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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

Juan C. Jara 1, and  Felipe Rivero 2,

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.
Keywords: non-autonomous dynamical systems, Non-autonomous prey-predator system, unbounded time-dependent coefficients, evolution processes., pullback attractor, permanece of solutions, non-autonomous logistic equation.
Mathematics Subject Classification: 92D15, 35B40, 37N25, 92D25, 34E9.
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

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