May  2020, 25(5): 1631-1647. doi: 10.3934/dcdsb.2019244

Stability and bifurcation analysis of Filippov food chain system with food chain control strategy

1. 

School of Mathematics and Information Science, Shaanxi Normal University, Xi'an 710062, China

2. 

Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut 71524, Egypt

* Corresponding author: Soliman A. A. Hamdallah

Received  March 2019 Revised  June 2019 Published  November 2019

In the present work, we introduce a control model to describe three species food chain interaction model composed of prey, middle predator, and top predator. The middle predator preys on prey and the top predator preys on middle predator. The control techniques of the exploited natural resources are used to modulate the harvesting effort to avoid high risks of extinction of the middle predator and keep stability of the food chain, by prohibiting fishing when the population density drops below a prescribed threshold. The behavior of the system stability of the regular, virtual, pseudo-equilibrium and tangent points are discussed. The complicated non-smooth dynamic behaviors (sliding and crossing segment and their domains) are analyzed. The bifurcation set of pseudo-equilibrium and the sliding crossing bifurcation have been investigated. Our analytical findings are verified through numerical investigations.

Citation: Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244
References:
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V. Acary and B. Brogliato, Numerical Methods For Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics, Springer-Verlag, New York, 2008. Google Scholar

[2]

A. Al-Khedhairi, The chaos and control of food chain model using nonlinear feedback, Appl. Math. Sci., 3 (2009), 591-604.   Google Scholar

[3]

M. BanerjeeN. Mukherjee and V. Volpert, Prey-Predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041.  Google Scholar

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S. P. BeraA. Maiti and G. P. Samanta, Dynamics of a food chain model with herd behaviour of the prey, Model. Earth Syst. Environ., 2 (2016), 131-140.  doi: 10.1007/s40808-016-0189-4.  Google Scholar

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R. CaseyH. de Jong and J. L. Gouze, Piecewise-linear models of genetic regulatory networks: Equilibria and their stability, J.Math.Biol., 52 (2006), 27-56.  doi: 10.1007/s00285-005-0338-2.  Google Scholar

[9]

X. Chen and W. Zhang, Normal forms of planar switching systems, Discrete Contin. Dyn. Syst.-A, 36 (2016), 6715-6736.  doi: 10.3934/dcds.2016092.  Google Scholar

[10]

A. ColomboN. D. BuonoL. Lopez and A. Pugliese, Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems, Discrete Contin. Dyn. Syst.-B, 23 (2018), 2911-2934.  doi: 10.3934/dcdsb.2018166.  Google Scholar

[11]

M. I. S. Costa and L. D. B. Faria, Integrated pest management: theoretical insights from a threshold policy, Neotropical Entomology, 39 (2010), 1-8.  doi: 10.1590/S1519-566X2010000100001.  Google Scholar

[12]

M. I. S. Costa and M. E. M. Meza, Application of a threshold policy in the management of multispecies fisheries and predator culling, Math. Medicine and Bio.: A Journal of the IMA, 23 (2006), 63-75.  doi: 10.1093/imammb/dql005.  Google Scholar

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M. di Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-708-4.  Google Scholar

[14]

M. di BernardoC. BuddA. R. ChampneysP. KowalczykA. B. NordmarkG. Olivar and P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.  doi: 10.1137/050625060.  Google Scholar

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M. di BernardoP. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170 (2002), 175-205.  doi: 10.1016/S0167-2789(02)00547-X.  Google Scholar

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L. Dieci and F. Difonzo, A comparison of Filippov sliding vector fields in codimension 2, J. Comput. Appl. Math., 262 (2014), 161-179.  doi: 10.1016/j.cam.2013.10.055.  Google Scholar

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A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231.   Google Scholar

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A. F. Filippov, Differential equations with discontinuous right-hand sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

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H. I. Freedman and P. Waltman, Mathematical analysis of some three species food-chain models, Math. Biosci., 33 (1977), 257-276.  doi: 10.1016/0025-5564(77)90142-0.  Google Scholar

