- Open Access
Global stability of a modified Leslie-Gower model with Beddington-DeAngelis functional response
© Yu; licensee Springer 2014
- Received: 21 October 2013
- Accepted: 24 February 2014
- Published: 13 March 2014
A predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response is studied. The local stability of the equilibria and the permanence of the system are investigated. By applying the fluctuation lemma, qualitative analysis and Lyapunov direct method, respectively, three sufficient conditions on the global asymptotic stability of a positive equilibrium are obtained.
MSC:34D23, 92D25, 34D20, 34D40.
- global asymptotic stability
- Lyapunov function
- Dulac function
- fluctuation lemma
where , stand for the population (the density) of the prey and the predator at time t, respectively. The parameters and are the intrinsic growth rates of the prey and the predator, respectively. measures the strength of competition among individuals of species x. The value is the carrying capacity of the prey in the absence of predation. The predator consumes the prey according to the functional response and carries capacity . The parameter is a measure of the food quantity that the prey provides converted to predator birth. The term is the Leslie-Gower term which measures the loss in the predator population due to rarity (per capita ) of its favorite food. Leslie model is a predator-prey model where the carrying capacity of the predator is proportional to the number of prey, stressing the fact that there are upper limits to the rates of increase in both prey x and predator y, which are not recognized in the Lotka-Volterra model. These upper limits can be approached under favorable conditions: for the predators, when the number of prey per predator is large; for the prey, when the number of predators (and perhaps the number of prey also) is small . For more details of the model, one can see [4–9] and the references cited therein. Holling  suggested three different kinds of functional response for different kinds of species to model the phenomena of predation, which made the standard Lotka-Volterra system more realistic. When , the functional response is called Holling-type II.
where , , , have the same meaning as in models (1.1). is the maximum value of the per capita reduction rate of x due to y, (respectively, ) measures the extent to which the environment provides protection to prey x (respectively, to the predator y). The authors studied the boundedness and global stability of positive equilibrium of system (1.2). Since then, system (1.2) and its non-autonomous versions have been studied by incorporating delay and impulses, harvesting and so on (see, for example, [12–29]). In , we studied the structure, linearized stability and the global asymptotic stability of equilibria of (1.2). Nindjin et al.  incorporated time delay to system (1.2) and studied the global stability and persistence of the delayed system by using the Lyapunov functional. Yafia et al.  and  further studied the occurrence of Hopf bifurcation at equilibrium by using the time delay as a parameter of bifurcation. Nindjin and Aziz-Alaoui  studied uniform persistence and global stability of three Leslie-Gower-type species food chain system. Gakkhar and Singh  studied the dynamic behaviors of a modified Leslie-Gower predator-prey system with seasonally varying parameters. Guo and Song , Song and Li  further considered the influence of impulsive effect. Zhu and Wang  obtained sufficient conditions for the existence and global attractivity of positive periodic solutions of system (1.2) with periodic coefficients. Liu and Wang  considered a stochastic predator-prey system with modified Leslie-Gower and Holling-type II schemes with Lévy jumps. The results showed that the Lévy jumps can change the properties of the population systems significantly. Kar and Ghorai  obtained local stability, global stability, influence of harvesting and bifurcation of a delayed predator-prey system in the presence of harvesting. Two stage-structured predator-prey models with modified Leslie-Gower and Holling-type II schemes were studied in [23–25]. Gupta and Chandra  discussed the permanence, stability and bifurcation (saddle-node bifurcation, transcritical, Hopf-Andronov and Bogdanov-Takens) of a modified Leslie-Gower prey-predator model with Michaelis-Menten type prey harvesting. Ji et al. [27, 28] showed there was a stationary distribution of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property. Lian and Xu  discussed the Hopf bifurcation of a predator-prey system with Holling type IV functional response and time delay.
As we all know, the functional response can be classified into two types: prey-dependent and predator-dependent. Prey-dependent depends on prey density only, while predator-dependent means that the functional response is a function of both the preys and the predators densities. Recently, the prey-dependent functional responses have been challenged by several ecologists. There is a growing explicit biological and physiological evidence [30–32] that in many situations, especially when the predator has to search for food (and therefore has to shave or compete for food), a more suitable general predator-prey theory should be predator-dependent. This is supported by numerous fields and laboratory experiments and observations [33, 34]. Starting from this argument and the traditional prey-dependent-only model, Arditi and Ginzburg  first proposed the ratio-dependent predator-prey model. Many authors have observed that the ratio-dependent models can exhibit much richer, more complicated and more reasonable or acceptable dynamics, but it has somewhat singular behavior at low densities which has been the source of controversy . For the ratio-dependent predator-prey models, one can refer to [36–39].
