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February  2021, 26(2): 943-962. doi: 10.3934/dcdsb.2020148

A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator

1. 

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia, Facultad de Educación, Universidad de Las Américas, Av. Manuel Montt 948, Santiago, Chile

2. 

Department of Computer Science, The University of South Dakota, Vermillion, SD 57069, South Dakota, USA

3. 

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, GP Campus, Brisbane, Queensland 4001 Australia

* Corresponding author: Claudio Arancibia-Ibarra

Received  May 2019 Revised  November 2019 Published  February 2021 Early access  May 2020

We study a predator-prey model with Holling type Ⅰ functional response, an alternative food source for the predator, and multiple Allee effects on the prey. We show that the model has at most two equilibrium points in the first quadrant, one is always a saddle point while the other can be a repeller or an attractor. Moreover, there is always a stable equilibrium point that corresponds to the persistence of the predator population and the extinction of the prey population. Additionally, we show that when the parameters are varied the model displays a wide range of different bifurcations, such as saddle-node bifurcations, Hopf bifurcations, Bogadonov-Takens bifurcations and homoclinic bifurcations. We use numerical simulations to illustrate the impact changing the predation rate, or the non-fertile prey population, and the proportion of alternative food source have on the basins of attraction of the stable equilibrium point in the first quadrant (when it exists). In particular, we also show that the basin of attraction of the stable positive equilibrium point in the first quadrant is bigger when we reduce the depensation in the model.

Citation: Claudio Arancibia-Ibarra, José Flores, Michael Bode, Graeme Pettet, Peter van Heijster. A modified May–Holling–Tanner predator-prey model with multiple Allee effects on the prey and an alternative food source for the predator. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 943-962. doi: 10.3934/dcdsb.2020148
References:
[1]

P. AguirreE. González-Olivares and E. Sáez, Two limit cycles in a Leslie–Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416.  doi: 10.1016/j.nonrwa.2008.01.022.

[2]

W. Allee, The Social Life of Animals, WW Norton & Co, New York, 1938.

[3]

M. Andersson and S. Erlinge, Influence of predation on rodent populations, Oikos, 29 (1977), 591-597.  doi: 10.2307/3543597.

[4]

E. AnguloG. RoemerL. BerecJ. Gascoigne and F. Courchamp, Double Allee effects and extinction in the island fox, Conservation Biology, 21 (2007), 1082-1091.  doi: 10.1111/j.1523-1739.2007.00721.x.

[5]

C. Arancibia-Ibarra, The basins of attraction in a modified May–Holling–Tanner predator-prey model with Allee effect, Nonlinear Analysis, 185 (2019), 15-28.  doi: 10.1016/j.na.2019.03.004.

[6]

C. Arancibia-Ibarra, J. Flores, G. Pettet and P. van Heijster, A Holling–Tanner predator-prey model with strong Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 16 pp. doi: 10.1142/S0218127419300325.

[7]

C. Arancibia-Ibarra and E. González-Olivares, A modified Leslie–Gower predator-prey model with hyperbolic functional response and Allee effect on prey, BIOMAT 2010 International Symposium on Mathematical and Computational Biology, (2011), 146–162. doi: 10.1142/9789814343435_0010.

[8]

C. Arancibia-Ibarra and E. González-Olivares, The Holling–Tanner model considering an alternative food for predator, Proceedings of the 2015 International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE, 2015 (2015), 130-141. 

[9]

M. Aziz-Alaoui and M. Daher, Boundedness and global stability for a predator-prey model with modified Leslie–Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.

[10]

C. BakerA. Gordon and M. Bode, Ensemble ecosystem modelling for predicting ecosystem response to predator reintroduction, Conservation Biology, 31 (2017), 376-384. 

[11]

L. BerecE. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007), 185-191.  doi: 10.1016/j.tree.2006.12.002.

[12]

M. BimlerD. StoufferH. Lai and M. Mayfield, Accurate predictions of coexistence in natural systems require the inclusion of facilitative interactions and environmental dependency, Journal of Ecology, 106 (2018), 1839-1852.  doi: 10.1111/1365-2745.13030.

