# 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. doi: 10.2307/2937172. [2] H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predation: An abstraction that works,, Ecology, 76 (1995), 995. doi: 10.2307/1939362. [3] W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, "Principles of Animal Ecology", W. B. Saunders, (1949). [4] R. Arditi and A. A. Berryman, The biological control paradox,, Trends in Ecology and Evolution, 6 (1991). doi: 10.1016/0169-5347(91)90148-Q. [5] R. Arditi and L. R. Ginzburg, Coupling in Predator-prey dynamics: ratio-dependence,, Journal of Theoretical Biology, 139 (1989), 311. doi: 10.1016/S0022-5193(89)80211-5. [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. doi: 10.1086/285286. [7] R. Arditi and L. R. Ginzburg, "How Species Interact Altering the Standard View on Trophic Ecology,", Oxford University Press, (2012). [8] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption,, Ecology, 73 (1992), 1544. [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. doi: 10.1088/0951-7715/17/3/018. [10] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability ,, Nonlinearity, 18 (2005), 913. doi: 10.1088/0951-7715/18/2/022. [11] M. Banerjee and S. Petrovskii, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system,, Theoretical Ecology, 4 (2011), 37. doi: 10.1007/s12080-010-0073-1. [12] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", World Scientific Series on nonlinear science, (1989). doi: 10.1142/9789812798725. [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. [14] F. Berezovskaya, G. Karev and R. Arditi, Parametric analysis of the ratio-dependent predator-prey model,, Journal of Mathematical Biology, 43 (2001), 221. doi: 10.1007/s002850000078. [15] A. A. Berryman, The origins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. [16] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields, on the plane,, Selecta Math Sov., 1 (1981), 373. [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. doi: 10.1007/BF01075453. [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. doi: 10.1016/j.tpb.2006.12.003. [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. [20] S. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields,", Combridge University Press, (1994). doi: 10.1017/CBO9780511665639. [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. doi: 10.1137/0146043. [22] F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation,", Oxford University Press, (2008). [23] F.Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect,, Trends in Ecology and Evolution, 14 (1999), 405. [24] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interactions,, Ecology, 56 (1975), 881. doi: 10.2307/1936298. [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. doi: 10.1016/j.ecolmodel.2005.06.005. [26] B. Dennis, Allee effects: population growth, critical density, and the chance of extinction,, Natural Resource Modeling, 3 (1989), 481. [27] H. I. Freedman, "Deterministic Mathematical Models in Population Ecology,", Marcel Dekker, (1980). [28] W. M. Getz, Population dynamics: A per capita resource approach,, Journal of Theoretical Biology, 108 (1984), 623. doi: 10.1016/S0022-5193(84)80082-X. [29] L. R. Ginzburg and H. R. Akcakaya, Consequences of ratio-dependent predation for steady state properties of ecosystems,, Ecology, 73 (1992), 1536. doi: 10.2307/1940006. [30] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (1983). [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. [32] J. K. Hale, "Ordinary Differential Equations,", Krieger Publishing Co. Malabar, (1980). [33] M. Haque, Ratio-dependent predator-prey models of interacting populations,, Bulletin of Mathematical Biology, 71 (2009), 430. doi: 10.1007/s11538-008-9368-4. [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. doi: 10.1086/593357. [35] F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect,, Journal of Biological Dynamics, 4 (2010), 86. doi: 10.1080/17513750903026429. [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. doi: 10.1007/s002850100079. [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. doi: 10.3934/dcdsb.2009.11.893. [38] C. B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations,, Hilgardia, 27 (1958), 343. [39] M. Kot, "Elements of Mathematical Ecology,", Combridge University Press, (2001). doi: 10.1017/CBO9780511608520. [40] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 36 (1998), 389. doi: 10.1007/s002850050105. [41] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Applied Mathematical Sciences 112, (1995). [42] R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise,, Oikos, 83 (1998), 