Beddington-DeAngelis functional response was first proposed by Beddington and DeAngelis [40, 41]. Predator-prey model with Beddington-DeAngelis functional response has rich dynamical features, which can describe the species and the ecological systems more reasonably. Beddington-DeAngelis functional response is similar to the well-known Holling type II functional response but has an extra term cy in the denominator modeling mutual interference among predators and has some of the same qualitative features as the ratio-dependent form but avoids some of the singular behaviors of ratio-dependent models at low densities.
with initial conditions and . The parameters , p, α, a, b, c, , β and k are positive constants and have the same meaning as in model (1.2).
It is easy to see that both the first quadrant and the positive first quadrant are invariant for system (1.3). As a result, solutions to (1.3) with are all positive solutions.
The rest of this paper is organized as follows. In Section 2, we discuss the structure of nonnegative equilibria to (1.3) and their local stability, which motivates us to study permanence and global stability of (1.3) respectively in Section 3 and Section 4.
respectively. As a direct consequence of (2.1), we have the following result.
Both and are unstable.
is locally asymptotically stable if , while it is unstable if .
Hence, we have the following result.
Hence, the following proposition follows from (2.1).
holds, then the positive equilibrium is locally asymptotically stable.
Proposition 2.1 and Proposition 2.3 naturally motivate us to seek sufficient conditions on the global stability of and the unique positive equilibrium to (1.3). To achieve it, we need the bounds for positive solutions.
The following result can be proved by slightly modifying the proof of Lemma 3.2 of Chen  and it will play an important role in finding the bounds for positive solutions to (1.3).
where and .
Thus by letting . □
This gives by letting . □
Combing Proposition 3.1 with Proposition 3.2 gives the permanence of (1.3).
Theorem 3.1 Suppose that (H2) holds, then (1.3) is permanent.
The goal of this section is to establish sufficient conditions on the global asymptotic stability of equilibrium to (1.3). The first two results are proved by employing the fluctuation lemma, which is cited below for the convenience of the readers. See Hirsch et al.  or Tineo  for more details on the fluctuation lemma.
Lemma 4.1 (Fluctuation lemma)
and as ;
and as .
Then is globally asymptotically stable for system (1.3).
Proposition 3.1 again tells us that is bounded and . By Lemma 4.1, there exist sequences , such that , , , and , as .
It follows from (4.3) and (4.6) that is globally asymptotically stable. □
where is defined in Proposition 3.2. Then system (1.3) has a unique positive equilibrium which is globally asymptotically stable.
Proof Note that (H2) implies (H0), thus (1.3) has a unique positive equilibrium according to Proposition 2.2. Let be any positive solution of (1.3). By the results in Section 3, (>0) and .
That is, is a positive equilibrium of (1.3). This completes the proof as the positive equilibrium is unique and so the unique equilibrium point is globally asymptotically stable. □
The following results in this section are proved by qualitative method and applying the Lyapunov direct method with the Lyapunov function.
Theorem 4.3 Assume that (H0) and (H1) hold, then (1.3) has a unique positive equilibrium which is globally asymptotically stable.
where is the vector field of (1.3). By the positivity of x, y, it is easy to obtain that if (H1) holds. Then, by the Dulac criteria, (1.3) admits no limit cycles or separatrix cycles. Proposition 2.3 shows that is locally asymptotically stable when (H1) holds. On the other hand, (1.3) admits only four equilibria () and . Also, Proposition 2.1 shows that () are all unstable when holds. So, according to Proposition 3.1 and the Poincaré-Bendixson theorem, is globally asymptotically stable. □
hold, then (1.3) has a unique positive equilibrium which is globally asymptotically stable.
