October  2015, 20(8): 2663-2690. doi: 10.3934/dcdsb.2015.20.2663

Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays

1. 

Departamento de Mátematicas, Universidad Centro Occidental Lisandro Alvarado, Barquisimeto, Estado Lara, Venezuela, Venezuela

Received  October 2014 Revised  April 2015 Published  August 2015

The qualitative properties of certain type of nonautonomous competitive Lotka-Volterra systems with infinite delay are considered.
    By constructing suitable Lyapunov-type functional, we establish a series of easily verifiable algebraic conditions on the coefficients and the kernels, which are sufficient to ensure the extinction and survival of a determined number of species. The surviving part stabilizes around any solution of a subsystem of the systems in study. These conditions also guarantee the persistence, extreme stability and asymptotic behavior of the systems.
Citation: Francisco Montes de Oca, Liliana Pérez. Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems with infinite delays. Discrete & Continuous Dynamical Systems - B, 2015, 20 (8) : 2663-2690. doi: 10.3934/dcdsb.2015.20.2663
References:
[1]

S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems,, Proc. Amer. Math. Soc., 127 (1999), 2905. doi: 10.1090/S0002-9939-99-05083-2.

[2]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations,, Proc. Amer. Math. Soc., 117 (1993), 199. doi: 10.1090/S0002-9939-1993-1143013-3.

[3]

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.

[4]

A. Battauz and F. Zanolin, Coexistence states for periodic competitive Kolmogorov systems,, J. Math. Anal. Appl., 219 (1998), 179. doi: 10.1006/jmaa.1997.5726.

[5]

A. Bermand and R. Plemmons, Nonnegative Matrices in The Mathematical Sciences,, Classics in Applied Mathematics, (1979).

[6]

R. S. Cantrell and C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion,, Houston J. Math., 13 (1987), 337.

[7]

F. Chen, Z. Li and Y. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls,, Nonlinear Analysis: Real World Applications, 8 (2007), 680. doi: 10.1016/j.nonrwa.2006.02.006.

[8]

C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion,, SIAM J. Appl. Math., 44 (1984), 1112. doi: 10.1137/0144080.

[9]

L. Dung and H. L. Smith, Steady states of models of microbial growth and competition with chemotaxis,, J Math. Anal. Appl., 229 (1999), 295. doi: 10.1006/jmaa.1998.6167.

[10]

L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor,, Journal of Differential Equations, 130 (1996), 59. doi: 10.1006/jdeq.1996.0132.

[11]

C. Feng, On the existence and uniqueness of almost periodic solutions for delay Logistic equations,, Applied Mathematics and Computation, 136 (2003), 487. doi: 10.1016/S0096-3003(02)00072-3.

[12]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of population dynamics,, Mathematics and Its Applications, (1992). doi: 10.1007/978-94-015-7920-9.

[13]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, 99 (1993). doi: 10.1007/978-1-4612-4342-7.

[14]

X. He, Almost periodic solutions of a competition system with dominated infinite delays,, Tohoku Math. J., 50 (1998), 71. doi: 10.2748/tmj/1178225015.

[15]

Z. Hou, Permanence, global attraction and stability,, in Lotka-Volterra and Related Systems, (2013), 1.

[16]

H. Hu, Z. Teng and S. Gao, Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls,, Nonlinear Analysis: Real World Applications, 10 (2009), 2508. doi: 10.1016/j.nonrwa.2008.05.011.

[17]

H. Hu, Z. Teng and H. Jiang, On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls,, Nonlinear Analysis: Real World Applications, 10 (2009), 1803. doi: 10.1016/j.nonrwa.2008.02.017.

[18]

F. Montes de Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay,, Nonlinear Analysis: Theory, 75 (2012), 758. doi: 10.1016/j.na.2011.09.009.

[19]

F. Montes de Oca and M. Vivas, Extinction in a two dimensional Lotka-Volterra systems with infinite delay,, Nonlinear Analysis: Real World Applications, 7 (2006), 1042. doi: 10.1016/j.nonrwa.2005.09.005.

[20]

F. Montes de Oca and M. L. Zeeman, Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems,, J. Math. Anal. Appl., 192 (1995), 360. doi: 10.1006/jmaa.1995.1177.

[21]

C. Shi, Z. Li and F. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,, Nonlinear Analysis: Real World Applications, 13 (2012), 2214. doi: 10.1016/j.nonrwa.2012.01.016.

[22]

Z. Teng, On the nonautonomous Lotka-Volerra N-species competing systems,, Appl. Math.Comp. 114 (2000), 114 (2000), 175.

[23]

A. Tineo, On the asymptotic behaviour of some population models,, J. Math. Anal. Appl., 167 (1992), 516. doi: 10.1016/0022-247X(92)90222-Y.

[24]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems,, Results Math., 21 (1992), 224. doi: 10.1007/BF03323081.

[25]

J. Zhao and J. Tiang, Average conditions for permanence and extinction in nonautonomous Lotka-Voltera systems,, JMAA, 299 (2004), 663. doi: 10.1016/j.jmaa.2004.06.019.

