Persistence and Extinction in Two Species Reaction–Diffusion Systems with Delays

Abstract Both uniform persistence and global extinction are established for two species predator–prey and competition reaction–diffusion systems with delays in terms of the principal eigenvalues of the scalar elliptic eigenvalue problems by appealing to the theories of abstract persistence, asymptotically autonomous semiflows, and monotone dynamical systems.

[1]  Shigui Ruan,et al.  Uniform persistence and flows near a closed positively invariant set , 1994 .

[2]  Josef Hofbauer,et al.  Uniform persistence and repellors for maps , 1989 .

[3]  Jack K. Hale,et al.  Persistence in infinite-dimensional systems , 1989 .

[4]  P. Hess,et al.  Periodic-Parabolic Boundary Value Problems and Positivity , 1991 .

[5]  Chris Cosner,et al.  Practical persistence in ecological models via comparison methods , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[6]  Yang Kuang,et al.  Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks , 1993, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[7]  J. Hale Asymptotic Behavior of Dissipative Systems , 1988 .

[8]  Z. Teng,et al.  Persistence in dynamical systems , 1990 .

[9]  Xiao-Qiang Zhao Asymptotic Behavior for Asymptotically Periodic Semiflows with Applications , 1996 .

[10]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[11]  Wenzhang Huang,et al.  Global Dynamics for a Reaction–Diffusion Equation with Time Delay☆ , 1998 .

[12]  E. N. Dancer,et al.  Stability of fixed points for order-preserving discrete-time dynamical systems. , 1991 .

[13]  Stephen A. Gourley,et al.  A predator-prey reaction-diffusion system with nonlocal effects , 1996 .

[14]  C. Travis,et al.  Existence and stability for partial functional differential equations , 1974 .

[15]  Julián López-Gómez On the structure of the permanence region for competing species models with general diffusivities and transport effects , 1996 .

[16]  J. Furter,et al.  Diffusion-mediated permanence problem for a heterogeneous Lotka–Volterra competition model , 1997, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[17]  H. I. Freedman,et al.  Persistence in discrete semidynamical systems , 1989 .

[18]  Jack K. Hale,et al.  Large Diffusivity and Asymptotic Behavior in Parabolic Systems. , 1986 .

[19]  F. M. Arscott,et al.  PERIODIC‐PARABOLIC BOUNDARY VALUE PROBLEMS AND POSITIVITY , 1992 .

[20]  Xinzhi Liu Comparison methods and stability theory , 1994 .

[21]  Stephen A. Gourley,et al.  Parameter domains for instability of uniform states in systems with many delays , 1997 .

[22]  H. I. Freedman,et al.  Global Asymptotics in Some Quasimonotone Reaction-Diffusion Systems with Delays , 1997 .

[23]  Horst R. Thieme,et al.  Persistence under relaxed point-dissipativity (with application to an endemic model) , 1993 .

[24]  R. Martin,et al.  Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence. , 1991 .

[25]  Horst R. Thieme,et al.  Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations , 1992 .

[26]  Hal L. Smith,et al.  Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions , 1995 .

[27]  Robert Stephen Cantrell,et al.  Permanence in ecological systems with spatial heterogeneity , 1993, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[28]  Jianhong Wu Theory and Applications of Partial Functional Differential Equations , 1996 .

[29]  E O Powell,et al.  Theory of the chemostat. , 1965, Laboratory practice.

[30]  Chris Cosner,et al.  Models for Predator-Prey Systems at Multiple Scales , 1996, SIAM Rev..

[31]  Paul Waltman,et al.  Uniformly persistent systems , 1986 .

[32]  M. Hirsch Stability and convergence in strongly monotone dynamical systems. , 1988 .

[33]  K. Schmitt,et al.  Permanence and the dynamics of biological systems. , 1992, Mathematical biosciences.

[34]  Nicholas F. Britton,et al.  Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model , 1990 .

[35]  Yoshio Yamada,et al.  Stability of steady states for prey-predator diffusion equations with homogeneous dirichlet conditions , 1990 .

[36]  H. I. Freedman,et al.  Uniform Persistence in Functional Differential Equations , 1995 .

[37]  Xiao-Qiang Zhao,et al.  Permanence in Kolmogorov periodic predator-prey models with diffusion , 1994 .

[38]  Maria Giovanna Garroni,et al.  Green Functions for Second Order Parabolic Integro-Differential Problems , 1993 .

[39]  Xin Lu,et al.  Harmless Delays for Permanence in a Class of Population Models with Diffusion Effects , 1997 .

[40]  Hal L. Smith,et al.  Abstract functional-differential equations and reaction-diffusion systems , 1990 .

[41]  C. V. Pao,et al.  Dynamics of Nonlinear Parabolic Systems with Time Delays , 1996 .

[42]  S. Ruan,et al.  A generalization of the Butler-McGehee lemma and its applications in persistence theory , 1996, Differential and Integral Equations.