Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator

Alexander Pimenov; Dmitrii Rachinskii

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 285-298
  • ISSN: 0862-7959

Abstract

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Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.

How to cite

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Pimenov, Alexander, and Rachinskii, Dmitrii. "Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator." Mathematica Bohemica 139.2 (2014): 285-298. <http://eudml.org/doc/261900>.

@article{Pimenov2014,
abstract = {Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.},
author = {Pimenov, Alexander, Rachinskii, Dmitrii},
journal = {Mathematica Bohemica},
keywords = {robust homoclinic; orbit Preisach operator; operator-differential equations; predator-prey model; operator-differential equations; Preisach operator; predator-prey model; two-patch model; robust homoclinic loop},
language = {eng},
number = {2},
pages = {285-298},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator},
url = {http://eudml.org/doc/261900},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Pimenov, Alexander
AU - Rachinskii, Dmitrii
TI - Homoclinic orbits in a two-patch predator-prey model with Preisach hysteresis operator
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 285
EP - 298
AB - Systems of operator-differential equations with hysteresis operators can have unstable equilibrium points with an open basin of attraction. Such equilibria can have homoclinic orbits attached to them, and these orbits are robust. In this paper a population dynamics model with hysteretic response of the prey to variations of the predator is introduced. In this model the prey moves between two patches, and the derivative of the Preisach operator is used to describe the hysteretic flow between the patches. A numerical example of a robust homoclinic loop is presented, and a mechanism creating this homoclinic trajectory is discussed.
LA - eng
KW - robust homoclinic; orbit Preisach operator; operator-differential equations; predator-prey model; operator-differential equations; Preisach operator; predator-prey model; two-patch model; robust homoclinic loop
UR - http://eudml.org/doc/261900
ER -

