# American Institute of Mathematical Sciences

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  May 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244
##### References:
 [1] V. Acary and B. Brogliato, Numerical Methods For Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics, Springer-Verlag, New York, 2008. [2] A. Al-Khedhairi, The chaos and control of food chain model using nonlinear feedback, Appl. Math. Sci., 3 (2009), 591-604. [3] M. Banerjee, N. Mukherjee and V. Volpert, Prey-Predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041. [4] S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Bifurcations, Chaos, Control, and Applications, John Wiley and Sons, 2001. [5] S. P. Bera, A. 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. [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. [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. [8] R. Casey, H. 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. [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. [10] A. Colombo, N. D. Buono, L. 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. [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. [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. [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. [14] M. di Bernardo, C. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. Olivar and P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.  doi: 10.1137/050625060. [15] M. di Bernardo, P. 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. [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. [17] A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231. [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. [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. [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. [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. [22] A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591. [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. [24] Y. A. Kuznetsov, S. 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. [25] W. S. Madasanjaya, M. Mamat, Z. Salleh, I. 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. [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. [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. [28] J. M. Schumacher, Time-scaling symmetry and Zeno solutions, Automatica, 45 (2009), 1237-1242.  doi: 10.1016/j.automatica.2008.12.008. [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. [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. [31] S. Tang, J. H. Liang, Y. 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. [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. [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. [34] F. Tao, B. Kang, B. 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. [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.

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##### References:
 [1] V. Acary and B. Brogliato, Numerical Methods For Nonsmooth Dynamical Systems, Applications in Mechanics and Electronics, Springer-Verlag, New York, 2008. [2] A. Al-Khedhairi, The chaos and control of food chain model using nonlinear feedback, Appl. Math. Sci., 3 (2009), 591-604. [3] M. Banerjee, N. Mukherjee and V. Volpert, Prey-Predator model with a nonlocal bistable dynamics of prey, Mathematics, 6 (2018), 1-13.  doi: 10.3390/math6030041. [4] S. Banerjee and G. Verghese, Nonlinear Phenomena in Power Electronics: Bifurcations, Chaos, Control, and Applications, John Wiley and Sons, 2001. [5] S. P. Bera, A. 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. [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. [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. [8] R. Casey, H. 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. [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. [10] A. Colombo, N. D. Buono, L. 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. [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. [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. [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. [14] M. di Bernardo, C. Budd, A. R. Champneys, P. Kowalczyk, A. B. Nordmark, G. Olivar and P. T. Piiroinen, Bifurcations in nonsmooth dynamical systems, SIAM Review, 50 (2008), 629-701.  doi: 10.1137/050625060. [15] M. di Bernardo, P. 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. [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. [17] A. F. Filippov, Differential equations with discontinuous right-hand side, American Mathematical Society Translations, 2 (1964), 199-231. [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. [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. [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. [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. [22] A. Hastings and T. Powell, Chaos in a three-species food chain, Ecology, 72 (1991), 896-903.  doi: 10.2307/1940591. [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. [24] Y. A. Kuznetsov, S. 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. [25] W. S. Madasanjaya, M. Mamat, Z. Salleh, I. 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. [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. [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. [28] J. M. Schumacher, Time-scaling symmetry and Zeno solutions, Automatica, 45 (2009), 1237-1242.  doi: 10.1016/j.automatica.2008.12.008. [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. [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. [31] S. Tang, J. H. Liang, Y. 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. [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. [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. [34] F. Tao, B. Kang, B. 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. [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.
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 $C\; c_2> a_3,\;c2<a_3+E$
Bifurcation of pseudo-equilibrium
Bifurcation of pseudo-equilibrium
Limit cycle without sliding segment
Limit cycle with sliding segment $\Sigma^S_1$
Stable crossing sliding cycle
The phase portrait of Filippov system (3). We choose $a_3 = 0.0135,\;E = 0.02$
The phase portrait of Filippov system (3). We choose $a_3 = 0.0124,\;E = 0.02$
The phase portrait of Filippov system (3). We choose $a_3 = 0.0122,\;E = 0.0002$
The phase portrait of Filippov system (3). We choose $a_3 = 0.0123,\;E = 0.0002$
The phase portrait of Filippov system (3). We choose $a_3 = 0.01,\;E = 0.02$
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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