Predator-prey system with strong Allee effect in prey. (English) Zbl 1232.92076
Summary: Global bifurcation analysis of a class of general predator-prey models with strong Allee effect in the prey population is given in detail. We show the existence of a point-to-point heteroclinic orbit loop, consider Hopf bifurcations, and prove the existence/uniqueness and nonexistence of limit cycles for an appropriate range of parameters. For a unique parameter value, a threshold curve separates the overexploitation and coexistence (successful invasion of predators) regions of initial conditions. Our rigorous results justify some recent ecological observations, and practical ecological examples are used to demonstrate our theoretical work.
MSC:
| 92D40 | Ecology |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 37N25 | Dynamical systems in biology |
Software:
MATCONTReferences:
| [1] | Aguirre P, González-Olivares E, Sáez E (2009) Three limit cycles in a Leslie-Gower predator-prey model with additive Allee effect. SIAM J Appl Math 69(5): 1244–1262 · Zbl 1184.92046 · doi:10.1137/070705210 |
| [2] | Albrecht F, Gatzke H, Wax N (1973) Stable limit cycles in prey-predator populations. Science 181: 1074–1075 · doi:10.1126/science.181.4104.1073 |
| [3] | Albrecht F, Gatzke H, Haddad A, Wax N (1974) The dynamics of two interacting populations. J Math Anal Appl 46: 658–670 · Zbl 0281.92012 · doi:10.1016/0022-247X(74)90267-4 |
| [4] | Allee WC (1931) Animal aggregations: a study in general sociology. University of Chicago Press |
| [5] | Alexander JC, Yorke JA (1978) Global bifurcations of periodic orbits. Am J Math 100: 263–292 · Zbl 0386.34040 · doi:10.2307/2373851 |
| [6] | Arditi R, Ginzburg LR (1989) Coupling in predator-prey dynamics: ratio-dependence. J Theor Biol 139: 311–326 · doi:10.1016/S0022-5193(89)80211-5 |
| [7] | Bazykin AD (1998) Nonlinear dynamics of interacting populations. World Scientific Series on nonlinear science. Series A: monographs and treatises, 11. World Scientific Publishing Co., Inc., River Edge |
| [8] | Berec L, Angulo E, Courchamp F (2007) Multiple Allee effects and population management. Trends Ecol Evol 22: 185–191 · doi:10.1016/j.tree.2006.12.002 |
| [9] | Boukal SD, Berec L (2002) Single-species models of the Allee effect: extinction boundaries, sex ratios and mate encounters. J Theor Biol 218: 375–394 · doi:10.1006/jtbi.2002.3084 |
| [10] | Boukal SD, Sabelis WM, Berec L (2007) How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses. Theor Popul Biol 72: 136–147 · Zbl 1123.92034 · doi:10.1016/j.tpb.2006.12.003 |
| [11] | Burgman MA, Ferson S, Akcakaya HR (1993) Risk assessment in conservation biology. Chapman and Hall, London |
| [12] | Cheng KS (1981) Uniqueness of a limit cycle for a predator-prey system. SIAM J Math Anal 12: 541–548 · Zbl 0471.92021 · doi:10.1137/0512047 |
| [13] | Chow SN, Mallet-Paret J (1978) The Fuller index and global Hopf bifurcation. J Differ Equ 29: 66–85 · Zbl 0369.34020 · doi:10.1016/0022-0396(78)90041-4 |
| [14] | Conway ED, Smoller JA (1986) Global analysis of a system of predator-prey equations. SIAM J Appl Math 46: 630–642 · Zbl 0608.92016 · doi:10.1137/0146043 |
| [15] | Courchamp F, Clutton-Brock T, Grenfell B (1999) Inverse density dependence and the Allee effect. Trends Ecol Evol 14: 405–410 · doi:10.1016/S0169-5347(99)01683-3 |
| [16] | Courchamp F, Berec L, Gascoigne J (2008) Allee effects in ecology and conservation. Oxford University Press, Oxford |
| [17] | Dennis B (1989) Allee effect: population growth, critical density, and chance of extinction. Nat Resour Model 3: 481–538 · Zbl 0850.92062 |
| [18] | Dhooge A, Govaerts W, Kuznetsov YA (2003) MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs. ACM Trans Math Softw 29: 141–164 · Zbl 1070.65574 · doi:10.1145/779359.779362 |
| [19] | Edelstein KL (1998) Mathematical models in biology. Random House, New York |
| [20] | Gascoigne JC, Lipcius RN (2004) Allee effects driven by predation. J Appl Ecol 41: 801–810 · doi:10.1111/j.0021-8901.2004.00944.x |
