Positive Periodic Solutions of a Class of Non–autonomous Single Species Population Model with Delays and Feedback Control

AbstractWith the help of a continuation theorem based on Gaines and Mawhin's coincidence degree, several verifiable criteria are established for the global existence of positive periodic solutions of a class of non–autonomous single species population model with delays (both state–dependent delays and continuous delays) and feedback control. After that, by constructing a suitable Lyapunov functional, sufficient conditions which guarantee the existence of a unique globally asymptotic stable positive periodic solution of a kind of nonlinear feedback control ecosystem are obtained. Our results extend and improve the existing results, and have further applications in population dynamics.

[1]  Zhidong Teng,et al.  Nonautonomous Lotka–Volterra Systems with Delays , 2002 .

[2]  Ke Wang,et al.  Periodicity in a “Food-limited” Population Model with Toxicants and Time Delays , 2002 .

[3]  J. Milton,et al.  Insight into the transfer function, gain, and oscillation onset for the pupil light reflex using nonlinear delay-differential equations , 1989, Biological Cybernetics.

[4]  Wan-Tong Li,et al.  Positive periodic solutions of a class of delay differential system with feedback control , 2004, Appl. Math. Comput..

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

[6]  Fan Meng,et al.  Periodic Solutions of Single Population Model with Hereditary Effects , 2000 .

[7]  J. Mallet-Paret,et al.  Boundary layer phenomena for differential-delay equations with state-dependent time lags, I. , 1992 .

[8]  Yang Kuang,et al.  Periodic solutions in periodic state-dependent delay equations and population models , 2001 .

[9]  Peixuan Weng.  Existence and global stability of positive periodic solution of periodic integrodifferential systems with feedback controls , 2000 .

[10]  Ping Liu,et al.  Positive periodic solutions of a class of functional differential systems with feedback controls , 2004 .

[11]  Li Xiao.  POSITIVE PERIODIC SOLUTION OF SINGLE SPECIES MODEL WITH FEEDBACK REGULATION AND INFINITE DELAY , 2002 .

[12]  M. Birkner,et al.  Blow-up of semilinear PDE's at the critical dimension. A probabilistic approach , 2002 .

[13]  Guihong Fan.  Existence of positive periodic solution for a single species model with state dependent delay , 2002 .

[14]  Wendi Wang,et al.  The effect of dispersal on population growth with stage-structure☆ , 2000 .

[15]  Lansun Chen,et al.  Permanence and extinction in logistic and Lotka-Volterra systems with diffusion , 2001 .

[16]  Ravi P. Agarwal,et al.  Periodicity and Stability in Periodic n-Species Lotka-Volterra Competition System with Feedback Controls and Deviating Arguments , 2003 .

[17]  Reflective function and periodic solution of differential systems , 2002 .

[18]  杨帆,et al.  EXISTENCE AND GLOBAL ATTRACTIVITY OF POSITIVE PERIODIC SOLUTION OF A LOGISTIC GROWTH SYSTEM WITH FEEDBACK CONTROL AND DEVIATING ARGUMENTS , 2001 .

[19]  N. Rose,et al.  Differential Equations With Applications , 1967 .

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

[21]  O. Arino,et al.  Existence of Periodic Solutions for a State Dependent Delay Differential Equation , 2000 .

[22]  Michael C. Mackey,et al.  Feedback, delays and the origin of blood cell dynamics , 1990 .

[23]  Zhidong Teng,et al.  Uniform persistence and existence of strictly positive solutions in nonautonomous Lotka-Volterra competitive systems with delays☆ , 1999 .

[24]  Yang Kuang,et al.  Periodic Solutions of Periodic Delay Lotka–Volterra Equations and Systems☆ , 2001 .

[25]  Yongkun Li,et al.  Existence and global attractivity of a positive periodic solution of a class of delay differential equation , 1998 .

[26]  André Longtin,et al.  Complex oscillations in the human pupil light reflex with “mixed” and delayed feedback , 1988 .

[27]  R. Gaines,et al.  Coincidence Degree and Nonlinear Differential Equations , 1977 .

[28]  Yuming Chen.  Periodic solutions of a delayed, periodic logistic equation , 2003, Appl. Math. Lett..

[29]  K. Gopalsamy,et al.  FEEDBACK REGULATION OF LOGISTIC GROWTH , 1993 .

[30]  A Longtin,et al.  Modelling autonomous oscillations in the human pupil light reflex using non-linear delay-differential equations. , 1989, Bulletin of mathematical biology.

[31]  Yongkun Li,et al.  Periodic Solutions for Delay Lotka–Volterra Competition Systems , 2000 .

[32]  Jurang Yan,et al.  Global attractivity and oscillation in a nonlinear delay equation , 2001 .

[33]  André Longtin.  Nonlinear oscillations, noise and chaos in neural delayed feedback , 1989 .

[34]  Fengde Chen.  Positive periodic solutions of neutral Lotka-Volterra system with feedback control , 2005, Appl. Math. Comput..

[35]  Xiaoxin Chen,et al.  Sufficient conditions for the existence positive periodic solutions of a class of neutral delay models with feedback control , 2004, Appl. Math. Comput..

[36]  John Mallet-Paret,et al.  Boundary layer phenomena for differential-delay equations with state-dependent time lags: III , 2003 .

[37]  O. Arino,et al.  Existence of Periodic Solutions for Delay Differential Equations with State Dependent Delay , 1998 .

[38]  Yongkun Li,et al.  Existence of positive periodic solutions for a periodic logistic equation , 2003, Appl. Math. Comput..