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Backward bifurcation of predator–prey model with anti-predator behaviors
- Guangyao Tang^{1} and
- Wenjie Qin^{2}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-019-1944-4
© The Author(s) 2019
- Received: 9 August 2018
- Accepted: 1 January 2019
- Published: 10 January 2019
Abstract
In this study, we consider a predator–prey model with stage structure and anti-predator behavior such that the adult prey can counterattack their predators. We first investigate the existence and stability of the equilibria. Especially, we verify that there can exist at most one positive equilibrium, which is always stable whenever it exists, if the predator only feed on one age class. We then prove that the system can undergo either a forward bifurcation or a backward bifurcation. Numerical analyses show that anti-predator behavior is beneficial to the growth of prey population, especially helps the equilibrium level of the prey population increase, by enhancing the pressure on the predator. Moreover, anti-predator behavior makes the coexistence of the predator and prey less likely by shrinking the coexistence region with respect to the initial conditions or weakening the existence and stability of the positive equilibrium.
Keywords
- Predator–prey system
- Anti-predator behavior
- Stage structure
- Backward bifurcation
1 Introduction
Although biologists routinely label the animals as predator or prey, there are many examples of role reversals in predators and prey (anti-predator behaviors) [1–3]. That is, juvenile prey that escape from predation and become adult can counterattack juvenile predators, and adults just kill the juveniles but do not consume them, which can serve to reduce future predation risk [4]. Therefore, it is very important to evaluate the cyclic dominance for predator–prey interactions when anti-predator behaviors occur.
The dynamical relationship between predators and their preys has been considered in depth in a mass of studies [5–10]. In these studies, both predators and preys are assumed to be homogeneous. However, the anti-predator behavior of adult prey for juvenile predators indicates that we should take the age class structure of both predators and preys into consideration. Actually, it has been recognized for a long time that the age class structure of both predators and preys has a great influence on the dynamics of the interactions between these species [11, 12]. For example, Dörner et al. reported that Perca fluviatilis play an essential role in structuring the fish community because it is important in controlling the juvenile fish abundance [13].
Many studies investigated the dynamics of the predator–prey system with stage structure. In 1990, Aiello et al. studied a single specie model with stage structure [14]. Then the dynamics of the single population with stage structure, especially the bifurcation phenomenon, has been discussed in depth in [15–17]. By dividing the prey into multure and immature subpopulations, many researchers investigated the predator–prey system through autonomous models [18, 19], periodical models [20–22], delay differential equations [23–25] and partial differential equations [26]. Many other researchers also considered how the stage structure for predator [27–31] or for both predator and prey [32–34] affects the dynamics of the predator–prey system. Particularly, in 2000, Zhang et al. proposed a basic predator–prey model with stage structure for the prey [35], where they assume that the birth rate of the adult predator is proportional to its existing population. In 2015, Falconi et al. considered the carrying capacity of the habitat for the juvenile class [36]. Recently, Coast et al. introduced an exponential density dependence for the fecundity of adult preys [37] with a particular focus on the impact of culling predators for the prey. In this study, the authors have assumed the density of the predator population to be a constant.
The paper is organized as follows. In Sect. 2, we mainly discuss the existence and stability of the equilibria, especially, discuss the existence and stability of the equilibria for two subcases that the predator only feed on the juvenile prey or the adult prey. In Sect. 3, we prove that the proposed model can undergo a forward bifurcation or a backward bifurcation, which are the common bifurcations in epidemic systems discussed in [40, 41]. In Sect. 4, we provide some numerical simulations to show how the anti-predator behavior affects the dynamics of the predator–prey system. In Sect. 5, we make the conclusion and discussion of this study.
2 The existence and stability for the equilibria
Theorem 2.1
System (1) always has a trivial equilibrium \(E_{0}\), and it is locally asymptotically stable when \(R_{0}<1\). Meanwhile, if the inequality \(R_{0}>1\) holds true, there exists a predator-extinction equilibrium Ê. Furthermore, if \(\eta >\eta ^{*}\) (or \(\eta <\eta ^{*}\) and \(1< R_{0}< R^{*}\)), then the predator-extinction equilibrium Ê is locally asymptotically stable.
Theorem 2.2
3 Bifurcation analysis
Theorem 3.1
If \(\varLambda <0\), then there is a stable positive equilibrium near Ê for \(\delta ^{*}-\varepsilon _{1}< \delta <\delta ^{*}\) and system (1) undergoes a forward bifurcation at \(\delta =\delta ^{*}\). If \(\varLambda >0\), then there is an unstable positive equilibrium near Ê for \(\delta ^{*}<\delta < \delta ^{*}+\varepsilon _{2}\) and system (1) undergo a backward bifurcation at \(\delta =\delta ^{*}\).
