On a periodic multi-species ecological model

A periodic predator-prey model with m-predators and n-preys is proposed in this paper, which can be seen as the modification of the traditional Lotka-Volterra model. By using comparison theorem, the ultimately bounded region of the system is obtained. By using comparison theorem and Brouwer fixed point theorem, sufficient conditions with guarantee the existence of a positive periodic solution of the system is obtained. Finally, by constructing a suitable Lyapunov function, some sufficient conditions are obtained for the existence of a unique globally attractive periodic solution of system. The results obtained here generalized the main results of [J.D. Zhao, W.C. Chen, Global asymptotic stability of a periodic ecological model, Applied Mathematics and Computation, 147(3) (2004), 881-892].

[1]  Zhien Ma,et al.  Periodic solutions for delay differential equations model of plankton allelopathy , 2002 .

[2]  K. Gopalsamy,et al.  Global asymptotic stability in a periodic Lotka-Volterra system , 1985, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[3]  M E Gilpin,et al.  Schoener's model and Drosophila competition. , 1976, Theoretical population biology.

[4]  Jinlin Shi,et al.  Periodicity in a logistic type system with several delays , 2004 .

[5]  Ke Wang,et al.  Global periodic solutions of a generalized n-species Gilpin-Ayala competition model☆ , 2000 .

[6]  Yang Pinghua,et al.  Global Attractivity of the Periodic Lotka–Volterra System☆ , 1999 .

[7]  John Maynard Smith.  Models in ecology , 1974 .

[8]  Xiao-Qiang Zhao,et al.  The qualitative analysis of n-species Lotka-Volterra periodic competition systems , 1991 .

[9]  Joydev Chattopadhyay,et al.  Effect of toxic substances on a two-species competitive system , 1996 .

[10]  Jiandong Zhao,et al.  Global asymptotic stability of a periodic ecological model , 2004, Appl. Math. Comput..

[11]  K. Gopalsamy,et al.  Exchange of equilibria in two species Lotka-Volterra competition models , 1982, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[12]  Jinlin Shi,et al.  Periodicity in a food-limited population model with toxicants and state dependent delays☆ , 2003 .

[13]  Jinde Cao,et al.  Existence and global attractivity of an almost periodic ecological model , 2004, Appl. Math. Comput..

[14]  M E Gilpin,et al.  Competition between species: theoretical models and experimental tests. , 1973, Theoretical population biology.

[15]  Alan A. Berryman,et al.  The Orgins and Evolution of Predator‐Prey Theory , 1992 .

[16]  M. Gilpin,et al.  Global models of growth and competition. , 1973, Proceedings of the National Academy of Sciences of the United States of America.

[17]  Antonio Tineo,et al.  A, Different Consideraton about the Globally Asymptotically Stable Solution of the Periodic n-Competing Species Problem* , 1991 .