According to (4.13) and (4.14), strictly for all except the positive equilibrium , where . Thus, satisfies Lyapunov’s asymptotic stability theorem, and the positive equilibrium of system (1.3) is globally asymptotically stable. This ends the proof of Theorem 4.4. □
In this paper, we consider a predator-prey model with modified Leslie-Gower and Beddington-DeAngelis functional response. We discuss the structure of nonnegative equilibria and their local stability. Also, the permanence of the system is investigated. By applying the fluctuation lemma, qualitative analysis and Lyapunov direct method, respectively, three sufficient conditions on the global asymptotic stability of a positive equilibrium are obtained. Compare Theorem 4.2 with Theorem 4.3. Since (H2) contains (H0), what will happen when (H0) and (H4) hold? This is a further problem, which can be studied in the future.
The author would like to thank the two anonymous referees for their constructive suggestions on improving the presentation of the paper. This research is supported by the Foundation of Fujian Education Bureau (JA13365).
- Berryman AA: The origins and evolution of predator-prey theory. Ecology 1992, 75: 1530–1535.View ArticleGoogle Scholar
- Leslie PH: Some further notes on the use of matrices in population mathematics. Biometrika 1948, 35: 213–245. 10.1093/biomet/35.3-4.213MathSciNetView ArticleGoogle Scholar
- Leslie PH: A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 1958, 45: 16–31. 10.1093/biomet/45.1-2.16MathSciNetView ArticleGoogle Scholar
- Korobeinikov A: A Lyapunov function for Leslie-Gower predator-prey models. Appl. Math. Lett. 2001, 14: 697–699. 10.1016/S0893-9659(01)80029-XMathSciNetView ArticleGoogle Scholar
- Hsu SB, Hwang TW: Global stability for a class of predator-prey systems. SIAM J. Appl. Math. 1995, 55: 763–783. 10.1137/S0036139993253201MathSciNetView ArticleGoogle Scholar
- Hsu SB, Hwang TW: Uniqueness of limit cycles for a predator-prey system of Holling and Leslie type. Can. Appl. Math. Q. 1998, 6: 91–117.MathSciNetGoogle Scholar
- Hsu SB, Hwang TW: Hopf bifurcation analysis for a predator-prey system of Holling and Leslie type. Taiwan. J. Math. 1999, 3: 35–53.MathSciNetGoogle Scholar
- Yuan S, Song Y: Bifurcation and stability analysis for a delayed Leslie-Gower predator-prey system. IMA J. Appl. Math. 2009, 74: 574–603. 10.1093/imamat/hxp013MathSciNetView ArticleGoogle Scholar
- May RM: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton; 2001.Google Scholar
- Holling CS: The functional response of predator to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 1965, 45: 1–60.View ArticleGoogle Scholar
- Aziz-Alaoui MA, Daher Okiye M: Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Appl. Math. Lett. 2003, 16: 1069–1075. 10.1016/S0893-9659(03)90096-6MathSciNetView ArticleGoogle Scholar
- Yu SB: Global asymptotic stability of a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Discrete Dyn. Nat. Soc. 2012., 2012: Article ID 208167Google Scholar
- Nindjin AF, Aziz-Alaoui MA, Cadivel M: Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay. Nonlinear Anal., Real World Appl. 2006, 7: 1104–1118. 10.1016/j.nonrwa.2005.10.003MathSciNetView ArticleGoogle Scholar
- Yafia R, El Adnani F, Talibi Alaoui H: Stability of limit cycle in a predator-prey model with modified Leslie-Gower and Holling-type II schemes with time delay. Appl. Math. Sci. 2007, 1: 119–131.MathSciNetGoogle Scholar
- Yafia R, El Adnani F, Talibi Alaoui H: Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes. Nonlinear Anal., Real World Appl. 2008, 9: 2055–2067. 10.1016/j.nonrwa.2006.12.017MathSciNetView ArticleGoogle Scholar
- Nindjin AF, Aziz-Alaoui MA: Persistence and global stability in a delayed Leslie-Gower type three species food chain. J. Math. Anal. Appl. 2008, 340: 340–357. 10.1016/j.jmaa.2007.07.078MathSciNetView ArticleGoogle Scholar
