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doi: 10.3934/dcdsb.2022094
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Bifurcation and overexploitation in Rosenzweig-MacArthur model

1. 

Department of Mathematics, Hangzhou Normal University, Zhejiang 311121, China

2. 

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

3. 

Department of Mathematics, Columbus State University, Columbus, GA 31907, USA

* Corresponding author: Yancong Xu

Received  December 2020 Revised  February 2022 Early access May 2022

Fund Project: The second author is supported by the National NSF of China (No. 11671114) and NSF of Zhejiang Province(LY20A010002); The third author is supported by NSF of Shanghai (20ZR1440600, 20JC1413800)

In this paper, we propose a Rosenzweig–MacArthur predator-prey model with strong Allee effect and trigonometric functional response. The local and global stability of equilibria is studied, and the existence of bifurcation is determined in terms of the carrying capacity of the prey, the death rate of the predator and the Allee effect. An analytic expression is employed to determine the criticality and codimension of Hopf bifurcation. The existence of supercritical Hopf bifurcation and the non-existence of Bogdanov–Takens bifurcation at the positive equilibrium are proved. A point-to-point heteroclinic cycle is also found. Biologically speaking, such a heteroclinic cycle always indicates the collapse of the system after the invasion of the predator, i.e., overexploitation occurs. It is worth pointing out that heteroclinic relaxation cycles are driven by either the strong Allee effect or the high per capita death rate. In addition, numerical simulations are given to demonstrate the theoretical results.

Citation: Xiaoqing Lin, Yancong Xu, Daozhou Gao, Guihong Fan. Bifurcation and overexploitation in Rosenzweig-MacArthur model. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022094
References:
[1]

C. D. Aline and J. A. Prevedello, The importance of protected areas for overexploited plants: Evidence from a biodiversity hotspot, Biological Conservation, 243 (2020), 108482.  doi: 10.1016/j.biocon.2020.108482.

[2] W. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, USA, 1931.  doi: 10.5962/bhl.title.7313.
[3]

L. BerecV. Bernhauerova and B. Boldin, Evolution of mate-finding Allee effect in prey, J. Theoret. Biol., 441 (2018), 9-18.  doi: 10.1016/j.jtbi.2017.12.024.

[4]

S. BiswasS. K. SasmalS. SamantaM. SaifuddinQ. J. A. Khan and J. Chattopadhyay, A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect, Math. Biosci., 263 (2015), 198-208.  doi: 10.1016/j.mbs.2015.02.013.

[5]

C. Castillo-Chavez, et al., Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, 2002.

[6]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[7]

M. H. Cortez and P. A. Abrams, Hydra effects in stable communities and their implications for system dynamics, Ecology, 97 (2016), 1135-1145.  doi: 10.1890/15-0648.1.

[8]

E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Technical report, Concordia University, 2009.

[9]

K. J. DuffyK. L. Patrick and S. D. Johnson, Does the likelihood of an Allee effect on plant fecundity depend on the type of pollinator?, Journal of Ecology, 101 (2013), 953-962.  doi: 10.1111/1365-2745.12104.

[10]

N. T. Fadai and M. J. Simpson, Population dynamics with threshold effects give rise to a diverse family of Allee effects, Bull. Math. Biol., 82 (2020), 74, 22 pp. doi: 10.1007/s11538-020-00756-5.

[11]

G. F. Fussmann and B. Blasius, Community response to enrichment is highly sensitive to model structure, Biology Letters, 1 (1992), 9-12.  doi: 10.1098/rsbl.2004.0246.

[12]

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

[13]

D. W. GoodsmanD. KochC. WhitehouseM. L. EvendenB. J. Cooke and M. A. Lewis, Aggregation and a strong Allee effect in a cooperative outbreak insect, Ecological Applications, 26 (2016), 2623-2636.  doi: 10.1002/eap.1404.

[14]

W. M. HirschH. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1995), 733-753.  doi: 10.1002/cpa.3160380607.

[15]

P. IlariaB. UgoE. A. Toufic and L. Alessandro, Dynamic patterns of overexploitation in fisheries, Ecological Modelling, 359 (2017), 285-292.  doi: 10.1016/j.ecolmodel.2017.06.009.

[16]

A. D. Jassby and T. Platt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnology and Oceanography, 21 (1976), 540-547. 

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Springer-Verlag, New York, 1998.