353. [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. doi: 10.1137/060662460. [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. doi: 10.1016/j.jde.2007.10.034. [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, (2008). [46] R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton Univ.Press, (1974). [47] P. Matson and A. Berryman, Special Feature: Ratio-dependent predator-prey theory,, Ecology, 73 (1992). doi: 10.2307/1940004. [48] M. A. McCarthy, The Allee effect, finding mates and theoretical models,, Ecological Modelling, 103 (1997), 99. doi: 10.1016/S0304-3800(97)00104-X. [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. doi: 10.1046/j.1461-0248.2002.00324.x. [50] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation systems in ecological time,, Science, 171 (1969), 385. [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. doi: 10.1007/s00285-007-0153-z. [52] L. B. Slobodkin, A summary of the special feature and comments on its theoretical context and importance,, Ecology, 73 (1992), 1564. doi: 10.2307/1940009. [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. doi: 10.1016/S0169-5347(99)01684-5. [54] Y. Tang and W. Zhang, Heteroclinic bifurcation in a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 50 (2005), 699. doi: 10.1007/s00285-004-0307-1. [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. doi: 10.1080/17513750802376313. [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. doi: 10.1016/j.mbs.2007.02.006. [57] P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models,, Ecology, 38 (1957), 136. doi: 10.2307/1932137. [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). doi: 10.1088/1742-5468/2010/11/P11036. [59] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey,, Journal of Mathematical Biology, 62 (2011), 291. doi: 10.1007/s00285-010-0332-1. [60] D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 43 (2001), 268. doi: 10.1007/s002850100097. [61] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equations,", Translations of Mathematical Monographs 101, (1992). [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. doi: 10.1137/S0036139901397285.

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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. doi: 10.2307/2937172. [2] H. R. Akcakaya, R. Arditi and L. R. Ginzburg, Ratio-dependent predation: An abstraction that works,, Ecology, 76 (1995), 995. doi: 10.2307/1939362. [3] W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, "Principles of Animal Ecology", W. B. Saunders, (1949). [4] R. Arditi and A. A. Berryman, The biological control paradox,, Trends in Ecology and Evolution, 6 (1991). doi: 10.1016/0169-5347(91)90148-Q. [5] R. Arditi and L. R. Ginzburg, Coupling in Predator-prey dynamics: ratio-dependence,, Journal of Theoretical Biology, 139 (1989), 311. doi: 10.1016/S0022-5193(89)80211-5. [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. doi: 10.1086/285286. [7] R. Arditi and L. R. Ginzburg, "How Species Interact Altering the Standard View on Trophic Ecology,", Oxford University Press, (2012). [8] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption,, Ecology, 73 (1992), 1544. [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. doi: 10.1088/0951-7715/17/3/018. [10] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability ,, Nonlinearity, 18 (2005), 913. doi: 10.1088/0951-7715/18/2/022. [11] M. Banerjee and S. Petrovskii, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system,, Theoretical Ecology, 4 (2011), 37. doi: 10.1007/s12080-010-0073-1. [12] A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", World Scientific Series on nonlinear science, (1989). doi: 10.1142/9789812798725. [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. [14] F. Berezovskaya, G. Karev and R. Arditi, Parametric analysis of the ratio-dependent predator-prey model,, Journal of Mathematical Biology, 43 (2001), 221. doi: 10.1007/s002850000078. [15] A. A. Berryman, The origins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. [16] R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields, on the plane,, Selecta Math Sov., 1 (1981), 373. [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. doi: 10.1007/BF01075453. [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. doi: 10.1016/j.tpb.2006.12.003. [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. [20] S. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields,", Combridge University Press, (1994). doi: 10.1017/CBO9780511665639. [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. doi: 10.1137/0146043. [22] F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation,", Oxford University Press, (2008). [23] F.Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect,, Trends in Ecology and Evolution, 14 (1999), 405. [24] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interactions,, Ecology, 56 (1975), 881. doi: 10.2307/1936298. [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. doi: 10.1016/j.ecolmodel.2005.06.005. [26] B. Dennis, Allee effects: population growth, critical density, and the chance of extinction,, Natural Resource Modeling, 3 (1989), 481. [27] H. I. Freedman, "Deterministic Mathematical Models in Population Ecology,", Marcel Dekker, (1980). [28] W. M. Getz, Population dynamics: A per capita resource approach,, Journal of Theoretical Biology, 108 (1984), 623. doi: 10.1016/S0022-5193(84)80082-X. [29] L. R. Ginzburg and H. R. Akcakaya, Consequences of ratio-dependent predation for steady state properties of ecosystems,, Ecology, 73 (1992), 1536. doi: 10.2307/1940006. [30] J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (1983). [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. [32] J. K. Hale, "Ordinary Differential Equations,", Krieger Publishing Co. Malabar, (1980). [33] M. Haque, Ratio-dependent predator-prey models of interacting populations,, Bulletin of Mathematical Biology, 71 (2009), 430. doi: 10.1007/s11538-008-9368-4. [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. doi: 10.1086/593357. [35] F. M. Hilker, Population collapse to extinction: The catastrophic combination of parasitism and Allee effect,, Journal of Biological Dynamics, 4 (2010), 86. doi: 10.1080/17513750903026429. [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. doi: 10.1007/s002850100079. [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. doi: 10.3934/dcdsb.2009.11.893. [38] C. B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations,, Hilgardia, 27 (1958), 343. [39] M. Kot, "Elements of Mathematical Ecology,", Combridge University Press, (2001). doi: 10.1017/CBO9780511608520. [40] Y. Kuang and E. Beretta, Global qualitative analysis of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 36 (1998), 389. doi: 10.1007/s002850050105. [41] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Applied Mathematical Sciences 112, (1995). [42] R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise,, Oikos, 83 (1998), 353. [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. doi: 10.1137/060662460. [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. doi: 10.1016/j.jde.2007.10.034. [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, (2008). [46] R. M. May, "Stability and Complexity in Model Ecosystems,", Princeton Univ.Press, (1974). [47] P. Matson and A. Berryman, Special Feature: Ratio-dependent predator-prey theory,, Ecology, 73 (1992). doi: 10.2307/1940004. [48] M. A. McCarthy, The Allee effect, finding mates and theoretical models,, Ecological Modelling, 103 (1997), 99. doi: 10.1016/S0304-3800(97)00104-X. [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. doi: 10.1046/j.1461-0248.2002.00324.x. [50] M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation systems in ecological time,, Science, 171 (1969), 385. [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. doi: 10.1007/s00285-007-0153-z. [52] L. B. Slobodkin, A summary of the special feature and comments on its theoretical context and importance,, Ecology, 73 (1992), 1564. doi: 10.2307/1940009. [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. doi: 10.1016/S0169-5347(99)01684-5. [54] Y. Tang and W. Zhang, Heteroclinic bifurcation in a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 50 (2005), 699. doi: 10.1007/s00285-004-0307-1. [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. doi: 10.1080/17513750802376313. [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. doi: 10.1016/j.mbs.2007.02.006. [57] P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models,, Ecology, 38 (1957), 136. doi: 10.2307/1932137. [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). doi: 10.1088/1742-5468/2010/11/P11036. [59] J. Wang, J. Shi and J. Wei, Predator-prey system with strong Allee effect in prey,, Journal of Mathematical Biology, 62 (2011), 291. doi: 10.1007/s00285-010-0332-1. [60] D. Xiao and S. Ruan, Global dynamics of a ratio-dependent predator-prey system,, Journal of Mathematical Biology, 43 (2001), 268. doi: 10.1007/s002850100097. [61] Z. Zhang, T. Ding, W. Huang and Z. Dong, "Qualitative Theory of Differential Equations,", Translations of Mathematical Monographs 101, (1992). [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. doi: 10.1137/S0036139901397285.
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