[13]

T. Blows and N. Lloyd, The number of limit cycles of certain polynomial differential equations, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 98 (1984), 215-239.  doi: 10.1017/S030821050001341X.

[14]

C. Chicone, Ordinary Differential Equations with Applications, Second edition. Texts in Applied Mathematics, 34. Springer, New York, 2006.

[15]

F. Courchamp, L. Berec and J. Gascoigne, Allee effects in ecology and conservation, Oxford University Press, (2008).

[16]

F. CourchampT. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology & Evolution, 14 (1999), 405-410.  doi: 10.1016/S0169-5347(99)01683-3.

[17]

A. Dhooge, W. Govaerts and Y. Kuznetsov, Matcont: A matlab package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 141–164. doi: 10.1145/779359.779362.

[18]

S. Erlinge, Predation and noncyclicity in a microtine population in southern Sweden, Oikos, 50 (1987), 347-352.  doi: 10.2307/3565495.

[19]

J. Flores and E. González-Olivares, Dynamics of a predator–prey model with Allee effect on prey and ratio-dependent functional response, Ecological Complexity, 18 (2014), 59-66. 

[20]

J. Flores and E. González–Olivares, A modified Leslie–Gower predator-prey model with ratio–dependent functional response and alternative food for the predator, Mathematical Methods in the Applied Sciences, 40 (2017), 2313-2328.  doi: 10.1002/mma.4172.

[21]

V. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem, Mathematics and Its Applications, Springer Science Business Media, 2013. doi: 10.1007/978-1-4419-9168-3.

[22]

E. González-OlivaresB. González-YañezJ. Mena-LorcaA. Rojas-Palma and J. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Computers & Mathematics with Applications, 62 (2011), 3449-3463.  doi: 10.1016/j.camwa.2011.08.061.

[23]

I. HanskiL. Hansson and H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle, The Journal of Animal Ecology, 60 (1991), 353-367.  doi: 10.2307/5465.

[24]

I. HanskiH. HenttonenE. KorpimäkiL. Oksanen and P. Turchin, Small-rodent dynamics and predation, Ecology, 82 (2001), 1505-1520. 

[25]

L. Hansson, Competition between rodents in successional stages of taiga forests: Microtus agrestis vs. Clethrionomys glareolus, Oikos, 40 (1983), 258-266.  doi: 10.2307/3544590.

[26]

K. HarleyP. van HeijsterR. MarangellG. Pettet and M. Wechselberger, Existence of traveling wave solutions for a model of tumor invasion, SIAM Journal on Applied Dynamical Systems, 13 (2014), 366-396.  doi: 10.1137/130923129.

[27]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the european pine sawfly, Tenth International Congress of Entomology, 91 (1959), 293-320. 

[28]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM Journal on Applied Mathematics, 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.

[29]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete and Continuous Dynamical Systems Series B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101.

[30]

D. HooperF. ChapinJ. EwelA. HectorP. InchaustiS. LavorelJ. LawtonD. LodgeM. Loreau and S. Naeem, Effects of biodiversity on ecosystem functioning: A consensus of current knowledge, Ecological monographs, 75 (2005), 3-35.  doi: 10.1890/04-0922.

[31]

A. Korobeinikov, A Lyapunov function for Leslie–Gower predator-prey models, Applied Mathematics Letters, 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[32]

A. KramerL. Berec and J. Drake, Allee effects in ecology and evolution, Journal of Animal Ecology, 87 (2018), 7-10. 

[33]

R. Levins, Discussion paper: The qualitative analysis of partially specified systems, Annals of the New York Academy of Sciences, 231 (1974), 123-138.  doi: 10.1111/j.1749-6632.1974.tb20562.x.

[34]

M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 33-58. 

[35]

A. Lotka, Contribution to the theory of periodic reactions, The Journal of Physical Chemistry, 14 (1910), 271-274.  doi: 10.1021/j150111a004.