[18]

M. Y. LiW. LiuC. Shan and Y. Yi, Turning points and relaxation oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.

[19]

J. L. OrrockR. D. Holt and M. L. Baskett, Refuge-mediated apparent competition in plant-consumer interactions, Ecology Letters, 13 (2010), 11-20.  doi: 10.1111/j.1461-0248.2009.01412.x.

[20]

S. V. PetrovskiiA. 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.

[21]

L. A. D. RodriguesD. C. Mistro and S. Petrovskii, Pattern formation in a space- and time-discrete predator-prey system with a strong Allee effect, Theoretical Ecology, 5 (2012), 341-362.  doi: 10.1007/s12080-011-0139-8.

[22]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272.

[23]

B. Sandstede and Y. Xu, Snakes and isolas in non-reversible conservative systems, Dyn. Syst., 27 (2012), 317-329.  doi: 10.1080/14689367.2012.691961.

[24]

S. K. Sasmal and J. Chattopadhyay, An eco-epidemiological system with infected prey and predator subject to the weak Allee effect, Math. Biosci., 246 (2013), 260-271.  doi: 10.1016/j.mbs.2013.10.005.

[25]

G. Seo and G. S. K. Wolkowicz, Sensitivity of the dynamics of the general Rosenzweig–MacArthur model to the mathematical form of the functional response: A bifurcation theory approach, J. Math. Biol., 76 (2018), 1873-1906.  doi: 10.1007/s00285-017-1201-y.

[26]

D. K. Sorenson and M. H. Cortez, How intra-stage and inter-stage competition affect overcompensation in density and hydra effects in single-species, stage-structured models, Theoretical Ecology, 14 (2020), 23-39.  doi: 10.1007/s12080-020-00488-1.

[27]

D. Start and B. Gilbert, Plant sex alters Allee effects in aggregating plant parasites, Oikos, 127 (2018), 792-802.  doi: 10.1111/oik.04405.

[28]

P. A. StephensW. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.  doi: 10.2307/3547011.

[29]

J. Sugie and Y. Saito, Uniqueness of limit cycles in a Rosenzweig–MacArthur model with prey immigration, SIAM J. Appl. Math., 72 (2012), 299-316.  doi: 10.1137/11084008X.

[30]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[31]

G. A. K. van VoornL. HemerikM. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469.  doi: 10.1016/j.mbs.2007.02.006.

[32]

M. Verma, Modeling the effect of rarity value on the exploitation of a wildlife species subjected to the Allee effect, Nat. Resour. Model., 29 (2016), 470-494.  doi: 10.1111/nrm.12100.

[33]

J. WangJ. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.  doi: 10.1007/s00285-010-0332-1.

[34]

Y. Xu, Z. Zhu, Y. Yang and F. Meng, Vectored immunoprophylaxis and cell-to-cell transmission in HIV dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050185, 19 pp. doi: 10.1142/S0218127420501850.

[35]

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vibration, 211 (1998), 19-38.  doi: 10.1006/jsvi.1997.1347.

[36]

W. J. ZhangM. W. Lindi and P. Yu, Viral blips may not need a trigger: How transient viremia can arise in deterministic in-host models, SIAM Review, 56 (2014), 127-155.  doi: 10.1137/130937421.

show all references

References:
[1]

C. D. Aline and J. A. Prevedello, The importance of protected areas for overexploited plants: Evidence from a biodiversity hotspot, Biological Conservation, 243 (2020), 108482.  doi: 10.1016/j.biocon.2020.108482.

[2] W. Allee, Animal Aggregations: A Study in General Sociology, University of Chicago Press, Chicago, USA, 1931.  doi: 10.5962/bhl.title.7313.
[3]

L. BerecV. Bernhauerova and B. Boldin, Evolution of mate-finding Allee effect in prey, J. Theoret. Biol., 441 (2018), 9-18.  doi: 10.1016/j.jtbi.2017.12.024.

[4]

S. BiswasS. K. SasmalS. SamantaM. SaifuddinQ. J. A. Khan and J. Chattopadhyay, A delayed eco-epidemiological system with infected prey and predator subject to the weak Allee effect, Math. Biosci., 263 (2015), 198-208.  doi: 10.1016/j.mbs.2015.02.013.

[5]

C. Castillo-Chavez, et al., Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction, Springer-Verlag, New York, 2002.

[6]

S. N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer, New York, 1982.

[7]

M. H. Cortez and P. A. Abrams, Hydra effects in stable communities and their implications for system dynamics, Ecology, 97 (2016), 1135-1145.  doi: 10.1890/15-0648.1.

