# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 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), 377-387. doi: 10.1016/0022-247X(87)90116-8. [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), 1407-1425. doi: 10.3934/dcdsb.2012.17.1407. [3] T. A. Burton and V. Hutson, Permanence for nonautonomous predator-prey systems, Differential Integral Equations, 4 (1991), 1269-1280. [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), 1967-1976. doi: 10.1016/j.na.2009.09.037. [5] D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems, World Scientific Publishing Co. Pte. Ltd, 2004. doi: 10.1142/9789812563088. [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, 49, America Mathematical Society, 2002. [7] B. D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change, Math. Biosci, 45 (1979), 159-173. doi: 10.1016/0025-5564(79)90057-9. [8] T. G. Hallam and C. E. Clarck, Nonautonomous logistic equations as models of populations in a deteriorating environment, J. Theoret. Biol., 93 (1981), 301-311. doi: 10.1016/0022-5193(81)90106-5. [9] J. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 9 (1989), 388-395. doi: 10.1137/0520025. [10] V. Hutson and K. Schmitt, Permanence in dynamical systems, Math. Biosci., 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B. [11] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical system, Stochastics and Dynamics, 3 (2003), 101-112. doi: 10.1142/S0219493703000632. [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), 2179-2216. doi: 10.1137/080721790. [13] J. A. Langa, J. C. Robinson and A. Suárez, Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system, Nonlinearity, 16 (2003), 1277-1293. doi: 10.1088/0951-7715/16/4/305. [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), 414-445. doi: 10.1016/j.jde.2010.04.001. [15] M. N. Nkashama, Dynamics of logistic equations with non-autonomous bounded coefficients, Electron. J. Differential Equations, 2000 (2000), 8 pp. (electronic). [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. [17] G. R. Sell, Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283. [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), 3475-3483. doi: 10.1090/S0002-9939-2011-11124-9. [19] R. R. Vance and E. A. Coddington, A nonautonomous model of population growth, J. Math. Biol., 27 (1989), 491-506. doi: 10.1007/BF00288430.

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##### References:
 [1] S. Ahmad, Convergence and ultimate bounds of solutions of the nonautonomous Volterra-Lotka competition equations, J. Math. Anal. Appl., 127 (1987), 377-387. doi: 10.1016/0022-247X(87)90116-8. [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), 1407-1425. doi: 10.3934/dcdsb.2012.17.1407. [3] T. A. Burton and V. Hutson, Permanence for nonautonomous predator-prey systems, Differential Integral Equations, 4 (1991), 1269-1280. [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), 1967-1976. doi: 10.1016/j.na.2009.09.037. [5] D. N. Cheban, Global Attractors of Non-Autonomous Dissipative Dynamical Systems, World Scientific Publishing Co. Pte. Ltd, 2004. doi: 10.1142/9789812563088. [6] V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, Colloquium Publications, 49, America Mathematical Society, 2002. [7] B. D. Coleman, Nonautonomous logistic equations as models of the adjustment of populations to environmental change, Math. Biosci, 45 (1979), 159-173. doi: 10.1016/0025-5564(79)90057-9. [8] T. G. Hallam and C. E. Clarck, Nonautonomous logistic equations as models of populations in a deteriorating environment, J. Theoret. Biol., 93 (1981), 301-311. doi: 10.1016/0022-5193(81)90106-5. [9] J. Hale and P. Waltman, Persistence in infinite dimensional systems, SIAM J. Math. Anal., 9 (1989), 388-395. doi: 10.1137/0520025. [10] V. Hutson and K. Schmitt, Permanence in dynamical systems, Math. Biosci., 111 (1992), 1-71. doi: 10.1016/0025-5564(92)90078-B. [11] P. E. Kloeden, Pullback attractors of nonautonomous semidynamical system, Stochastics and Dynamics, 3 (2003), 101-112. doi: 10.1142/S0219493703000632. [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), 2179-2216. doi: 10.1137/080721790. [13] J. A. Langa, J. C. Robinson and A. Suárez, Forwards and pullback behaviour of a non-autonomous Lotka-Volterra system, Nonlinearity, 16 (2003), 1277-1293. doi: 10.1088/0951-7715/16/4/305. [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), 414-445. doi: 10.1016/j.jde.2010.04.001. [15] M. N. Nkashama, Dynamics of logistic equations with non-autonomous bounded coefficients, Electron. J. Differential Equations, 2000 (2000), 8 pp. (electronic). [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. [17] G. R. Sell, Nonautonomous differential equations and dynamical systems, Trans. Amer. Math. Soc., 127 (1967), 241-283. [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), 3475-3483. doi: 10.1090/S0002-9939-2011-11124-9. [19] R. R. Vance and E. A. Coddington, A nonautonomous model of population growth, J. Math. Biol., 27 (1989), 491-506. doi: 10.1007/BF00288430.
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