Persistence and Global Stability in a Population Model

A difference equation modelling the dynamics of a population undergoing a density-dependent harvesting is considered. A sufficient condition is established for all positive solutions of the corresponding discrete dynamic system to converge eventually to the positive equilibrium. Elementary methods of differential calculus are used. The result of this article provides a generalization of a result known for a simpler special model with no harvesting.

[1]  G. Seifert.  On an interval map associated with a delay logistic equation with discontinuous delays , 1991 .

[2]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[3]  R. May,et al.  Biological populations obeying difference equations: stable points, stable cycles, and chaos. , 1975, Journal of theoretical biology.

[4]  Jack K. Hale,et al.  Studies in Ordinary Differential Equations , 1977 .

[5]  Anatoli F. Ivanov,et al.  Oscillations in Singularly Perturbed Delay Equations , 1992 .

[6]  R. May,et al.  Bifurcations and Dynamic Complexity in Simple Ecological Models , 1976, The American Naturalist.

[7]  E. Soewono,et al.  Period doubling and Chaotic behavior of solutions to y′ t) μy(t)(1 − y(δ[(t+α)/δ])) , 1991 .

[8]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[9]  B. S. Goh,et al.  Stability results for delayed-recruitment models in population dynamics , 1984 .

[10]  V. Kocić,et al.  Global Behavior of Nonlinear Difference Equations of Higher Order with Applications , 1993 .

[11]  Joseph Wiener,et al.  Generalized Solutions of Functional Differential Equations , 1993 .

[12]  K. Cooke,et al.  A nonlinear equation with piecewise continuous argument , 1988, Differential and Integral Equations.

[13]  J. Guckenheimer,et al.  The dynamics of density dependent population models , 1977, Journal of mathematical biology.

[14]  James T. Sandefur,et al.  Discrete dynamical systems - theory and applications , 1990 .

[15]  Akitaka Dohtani,et al.  Occurrence of chaos in higher-dimensional discrete-time systems , 1992 .

[16]  P Cull,et al.  Global stability of population models. , 1981, Bulletin of mathematical biology.

[17]  Paul Cull.  Global stability of population models , 1981 .