[36]

M. I. G. and X. Lambin, The impact of weasel predation on cyclic field-vole survival: The specialist predator hypothesis contradicted, Journal of Animal Ecology, 71 (2002), 946-956. 

[37]

N. Martínez-Jeraldo and P. Aguirre, Allee effect acting on the prey species in a Leslie–Gower predation model, Nonlinear Analysis: Real World Applications, 45 (2019), 895-917.  doi: 10.1016/j.nonrwa.2018.08.009.

[38] R. May, Stability and Complexity in Model Ecosystems, Princeton university press, 2001. 
[39]

R. MonclusD. von HolstD. Blumstein and H. Rödel, Long-term effects of litter sex ratio on female reproduction in two iteroparous mammals, Functional Ecology, 28 (2014), 954-962.  doi: 10.1111/1365-2435.12231.

[40]

R. Ostfeld and C. Canham, Density-dependent processes in meadow voles: An experimental approach, Ecology, 76 (1995), 521-532.  doi: 10.2307/1941210.

[41]

L. Perko, Differential Equations and Dynamical Systems, Springer New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[42]

S. Prager and W. Reiners, Historical and emerging practices in ecological topology, Ecological Complexity, 6 (2009), 160-171.  doi: 10.1016/j.ecocom.2008.11.001.

[43]

B. RaymondJ. McInnesJ. D. nd S. Way and D. Bergstrom, Qualitative modelling of invasive species eradication on subantarctic Macquarie Island, Journal of Applied Ecology, 48 (2011), 181-191.  doi: 10.1111/j.1365-2664.2010.01916.x.

[44]

P. RouxJ. Shaw and S. Chown, Ontogenetic shifts in plant interactions vary with environmental severity and affect population structure, New Phytologist, 200 (2013), 241-250. 

[45]

E. Sáez and E. González-Olivares, Dynamics on a predator-prey model, SIAM Journal on Applied Mathematics, 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.

[46]

X. Santos and M. Cheylan, Taxonomic and functional response of a Mediterranean reptile assemblage to a repeated fire regime, Biological Conservation, 168 (2013), 90-98.  doi: 10.1016/j.biocon.2013.09.008.

[47]

P. Stephens and W. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.

[48] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton university press, 2003. 
[49]

A. Verdy, Modulation of predator-prey interactions by the Allee effect, Ecological Modelling, 221 (2010), 1098-1107.  doi: 10.1016/j.ecolmodel.2010.01.005.

[50]

G. VoornL. HemerikM. Boer and B. 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.

[51]

S. Wood and M. Thomas, Super-sensitivity to structure in biological models, Proceedings of the Royal Society of London. Series B: Biological Sciences, 266 (1999), 565-570.  doi: 10.1098/rspb.1999.0673.

[52]

D. Xiao and S. Ruan, Bogdanov–Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Institute Communications, 21 (1999), 493-506. 

[53]

Z. Yue, X. Wang and H. Liu, Complex dynamics of a diffusive Holling–Tanner predator-prey model with the Allee effect, Abstract and Applied Analysis, 2013 (2013), 12 pp. doi: 10.1155/2013/270191.

[54]

Z. ZhaoL. Yang and L. Chen, Impulsive perturbations of a predator-prey system with modified Leslie–Gower and Holling type Ⅱ schemes, Journal of Applied Mathematics and Computing, 35 (2011), 119-134.  doi: 10.1007/s12190-009-0346-2.

[55]

J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type Ⅱ functional response, Applied Mathematics and Computation, 217 (2010), 3542-3556.  doi: 10.1016/j.amc.2010.09.029.

show all references

References:
[1]

P. AguirreE. González-Olivares and E. Sáez, Two limit cycles in a Leslie–Gower predator-prey model with additive Allee effect, Nonlinear Analysis: Real World Applications, 10 (2009), 1401-1416.  doi: 10.1016/j.nonrwa.2008.01.022.

[2]

W. Allee, The Social Life of Animals, WW Norton & Co, New York, 1938.

[3]

M. Andersson and S. Erlinge, Influence of predation on rodent populations, Oikos, 29 (1977), 591-597.  doi: 10.2307/3543597.