- Gakkhar S, Singh B: Dynamics of modified Leslie-Gower-type prey-predator model with seasonally varying parameters. Chaos Solitons Fractals 2006, 27: 1239–1255. 10.1016/j.chaos.2005.04.097MathSciNetView ArticleGoogle Scholar
- Guo HJ, Song XY: An impulsive predator-prey system with modified Leslie-Gower and Holling type II schemes. Chaos Solitons Fractals 2008, 36: 1320–1331. 10.1016/j.chaos.2006.08.010MathSciNetView ArticleGoogle Scholar
- Song XY, Li YF: Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect. Nonlinear Anal., Real World Appl. 2008, 9: 64–79. 10.1016/j.nonrwa.2006.09.004MathSciNetView ArticleGoogle Scholar
- Zhu Y, Wang K: Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes. J. Math. Anal. Appl. 2011, 384: 400–408. 10.1016/j.jmaa.2011.05.081MathSciNetView ArticleGoogle Scholar
- Liu M, Wang K: Dynamics of a Leslie-Gower Holling-type II predator-prey system with Lévy jumps. Nonlinear Anal. 2013, 85: 204–213.MathSciNetView ArticleGoogle Scholar
- Kar TK, Ghorai A: Dynamic behaviour of a delayed predator-prey model with harvesting. Appl. Math. Comput. 2011, 217: 9085–9104. 10.1016/j.amc.2011.03.126MathSciNetView ArticleGoogle Scholar
- Huo H, Wang X, Chavez C: Dynamics of a stage-structured Leslie-Gower predator-prey model. Math. Probl. Eng. 2011., 2011: Article ID 149341Google Scholar
- Li Z, Han M, Chen F: Global stability of a stage-structured predator-prey model with modified Leslie-Gower and Holling-type II schemes. Int. J. Biomath. 2012., 5: Article ID 1250057Google Scholar
- Liu C, Zhang Q, Huang J: Stability analysis of a harvested prey-predator model with stage structure and maturation delay. Math. Probl. Eng. 2013., 2013: Article ID 329592 10.1155/2013/329592Google Scholar
- Gupta RP, Chandra P: Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting. J. Math. Anal. Appl. 2013, 398: 278–295. 10.1016/j.jmaa.2012.08.057MathSciNetView ArticleGoogle Scholar
- Ji CY, Jiang DQ, Shi NZ: Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 2009, 359: 482–498. 10.1016/j.jmaa.2009.05.039MathSciNetView ArticleGoogle Scholar
- Ji CY, Jiang DQ, Shi NZ: A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 2011, 377: 435–440. 10.1016/j.jmaa.2010.11.008MathSciNetView ArticleGoogle Scholar
- Lian F, Xu Y: Hopf bifurcation analysis of a predator-prey system with Holling type IV functional response and time delay. Appl. Math. Comput. 2009, 215: 1484–1495. 10.1016/j.amc.2009.07.003MathSciNetView ArticleGoogle Scholar
- Arditi R, Saiah H: Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 1992, 73: 1544–1551. 10.2307/1940007View ArticleGoogle Scholar
- Arditi R, Ginzburg LR, Akcakaya HR: Variation in plankton densities among lakes: a case for ratio-dependent models. Am. Nat. 1991, 138: 1287–1296. 10.1086/285286View ArticleGoogle Scholar
- Gutierrez AP: The physiological basis of ratio-dependent predator-prey theory: A metabolic pool model of Nicholson’s blowflies as an example. Ecology 1992, 73: 1552–1563. 10.2307/1940008View ArticleGoogle Scholar
- Arditi R, Ginzburg LR: Coupling in predator-prey dynamics: ratio-dependence. J. Theor. Biol. 1989, 139: 311–326. 10.1016/S0022-5193(89)80211-5View ArticleGoogle Scholar
- Arditi R, Perrin N, Saiah H: Functional response and heterogeneities: an experiment test with cladocerans. Oikos 1991, 60: 69–75. 10.2307/3544994View ArticleGoogle Scholar
- Yao Z, Xie S, Yu N: Dynamics of cooperative predator-prey system with impulsive effects and Beddington-DeAngelis functional response. J. Egypt. Math. Soc. 2013, 21: 213–223. 10.1016/j.joems.2013.04.008MathSciNetView ArticleGoogle Scholar