Remark 3.2
Remark 3.3
4 Numerical simulations
Because \(R_{0}\) is independent on the parameters δ and η, they cannot change the stability of the trivial equilibrium. Thus, we then let b be a bifurcation parameter and fix \(\delta =2\) and \(\eta =0.01\), Figs. 3(D)–(F) show that system (1) first undergoes a forward bifurcation at \(b=1.25\) with the trivial equilibrium losing its stability and a stable predator-extinction equilibrium emerging. When b continues increases to 43.56, a saddle-node bifurcation occurs and then there exist an unstable positive equilibrium and a stable positive equilibrium (which is bistable with the predator-extinction equilibrium). Then, system (1) undergoes a backward bifurcation (transcritical bifurcation) at \(b=68.2\), here, the predator-extinction equilibrium loses its stability while the unstable positive equilibrium disappears. As we can see from Figs. 3(D)–(F), increasing b can increase the population of the prey. However, different from the anti-predator behavior, increase the birth rate of the prey can also increase the population of the predator.
5 Conclusion and discussion
This paper proposed a predator–prey model with stage structure for prey such that the adult prey can counterattack their predators. Firstly, the existence and the stability of the equilibria was discussed through exploring the characteristic equations. It found that there is always a trivial equilibrium which is stable when \(R_{0}<1\) and becomes unstable if \(R_{0}>1\). Correspondingly, there emerges a predator-extinction equilibrium when \(R_{0}>1\). Meanwhile, it is verified that system (1) undergoes a forward bifurcation at \(R_{0}=1\). If \(\eta <\eta ^{*}\) holds true, the predator-extinction equilibrium has single zero eigenvalue while the other two eigenvalues have negative real parts at \(R_{0}=R^{*}\). Based on this condition, system (1) may undergo either a forward bifurcation or a backward bifurcation by choosing the death rate of the predator as a bifurcation parameter. Furthermore, we also discussed the existence and stability of the equilibria for two special cases. The results show that if the predator only feeds on one age class, the backward bifurcation could not happen, and the system can have at most one positive equilibrium, which is stable whenever it exists.
Numerical analysis shows that the predator can coexist with the prey in the term of a stable positive equilibrium if the rate of anti-predator behavior is relatively small, as shown in Figs. 3(A)–(C). As expected, the anti-predator behavior is beneficial to the growth of both the juvenile and the adult prey population through inhibiting the growth of the predator population. Also, the anti-predator behavior can weaken the stability of the positive equilibrium, while it enhances the stability of the predator-extinction equilibrium in terms of increasing the attraction area. These results are in agreement with the main results obtained in [38]. Correspondingly, when η exceeds the backward bifurcation point, the positive equilibrium is bistable with the predator-extinction equilibrium. We then showed that anti-predator behavior can make the coexistence of the prey and predator less likely by shrinking the stable region of the positive equilibrium when it is bistable with the predator-extinction equilibrium. Moreover, if the prey can further improve their anti-predator behavior, the predator population would become extinct with the stable positive equilibrium disappearing. It should be noticed that the impact of the anti-predator behavior also depends on the characters of the predator–prey system. For example, if the birth rate of the adult prey is not enough to support the coexistence of the prey and the predator, then the anti-predator behavior would not affect their dynamics.
Our model uses the simple bilinear terms to represent the anti-predator behavior and the functional response of the predation. The dynamics of the predator–prey system can be very complex if we take the other Holling type functional response into consideration. Our model should also incorporate the seasonal factor and the delay effect between the reproduction and the predation of the predator when analyzing the impact of the anti-predator behavior. Addressing these issues needs more future work.
Declarations
Acknowledgements
We would like to thank the anonymous referees for their helpful comments and the Editor for his constructive suggestions, which greatly improved the presentation of this paper.
Availability of data and materials
Data sharing is not applicable to this article as no data sets were generated or analysed during the current study.
Funding
This work was partially supported by National Natural Science Foundation of China (No.: 11761031, Guangyao Tang & No.: 11601268, Wenjie Qin), Educational Commission of Hubei Province (No.: Q20161212, Wenjie Qin), and the Youth Foundation of Hubei University for Nationalities(No.: MY2017Q007).
Authors’ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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