Permanence and Globally Asymptotic Stability of Cooperative System Incorporating Harvesting

DOI: 10.4236/apm.2013.37082   PDF   HTML   XML   2,837 Downloads   4,523 Views   Citations


The stability of a kind of cooperative models incorporating harvesting is considered in this paper. By analyzing the characteristic roots of the models and constructing suitable Lyapunov functions, we prove that nonnegative equilibrium points of the models are globally asymptotically stable. Further, the corresponding nonautonomous cooperative models have a unique asymptotically periodic solution, which is uniformly asymptotically stable. An example is given to illustrate the effectiveness of our results.

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F. Wei and C. Li, "Permanence and Globally Asymptotic Stability of Cooperative System Incorporating Harvesting," Advances in Pure Mathematics, Vol. 3 No. 7, 2013, pp. 627-632. doi: 10.4236/apm.2013.37082.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] J. Rebaza, “Dynamics of Prey Threshold Harvesting and Refuge,” Journal of Computational and Applied Mathematics, Vol. 236, No. 7, 2012, pp. 1743-1752.
[2] X. Y. Zhang and K. Wang, “Almost Periodic Solution for n-Species Cooperation System with Time Delay,” Journal of Northeast Normal University (Natural Science Edition), Vol. 34, No. 3, 2002, in Chinese.
[3] Y. Nakata and Y. Muroya, “Permanence for Nonautonomous Lotka Volterra Cooperative Systems with Delays,” Nonlinear Analysis: Real World Applications, Vol. 11, No. 1, 2010, pp. 528-534.
[4] X. Y. Song and L. S. Chen, “Global Asymptotic Stability of a Two Species Competitive System with Stage Structure and Harvesting,” Communications in Nonlinear Science & Numerical Simulation, Vol. 6, No. 2, 2001, pp. 81-87.
[5] T. K. Kar and H. Matsuda, “Global Dynamics and Controllability of a Harvested Prey-Predator System with Holling Type III Functional Response,” Nonlinear Analysis: Hybrid Systems, Vol. 1, No. 1, 2007, pp. 59-67.
[6] D. W. Hu and Z. Q. Zhang, “Four Positive Periodic Solutions to a Lotka-Volterra Cooperative System with Harvesting Terms,” Nonlinear Analysis: Real World Applications, Vol. 11, No. 2, 2010, pp. 1115-1121.
[7] G. Y. Chen and Z. D. Teng, “On the Stability in a Discrete Two-Species Competition System,” Journal of Applied Mathematics and Computing, Vol. 38, No. 1-2, 2012, pp. 25-39.
[8] T. K. Kar and K. Chakraborty, “Effort Dynamics in a Prey-Predator Model with Harvesting,” International Journal of Information and Systems Sciences, Vol. 6, No. 3, 2010, pp. 318-332.

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