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T. Gard, Top predator persistence in differential equations models of food chains: The effects of omnivory and external forcing of lower trophic levels, J. Math. Biology, 14 (1982), 285-299.  doi: 10.1007/BF00275394.  Google Scholar

[21]

K. Gupta and S. Gakkhar, The Filippov approach for predator-prey system involving mixed type of functional responses, Differential Eq. and Dynamical Syst., (2016), 1-21. doi: 10.1007/s12591-016-0322-x.  Google Scholar

[22]

A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.  Google Scholar

[23]

M. R. Jeffrey, Dynamics at switching intersection: Hierarchy, isonomy and multiple-sliding, SIAM J. Appl. Dyn. Syst., 13 (2014), 1082-1105.  doi: 10.1137/13093368X.  Google Scholar

[24]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.  Google Scholar

[25]

W. S. MadasanjayaM. MamatZ. SallehI. Mohd and N. M. Mohamad, Numerical simulation dynamical model of three species food chain with Holling type-Ⅱ functional response, Malays. J. Math. Sci., 5 (2011), 1-12.   Google Scholar

[26]

K. Popp and P. Stelter, Stick-slip vibrations and chaos, Phil. Tran.: Phys. Scie. and Eng., 332 (1990), 89-105.  doi: 10.1098/rsta.1990.0102.  Google Scholar

[27]

F. D. Rossa and F. Dercole, The transition from persistence to nonsmooth-fold scenarios in relay control system, IFAC Proceedings Volumes, 44 (2011), 13287-13292.  doi: 10.3182/20110828-6-IT-1002.01354.  Google Scholar

[28]

J. M. Schumacher, Time-scaling symmetry and Zeno solutions, Automatica, 45 (2009), 1237-1242.  doi: 10.1016/j.automatica.2008.12.008.  Google Scholar

[29]

Z. Shuwen and T. Dejun, Permanence in a food chain system with impulsive perturbations, Chaos Solitons Fractals, 40 (2009), 392-400.  doi: 10.1016/j.chaos.2007.07.074.  Google Scholar

[30]

Z. Shuwen and C. Lansun, A Holling Ⅱ functional response food chain model with impulsive perturbations, Chaos, Solitons and Fractals, 24 (2005), 1269-1278.  doi: 10.1016/j.chaos.2004.09.051.  Google Scholar

[31]

S. TangJ. H. LiangY. N. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math., 72 (2012), 1061-1080.  doi: 10.1137/110847020.  Google Scholar

[32]

S. Tang, G. Tang and W. Qin, Codimension-1 sliding bifurcations of a Filippov pest growth model with threshold policy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014). doi: 10.1142/S0218127414501223.  Google Scholar

[33]

Y. Tang, Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems, Discrete Contin. Dyn. Syst.-A, 38 (2018), 2029-2046.  doi: 10.3934/dcds.2018082.  Google Scholar

[34]

F. TaoB. KangB. Liu and L. Qu, Threshold strategy for nonsmooth Filippov stage-structured pest growth models, Math. Probl. Eng., 1 (2019), 1-7.  doi: 10.1155/2019/9742197.  Google Scholar

[35] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational biology, Princeton University Press, Princeton, 2003.  doi: 10.2307/j.ctv301f9v.  Google Scholar

show all references

References:
[1]

V. Acary and B. Brogliato, Numerical Methods For Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics, Springer-Verlag, New York, 2008. Google Scholar

[2]

A. Al-Khedhairi, The chaos and control of food chain model using nonlinear feedback, Appl. Math. Sci., 3 (2009), 591-604.   Google Scholar

[3]

M. BanerjeeN. Mukherjee and V. Volpert, Prey-Predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041.  Google Scholar

[4]

S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Bifurcations, Chaos, Control, and Applications, John Wiley and Sons, 2001. Google Scholar

[5]

S. P. BeraA. Maiti and G. P. Samanta, Dynamics of a food chain model with herd behaviour of the prey, Model. Earth Syst. Environ., 2 (2016), 131-140.  doi: 10.1007/s40808-016-0189-4.  Google Scholar

[6]