| [21] | González-Olivares E, González-Yaez B, Sáez E, Szántó I (2006) On the number of limit cycles in a predator prey model with non-monotonic functional response. Discrete Contin Dyn Syst B 6(3): 525–534 · Zbl 1092.92045 · doi:10.3934/dcdsb.2006.6.525 |
| [22] | Gruntfest Y, Arditi R, Dombrovsky Y (1997) A fragmented population in a varying environment. J Theor Biol 185: 539–547 · doi:10.1006/jtbi.1996.0358 |
| [23] | Hassard BD, Kazarinoff ND, Wan Y (1981) Theory and applications of Hopf bifurcation. London Math Soc Lecture Note Ser, vol 41. Cambridge University Press, Cambridge · Zbl 0474.34002 |
| [24] | Hilker FM, Langlais M, Malchow H (2009) The Allee effect and infectious diseases: extinction, multistability, and the (dis-)appearance of oscillations. Am Nat 173: 72–88 · doi:10.1086/593357 |
| [25] | Holling CS (1959) The components of predation as revealed by a study of small mammal predation of the European Pine Sawfly. Can Entomol 91: 293–320 · doi:10.4039/Ent91293-5 |
| [26] | Hsu SB (1978) On global stability of a predator-prey system. Math Biosci 39: 1–10 · Zbl 0383.92014 · doi:10.1016/0025-5564(78)90025-1 |
| [27] | Hsu SB (2006) Ordinary differential equations with applications. Series on Applied Mathematics, vol 16. World Scientific Publishing Co., Hackensack |
| [28] | Hsu SB, Shi J (2009) Relaxation oscillator profile of limit cycle in predator-prey system. Discrete Contin Dyn Syst B 11(4): 893–911 · Zbl 1176.34049 · doi:10.3934/dcdsb.2009.11.893 |
| [29] | Hsu SB, Hwang TW, Kuang Y (2001) Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system. J Math Biol 42: 489–506 · Zbl 0984.92035 · doi:10.1007/s002850100079 |
| [30] | Ivlev VS (1955) Experimental ecology of the feeding of fishes. Yale University Press |
| [31] | Jacobs J (1984) Cooperation, optimal density and low density thresholds: yet another modification of the logistic model. Oecologia 64: 389–395 · doi:10.1007/BF00379138 |
| [32] | Jiang J, Shi J (2009) Bistability dynamics in some structured ecological models. In: Spatial ecology. CRC Press, Boca Raton |
| [33] | Kazarinov N, van den Driessche P (1978) A model predator-prey systems with functional response. Math Biosci 39: 125–134 · Zbl 0382.92007 · doi:10.1016/0025-5564(78)90031-7 |
| [34] | Kuang Y, Beretta E (1998) Global qualitative analysis of a ratio-dependent predator-prey system. J Math Biol 36: 389–406 · Zbl 0895.92032 · doi:10.1007/s002850050105 |
| [35] | Kuang Y, Freedman HI (1988) Uniqueness of limit cycles in Gause-type models of predator-prey systems. Math Biosci 88: 67–84 · Zbl 0642.92016 · doi:10.1016/0025-5564(88)90049-1 |
| [36] | Kuznetsov YA (2004) Elements of applied bifurcation theory. Appl Math Sci, vol 112. Springer-Verlag, New York · Zbl 1082.37002 |
| [37] | Lande R (1987) Extinction thresholds in demographic models of territorial populations. Am Nat 130: 624–635 · doi:10.1086/284734 |
| [38] | Lewis MA, Kareiva P (1993) Allee dynamics and the spread of invading organisms. Theor Popul Biol 43: 141–158 · Zbl 0769.92025 · doi:10.1006/tpbi.1993.1007 |
| [39] | Lotka AJ (1925) Elements of physical biology. Williams and Wilkins, Baltimore · JFM 51.0416.06 |
| [40] | Malchow H, Petrovskii SV, Venturino E (2008) Spatiotemporal patterns in ecology and epidemiology. Theory, models, and simulation. Chapman & Hall/CRC Mathematical and Computational Biology Series. Chapman & Hall, Boca Raton · Zbl 1298.92004 |
| [41] | May RM (1972) Limit cycles in predator-prey communities. Science 177: 900–902 · doi:10.1126/science.177.4052.900 |
| [42] | Morozov A, Petrovskii S, Li B-L (2004) Bifurcations and chaos in a predator-prey system with the Allee effect. Proc R Soc Lond B Biol Sci 271: 1407–1414 · doi:10.1098/rspb.2004.2733 |
| [43] | Morozov A, Petrovskii S, Li B-L (2006) Spatiotemporal complexity of patchy invasion in a predator-prey system with the Allee effect. J Theor Biol 238(1): 18–35 · doi:10.1016/j.jtbi.2005.05.021 |
| [44] | Owen MR, Lewis MA (2001) How predation can slow, stop or reverse a prey invasion. Bull Math Biol 63: 655–684 · Zbl 1323.92181 · doi:10.1006/bulm.2001.0239 |