[4]

E. AnguloG. RoemerL. BerecJ. Gascoigne and F. Courchamp, Double Allee effects and extinction in the island fox, Conservation Biology, 21 (2007), 1082-1091.  doi: 10.1111/j.1523-1739.2007.00721.x.

[5]

C. Arancibia-Ibarra, The basins of attraction in a modified May–Holling–Tanner predator-prey model with Allee effect, Nonlinear Analysis, 185 (2019), 15-28.  doi: 10.1016/j.na.2019.03.004.

[6]

C. Arancibia-Ibarra, J. Flores, G. Pettet and P. van Heijster, A Holling–Tanner predator-prey model with strong Allee effect, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 16 pp. doi: 10.1142/S0218127419300325.

[7]

C. Arancibia-Ibarra and E. González-Olivares, A modified Leslie–Gower predator-prey model with hyperbolic functional response and Allee effect on prey, BIOMAT 2010 International Symposium on Mathematical and Computational Biology, (2011), 146–162. doi: 10.1142/9789814343435_0010.

[8]

C. Arancibia-Ibarra and E. González-Olivares, The Holling–Tanner model considering an alternative food for predator, Proceedings of the 2015 International Conference on Computational and Mathematical Methods in Science and Engineering CMMSE, 2015 (2015), 130-141. 

[9]

M. Aziz-Alaoui and M. Daher, Boundedness and global stability for a predator-prey model with modified Leslie–Gower and Holling-type Ⅱ schemes, Appl. Math. Lett., 16 (2003), 1069-1075.  doi: 10.1016/S0893-9659(03)90096-6.

[10]

C. BakerA. Gordon and M. Bode, Ensemble ecosystem modelling for predicting ecosystem response to predator reintroduction, Conservation Biology, 31 (2017), 376-384. 

[11]

L. BerecE. Angulo and F. Courchamp, Multiple Allee effects and population management, Trends in Ecology & Evolution, 22 (2007), 185-191.  doi: 10.1016/j.tree.2006.12.002.

[12]

M. BimlerD. StoufferH. Lai and M. Mayfield, Accurate predictions of coexistence in natural systems require the inclusion of facilitative interactions and environmental dependency, Journal of Ecology, 106 (2018), 1839-1852.  doi: 10.1111/1365-2745.13030.

[13]

T. Blows and N. Lloyd, The number of limit cycles of certain polynomial differential equations, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 98 (1984), 215-239.  doi: 10.1017/S030821050001341X.

[14]

C. Chicone, Ordinary Differential Equations with Applications, Second edition. Texts in Applied Mathematics, 34. Springer, New York, 2006.

[15]

F. Courchamp, L. Berec and J. Gascoigne, Allee effects in ecology and conservation, Oxford University Press, (2008).

[16]

F. CourchampT. Clutton-Brock and B. Grenfell, Inverse density dependence and the Allee effect, Trends in Ecology & Evolution, 14 (1999), 405-410.  doi: 10.1016/S0169-5347(99)01683-3.

[17]

A. Dhooge, W. Govaerts and Y. Kuznetsov, Matcont: A matlab package for numerical bifurcation analysis of odes, ACM Transactions on Mathematical Software (TOMS), 29 (2003), 141–164. doi: 10.1145/779359.779362.

[18]

S. Erlinge, Predation and noncyclicity in a microtine population in southern Sweden, Oikos, 50 (1987), 347-352.  doi: 10.2307/3565495.

[19]

J. Flores and E. González-Olivares, Dynamics of a predator–prey model with Allee effect on prey and ratio-dependent functional response, Ecological Complexity, 18 (2014), 59-66. 

[20]

J. Flores and E. González–Olivares, A modified Leslie–Gower predator-prey model with ratio–dependent functional response and alternative food for the predator, Mathematical Methods in the Applied Sciences, 40 (2017), 2313-2328.  doi: 10.1002/mma.4172.

