December  2021, 29(6): 3995-4008. doi: 10.3934/era.2021069

Variations on Lyapunov's stability criterion and periodic prey-predator systems

Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain

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

Received  June 2021 Published  December 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, 2021, 29 (6) : 3995-4008. 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. DingH. 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-GómezR. 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

show all references

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. DingH. 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-GómezR. 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

[1]

Wei Feng, Michael T. Cowen, Xin Lu. Coexistence and asymptotic stability in stage-structured predator-prey models. Mathematical Biosciences & Engineering, 2014, 11 (4) : 823-839. doi: 10.3934/mbe.2014.11.823

[2]

Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

[3]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301

[4]

Zhifu Xie. Turing instability in a coupled predator-prey model with different Holling type functional responses. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1621-1628. doi: 10.3934/dcdss.2011.4.1621

[5]

Hao Zhang, Hirofumi Izuhara, Yaping Wu. Asymptotic stability of two types of traveling waves for some predator-prey models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (4) : 2323-2342. doi: 10.3934/dcdsb.2021046

[6]

Yu Ma, Chunlai Mu, Shuyan Qiu. Boundedness and asymptotic stability in a two-species predator-prey chemotaxis model. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021218

[7]

J. Gani, R. J. Swift. Prey-predator models with infected prey and predators. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5059-5066. doi: 10.3934/dcds.2013.33.5059

[8]

Mary Ballyk, Ross Staffeldt, Ibrahim Jawarneh. A nutrient-prey-predator model: Stability and bifurcations. Discrete & Continuous Dynamical Systems - S, 2020, 13 (11) : 2975-3004. doi: 10.3934/dcdss.2020192

[9]

Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127

[10]

Mihaela Negreanu. Global existence and asymptotic behavior of solutions to a chemotaxis system with chemicals and prey-predator terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (9) : 3335-3356. doi: 10.3934/dcdsb.2020064

[11]

Na Min, Mingxin Wang. Hopf bifurcation and steady-state bifurcation for a Leslie-Gower prey-predator model with strong Allee effect in prey. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 1071-1099. doi: 10.3934/dcds.2019045

[12]

Shanshan Chen, Jianshe Yu. Stability and bifurcation on predator-prey systems with nonlocal prey competition. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 43-62. doi: 10.3934/dcds.2018002

[13]

Isam Al-Darabsah, Xianhua Tang, Yuan Yuan. A prey-predator model with migrations and delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (3) : 737-761. doi: 10.3934/dcdsb.2016.21.737

[14]

Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913

[15]

R. P. Gupta, Peeyush Chandra, Malay Banerjee. Dynamical complexity of a prey-predator model with nonlinear predator harvesting. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 423-443. doi: 10.3934/dcdsb.2015.20.423

[16]

Malay Banerjee, Nayana Mukherjee, Vitaly Volpert. Prey-predator model with nonlocal and global consumption in the prey dynamics. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2109-2120. doi: 10.3934/dcdss.2020180

[17]

Leonid Braverman, Elena Braverman. Stability analysis and bifurcations in a diffusive predator-prey system. Conference Publications, 2009, 2009 (Special) : 92-100. doi: 10.3934/proc.2009.2009.92

[18]

Sílvia Cuadrado. Stability of equilibria of a predator-prey model of phenotype evolution. Mathematical Biosciences & Engineering, 2009, 6 (4) : 701-718. doi: 10.3934/mbe.2009.6.701

[19]

Zhong Li, Maoan Han, Fengde Chen. Global stability of a predator-prey system with stage structure and mutual interference. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 173-187. doi: 10.3934/dcdsb.2014.19.173

[20]

Yinshu Wu, Wenzhang Huang. Global stability of the predator-prey model with a sigmoid functional response. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1159-1167. doi: 10.3934/dcdsb.2019214

2020 Impact Factor: 1.833

Metrics

  • PDF downloads (98)
  • HTML views (166)
  • Cited by (0)

Other articles
by authors

[Back to Top]