# American Institute of Mathematical Sciences

November  2013, 18(9): 2283-2313. doi: 10.3934/dcdsb.2013.18.2283

## Dynamics of a ratio-dependent predator-prey system with a strong Allee effect

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China 2 Department of Mathematics, University of Louisville, Louisville, KY 40292

Received  September 2012 Revised  May 2013 Published  September 2013

A ratio-dependent predator-prey model with a strong Allee effect in prey is studied. We show that the model has a Bogdanov-Takens bifurcation that is associated with a catastrophic crash of the predator population. Our analysis indicates that an unstable limit cycle bifurcates from a Hopf bifurcation, and it disappears due to a homoclinic bifurcation which can lead to different patterns of global population dynamics in the model. We study the heteroclinic orbits and determine all possible phase portraits when the Bogdanov-Takens bifurcation occurs. We also provide the conditions for nonexistence of limit cycle under which the global dynamics of the model can be determined.
Citation: Yujing Gao, Bingtuan Li. Dynamics of a ratio-dependent predator-prey system with a strong Allee effect. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2283-2313. doi: 10.3934/dcdsb.2013.18.2283
##### References:
 [1] H. R. Akcakaya, Population cycles of mammals, evidence for a ratio-dependent predation hypothesis, Ecological Monographs, 62 (1992), 119-142. doi: 10.2307/2937172.  Google Scholar [2] H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predation: An abstraction that works, Ecology, 76 (1995), 995-1004. doi: 10.2307/1939362.  Google Scholar [3] W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, "Principles of Animal Ecology" W. B. Saunders, Philadelphia, Pennsylvania, USA 1949. Google Scholar [4] R. Arditi and A. A. Berryman, The biological control paradox, Trends in Ecology and Evolution, 6 (1991), 32. doi: 10.1016/0169-5347(91)90148-Q.  Google Scholar [5] R. Arditi and L. R. Ginzburg, Coupling in Predator-prey dynamics: ratio-dependence, Journal of Theoretical Biology, 139 (1989), 311-326. doi: 10.1016/S0022-5193(89)80211-5.  Google Scholar [6] R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: A case for ratio-dependent predation models, American Naturalist, 138 (1991), 1287-1296. doi: 10.1086/285286.  Google Scholar [7] R. Arditi and L. R. Ginzburg, "How Species Interact Altering the Standard View on Trophic Ecology," Oxford University Press, Oxford, UK 2012. Google Scholar [8] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. Google Scholar [9] O. Arino, A. El abdllaoui, J. Mikram and J. Chattopadhyay, Infection in prey population may act as a biological control in ratio-dependent predator-prey models, Nonlinearity, 17 (2004), 1101-1116. doi: 10.1088/0951-7715/17/3/018.  Google Scholar [10] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability , Nonlinearity, 18 (2005), 913-936 doi: 10.1088/0951-7715/18/2/022.  Google Scholar [11] M. Banerjee and S. Petrovskii, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, Theoretical Ecology, 4 (2011), 37-53. doi: 10.1007/s12080-010-0073-1.  Google Scholar [12] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations," World Scientific Series on nonlinear science, Series A: monographs and treatises, 11, World Scientific Publishing Co., Inc., River Edge, 1989. doi: 10.1142/9789812798725.  Google Scholar [13] L. Berec, D. S. Boukal and M .Berec, Linking the Allee effect, sexual reproduction, and temperaturedependent sex determination via spatial dynamics, American Naturalist, 157(2001), 217-230. Google Scholar [14] F. Berezovskaya, G. Karev and R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, Journal of Mathematical Biology, 43 (2001), 221-246. doi: 10.1007/s002850000078.  Google Scholar [15] A. A. Berryman, The origins and evolution of predator-prey theory, Ecology,73 (1992), 1530-1535. Google Scholar [16] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields, on the plane, Selecta Math Sov., 1 (1981), 373-388. Google Scholar [17] R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Functional Analysis and Its Applications, 9 (1975), 144-145. doi: 10.1007/BF01075453.  