[21]

V. Gaiko, Global Bifurcation Theory and Hilbert's Sixteenth Problem, Mathematics and Its Applications, Springer Science Business Media, 2013. doi: 10.1007/978-1-4419-9168-3.

[22]

E. González-OlivaresB. González-YañezJ. Mena-LorcaA. Rojas-Palma and J. Flores, Consequences of double Allee effect on the number of limit cycles in a predator-prey model, Computers & Mathematics with Applications, 62 (2011), 3449-3463.  doi: 10.1016/j.camwa.2011.08.061.

[23]

I. HanskiL. Hansson and H. Henttonen, Specialist predators, generalist predators, and the microtine rodent cycle, The Journal of Animal Ecology, 60 (1991), 353-367.  doi: 10.2307/5465.

[24]

I. HanskiH. HenttonenE. KorpimäkiL. Oksanen and P. Turchin, Small-rodent dynamics and predation, Ecology, 82 (2001), 1505-1520. 

[25]

L. Hansson, Competition between rodents in successional stages of taiga forests: Microtus agrestis vs. Clethrionomys glareolus, Oikos, 40 (1983), 258-266.  doi: 10.2307/3544590.

[26]

K. HarleyP. van HeijsterR. MarangellG. Pettet and M. Wechselberger, Existence of traveling wave solutions for a model of tumor invasion, SIAM Journal on Applied Dynamical Systems, 13 (2014), 366-396.  doi: 10.1137/130923129.

[27]

C. S. Holling, The components of predation as revealed by a study of small mammal predation of the european pine sawfly, Tenth International Congress of Entomology, 91 (1959), 293-320. 

[28]

S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM Journal on Applied Mathematics, 55 (1995), 763-783.  doi: 10.1137/S0036139993253201.

[29]

J. HuangY. Gong and S. Ruan, Bifurcation analysis in a predator-prey model with constant-yield predator harvesting, Discrete and Continuous Dynamical Systems Series B, 18 (2013), 2101-2121.  doi: 10.3934/dcdsb.2013.18.2101.

[30]

D. HooperF. ChapinJ. EwelA. HectorP. InchaustiS. LavorelJ. LawtonD. LodgeM. Loreau and S. Naeem, Effects of biodiversity on ecosystem functioning: A consensus of current knowledge, Ecological monographs, 75 (2005), 3-35.  doi: 10.1890/04-0922.

[31]

A. Korobeinikov, A Lyapunov function for Leslie–Gower predator-prey models, Applied Mathematics Letters, 14 (2001), 697-699.  doi: 10.1016/S0893-9659(01)80029-X.

[32]

A. KramerL. Berec and J. Drake, Allee effects in ecology and evolution, Journal of Animal Ecology, 87 (2018), 7-10. 

[33]

R. Levins, Discussion paper: The qualitative analysis of partially specified systems, Annals of the New York Academy of Sciences, 231 (1974), 123-138.  doi: 10.1111/j.1749-6632.1974.tb20562.x.

[34]

M. Liermann and R. Hilborn, Depensation: Evidence, models and implications, Fish and Fisheries, 2 (2001), 33-58. 

[35]

A. Lotka, Contribution to the theory of periodic reactions, The Journal of Physical Chemistry, 14 (1910), 271-274.  doi: 10.1021/j150111a004.

[36]

M. I. G. and X. Lambin, The impact of weasel predation on cyclic field-vole survival: The specialist predator hypothesis contradicted, Journal of Animal Ecology, 71 (2002), 946-956. 

[37]

N. Martínez-Jeraldo and P. Aguirre, Allee effect acting on the prey species in a Leslie–Gower predation model, Nonlinear Analysis: Real World Applications, 45 (2019), 895-917.  doi: 10.1016/j.nonrwa.2018.08.009.

[38] R. May, Stability and Complexity in Model Ecosystems, Princeton university press, 2001. 
[39]

R. MonclusD. von HolstD. Blumstein and H. Rödel, Long-term effects of litter sex ratio on female reproduction in two iteroparous mammals, Functional Ecology, 28 (2014), 954-962.  doi: 10.1111/1365-2435.12231.

