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

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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

Abstract
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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.

Keywords: Hopf bifurcation, Bogdanov-Takens bifurcation, limit cycle., heteroclinic orbit, Ratio-dependent predator-prey system, Allee effect.

Mathematics Subject Classification: Primary: 34C05, 34D20; Secondary: 92D4.

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.

show all references

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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