| [45] | Petrovskii SV, Morozov A, Venturino E (2002) Allee effect makes possible patchy invasion in a predator-prey system. Ecol Lett 5: 345–352 · doi:10.1046/j.1461-0248.2002.00324.x |
| [46] | Petrovskii S, Morozov A, Li B-L (2005) Regimes of biological invasion in a predator-prey system with the Allee effect. Bull Math Biol 67(3): 637–661 · Zbl 1334.92363 · doi:10.1016/j.bulm.2004.09.003 |
| [47] | Polking JA, Arnold D (2003) Ordinary differential equations using MATLAB, 3rd edn. Prentice-Hall, Englewood Cliffs |
| [48] | Rosenzweig ML (1969) Why the prey curve has a hump? Am Nat 103: 81–87 · doi:10.1086/282584 |
| [49] | Rosenzweig LM (1971) Paradox of enrichment: destabilization of exploitation ecosystems in ecological time. Science 171(3969): 385–387 · doi:10.1126/science.171.3969.385 |
| [50] | Rosenzweig ML, MacArthur R (1963) Graphical representation and stability conditions of predator-prey interactions. Am Nat 97: 209–223 · doi:10.1086/282272 |
| [51] | Ruan S, Xiao D (2000) Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J Appl Math 61: 1445–1472 · Zbl 0986.34045 |
| [52] | Shi J, Shivaji R (2006) Persistence in reaction diffusion models with weak Allee effect. J Math Biol 52: 807–829 · Zbl 1110.92055 · doi:10.1007/s00285-006-0373-7 |
| [53] | Stephens PA, Sutherland WJ (1999) Consequences of the Allee effect for behaviour, ecology and conservation. Trends Ecol Evol 14: 401–405 · doi:10.1016/S0169-5347(99)01684-5 |
| [54] | Stephens PA, Sutherland WJ, Freckleton RP (1999) What is the Allee effect? Oikos 87: 185–190 · doi:10.2307/3547011 |
| [55] | Takeuchi Y (1996) Global dynamical properties of Lotka-Volterra systems. World Scientific, Singapore · Zbl 0844.34006 |
| [56] | Thieme HR, Dhirasakdanon T, Han Z, Trevino R (2009) Species decline and extinction: synergy of infectious disease and Allee effect? J Biol Dyn 3: 305–323 · Zbl 1342.92206 · doi:10.1080/17513750802376313 |
| [57] | Turchin P (2003) Complex population dynamics: a theoretical/empirical synthesis. Princeton University Press · Zbl 1062.92077 |
| [58] | van Voorn GAK, Hemerik L, Boer MP, Kooi BW (2007) Heteroclinic orbits indicate overexploitation in predator prey systems with a strong Allee effect. Math Biosci 209: 451–469 · Zbl 1126.92062 · doi:10.1016/j.mbs.2007.02.006 |
| [59] | Volterra V (1926) Fluctuations in the abundance of species, considered mathmatically. Nature 118: 558 · JFM 52.0453.03 · doi:10.1038/118558a0 |
| [60] | Wang ME, Kot M (2001) Speeds of invasion in a model with strong or weak Allee effects. Math Biosci 171: 83–97 · Zbl 0978.92033 · doi:10.1016/S0025-5564(01)00048-7 |
| [61] | Wiggins S (1990) Introduction to applied nonlinear dynamical systems and chaos. Texts Appl Math, vol 2. Springer, New York · Zbl 0701.58001 |
| [62] | Wilson EO, Bossert WH (1971) A primer of population biology. Sinauer Assoxiates, Sunderland |
| [63] | Wu J (1998) Symmetric functional-differential equations and neural networks with memory. Trans Am Math Soc 350: 4799–4838 · Zbl 0905.34034 · doi:10.1090/S0002-9947-98-02083-2 |
| [64] | Xiao D, Ruan S (2001) Global dynamics of a ratio-dependent predator-prey system. J Math Biol 43: 268–290 · Zbl 1007.34031 · doi:10.1007/s002850100097 |
| [65] | Xiao D, Zhang Z (2003) On the uniqueness and nonexistence of limit cycles for predator-prey system. Nonlinearity 16: 1–17 · Zbl 1042.34060 · doi:10.1088/0951-7715/16/3/321 |
| [66] | Xiao D, Zhang Z (2008) On the existence and uniqueness of limit cycles for generalized Liénard systems. J Math Anal Appl 343: 299–309 · Zbl 1143.34020 · doi:10.1016/j.jmaa.2008.01.059 |
| [67] | Zeng X, Zhang Z, Gao S (1994) On the uniqueness of the limit cycle of the generalized Liénard equation. Bull Lond Math Soc 26: 213–247 · Zbl 0805.34031 · doi:10.1112/blms/26.3.213 |
| [68] | Zhang ZF (1986) Proof of the uniqueness theorem of limit cycles of generalized Liénard equations. Appl Anal 23(1–2): 63–76 · Zbl 0595.34033 · doi:10.1080/00036818608839631 |
| [69] | Zhou SR, Liu YF, Wang G (2005) The stability of predator-prey systems subject to the Allee effects. Theor Popul Biol 67(1): 23–31 · Zbl 1072.92060 · doi:10.1016/j.tpb.2004.06.007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
