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Evolutionary branching patterns in predator-prey structured populations
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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. Arditi and L. R. Ginzburg, "How Species Interact Altering the Standard View on Trophic Ecology," Oxford University Press, Oxford, UK 2012. |
[8] |
R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. A. Berryman, The origins and evolution of predator-prey theory, Ecology,73 (1992), 1530-1535. |
[16] |
R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields, on the plane, Selecta Math Sov., 1 (1981), 373-388. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford University Press, Oxford, UK 2008. |
[23] |
F.Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405-410. |
[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. |
[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. |
[26] |
B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538. |
[27] |
H. I. Freedman, "Deterministic Mathematical Models in Population Ecology," Marcel Dekker, New York, 1980. |
[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. |
[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. |
[30] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, New York, 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), 1552-1563. |
[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), 430-452.
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), 72-88.
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), 86-101.
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), 489-506.
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), 893-911.
doi: 10.3934/dcdsb.2009.11.893. |
[38] |
C. B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383. |
[39] |
M. Kot, "Elements of Mathematical Ecology," Combridge University Press, Cambridge, 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), 389-406.
doi: 10.1007/s002850050105. |
[41] |
Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences 112, Springer Verlag, New York 1995. |
[42] |
R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise, Oikos, 83 (1998), 353-358. |
[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. |
[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. |
[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. |
[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), 1592.
doi: 10.2307/1940004. |
[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. |
[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. |
[50] |
M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation systems in ecological time, Science, 171 (1969), 385-387. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[57] |
P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.
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), 11036.
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), 291-331.
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), 268-290.
doi: 10.1007/s002850100097. |
[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. |
[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. |
show all references
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[7] |
R. Arditi and L. R. Ginzburg, "How Species Interact Altering the Standard View on Trophic Ecology," Oxford University Press, Oxford, UK 2012. |
[8] |
R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 73 (1992), 1544-1551. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[15] |
A. A. Berryman, The origins and evolution of predator-prey theory, Ecology,73 (1992), 1530-1535. |
[16] |
R. Bogdanov, Bifurcations of a limit cycle for a family of vector fields, on the plane, Selecta Math Sov., 1 (1981), 373-388. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
F. Courchamp, L. Berec and J. Gascoigne, "Allee Effects in Ecology and Conservation," Oxford University Press, Oxford, UK 2008. |
[23] |
F.Courchamp, T. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology and Evolution, 14 (1999), 405-410. |
[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. |
[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. |
[26] |
B. Dennis, Allee effects: population growth, critical density, and the chance of extinction, Natural Resource Modeling, 3 (1989), 481-538. |
[27] |
H. I. Freedman, "Deterministic Mathematical Models in Population Ecology," Marcel Dekker, New York, 1980. |
[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. |
[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. |
[30] |
J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, New York, 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), 1552-1563. |
[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), 430-452.
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), 72-88.
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), 86-101.
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), 489-506.
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), 893-911.
doi: 10.3934/dcdsb.2009.11.893. |
[38] |
C. B. Huffaker, Experimental studies on predation: dispersion factors and predator-prey oscillations, Hilgardia, 27 (1958), 343-383. |
[39] |
M. Kot, "Elements of Mathematical Ecology," Combridge University Press, Cambridge, 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), 389-406.
doi: 10.1007/s002850050105. |
[41] |
Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Applied Mathematical Sciences 112, Springer Verlag, New York 1995. |
[42] |
R. Lande, Demographic stochasticity and Allee effect on a scale with isotrophic noise, Oikos, 83 (1998), 353-358. |
[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. |
[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. |
[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. |
[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), 1592.
doi: 10.2307/1940004. |
[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. |
[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. |
[50] |
M. L. Rosenzweig, Paradox of enrichment: Destabilization of exploitation systems in ecological time, Science, 171 (1969), 385-387. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[57] |
P. J. Wangersky and W. J. Cunningham, Time lag in prey-predator population models, Ecology, 38 (1957), 136-139.
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), 11036.
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), 291-331.
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), 268-290.
doi: 10.1007/s002850100097. |
[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. |
[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. |
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