October  2014, 34(10): 4127-4137. doi: 10.3934/dcds.2014.34.4127

Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients

1. 

Departamento de Ecuaciones Diferenciales y Análisis Numérico, Facultad de Matemáticas, Universidad de Sevilla, Calle Tarfia s/n, 41012-Seville, Spain

2. 

Departamento de Matemática y Mecánica, I.I.M.A.S - U.N.A.M., Apdo. Postal 20-726, 01000 México D. F., Mexico

Received  September 2013 Revised  October 2013 Published  April 2014

In this work we study the asymptotic behaviour of the following prey-predator system \begin{equation*} \left\{ \begin{split} &A'=\alpha f(t)A-\beta g(t)A^2-\gamma AP\\ &P'=\delta h(t)P-\lambda m(t)P^2+\mu AP, \end{split} \right. \end{equation*} where functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ are not necessarily bounded above. We also prove the existence of the pullback attractor and the permanence of solutions for any positive initial data and initial time, making a previous study of a logistic equation with unbounded terms, where one of them can be negative for a bounded interval of time. The analysis of a non-autonomous logistic equation with unbounded coefficients is also needed to ensure the permanence of the model.
Citation: Juan C. Jara, Felipe Rivero. Asymptotic behaviour for prey-predator systems and logistic equations with unbounded time-dependent coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4127-4137. doi: 10.3934/dcds.2014.34.4127
References:
[1]

S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations,, J. Math. Anal. Appl., 127 (1987), 377. doi: 10.1016/0022-247X(87)90116-8. Google Scholar

[2]

J. Balbus and J. Mierczyński, Time-averaging and permanence in nonautonomous competitive systems of PDE's via Vance-Coddington estiamtes,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1407. doi: 10.3934/dcdsb.2012.17.1407. Google Scholar

[3]

T. A. Burton and V. Hutson, Permanence for nonautonomous predator-prey systems,, Differential Integral Equations, 4 (1991), 1269. Google Scholar

[4]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes,, Nonlinear Analysis, 72 (2010), 1967. doi: 10.1016/j.na.2009.09.037. Google Scholar

[5]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems,, World Scientific Publishing Co. Pte. Ltd, (2004). doi: 10.1142/9789812563088. Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (2002). Google Scholar

[7]

B. D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change,, Math. Biosci, 45 (1979), 159. doi: 10.1016/0025-5564(79)90057-9. Google Scholar

[8]

T. G. Hallam and C. E. Clarck, Nonautonomous logistic equations as models of populations in a deteriorating environment,, J. Theoret. Biol., 93 (1981), 301. doi: 10.1016/0022-5193(81)90106-5. Google Scholar

[9]

J. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 9 (1989), 388. doi: 10.1137/0520025. Google Scholar

[10]

V. Hutson and K. Schmitt, Permanence in dynamical systems,, Math. Biosci., 111 (1992), 1. doi: 10.1016/0025-5564(92)90078-B. Google Scholar

[11]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical system,, Stochastics and Dynamics, 3 (2003), 101. doi: 10.1142/S0219493703000632. Google Scholar

[12]

J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal and A. Suárez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion,, SIAM J. Math. Anal., 40 (2009), 2179. doi: 10.1137/080721790. Google Scholar

[13]

J. A. Langa, J. C. Robinson and A. Suárez, Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system,, Nonlinearity, 16 (2003), 1277. doi: 10.1088/0951-7715/16/4/305. Google Scholar

[14]

J. A. Langa, A. Rodríguez-Bernal and S. Suárez, On the long time behaviour of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method,, J. Differential Equations, 249 (2010), 414. doi: 10.1016/j.jde.2010.04.001. Google Scholar

[15]

M. N. Nkashama, Dynamics of logistic equations with non-autonomous bounded coefficients,, Electron. J. Differential Equations, 2000 (2000). Google Scholar

[16]

J. C. Robinson, Infinite-Dimensional Dynamical System. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Text in Applied Mathematics, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[17]

G. R. Sell, Nonautonomous differential equations and dynamical systems,, Trans. Amer. Math. Soc., 127 (1967), 241. Google Scholar

[18]

J. Sugie, Y. Saito and M. Fan, Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment,, Proc. Amer. Math. Soc., 139 (2011), 3475. doi: 10.1090/S0002-9939-2011-11124-9. Google Scholar

[19]

R. R. Vance and E. A. Coddington, A nonautonomous model of population growth,, J. Math. Biol., 27 (1989), 491. doi: 10.1007/BF00288430. Google Scholar

show all references

References:
[1]

S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations,, J. Math. Anal. Appl., 127 (1987), 377. doi: 10.1016/0022-247X(87)90116-8. Google Scholar

[2]

J. Balbus and J. Mierczyński, Time-averaging and permanence in nonautonomous competitive systems of PDE's via Vance-Coddington estiamtes,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 1407. doi: 10.3934/dcdsb.2012.17.1407. Google Scholar

[3]

T. A. Burton and V. Hutson, Permanence for nonautonomous predator-prey systems,, Differential Integral Equations, 4 (1991), 1269. Google Scholar

[4]

T. Caraballo, A. N. Carvalho, J. A. Langa and F. Rivero, Existence of pullback attractors for pullback asymptotically compact processes,, Nonlinear Analysis, 72 (2010), 1967. doi: 10.1016/j.na.2009.09.037. Google Scholar

[5]