[40]

R. Ostfeld and C. Canham, Density-dependent processes in meadow voles: An experimental approach, Ecology, 76 (1995), 521-532.  doi: 10.2307/1941210.

[41]

L. Perko, Differential Equations and Dynamical Systems, Springer New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[42]

S. Prager and W. Reiners, Historical and emerging practices in ecological topology, Ecological Complexity, 6 (2009), 160-171.  doi: 10.1016/j.ecocom.2008.11.001.

[43]

B. RaymondJ. McInnesJ. D. nd S. Way and D. Bergstrom, Qualitative modelling of invasive species eradication on subantarctic Macquarie Island, Journal of Applied Ecology, 48 (2011), 181-191.  doi: 10.1111/j.1365-2664.2010.01916.x.

[44]

P. RouxJ. Shaw and S. Chown, Ontogenetic shifts in plant interactions vary with environmental severity and affect population structure, New Phytologist, 200 (2013), 241-250. 

[45]

E. Sáez and E. González-Olivares, Dynamics on a predator-prey model, SIAM Journal on Applied Mathematics, 59 (1999), 1867-1878.  doi: 10.1137/S0036139997318457.

[46]

X. Santos and M. Cheylan, Taxonomic and functional response of a Mediterranean reptile assemblage to a repeated fire regime, Biological Conservation, 168 (2013), 90-98.  doi: 10.1016/j.biocon.2013.09.008.

[47]

P. Stephens and W. Sutherland, Consequences of the Allee effect for behaviour, ecology and conservation, Trends in Ecology & Evolution, 14 (1999), 401-405.  doi: 10.1016/S0169-5347(99)01684-5.

[48] P. Turchin, Complex Population Dynamics: A Theoretical/Empirical Synthesis, Princeton university press, 2003. 
[49]

A. Verdy, Modulation of predator-prey interactions by the Allee effect, Ecological Modelling, 221 (2010), 1098-1107.  doi: 10.1016/j.ecolmodel.2010.01.005.

[50]

G. VoornL. HemerikM. Boer and B. 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.

[51]

S. Wood and M. Thomas, Super-sensitivity to structure in biological models, Proceedings of the Royal Society of London. Series B: Biological Sciences, 266 (1999), 565-570.  doi: 10.1098/rspb.1999.0673.

[52]

D. Xiao and S. Ruan, Bogdanov–Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Institute Communications, 21 (1999), 493-506. 

[53]

Z. Yue, X. Wang and H. Liu, Complex dynamics of a diffusive Holling–Tanner predator-prey model with the Allee effect, Abstract and Applied Analysis, 2013 (2013), 12 pp. doi: 10.1155/2013/270191.

[54]

Z. ZhaoL. Yang and L. Chen, Impulsive perturbations of a predator-prey system with modified Leslie–Gower and Holling type Ⅱ schemes, Journal of Applied Mathematics and Computing, 35 (2011), 119-134.  doi: 10.1007/s12190-009-0346-2.

[55]

J. Zu and M. Mimura, The impact of Allee effect on a predator-prey system with Holling type Ⅱ functional response, Applied Mathematics and Computation, 217 (2010), 3542-3556.  doi: 10.1016/j.amc.2010.09.029.

