while those of the other species neither benefit nor are harmed. An example of it is that remora are specially adapted to attach themselves to larger fish that provide locomotion and food. In the last decades, commensalism has attracted the attention of many researchers ([1–16]). Complicated dynamics have been found in the study. For example, in [3], Lin considered the effects of partial closure and harvesting. Depending on the size of the harvesting area, species can go extinct, partially survive, or become permanent. He also showed in [4] that the final density of the species increases as the Allee effect increases. This is quite different from results for predator-prey system with Allee effect.
Recently, Chen and Wu [5] proposed the following two species commensal symbiosis models with nonmonotonic functional response:where are all positive constants. System (1) admits four nonnegative equilibria, , , , and , where The stability of the equilibria is summarized as follows (see Theorem 2.1 and 2.2 in [5] for more detail).
Theorem A. , , and are unstable; is globally asymptotically stable.
When the interaction between the species is ignored, the growth for both species is described by traditional logistic equations. Indeed, without the presence of , the growth of the first species takes the formwhere is the intrinsic growth rate and is the density-dependent coefficient or the interspecific competition coefficient. However, in most situations, the intrinsic growth rate is not always constant. One model incorporating nonconstant intrinsic growth rate is the following density-dependent model:For more details, see [6–8]. Combining this with (1), we propose the following commensalism model:where () and , and are all positive constants. Here and are the densities of the first and second species at time , respectively. and stand for the total resources available per-unit-time for species and , respectively.
The aim of this paper is to investigate the attractivity of equilibria of (5). The main tool is the differential inequality theory or comparison principle. To the best of our knowledge, this is the first time to use differential inequality in this direction for ecosystems. The rest of the paper is arranged as follows. In Section 2, we obtain the existence and global attractivity of equilibria of system (5). Section 3 is devoted to illustrating the feasibility of the main results through numeric simulations. We end this paper by a brief discussion.
2. The Main Result
We first consider the existence of equilibria of (5). An equilibrium of (5) satisfies the equilibrium equations,If then (7) only has the unique nonnegative solution while if , then, besides , (7) also has a unique positive solution . Substituting into (6), we see that if , then is the only nonnegative solution while if , besides , (6) also has a unique positive solution . Similarly, substituting into (6), we can get that is the only nonnegative solution if while, besides , (6) also has a unique positive solution if , where . In summary, we have obtained the following result.
Proposition 1. The following statements on equilibria of (5) are valid. (i)If and then there is only the trivial equilibrium .(ii)If and then, besides , there is also the nontrivial boundary equilibrium .(iii)If and then there are only the two equilibria and .(iv)If and then there are only three equilibria , , and .(v)If and then there are only four equilibria , , , and .
Before analyzing the stability of the equilibria of (5), we first consider the dynamic behavior of the following equation: with . Clearly, every such solution of (8) is nonnegative.
Lemma 2. The following statements on (8) hold. (i)If then the unique positive equilibrium is globally attractive in .(ii)If then the equilibrium is globally attractive in .
Proof. DenoteNote that(i) When , it is easy to see that only has the unique positive zero . Observe that for and for . It follows easily that if ; that is, is globally attractive in .
(ii) When , clearly can not have positive zero and hence is the only equilibrium. As for , we obtain . This completes the proof.
Now we are ready to study the attractivity of the equilibria of (5).
Theorem 3. (i) Assume that and . Then is globally attractive in .
(ii) Suppose that and . Then is globally attractive in .
(iii) Assume that and . Then is globally attractive in .
(iv) Assume that and . Then the unique positive equilibrium is globally attractive in .
Proof. First, assume that . By Lemma 2 (ii), we have for .
(i) As , we can choose small enough so that . For this , there exists such that for . This, together with the first equation of (5), givesIt follows from the choice of , Lemma 2 (ii), and the comparison principle that . Therefore, ; that is, is globally attractive in .
(ii) On the one hand, for any such that , there exists such that for . Thus it follows from the first equation of (5) thatNote that . By comparison principle and Lemma 2 (i), Letting gives . On the other hand, note thatAgain, by comparison principle and Lemma 2 (i), we have . It follows that . In summary, ; namely, is globally attractive in .
Now suppose that . Then for by Lemma 2 (i).
(iii) Since , we choose sufficiently small so that , where . For this , there exists such thatThis, combined with the first equation of (5), givesThen by the choice of , Lemma 2 (ii), and the comparison principle. Thus we have shown ; that is, is globally attractive in .
(iv) This time . For any such that and , there exists such thatwhere . Again, employing the first equation of (5), we havefor . Note that . Applying Lemma 2 (i) and comparison principle again, we havewhere Letting , we get and hence . It follows that and so is globally attractive in . This completes the proof.
3. Numeric Simulations
In this section, we provide numeric simulations to illustrate the four situations in Theorem 3.
Example 4. Let , , , , , , , , , , , and . Then (5) becomesClearly, and . By Theorem 3(i), the boundary equilibrium is globally attractive. Figure 1 strongly supports it.
Example 5. ConsiderCorresponding to (5), . Obviously, and . It follows from Theorem 3 (iv) that is globally attractive (see Figure 2).
Example 6. Considerthat is, we take in (5). This time, and . Therefore, is globally attractive by Theorem 3 (ii), which is illustrated by Figure 3.
Example 7. Finally, let in (5); that is, considerNote that . We can calculate that and . Then we see that . Thus it follows from Theorem 3 (iii) that is globally attractive (see Figure 4).
4. Discussion
In this paper, inspired by the work in [17–19], we proposed a commensalism model under the assumption that the intrinsic growth rates of both species are density-dependent. The model can have at most four equilibria. For the first time, differential inequality has been applied to obtain the global attractivity of equilibria of such ecosystem models. Depending on the availability of resources, each of the possible equilibria can be globally attractive. This implies that density-dependent birth rates play an important role in the dynamics. Though the dynamics can be complicated, from the point view of commensalism, commensalism can be established only when resources for both species are large enough (see Theorem 3 (iv)). Hence, these results agree with those of Chen and Wu [5] (see Theorem A in Introduction).
As we know, delay always exists in many biological processes. We will leave the effect of delay on the dynamics for future study.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
The authors would like to thank Professor Yuming Chen of Wilfrid Laurier University for modifying the language of the article and Dr. Xiaofeng Chen for useful discussion about the mathematical modeling. The research was supported by the National Natural Science Foundation of China under Grant 11601085, the Natural Science Foundation of Fujian Province (2017J01400), Guangxi College Enhancing Youths Capacity Project (2017KY0599), the Science Research Development Fund of Youths Researchers of Guangxi University of Finance and Economics (2017QNB18) and the Basic Ability Promotion Plan of Youth in Guangxi Universities (KY2016YB404).