Google Scholar [18] D. S. Boukal, M. W. Sabelis and L. Berec, How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology, 72 (2007), 136-147. doi: 10.1016/j.tpb.2006.12.003.  Google Scholar [19] J. Charles, Studies on the Biologies of Two Mite Species, Predator and Prey, Including Some Effects of Gamma Radiation on Selected DevelopmentalStages, Ecology, 40 (1959), 572-579. Google Scholar [20] S. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields," Combridge University Press, New York, 1994. doi: 10.1017/CBO9780511665639.  Google Scholar [21] E. D. Conway and J. A. Smoller, Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630-642. doi: 10.1137/0146043.  Google Scholar [22] F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford University Press, Oxford, UK 2008. Google Scholar [23] F.Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405-410. Google Scholar [24] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interactions, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298.  Google Scholar [25] D. L. DeAngelis and J. N. Holland, Emergence of ratio-dependent and predator-dependent functional responses for pollination mutualism and seed parasitism, Ecological Modelling, 191 (2006), 551-556. doi: 10.1016/j.ecolmodel.2005.06.005.  Google Scholar [26] B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.  Google Scholar [27] H. I. Freedman, "Deterministic Mathematical Models in Population Ecology," Marcel Dekker, New York, 1980.  Google Scholar [28] W. M. Getz, Population dynamics: A per capita resource approach, Journal of Theoretical Biology, 108 (1984), 623-643. doi: 10.1016/S0022-5193(84)80082-X.  Google Scholar [29] L. R. Ginzburg and H. R. Akcakaya, Consequences of ratio-dependent predation for steady state properties of ecosystems, Ecology, 73 (1992), 1536-1543. doi: 10.2307/1940006.  Google Scholar [30] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983.  Google Scholar [31] A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson's blowflies as an example, Ecology, 73 (1992), 1552-1563. Google Scholar [32] J. K. Hale, "Ordinary Differential Equations," Krieger Publishing Co. Malabar, 1980.  Google Scholar [33] M. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71 (2009), 430-452. doi: 10.1007/s11538-008-9368-4.  Google Scholar [34] F. M. Hilker, M. Langlais and H. Malchow, The Allee effect and infectious diseases: extinction, multistability, and the (dis-)appearance of oscillations, American Naturalist, 173 (2009), 72-88. doi: 10.1086/593357.  Google Scholar [35] F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, Journal of Biological Dynamics, 4 (2010), 86-101. doi: 10.1080/17513750903026429.  Google Scholar [36] S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42 (2001), 489-506. doi: 10.1007/s002850100079.  Google Scholar [37] S. B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete and Continuous Dynamical Systems - Series B, 11 (2009), 893-911. doi: 10.3934/dcdsb.2009.11.893.  Google Scholar [38] C. B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383. Google Scholar [39] M. Kot, "Elements of Mathematical Ecology," Combridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511608520.  Google Scholar [40] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36 (1998), 389-406. doi: 10.1007/s002850050105.  Google Scholar [41] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences 112, Springer Verlag, New York 1995.  Google Scholar [42] R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise, Oikos, 83 (1998), 353-358. Google Scholar [43] B. Li and Y. Kuang, Heteroclinic bifercation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM Journal on Applied Mathematics, 67 (2007), 1453-1464. doi: 10.1137/060662460.  Google Scholar [44] R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis, Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, Journal of Differential Equations, 245 (2008), 442-467. doi: 10.1016/j.jde.2007.10.034.  Google Scholar [45] H. Malchow, S. V. Petrovskii and E. Venturino, "Spatiotemporal Patterns in Ecology and Epidemiology, Theory, Models, and Simulation," Chapman $&$ Hall/CRC Mathematical and Computational Biology Series, Chapman $&$ Hall, Boca Raton, 2008.  