[8]

E. J. Doedel and B. E. Oldeman, AUTO-07P: Continuation and Bifurcation Software for Ordinary Differential Equations, Technical report, Concordia University, 2009.

[9]

K. J. DuffyK. L. Patrick and S. D. Johnson, Does the likelihood of an Allee effect on plant fecundity depend on the type of pollinator?, Journal of Ecology, 101 (2013), 953-962.  doi: 10.1111/1365-2745.12104.

[10]

N. T. Fadai and M. J. Simpson, Population dynamics with threshold effects give rise to a diverse family of Allee effects, Bull. Math. Biol., 82 (2020), 74, 22 pp. doi: 10.1007/s11538-020-00756-5.

[11]

G. F. Fussmann and B. Blasius, Community response to enrichment is highly sensitive to model structure, Biology Letters, 1 (1992), 9-12.  doi: 10.1098/rsbl.2004.0246.

[12]

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

[13]

D. W. GoodsmanD. KochC. WhitehouseM. L. EvendenB. J. Cooke and M. A. Lewis, Aggregation and a strong Allee effect in a cooperative outbreak insect, Ecological Applications, 26 (2016), 2623-2636.  doi: 10.1002/eap.1404.

[14]

W. M. HirschH. Hanisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1995), 733-753.  doi: 10.1002/cpa.3160380607.

[15]

P. IlariaB. UgoE. A. Toufic and L. Alessandro, Dynamic patterns of overexploitation in fisheries, Ecological Modelling, 359 (2017), 285-292.  doi: 10.1016/j.ecolmodel.2017.06.009.

[16]

A. D. Jassby and T. Platt, Mathematical formulation of the relationship between photosynthesis and light for phytoplankton, Limnology and Oceanography, 21 (1976), 540-547. 

[17]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Springer-Verlag, New York, 1998.

[18]

M. Y. LiW. LiuC. Shan and Y. Yi, Turning points and relaxation oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2016), 663-687.  doi: 10.1137/15M1038785.

[19]

J. L. OrrockR. D. Holt and M. L. Baskett, Refuge-mediated apparent competition in plant-consumer interactions, Ecology Letters, 13 (2010), 11-20.  doi: 10.1111/j.1461-0248.2009.01412.x.

[20]

S. V. PetrovskiiA. 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.

[21]

L. A. D. RodriguesD. C. Mistro and S. Petrovskii, Pattern formation in a space- and time-discrete predator-prey system with a strong Allee effect, Theoretical Ecology, 5 (2012), 341-362.  doi: 10.1007/s12080-011-0139-8.

[22]

M. L. Rosenzweig and R. H. MacArthur, Graphical representation and stability conditions of predator-prey interactions, The American Naturalist, 97 (1963), 209-223.  doi: 10.1086/282272.

[23]

B. Sandstede and Y. Xu, Snakes and isolas in non-reversible conservative systems, Dyn. Syst., 27 (2012), 317-329.  doi: 10.1080/14689367.2012.691961.

[24]

S. K. Sasmal and J. Chattopadhyay, An eco-epidemiological system with infected prey and predator subject to the weak Allee effect, Math. Biosci., 246 (2013), 260-271.  doi: 10.1016/j.mbs.2013.10.005.

[25]

G. Seo and G. S. K. Wolkowicz, Sensitivity of the dynamics of the general Rosenzweig–MacArthur model to the mathematical form of the functional response: A bifurcation theory approach, J. Math. Biol., 76 (2018), 1873-1906.  doi: 10.1007/s00285-017-1201-y.

[26]

D. K. Sorenson and M. H. Cortez, How intra-stage and inter-stage competition affect overcompensation in density and hydra effects in single-species, stage-structured models, Theoretical Ecology, 14 (2020), 23-39.  doi: 10.1007/s12080-020-00488-1.

[27]

D. Start and B. Gilbert, Plant sex alters Allee effects in aggregating plant parasites, Oikos, 127 (2018), 792-802.  doi: 10.1111/oik.04405.

[28]

P. A. StephensW. J. Sutherland and R. P. Freckleton, What is the Allee effect?, Oikos, 87 (1999), 185-190.  doi: 10.2307/3547011.