B. Brogliato, Impact in Mechanical Systems - Analysis and Modelling, Lecture Notes in Physics, 551, Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-45501-9.  Google Scholar

[7]

B. Brogliato, Nonsmooth Mechanics - Models, Dynamics and Control, Communications and Control Engineering, Springer-Verlag, London, 1999. doi: 10.1007/978-3-319-28664-8.  Google Scholar

[8]

R. CaseyH. de Jong and J. L. Gouze, Piecewise-linear models of genetic regulatory networks: Equilibria and their stability, J.Math.Biol., 52 (2006), 27-56.  doi: 10.1007/s00285-005-0338-2.  Google Scholar

[9]

X. Chen and W. Zhang, Normal forms of planar switching systems, Discrete Contin. Dyn. Syst.-A, 36 (2016), 6715-6736.  doi: 10.3934/dcds.2016092.  Google Scholar

[10]

A. ColomboN. D. BuonoL. Lopez and A. Pugliese, Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems, Discrete Contin. Dyn. Syst.-B, 23 (2018), 2911-2934.  doi: 10.3934/dcdsb.2018166.  Google Scholar

[11]

M. I. S. Costa and L. D. B. Faria, Integrated pest management: theoretical insights from a threshold policy, Neotropical Entomology, 39 (2010), 1-8.  doi: 10.1590/S1519-566X2010000100001.  Google Scholar

[12]

M. I. S. Costa and M. E. M. Meza, Application of a threshold policy in the management of multispecies fisheries and predator culling, Math. Medicine and Bio.: A Journal of the IMA, 23 (2006), 63-75.  doi: 10.1093/imammb/dql005.  Google Scholar

[13]

M. di Bernardo, C. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Springer-Verlag, London, 2008. doi: 10.1007/978-1-84628-708-4.  Google Scholar

[14]

M. di BernardoC. BuddA. R. ChampneysP. KowalczykA. B. NordmarkG. Olivar and P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.  doi: 10.1137/050625060.  Google Scholar

[15]

M. di BernardoP. Kowalczyk and A. Nordmark, Bifurcations of dynamical systems with sliding: Derivation of normal-form mappings, Physica D, 170 (2002), 175-205.  doi: 10.1016/S0167-2789(02)00547-X.  Google Scholar

[16]

L. Dieci and F. Difonzo, A comparison of Filippov sliding vector fields in codimension 2, J. Comput. Appl. Math., 262 (2014), 161-179.  doi: 10.1016/j.cam.2013.10.055.  Google Scholar

[17]

A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231.   Google Scholar

[18]

A. F. Filippov, Differential equations with discontinuous right-hand sides, Mathematics and Its Applications, Kluwer Academic, Dordrecht, Netherlands, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[19]

H. I. Freedman and P. Waltman, Mathematical analysis of some three species food-chain models, Math. Biosci., 33 (1977), 257-276.  doi: 10.1016/0025-5564(77)90142-0.  Google Scholar

[20]

T. Gard, Top predator persistence in differential equations models of food chains: The effects of omnivory and external forcing of lower trophic levels, J. Math. Biology, 14 (1982), 285-299.  doi: 10.1007/BF00275394.  Google Scholar

[21]

K. Gupta and S. Gakkhar, The Filippov approach for predator-prey system involving mixed type of functional responses, Differential Eq. and Dynamical Syst., (2016), 1-21. doi: 10.1007/s12591-016-0322-x.  Google Scholar

[22]

A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591.  Google Scholar

[23]

M. R. Jeffrey, Dynamics at switching intersection: Hierarchy, isonomy and multiple-sliding, SIAM J. Appl. Dyn. Syst., 13 (2014), 1082-1105.  doi: 10.1137/13093368X.  Google Scholar

[24]

Y. A. KuznetsovS. Rinaldi and A. Gragnani, One-parameter bifurcations in planar Filippov systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 13 (2003), 2157-2188.  doi: 10.1142/S0218127403007874.  Google Scholar

[25]

W. S. MadasanjayaM. MamatZ. SallehI. Mohd and N. M. Mohamad, Numerical simulation dynamical model of three species food chain with Holling type-Ⅱ functional response, Malays. J. Math. Sci., 5 (2011), 1-12.   Google Scholar