References

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  1. Appelbe, B., Flynn, D., McNamara, H., O'Kane, P., Pimenov, A., Pokrovskii, A., skii, D. Rachin-, Zhezherun, A., 10.1109/MCS.2008.930923, Control Systems Magazine, IEEE 29 (2009), 44-69 DOI 10.1109/MCS.2008.930923. (2009) DOI10.1109/MCS.2008.930923
  2. Appelbe, B., Rachinskii, D., Zhezherun, A., 10.1016/j.physb.2007.08.034, Physica B: Condensed Matter 403 (2008), 301-304 DOI 10.1016/j.physb.2007.08.034. (2008) DOI10.1016/j.physb.2007.08.034
  3. Bertotti, G., Mayergoyz, I. D., Serpico, C., Nonlinear magnetization dynamics. Switching and relaxation phenomena, The Science of Hysteresis II. Physical Modeling, Micromagnetics, and Magnetization Dynamics G. Bertotti, I. D. Mayergoyz Elsevier, Amsterdam 435-565 (2006). (2006) Zbl1148.78001MR2307930
  4. Bertotti, G., Mayergoyz, I., eds., The Science of Hysteresis, Elsevier, Amsterdam (2006). (2006) MR2307931
  5. Brokate, M., Pokrovskii, A., Rachinskii, D., 10.1016/j.jmaa.2006.02.060, J. Math. Anal. Appl. 319 (2006), 94-109. (2006) Zbl1111.34035MR2217849DOI10.1016/j.jmaa.2006.02.060
  6. Brokate, M., Pokrovskii, A., Rachinskii, D., Rasskazov, O., Differential equations with hysteresis via a canonical example, The Science of Hysteresis I. Mathematical Modeling and Applications G. Bertotti, I. D. Mayergoyz Elsevier, Amsterdam 125-291 (2006). (2006) Zbl1142.34026MR2307931
  7. Brokate, M., Sprekels, J., 10.1007/978-1-4612-4048-8_5, Applied Mathematical Sciences 121 Springer, New York (1996). (1996) Zbl0951.74002MR1411908DOI10.1007/978-1-4612-4048-8_5
  8. Chiorino, G., Auger, P., Chassé, J.-L., Charles, S., 10.1016/S0025-5564(98)10082-2, Math. Biosci. 157 (1999), 189-216. (1999) MR1686474DOI10.1016/S0025-5564(98)10082-2
  9. Cross, R., McNamara, H., Pokrovskii, A., Rachinskii, D., 10.1016/j.physb.2007.08.017, Physica B: Condensed Matter 403 (2008), 231-236 DOI 10.1016/j.physb.2007.08.017. (2008) DOI10.1016/j.physb.2007.08.017
  10. Davino, D., Krejčí, P., Visone, C., 10.1088/0964-1726/22/9/095009, Smart Materials and Structures 22 (2013), 14 pages DOI 10.1088/0964-1726/22/9/095009. (2013) DOI10.1088/0964-1726/22/9/095009
  11. Diamond, P., Kuznetsov, N., Rachinskii, D., 10.1006/jdeq.2000.3916, J. Differ. Equations 175 (2001), 1-26. (2001) Zbl0984.34029MR1849221DOI10.1006/jdeq.2000.3916
  12. Diamond, P., Rachinskii, D., Yumagulov, M., Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity, Nonlinear Anal., Theory Methods Appl. 42 (2000), 1017-1031. (2000) Zbl0963.34034MR1780452
  13. Eleuteri, M., Kopfová, J., Krejčí, P., 10.1137/080718383, SIAM J. Math. Anal. 41 (2009), 435-464. (2009) MR2507458DOI10.1137/080718383
  14. Guardia, M., Seara, T. M., Teixeira, M. A., 10.1016/j.jde.2010.11.016, J. Differ. Equations 250 (2011), 1967-2023. (2011) Zbl1225.34046MR2763562DOI10.1016/j.jde.2010.11.016
  15. Harrison, G. W., 10.1007/BF02460019, Bull. Math. Biol. 48 (1986), 137-148. (1986) Zbl0585.92023MR0845634DOI10.1007/BF02460019
  16. Hodgkin, A. L., Huxley, A. F., 10.1113/jphysiol.1952.sp004764, J. Physiol. 117 (1952), 500-544 DOI 10.1007/BF02459568. (1952) DOI10.1113/jphysiol.1952.sp004764
  17. Krasnosel'skii, A., Rachinskii, D., 10.1007/s00030-002-8120-2, NoDEA, Nonlinear Differ. Equ. Appl. 9 (2002), 93-115. (2002) Zbl1013.34036MR1891697DOI10.1007/s00030-002-8120-2
  18. Krasnosel'skij, A., Rachinskij, D. I., On the continua of cycles in systems with hysteresis, Dokl. Math. 63 (2001), 339-344. (2001) Zbl1052.34052
  19. Krasnosel'skij, M. A., Pokrovskij, A. V., Systems with Hysteresis. Translated from the Russian, Springer, Berlin (1989). (1989) 
  20. Krejčí, P., Hysteresis, Convexity and Dissipation in Hyperbolic Equations, GAKUTO International Series. Mathematical Sciences and Applications 8 Gakkotosho, Tokyo (1996). (1996) MR2466538
  21. Krejčí, P., On Maxwell equations with the Preisach hysteresis operator: The one-dimensional time-periodic case, Apl. Mat. 34 (1989), 364-374. (1989) Zbl0701.35098MR1014077
  22. Krejčí, P., 10.1023/A:1022333500777, Appl. Math. 45 (2000), 439-468. (2000) Zbl1010.34038MR1800964DOI10.1023/A:1022333500777
  23. Krejčí, P., O'Kane, J. P., Pokrovskii, A., Rachinskii, D., 10.1088/1742-6596/268/1/012016, Journal of Physics: Conference Series 268 (2011), 19 pages DOI 10.1088/1742-6596/268/1/012016. (2011) DOI10.1088/1742-6596/268/1/012016
  24. Krejčí, P., O'Kane, J. P., Pokrovskii, A., Rachinskii, D., 10.1016/j.physd.2011.05.005, Physica D. Nonlinear Phenomena 241 (2012), 2010-2028. (2012) MR2994340DOI10.1016/j.physd.2011.05.005
  25. Kuhnen, K., Krejčí, P., 10.1109/TAC.2009.2012984, IEEE Trans. Automat. Control 54 (2009), 537-550. (2009) MR2191546DOI10.1109/TAC.2009.2012984
  26. Kuznetsov, Yu. A., 10.1007/978-1-4757-3978-7, Applied Mathematical Sciences 112 Springer, New York (2004). (2004) Zbl1082.37002MR2071006DOI10.1007/978-1-4757-3978-7
  27. Kuznetsov, Yu. A., Rinaldi, S., Gragnani, A., 10.1142/S0218127403007874, Int. J. Bifurcation Chaos Appl. Sci. Eng. 13 (2003), 2157-2188. (2003) Zbl1079.34029MR2012652DOI10.1142/S0218127403007874
  28. Mayergoyz, I. D., Mathematical Models of Hysteresis and Their Applications, Elsevier, Amsterdam (2003). (2003) MR1083150
  29. McCarthy, S., Rachinskii, D., Dynamics of systems with Preisach memory near equilibria, Math. Bohem. 139 (2014), 39-73. (2014) MR3231429
  30. Pimenov, A., Kelly, T. C., Korobeinikov, A., O'Callaghan, M. J., Pokrovskii, A. V., Rachinskii, D., 10.1051/mmnp/20127313, Math. Model. Nat. Phenom. 7 (2012), 204-226. (2012) MR2928740DOI10.1051/mmnp/20127313
  31. Pimenov, A., Rachinskii, D., 10.3934/dcdsb.2009.11.997, Discrete Contin. Dyn. Syst., Ser. B 11 (2009), 997-1018. (2009) Zbl1181.47075MR2505656DOI10.3934/dcdsb.2009.11.997
  32. Visintin, A., 10.1007/978-3-662-11557-2, Applied Mathematical Sciences 111 Springer, Berlin (1994). (1994) Zbl0820.35004MR1329094DOI10.1007/978-3-662-11557-2
  33. Visone, C., 10.1088/1742-6596/138/1/012028, Journal of Physics: Conference Series 138 (2008), 23 pages DOI 10.1088/1742-6596/138/1/012028. (2008) DOI10.1088/1742-6596/138/1/012028

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