[29]

J. Sugie and Y. Saito, Uniqueness of limit cycles in a Rosenzweig–MacArthur model with prey immigration, SIAM J. Appl. Math., 72 (2012), 299-316.  doi: 10.1137/11084008X.

[30]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.  doi: 10.1016/S0025-5564(02)00108-6.

[31]

G. A. K. van VoornL. HemerikM. P. Boer and B. W. Kooi, Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect, Math. Biosci., 209 (2007), 451-469.  doi: 10.1016/j.mbs.2007.02.006.

[32]

M. Verma, Modeling the effect of rarity value on the exploitation of a wildlife species subjected to the Allee effect, Nat. Resour. Model., 29 (2016), 470-494.  doi: 10.1111/nrm.12100.

[33]

J. WangJ. Shi and J. Wei, Predator-prey system with strong Allee effect in prey, J. Math. Biol., 62 (2011), 291-331.  doi: 10.1007/s00285-010-0332-1.

[34]

Y. Xu, Z. Zhu, Y. Yang and F. Meng, Vectored immunoprophylaxis and cell-to-cell transmission in HIV dynamics, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 30 (2020), 2050185, 19 pp. doi: 10.1142/S0218127420501850.

[35]

P. Yu, Computation of normal forms via a perturbation technique, J. Sound Vibration, 211 (1998), 19-38.  doi: 10.1006/jsvi.1997.1347.

[36]

W. J. ZhangM. W. Lindi and P. Yu, Viral blips may not need a trigger: How transient viremia can arise in deterministic in-host models, SIAM Review, 56 (2014), 127-155.  doi: 10.1137/130937421.

Figure 1.  Bifurcation diagram for system (4) showing the transition of equilibrium solutions $ E_0, $ $ E_1, $ $ E_2, $ and $ E_3, $ where solid and dotted lines/curves represent stable and unstable equilibrium solutions, respectively
Figure 2.  The possible relative position between the unstable manifold $ W^s(0,C) $ of the saddle point $ E_1(0,C) $ and the stable manifold $ W^u(0,D) $ of the saddle point $ E_2(0,D) $. (a) $ X_s<X_u; $ (b) $ X_s>X_u $
Figure 3.  Phase portraits of model (4). (a) $ A = 0.6666667 $ and $ B = 2.8181335. $ (b) $ A = 0.5263158 $ and $ B = 2.2248422. $ (c) $ A = 0.1010101 $ and $ B = 0.4269899. $
Figure 4.  Bifurcation diagram for model (24). Here $ HB $, $ TC_1 $ and $ TC_2 $ denote the supercritical Hopf bifurcation point and two transcritical bifurcation points, respectively. (a) $ R_0 $ vs $ X $. (b) $ R_0 $ vs $ Y. $
Figure 5.  Bifurcation diagram and phase portrait of model (4). Here $ HB $, $ TC_1 $, $ TC_2 $ and $ TC_3 $ denote the supercritical Hopf bifurcation point and three transcritical bifurcation points, respectively. (a) $ A $ vs $ X $. (b) $ A $ vs $ Y $. (c) $ A $ vs the period. (d) A family of stable limit cycles approach a heteroclinic cycle which connects the boundary equilibria $ E_1(0, 0.0148) $ and $ E_2(0, 1.5984) $
Figure 6.  Bifurcation diagram and phase portrait of model (4). Here $ HB $, $ TC_1 $ and $ TC_2 $ denote the supercritical Hopf bifurcation point and two transcritical bifurcation points. (a) $ D $ vs $ X $. (b) $ D $ vs $ Y $. (c) $ D $ vs the period. (d) Heteroclinic relaxation oscillations appear in phase portrait
Figure 7.  Bifurcation diagram and phase portrait of model (4), where $ HB, $ $ TC_1 $ and $ TC_2 $ represent the supercritical Hopf bifurcation point and two transcritical bifurcation points, respectively. (a) $ C $ vs $ X $. (b) $ C $ vs $ Y $. (c) Period against $ C $. (d) The phase portrait indicates that a family of limit cycles with amplitudes stretched to a heteroclinic cycle connecting $ E_1(0, 1.41234) $ and $ E_2(0, 1.5984) $
Figure 8.  (a) Bifurcation diagram ($ C $ vs Period) of limit cycles of model (4) as $ A = 0.9888, B = 0.4269899, D = 1.5984. $ (b) Heteroclinic relaxation oscillations occur for model (4) with a family of limit cycles approaching a heteroclinic cycle in the phase plane
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