Figure 1.  In the left panel, we show the per capita growth rate of the logistic function (blue line), the strong Allee effect with $ m = 0.1 $ (red curve), the weak Allee effect with $ m = -0.1 $ (orange curve), multiple Allee effects with $ m = 0.1 $ and $ b = 0.15 $ (grey curve) and multiple Allee effects with $ m = 0.1 $ and $ b = 0.05 $ (green curve). In the right panel, we show the size of the depensation region for the strong Allee effect (6) (red curve) and for the multiple Allee effects (5) (grey curve) as function of the non-fertile prey population $ b $. We observe that the depensation region for the multiple Allee effects is always smaller than the depensation region for the strong Allee effect
Figure 2.  The intersections of the functions $ p(u) $ (red line) and $ d(u) $ (blue lines) for three different possible cases: (a) If $ \Delta<0 $ (10) then $ p(u) $ and $ d(u) $ do not intersect, and (8) does not have positive equilibrium points; (b) If $ \Delta = 0 $ then $ p(u) $ and $ d(u) $ intersect in one point, and (8) has a unique positive equilibrium point; (c) If $ \Delta>0 $ then $ p(u) $ and $ d(u) $ intersect in two points, and (8) has two distinct positive equilibrium points
Figure 3.  Phase plane of system (8) and its invariant regions $ \Phi $ and $ \Gamma\backslash\Phi $
Figure 4.  For $ M = 0.05 $, $ B = 0.05 $, $ C = 0.5 $, $ Q = 0.8 $, and $ S = 0.175 $, such that $ \Delta<0 $ (10), the equilibrium point $ (0,C) $ is a global attractor for trajectories starting in the first quadrant. The blue (red) curve represents the prey (predator) nullcline
Figure 5.  Let the system parameter $ (M,B,C,Q) = (0.07,0.0645,0.32,0.736) $ be such that $ \Delta>0 $ (10). (a) If $ S = 0.15 $ such that $ C<C_{H} $, then the equilibrium point $ P_2 $ is stable. (b) If $ S = 0.05 $ such that $ C>C_{H} $, then the equilibrium point $ P_2 $ is unstable. The blue (red) curve represents the prey (predator) nullcline. The orange (light blue) region represents the basin of attraction of the equilibrium point $ (0,C) $ ($ P_2 $). Note that the same color conventions are used in the upcoming figures
Figure 6.  If $ M = 0.05 $, $ B = 0.05 $, $ S = 0.125 $ and $ Q = 0.60821818 $, then $ \Delta = 0 $. Therefore, the equilibrium point $ P_3 $ is (a) a saddle-node repeller if $ C>C_{SN} $ and (b) a saddle-node attractor if $ C<C_{SN} $
Figure 7.  For $ M = 0.05 $, $ B = 0.05 $, $ C = 0.58951256 $, $ S = 0.125 $ and $ Q = 0.60821818 $, such that $ \Delta = 0 $ and $ f(u_3) = C_{SN} $, the point $ (0,C) $ is an attractor and the equilibrium point $ P_3 $ is a cusp point
Figure 8.  The bifurcation diagram of system (8) for $ M = 0.05 $ and $ S = 0.071080895 $ fixed and created with the numerical bifurcation package MATCONT [17]. In the left panel $ B = 0.1 $ fixed and varying $ Q $ and $ C $ and in the right panel $ Q = 0.608 $ fixed and varying $ B $ and $ C $. The curve $ C_H $ represents the Hopf curve, $ C_{HOM} $ represents the homoclinic curve, $ C_{SN} $ represents the saddle-node curve, and $ BT $ represents the Bogdanov-Takens bifurcation.The corresponding phase planes for the different regions are shown in Figure 9
Figure 9.  The phase planes of system (8) for $ B = 0.1 $, $ M = 0.05 $, $ Q = 0.75 $ and $ S = 0.071080895 $ fixed and varying $ C $. This last parameter impacts the number of equilibrium points of system (8). The light blue area in the phase plane represent the basins of attraction of the equilibrium points $ P_2 $, while the orange area in the phase plane represent the basins of attraction of the equilibrium points $ (0,C) $
Figure 10.  The size of the basin of attraction of $ p_2 $, in units$ ^2 $, of the stable equilibrium point $ p_2 $ of system (7) considering strong Allee effect (red line) and multiple Allee effect (blue line) for varying the non-fertile population $ b $ and with other system parameters $ r = 14 $, $ K = 150 $, $ m = 15 $, $ q = 1.08 $, $ s = 1.25 $, $ n = 0.05 $ and $ c = 0.75 $ fixed. The blue dotted-dashed line represents the region where the stable manifold of the saddle equilibrium point $ p_1 $ connects with (K, 0) and the blue dashed line represent the region where the equilibrium point $ p_2 $ is surrounded by an unstable limit cycle
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