Google Scholar [46] R. M. May, "Stability and Complexity in Model Ecosystems," Princeton Univ.Press, 1974. Google Scholar [47] P. Matson and A. Berryman, Special Feature: Ratio-dependent predator-prey theory, Ecology, 73 (1992), 1592. doi: 10.2307/1940004.  Google Scholar [48] M. A. McCarthy, The Allee effect, finding mates and theoretical models, Ecological Modelling, 103 (1997), 99-102. doi: 10.1016/S0304-3800(97)00104-X.  Google Scholar [49] S. V. Petrovskii, A. Y. Morozov and E. Venturino, Allee effect makes possible patchy invasion in a predator-prey system, Ecology Letters, 5 (2002), 345-352. doi: 10.1046/j.1461-0248.2002.00324.x.  Google Scholar [50] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation systems in ecological time, Science, 171 (1969), 385-387. Google Scholar [51] S. Ruan, Y. Tang and W. Zhang, Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response, Journal of Mathematical Biology, 57 (2008), 223-241. doi: 10.1007/s00285-007-0153-z.  Google Scholar [52] L. B. Slobodkin, A summary of the special feature and comments on its theoretical context and importance, Ecology, 73 (1992), 1564-1566. doi: 10.2307/1940009.  Google Scholar [53] P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behavior, ecology and conservation, Trends in Ecology and Evolution, 14 (1999), 401-405. doi: 10.1016/S0169-5347(99)01684-5.  Google Scholar [54] Y. Tang and W. Zhang, Heteroclinic bifurcation in a ratio-dependent predator-prey system, Journal of Mathematical Biology, 50 (2005), 699-712. doi: 10.1007/s00285-004-0307-1.  Google Scholar [55] H. R. Thieme, T. Dhirasakdanon, Z. Han and R. Trevino, Species decline and extinction: synergy of infectious disease and Allee effect?, Journal of Biological Dynamics, 3 (2009), 305-323. doi: 10.1080/17513750802376313.  Google Scholar [56] George A. K. van Voorn, L. Hemerik, M. P. Boer and B. M. Kooi, Heteroclinic orbits indicate overexploitation in predator prey systems with a strong Allee effect, Mathematical Biosciences, 209 (2007), 451-469. doi: 10.1016/j.mbs.2007.02.006.  Google Scholar [57] P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139. doi: 10.2307/1932137.  Google Scholar [58] W. Wang, Y. Lin, F. Rao, L. Zhang and Y. Tan, Pattern selection in a ratio-dependent predator-prey model, Journal of Statistical Mechanics: Theory and Experiment, 11 (2010), 11036. doi: 10.1088/1742-5468/2010/11/P11036.  Google Scholar [59] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, Journal of Mathematical Biology, 62 (2011), 291-331. doi: 10.1007/s00285-010-0332-1.  Google Scholar [60] D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 43 (2001), 268-290. doi: 10.1007/s002850100097.  Google Scholar [61] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equations," Translations of Mathematical Monographs 101, American Mathematical Society, Providence 1992.  Google Scholar [62] H. Zhu, S. A. Campbell and G. S. K. Wolkowicz, Bifurcation analysis of a predator-prey system with nonmonotonic function response, SIAM Journal on Applied Mathematics, 63 (2002), 636-682. doi: 10.1137/S0036139901397285.  Google Scholar

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##### References:
 [1] H. R. Akcakaya, Population cycles of mammals, evidence for a ratio-dependent predation hypothesis, Ecological Monographs, 62 (1992), 119-142. doi: 10.2307/2937172.  Google Scholar [2] H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predation: An abstraction that works, Ecology, 76 (1995), 995-1004. doi: 10.2307/1939362.  Google Scholar [3] W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, "Principles of Animal Ecology" W. B. Saunders, Philadelphia, Pennsylvania, USA 1949. Google Scholar [4] R. Arditi and A. A. Berryman, The biological control paradox, Trends in Ecology and Evolution, 6 (1991), 32. doi: 10.1016/0169-5347(91)90148-Q.  Google Scholar [5] R. Arditi and L. R. Ginzburg, Coupling in Predator-prey dynamics: ratio-dependence, Journal of Theoretical Biology, 139 (1989), 311-326. doi: 10.1016/S0022-5193(89)80211-5.  