American Institute of Mathematical Sciences

Advanced
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

Yujing Gao 1, and  Bingtuan Li 2,

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.
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. Google Scholar

[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. Google Scholar

[3]

W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, "Principles of Animal Ecology", W. B. Saunders, (1949). Google Scholar

[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. Google Scholar

[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. 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. 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, (2012). Google Scholar

[8]

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption,, Ecology, 73 (1992), 1544. 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. 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. 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. doi: 10.1007/s12080-010-0073-1. Google Scholar

[12]

A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", World Scientific Series on nonlinear science, (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. 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. doi: 10.1007/s002850000078. Google Scholar

[15]

A. A. Berryman, The origins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. 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. 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. 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. 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. Google Scholar

[20]

S. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields,", Combridge University Press, (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. doi: 10.1137/0146043. Google Scholar

[22]

F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation,", Oxford University Press, (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. Google Scholar

[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. 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. 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. Google Scholar

[27]

H. I. Freedman, "Deterministic Mathematical Models in Population Ecology,", Marcel Dekker, (1980). Google Scholar

[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. 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. doi: 10.2307/1940006. Google Scholar

[30]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (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. 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. 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. 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. 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. 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. 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. Google Scholar

[39]

M. Kot, "Elements of Mathematical Ecology,", Combridge University Press, (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. doi: 10.1007/s002850050105. Google Scholar

[41]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Applied Mathematical Sciences 112, (1995). Google Scholar

[42]

R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise,, Oikos, 83 (1998), 353. 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. 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. 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, (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). doi: 10.2307/1940004. Google Scholar

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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, (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. doi: 10.1137/S0036139901397285. Google Scholar

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. Google Scholar

[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. Google Scholar

[3]

W. C. Allee, A. E. Emerson, O. Park, T. Park and K. P. Schmidt, "Principles of Animal Ecology", W. B. Saunders, (1949). Google Scholar

[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. Google Scholar

[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. 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. 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, (2012). Google Scholar

[8]

R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption,, Ecology, 73 (1992), 1544. 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. 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. 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. doi: 10.1007/s12080-010-0073-1. Google Scholar

[12]

A. D. Bazykin, "Nonlinear Dynamics of Interacting Populations,", World Scientific Series on nonlinear science, (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. 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. doi: 10.1007/s002850000078. Google Scholar

[15]

A. A. Berryman, The origins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. 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. 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. 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. 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. Google Scholar

[20]

S. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields,", Combridge University Press, (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. doi: 10.1137/0146043. Google Scholar

[22]

F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation,", Oxford University Press, (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. Google Scholar

[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. 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. 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. Google Scholar

[27]

H. I. Freedman, "Deterministic Mathematical Models in Population Ecology,", Marcel Dekker, (1980). Google Scholar

[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. 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. doi: 10.2307/1940006. Google Scholar

[30]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (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. 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. 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. 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. 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. 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. 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. Google Scholar

[39]

M. Kot, "Elements of Mathematical Ecology,", Combridge University Press, (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. doi: 10.1007/s002850050105. Google Scholar

[41]

Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory,", Applied Mathematical Sciences 112, (1995). Google Scholar

[42]

R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise,, Oikos, 83 (1998), 353. 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. 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. 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, (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). doi: 10.2307/1940004. Google Scholar

[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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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). 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. 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. 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, (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. doi: 10.1137/S0036139901397285. Google Scholar

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