[26]

K. Popp and P. Stelter, Stick-slip vibrations and chaos, Phil. Tran.: Phys. Scie. and Eng., 332 (1990), 89-105.  doi: 10.1098/rsta.1990.0102.  Google Scholar

[27]

F. D. Rossa and F. Dercole, The transition from persistence to nonsmooth-fold scenarios in relay control system, IFAC Proceedings Volumes, 44 (2011), 13287-13292.  doi: 10.3182/20110828-6-IT-1002.01354.  Google Scholar

[28]

J. M. Schumacher, Time-scaling symmetry and Zeno solutions, Automatica, 45 (2009), 1237-1242.  doi: 10.1016/j.automatica.2008.12.008.  Google Scholar

[29]

Z. Shuwen and T. Dejun, Permanence in a food chain system with impulsive perturbations, Chaos Solitons Fractals, 40 (2009), 392-400.  doi: 10.1016/j.chaos.2007.07.074.  Google Scholar

[30]

Z. Shuwen and C. Lansun, A Holling Ⅱ functional response food chain model with impulsive perturbations, Chaos, Solitons and Fractals, 24 (2005), 1269-1278.  doi: 10.1016/j.chaos.2004.09.051.  Google Scholar

[31]

S. TangJ. H. LiangY. N. Xiao and R. A. Cheke, Sliding bifurcations of Filippov two stage pest control models with economic thresholds, SIAM J. Appl. Math., 72 (2012), 1061-1080.  doi: 10.1137/110847020.  Google Scholar

[32]

S. Tang, G. Tang and W. Qin, Codimension-1 sliding bifurcations of a Filippov pest growth model with threshold policy, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014). doi: 10.1142/S0218127414501223.  Google Scholar

[33]

Y. Tang, Global dynamics and bifurcation of planar piecewise smooth quadratic quasi-homogeneous differential systems, Discrete Contin. Dyn. Syst.-A, 38 (2018), 2029-2046.  doi: 10.3934/dcds.2018082.  Google Scholar

[34]

F. TaoB. KangB. Liu and L. Qu, Threshold strategy for nonsmooth Filippov stage-structured pest growth models, Math. Probl. Eng., 1 (2019), 1-7.  doi: 10.1155/2019/9742197.  Google Scholar

[35] H. R. Thieme, Mathematics in Population Biology, Princeton Series in Theoretical and Computational biology, Princeton University Press, Princeton, 2003.  doi: 10.2307/j.ctv301f9v.  Google Scholar
Figure 1.  Illustration of existence of sliding segment of Filippov system (3): where $ A\;c_2< a_3,\;c2<a_3+E,B\;c_2> a_3,\;c2>a_3+E, $ and $ C\; c_2> a_3,\;c2<a_3+E $
Figure 2.  Bifurcation of pseudo-equilibrium
Figure 3.  Bifurcation of pseudo-equilibrium
Figure 4.  Limit cycle without sliding segment
Figure 5.  Limit cycle with sliding segment $ \Sigma^S_1 $
Figure 6.  Stable crossing sliding cycle
Figure 7.  The phase portrait of Filippov system (3). We choose $ a_3 = 0.0135,\;E = 0.02 $
Figure 8.  The phase portrait of Filippov system (3). We choose $ a_3 = 0.0124,\;E = 0.02 $
Figure 9.  The phase portrait of Filippov system (3). We choose $ a_3 = 0.0122,\;E = 0.0002 $
Figure 10.  The phase portrait of Filippov system (3). We choose $ a_3 = 0.0123,\;E = 0.0002 $
Figure 11.  The phase portrait of Filippov system (3). We choose $ a_3 = 0.01,\;E = 0.02 $
Figure 12.  The phase portrait of Filippov system (3). We choose $ a_3 = 0.01,E = 0.5,a_1 = 1, a_2 = 0.5, b_1 = 0.09, b_2 = 1.117283951, c_1 = 1.1, c_2 = 1.25, d = 10, and\; ET = 0.9. $
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