Google Scholar [6] R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: A case for ratio-dependent predation models, American Naturalist, 138 (1991), 1287-1296. doi: 10.1086/285286.  Google Scholar [7] R. Arditi and L. R. Ginzburg, "How Species Interact Altering the Standard View on Trophic Ecology," Oxford University Press, Oxford, UK 2012. Google Scholar [8] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. Google Scholar [9] O. Arino, A. El abdllaoui, J. Mikram and J. Chattopadhyay, Infection in prey population may act as a biological control in ratio-dependent predator-prey models, Nonlinearity, 17 (2004), 1101-1116. doi: 10.1088/0951-7715/17/3/018.  Google Scholar [10] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability , Nonlinearity, 18 (2005), 913-936 doi: 10.1088/0951-7715/18/2/022.  Google Scholar [11] M. Banerjee and S. Petrovskii, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, Theoretical Ecology, 4 (2011), 37-53. doi: 10.1007/s12080-010-0073-1.  Google Scholar [12] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations," World Scientific Series on nonlinear science, Series A: monographs and treatises, 11, World Scientific Publishing Co., Inc., River Edge, 1989. doi: 10.1142/9789812798725.  Google Scholar [13] L. Berec, D. S. Boukal and M .Berec, Linking the Allee effect, sexual reproduction, and temperaturedependent sex determination via spatial dynamics, American Naturalist, 157(2001), 217-230. Google Scholar [14] F. Berezovskaya, G. Karev and R. Arditi, Parametric analysis of the ratio-dependent predator-prey model, Journal of Mathematical Biology, 43 (2001), 221-246. doi: 10.1007/s002850000078.  Google Scholar [15] A. A. Berryman, The origins and evolution of predator-prey theory, Ecology,73 (1992), 1530-1535. Google Scholar [16] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields, on the plane, Selecta Math Sov., 1 (1981), 373-388. Google Scholar [17] R. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Functional Analysis and Its Applications, 9 (1975), 144-145. doi: 10.1007/BF01075453.  Google Scholar [18] D. S. Boukal, M. W. Sabelis and L. Berec, How predator functional responses and Allee effects in prey affect the paradox of enrichment and population collapses, Theoretical Population Biology, 72 (2007), 136-147. doi: 10.1016/j.tpb.2006.12.003.  Google Scholar [19] J. Charles, Studies on the Biologies of Two Mite Species, Predator and Prey, Including Some Effects of Gamma Radiation on Selected DevelopmentalStages, Ecology, 40 (1959), 572-579. Google Scholar [20] S. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields," Combridge University Press, New York, 1994. doi: 10.1017/CBO9780511665639.  Google Scholar [21] E. D. Conway and J. A. Smoller, Global analysis of a system of predator-prey equations, SIAM Journal on Applied Mathematics, 46 (1986), 630-642. doi: 10.1137/0146043.  Google Scholar [22] F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford University Press, Oxford, UK 2008. Google Scholar [23] F.Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405-410. Google Scholar [24] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interactions, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298.  Google Scholar [25] D. L. DeAngelis and J. N. Holland, Emergence of ratio-dependent and predator-dependent functional responses for pollination mutualism and seed parasitism, Ecological Modelling, 191 (2006), 551-556. doi: 10.1016/j.ecolmodel.2005.06.005.  Google Scholar [26] B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538.  Google Scholar [27] H. I. Freedman, "Deterministic Mathematical Models in Population Ecology," Marcel Dekker, New York, 1980.  Google Scholar [28] W. M. Getz, Population dynamics: A per capita resource approach, Journal of Theoretical Biology, 108 (1984), 623-643. doi: 10.1016/S0022-5193(84)80082-X.  Google Scholar [29] L. R. Ginzburg and H. R. Akcakaya, Consequences of ratio-dependent predation for steady state properties of ecosystems, Ecology, 73 (1992), 1536-1543. doi: 10.2307/1940006.  Google Scholar [30] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, New York, 1983.  