# American Institute of Mathematical Sciences

doi: 10.3934/era.2021069
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## Variations on Lyapunov's stability criterion and periodic prey-predator systems

Dedicated to Professor E. N. Dancer, on the occasion of his 75th birthday

Received  June 2021 Early access September 2021

A classical stability criterion for Hill's equation is extended to more general families of periodic two-dimensional linear systems. The results are motivated by the study of mechanical vibrations with friction and periodic prey-predator systems.

Citation: Rafael Ortega. Variations on Lyapunov's stability criterion and periodic prey-predator systems. Electronic Research Archive, doi: 10.3934/era.2021069
##### References:
 [1] Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489.  doi: 10.1006/jmaa.1994.1262.  Google Scholar [2] B. M. Brown, M. S. P. Eastham and K. M. Schmidt, Periodic Differential Operators, Birkhäuser, New York, 2013. doi: 10.1007/978-3-0348-0528-5.  Google Scholar [3] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar [4] E. N. Dancer, Turing instabilities for systems of two equations with periodic coefficients, Differential Integral Equations, 7 (1994), 1253-1264.   Google Scholar [5] J. P. Den Hartog, Mechanical Vibrations, Dover Pub., New York, 1985. Google Scholar [6] T. Ding, H. Huang and F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations, Discrete Contin. Dynam. Systems, 1 (1995), 103-117.  doi: 10.3934/dcds.1995.1.103.  Google Scholar [7] A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2), 9 (1907), 203-474.   Google Scholar [8] J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. in Differential Equations, 1 (1996), 403-423.   Google Scholar [9] W. Magnus and S. Winkler, Hill's Equation, Dover Pub., New York, 1979.  Google Scholar [10] G. Meng and M. Zhang, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Equations, 254 (2013), 2196-2232.  doi: 10.1016/j.jde.2012.12.001.  Google Scholar [11] R. Ortega, The first interval of stability of a periodic equation of Duffing type, Proc. Am. Math. Soc., 115 (1992), 1061-1067.  doi: 10.1090/S0002-9939-1992-1092925-7.  Google Scholar [12] R. Ortega, Periodic solutions of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), 491-518.  doi: 10.1006/jdeq.1996.0103.  Google Scholar [13] R. Ortega, Periodic Differential Equations in the Plane. A Topological Perspective, De Gruyter, Berlin, 2019. Google Scholar [14] R. Ortega and M. Zhang, Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119-132.  doi: 10.1017/S0308210500003796.  Google Scholar [15] C. Rebelo and C. Soresina, Coexistence in seasonally varying predator-prey systems with Allee effect, Nonlinear Anal. Real World Appl., 55 (2020), 103140, 21 pp. doi: 10.1016/j.nonrwa.2020.103140.  Google Scholar [16] A. Tineo, On the asymptotic behavior of some population models, II, J. Math. Anal. Appl., 197 (1996), 249-258.  doi: 10.1006/jmaa.1996.0018.  Google Scholar [17] W. Walter, Differential- Und Integral- Ungleichungen, Springer-Verlag, Berlin, 1964. doi: 10.1007/978-3-662-42030-0.  Google Scholar [18] M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^{\alpha}$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.  doi: 10.1090/S0002-9939-02-06462-6.  Google Scholar

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##### References:
 [1] Z. Amine and R. Ortega, A periodic prey-predator system, J. Math. Anal. Appl., 185 (1994), 477-489.  doi: 10.1006/jmaa.1994.1262.  Google Scholar [2] B. M. Brown, M. S. P. Eastham and K. M. Schmidt, Periodic Differential Operators, Birkhäuser, New York, 2013. doi: 10.1007/978-3-0348-0528-5.  Google Scholar [3] L. Cesari, Asymptotic Behavior and Stability Problems in Ordinary Differential Equations, Springer-Verlag, New York, 1971.  Google Scholar [4] E. N. Dancer, Turing instabilities for systems of two equations with periodic coefficients, Differential Integral Equations, 7 (1994), 1253-1264.   Google Scholar [5] J. P. Den Hartog, Mechanical Vibrations, Dover Pub., New York, 1985. Google Scholar [6] T. Ding, H. Huang and F. Zanolin, A priori bounds and periodic solutions for a class of planar systems with applications to Lotka-Volterra equations, Discrete Contin. Dynam. Systems, 1 (1995), 103-117.  doi: 10.3934/dcds.1995.1.103.  Google Scholar [7] A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. (2), 9 (1907), 203-474.   Google Scholar [8] J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. in Differential Equations, 1 (1996), 403-423.   Google Scholar [9] W. Magnus and S. Winkler, Hill's Equation, Dover Pub., New York, 1979.  Google Scholar [10] G. Meng and M. Zhang, Dependence of solutions and eigenvalues of measure differential equations on measures, J. Differential Equations, 254 (2013), 2196-2232.  doi: 10.1016/j.jde.2012.12.001.  Google Scholar [11] R. Ortega, The first interval of stability of a periodic equation of Duffing type, Proc. Am. Math. Soc., 115 (1992), 1061-1067.  doi: 10.1090/S0002-9939-1992-1092925-7.  Google Scholar [12] R. Ortega, Periodic solutions of a Newtonian equation: Stability by the third approximation, J. Differential Equations, 128 (1996), 491-518.  doi: 10.1006/jdeq.1996.0103.  Google Scholar [13] R. Ortega, Periodic Differential Equations in the Plane. A Topological Perspective, De Gruyter, Berlin, 2019. Google Scholar [14] R. Ortega and M. Zhang, Optimal bounds for bifurcation values of a superlinear periodic problem, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 119-132.  doi: 10.1017/S0308210500003796.  Google Scholar [15] C. Rebelo and C. Soresina, Coexistence in seasonally varying predator-prey systems with Allee effect, Nonlinear Anal. Real World Appl., 55 (2020), 103140, 21 pp. doi: 10.1016/j.nonrwa.2020.103140.  Google Scholar [16] A. Tineo, On the asymptotic behavior of some population models, II, J. Math. Anal. Appl., 197 (1996), 249-258.  doi: 10.1006/jmaa.1996.0018.  Google Scholar [17] W. Walter, Differential- Und Integral- Ungleichungen, Springer-Verlag, Berlin, 1964. doi: 10.1007/978-3-662-42030-0.  Google Scholar [18] M. Zhang and W. Li, A Lyapunov-type stability criterion using $L^{\alpha}$ norms, Proc. Amer. Math. Soc., 130 (2002), 3325-3333.  doi: 10.1090/S0002-9939-02-06462-6.  Google Scholar
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