[26]

J. Zhao, L. Fu and J. Ruan, Extinction in a nonautonomous competitive Lotka-Volterra system,, Appl. Math. Letters, 22 (2009), 766. doi: 10.1016/j.aml.2008.08.015.

show all references

References:
[1]

S. Ahmad, Extinction of species in nonautonomous Lotka-Volterra systems,, Proc. Amer. Math. Soc., 127 (1999), 2905. doi: 10.1090/S0002-9939-99-05083-2.

[2]

S. Ahmad, On the nonautonomous Volterra-Lotka competition equations,, Proc. Amer. Math. Soc., 117 (1993), 199. doi: 10.1090/S0002-9939-1993-1143013-3.

[3]

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.

[4]

A. Battauz and F. Zanolin, Coexistence states for periodic competitive Kolmogorov systems,, J. Math. Anal. Appl., 219 (1998), 179. doi: 10.1006/jmaa.1997.5726.

[5]

A. Bermand and R. Plemmons, Nonnegative Matrices in The Mathematical Sciences,, Classics in Applied Mathematics, (1979).

[6]

R. S. Cantrell and C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion,, Houston J. Math., 13 (1987), 337.

[7]

F. Chen, Z. Li and Y. Huang, Note on the permanence of a competitive system with infinite delay and feedback controls,, Nonlinear Analysis: Real World Applications, 8 (2007), 680. doi: 10.1016/j.nonrwa.2006.02.006.

[8]

C. Cosner and A. C. Lazer, Stable coexistence states in the Volterra-Lotka competition model with diffusion,, SIAM J. Appl. Math., 44 (1984), 1112. doi: 10.1137/0144080.

[9]

L. Dung and H. L. Smith, Steady states of models of microbial growth and competition with chemotaxis,, J Math. Anal. Appl., 229 (1999), 295. doi: 10.1006/jmaa.1998.6167.

[10]

L. Dung and H. L. Smith, A parabolic system modeling microbial competition in an unmixed bio-reactor,, Journal of Differential Equations, 130 (1996), 59. doi: 10.1006/jdeq.1996.0132.

[11]

C. Feng, On the existence and uniqueness of almost periodic solutions for delay Logistic equations,, Applied Mathematics and Computation, 136 (2003), 487. doi: 10.1016/S0096-3003(02)00072-3.

[12]

K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of population dynamics,, Mathematics and Its Applications, (1992). doi: 10.1007/978-94-015-7920-9.

[13]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations,, Applied Mathematical Sciences, 99 (1993). doi: 10.1007/978-1-4612-4342-7.

[14]

X. He, Almost periodic solutions of a competition system with dominated infinite delays,, Tohoku Math. J., 50 (1998), 71. doi: 10.2748/tmj/1178225015.

[15]

Z. Hou, Permanence, global attraction and stability,, in Lotka-Volterra and Related Systems, (2013), 1.

[16]

H. Hu, Z. Teng and S. Gao, Extinction in nonautonomous Lotka-Volterra competitive system with pure-delays and feedback controls,, Nonlinear Analysis: Real World Applications, 10 (2009), 2508. doi: 10.1016/j.nonrwa.2008.05.011.

[17]

H. Hu, Z. Teng and H. Jiang, On the permanence in non-autonomous Lotka-Volterra competitive system with pure-delays and feedback controls,, Nonlinear Analysis: Real World Applications, 10 (2009), 1803. doi: 10.1016/j.nonrwa.2008.02.017.

[18]

F. Montes de Oca and L. Pérez, Extinction in nonautonomous competitive Lotka-Volterra systems with infinite delay,, Nonlinear Analysis: Theory, 75 (2012), 758. doi: 10.1016/j.na.2011.09.009.

[19]

F. Montes de Oca and M. Vivas, Extinction in a two dimensional Lotka-Volterra systems with infinite delay,, Nonlinear Analysis: Real World Applications, 7 (2006), 1042. doi: 10.1016/j.nonrwa.2005.09.005.

[20]

F. Montes de Oca and M. L. Zeeman, Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems,, J. Math. Anal. Appl., 192 (1995), 360. doi: 10.1006/jmaa.1995.1177.

[21]

C. Shi, Z. Li and F. Chen, Extinction in a nonautonomous Lotka-Volterra competitive system with infinite delay and feedback controls,, Nonlinear Analysis: Real World Applications, 13 (2012), 2214. doi: 10.1016/j.nonrwa.2012.01.016.

[22]

Z. Teng, On the nonautonomous Lotka-Volerra N-species competing systems,, Appl. Math.Comp. 114 (2000), 114 (2000), 175.

[23]

A. Tineo, On the asymptotic behaviour of some population models,, J. Math. Anal. Appl., 167 (1992), 516. doi: 10.1016/0022-247X(92)90222-Y.

[24]

F. Zanolin, Permanence and positive periodic solutions for Kolmogorov competing species systems,, Results Math., 21 (1992), 224. doi: 10.1007/BF03323081.

[25]

J. Zhao and J. Tiang, Average conditions for permanence and extinction in nonautonomous Lotka-Voltera systems,, JMAA, 299 (2004), 663. doi: 10.1016/j.jmaa.2004.06.019.

[26]

J. Zhao, L. Fu and J. Ruan, Extinction in a nonautonomous competitive Lotka-Volterra system,, Appl. Math. Letters, 22 (2009), 766. doi: 10.1016/j.aml.2008.08.015.

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