- Beretta E, Kuang Y: Global analysis in some delayed ratio-dependent predator-prey systems. Nonlinear Anal. 1998, 32: 381–408. 10.1016/S0362-546X(97)00491-4MathSciNetView ArticleGoogle Scholar
- Liang Z, Pan H: Qualitative analysis of a ratio-dependent Holling-Tanner mode. J. Math. Anal. Appl. 2007, 334: 954–964. 10.1016/j.jmaa.2006.12.079MathSciNetView ArticleGoogle Scholar
- Saha T, Chakrabarti C: Dynamical analysis of a delayed ratio-dependent Holling-Tanner predator-prey model. J. Math. Anal. Appl. 2009, 358: 389–402. 10.1016/j.jmaa.2009.03.072MathSciNetView ArticleGoogle Scholar
- Liu J, Zhang Z, Fu M: Stability and bifurcation in a delayed Holling-Tanner predator-prey system with ratio-dependent functional response. J. Appl. Math. 2012., 2012: Article ID 384293Google Scholar
- Beddington JR: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 1975, 44: 331–340. 10.2307/3866View ArticleGoogle Scholar
- DeAngelis DL, Goldstein RA, O’Neill RV: A model for trophic interaction. Ecology 1975, 56: 881–892. 10.2307/1936298View ArticleGoogle Scholar
- Skalski GT, Gilliam JF: Functional responses with predator interference: viable alternatives to the Holling type II model. Ecology 2001, 82: 3083–3092. 10.1890/0012-9658(2001)082[3083:FRWPIV]2.0.CO;2View ArticleGoogle Scholar
- Lu Z, Liu X: Analysis of a predator-prey model with modified Holling-Tanner functional response and time delay. Nonlinear Anal., Real World Appl. 2008, 9: 641–650. 10.1016/j.nonrwa.2006.12.016MathSciNetView ArticleGoogle Scholar
- Zhang J: Bifurcation analysis of a modified Holling-Tanner predator-prey model with time delay. Appl. Math. Model. 2012, 36: 1219–1231. 10.1016/j.apm.2011.07.071MathSciNetView ArticleGoogle Scholar
- Cantrell R, Cosner C: On the dynamics of predator-prey models with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 2001, 257: 206–222. 10.1006/jmaa.2000.7343MathSciNetView ArticleGoogle Scholar
- Liu ZH, Yuan R: Stability and bifurcation in a delayed predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 2004, 296: 521–537. 10.1016/j.jmaa.2004.04.051MathSciNetView ArticleGoogle Scholar
- Dimitrov DT, Kojouharov HV: Complete mathematical analysis of predator-prey models with linear prey growth and Beddington-DeAngelis functional response. Appl. Math. Comput. 2005, 162: 523–538. 10.1016/j.amc.2003.12.106MathSciNetView ArticleGoogle Scholar
- Lin G, Hong Y: Delay induced oscillation in predator-prey system with Beddington-DeAngelis functional response. Appl. Math. Comput. 2007, 190: 1296–1311. 10.1016/j.amc.2007.02.012MathSciNetView ArticleGoogle Scholar
- Chen WY, Wang MX: Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion. Math. Comput. Model. 2005, 42: 31–44. 10.1016/j.mcm.2005.05.013View ArticleGoogle Scholar
- Xiao HB: Positive equilibrium and its stability of the Beddington-DeAngelis’s type predator-prey dynamical system. Appl. Math. J. Chin. Univ. Ser. B 2006, 21: 429–436. 10.1007/s11766-006-0007-2View ArticleGoogle Scholar
- Hwang T: Uniqueness of limit cycles of the predator-prey system with Beddington-DeAngelis functional response. J. Math. Anal. Appl. 2004, 290: 113–122. 10.1016/j.jmaa.2003.09.073MathSciNetView ArticleGoogle Scholar
- Chen FD: On a nonlinear non-autonomous predator-prey model with diffusion and distributed delay. J. Comput. Appl. Math. 2005, 180: 33–49. 10.1016/j.cam.2004.10.001MathSciNetView ArticleGoogle Scholar
- Hirsch W, Hanisch H, Gabriel J: Differential equation models of some parasitic infection: methods for the study of asymptotic behavior. Commun. Pure Appl. Math. 1985, 38: 733–753. 10.1002/cpa.3160380607MathSciNetView ArticleGoogle Scholar
- Tineo A: Asymptotic behaviour of positive solutions of the nonautonomous Lotka-Volterra competition equations. Differ. Integral Equ. 1993, 6: 419–457.MathSciNetGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.