D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems,, World Scientific Publishing Co. Pte. Ltd, (2004). doi: 10.1142/9789812563088. Google Scholar

[6]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, Colloquium Publications, (2002). Google Scholar

[7]

B. D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change,, Math. Biosci, 45 (1979), 159. doi: 10.1016/0025-5564(79)90057-9. Google Scholar

[8]

T. G. Hallam and C. E. Clarck, Nonautonomous logistic equations as models of populations in a deteriorating environment,, J. Theoret. Biol., 93 (1981), 301. doi: 10.1016/0022-5193(81)90106-5. Google Scholar

[9]

J. Hale and P. Waltman, Persistence in infinite dimensional systems,, SIAM J. Math. Anal., 9 (1989), 388. doi: 10.1137/0520025. Google Scholar

[10]

V. Hutson and K. Schmitt, Permanence in dynamical systems,, Math. Biosci., 111 (1992), 1. doi: 10.1016/0025-5564(92)90078-B. Google Scholar

[11]

P. E. Kloeden, Pullback attractors of nonautonomous semidynamical system,, Stochastics and Dynamics, 3 (2003), 101. doi: 10.1142/S0219493703000632. Google Scholar

[12]

J. A. Langa, J. C. Robinson, A. Rodríguez-Bernal and A. Suárez, Permanence and asymptotically stable complete trajectories for nonautonomous Lotka-Volterra models with diffusion,, SIAM J. Math. Anal., 40 (2009), 2179. doi: 10.1137/080721790. Google Scholar

[13]

J. A. Langa, J. C. Robinson and A. Suárez, Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system,, Nonlinearity, 16 (2003), 1277. doi: 10.1088/0951-7715/16/4/305. Google Scholar

[14]

J. A. Langa, A. Rodríguez-Bernal and S. Suárez, On the long time behaviour of non-autonomous Lotka-Volterra models with diffusion via the sub-supertrajectory method,, J. Differential Equations, 249 (2010), 414. doi: 10.1016/j.jde.2010.04.001. Google Scholar

[15]

M. N. Nkashama, Dynamics of logistic equations with non-autonomous bounded coefficients,, Electron. J. Differential Equations, 2000 (2000). Google Scholar

[16]

J. C. Robinson, Infinite-Dimensional Dynamical System. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors,, Cambridge Text in Applied Mathematics, (2001). doi: 10.1007/978-94-010-0732-0. Google Scholar

[17]

G. R. Sell, Nonautonomous differential equations and dynamical systems,, Trans. Amer. Math. Soc., 127 (1967), 241. Google Scholar

[18]

J. Sugie, Y. Saito and M. Fan, Global asymptotic stability for predator-prey systems whose prey receives time-variation of the environment,, Proc. Amer. Math. Soc., 139 (2011), 3475. doi: 10.1090/S0002-9939-2011-11124-9. Google Scholar

[19]

R. R. Vance and E. A. Coddington, A nonautonomous model of population growth,, J. Math. Biol., 27 (1989), 491. doi: 10.1007/BF00288430. Google Scholar

[1]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703

[2]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809

[3]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194

[4]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195

[5]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037

[6]

Yang Lu, Xia Wang, Shengqiang Liu. A non-autonomous predator-prey model with infected prey. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3817-3836. doi: 10.3934/dcdsb.2018082

[7]

Peter E. Kloeden, Jacson Simsen. Pullback attractors for non-autonomous evolution equations with spatially variable exponents. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2543-2557. doi: 10.3934/cpaa.2014.13.2543

[8]

Aniello Buonocore, Luigia Caputo, Enrica Pirozzi, Amelia G. Nobile. A non-autonomous stochastic predator-prey model. Mathematical Biosciences & Engineering, 2014, 11 (2) : 167-188. doi: 10.3934/mbe.2014.11.167

[9]

Flank D. M. Bezerra, Vera L. Carbone, Marcelo J. D. Nascimento, Karina Schiabel. Pullback attractors for a class of non-autonomous thermoelastic plate systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3553-3571. doi: 10.3934/dcdsb.2017214

[10]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639

[11]

Sampurna Sengupta, Pritha Das, Debasis Mukherjee. Stochastic non-autonomous Holling type-Ⅲ prey-predator model with predator's intra-specific competition. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3275-3296. doi: 10.3934/dcdsb.2018244

[12]

Mahesh G. Nerurkar. Spectral and stability questions concerning evolution of non-autonomous linear systems. Conference Publications, 2001, 2001 (Special) : 270-275. doi: 10.3934/proc.2001.2001.270

[13]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087

[14]

Grzegorz Łukaszewicz, James C. Robinson. Invariant measures for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4211-4222. doi: 10.3934/dcds.2014.34.4211

[15]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120

[16]

Michael Dellnitz, Christian Horenkamp. The efficient approximation of coherent pairs in non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (9) : 3029-3042. doi: 10.3934/dcds.2012.32.3029

[17]

Felipe Rivero. Time dependent perturbation in a non-autonomous non-classical parabolic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 209-221. doi: 10.3934/dcdsb.2013.18.209

[18]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281

[19]

Suping Wang, Qiaozhen Ma. Existence of pullback attractors for the non-autonomous suspension bridge equation with time delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019221

[20]

Xianlong Fu. Approximate controllability of semilinear non-autonomous evolution systems with state-dependent delay. Evolution Equations & Control Theory, 2017, 6 (4) : 517-534. doi: 10.3934/eect.2017026

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]