Google Scholar [31] A. P. Gutierrez, The physiological basis of ratio-dependent predator-prey theory: a metabolic pool model of Nicholson's blowflies as an example, Ecology, 73 (1992), 1552-1563. Google Scholar [32] J. K. Hale, "Ordinary Differential Equations," Krieger Publishing Co. Malabar, 1980.  Google Scholar [33] M. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, 71 (2009), 430-452. doi: 10.1007/s11538-008-9368-4.  Google Scholar [34] F. M. Hilker, M. Langlais and H. Malchow, The Allee effect and infectious diseases: extinction, multistability, and the (dis-)appearance of oscillations, American Naturalist, 173 (2009), 72-88. doi: 10.1086/593357.  Google Scholar [35] F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect, Journal of Biological Dynamics, 4 (2010), 86-101. doi: 10.1080/17513750903026429.  Google Scholar [36] S. B. Hsu, T. W. Hwang and Y. Kuang, Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system, Journal of Mathematical Biology, 42 (2001), 489-506. doi: 10.1007/s002850100079.  Google Scholar [37] S. B. Hsu and J. Shi, Relaxation oscillation profile of limit cycle in predator-prey system, Discrete and Continuous Dynamical Systems - Series B, 11 (2009), 893-911. doi: 10.3934/dcdsb.2009.11.893.  Google Scholar [38] C. B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383. Google Scholar [39] M. Kot, "Elements of Mathematical Ecology," Combridge University Press, Cambridge, 2001. doi: 10.1017/CBO9780511608520.  Google Scholar [40] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system, Journal of Mathematical Biology, 36 (1998), 389-406. doi: 10.1007/s002850050105.  Google Scholar [41] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences 112, Springer Verlag, New York 1995.  Google Scholar [42] R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise, Oikos, 83 (1998), 353-358. Google Scholar [43] B. Li and Y. Kuang, Heteroclinic bifercation in the Michaelis-Menten-type ratio-dependent predator-prey system, SIAM Journal on Applied Mathematics, 67 (2007), 1453-1464. doi: 10.1137/060662460.  Google Scholar [44] R. Liu, Z. Feng, H. Zhu and D. L. DeAngelis, Bifurcation analysis of a plant-herbivore model with toxin-determined functional response, Journal of Differential Equations, 245 (2008), 442-467. doi: 10.1016/j.jde.2007.10.034.  Google Scholar [45] H. Malchow, S. V. Petrovskii and E. Venturino, "Spatiotemporal Patterns in Ecology and Epidemiology, Theory, Models, and Simulation," Chapman $&$ Hall/CRC Mathematical and Computational Biology Series, Chapman $&$ Hall, Boca Raton, 2008.  Google Scholar [46] R. M. May, "Stability and Complexity in Model Ecosystems," Princeton Univ.Press, 1974. Google Scholar [47] P. Matson and A. Berryman, Special Feature: Ratio-dependent predator-prey theory, Ecology, 73 (1992), 1592. doi: 10.2307/1940004.  Google Scholar [48] M. A. McCarthy, The Allee effect, finding mates and theoretical models, Ecological Modelling, 103 (1997), 99-102. doi: 10.1016/S0304-3800(97)00104-X.  Google Scholar [49] S. V. Petrovskii, A. Y. Morozov and E. Venturino, Allee effect makes possible patchy invasion in a predator-prey system, Ecology Letters, 5 (2002), 345-352. doi: 10.1046/j.1461-0248.2002.00324.x.  Google Scholar [50] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation systems in ecological time, Science, 171 (1969), 385-387. Google Scholar [51] S. Ruan, Y. Tang and W. Zhang, Computing the heteroclinic bifurcation curves in predator-prey systems with ratio-dependent functional response, Journal of Mathematical Biology, 57 (2008), 223-241. doi: 10.1007/s00285-007-0153-z.  Google Scholar [52] L. B. Slobodkin, A summary of the special feature and comments on its theoretical context and importance, Ecology, 73 (1992), 1564-1566. doi: 10.2307/1940009.  Google Scholar [53] P. A. Stephens and W. J. Sutherland, Consequences of the Allee effect for behavior, ecology and conservation, Trends in Ecology and Evolution, 14 (1999), 401-405. doi: 10.1016/S0169-5347(99)01684-5.  Google Scholar [54] Y. Tang and W. Zhang, Heteroclinic bifurcation in a ratio-dependent predator-prey system, Journal of Mathematical Biology, 50 (2005), 699-712. doi: 10.1007/s00285-004-0307-1.  Google Scholar [55] H. R. Thieme, T. Dhirasakdanon, Z. Han and R. 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