Predator-prey models with delay and prey harvesting

Abstract. It is known that predator-prey systems with constant rate harvesting exhibit very rich dynamics. On the other hand, incorporating time delays into predator-prey models could induce instability and bifurcation. In this paper we are interested in studying the combined effects of the harvesting rate and the time delay on the dynamics of the generalized Gause-type predator-prey models and the Wangersky-Cunningham model. It is shown that in these models the time delay can cause a stable equilibrium to become unstable and even a switching of stabilities, while the harvesting rate has a stabilizing effect on the equilibrium if it is under the critical harvesting level. In particular, one of these models loses stability when the delay varies and then regains its stability when the harvesting rate is increased. Computer simulations are carried to explain the mathematical conclusions.

[1]  Peter J. Wangersky,et al.  Time Lag in Prey‐Predator Population Models , 1957 .

[2]  Mark Bartlett,et al.  ON THEORETICAL MODELS FOR COMPETITIVE AND PREDATORY BIOLOGICAL SYSTEMS , 1957 .

[3]  J. Dieudonne.  Foundations of Modern Analysis , 1969 .

[4]  Elliott W. Montroll,et al.  Nonlinear Population Dynamics. (Book Reviews: On the Volterra and Other Nonlinear Models of Interacting Populations) , 1971 .

[5]  Scott A. Boorman,et al.  On the Volterra and Other Nonlinear Models of Interacting Populations. N. S. Goel, S. C. Maitra, and E. W. Montroll. Academic Press, New York, 1971. viii, 146 pp., illus. $5.50. Reviews of Modern Physics Monographs , 1972 .

[6]  Robert M. May,et al.  Time‐Delay Versus Stability in Population Models with Two and Three Trophic Levels , 1973 .

[7]  Colin W. Clark,et al.  Mathematical Bioeconomics: The Optimal Management of Renewable Resources. , 1993 .

[8]  Jim M Cushing,et al.  Integrodifferential Equations and Delay Models in Population Dynamics , 1977 .

[9]  Fred Brauer,et al.  Stability of some population models with delay , 1977 .

[10]  Jim M Cushing,et al.  Integrodifferential Equations and Delay Models in Population Dynamics. , 1978 .

[11]  N. Macdonald.  Time lags in biological models , 1978 .

[12]  A. C. Soudack,et al.  Stability regions in predator-prey systems with constant-rate prey harvesting , 1979 .

[13]  A. C. Soudack,et al.  Stability regions and transition phenomena for harvested predator-prey systems , 1979 .

[14]  Herbert W. Hethcote,et al.  Stability analysis for models of diseases without immunity , 1981, Journal of mathematical biology.

[15]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[16]  Jim M Cushing,et al.  STABILITY AND MATURATION PERIODS IN AGE STRUCTURED POPULATIONS , 1981 .

[17]  K. Cooke,et al.  Discrete delay, distributed delay and stability switches , 1982 .

[18]  H. I. Freedman.  Deterministic mathematical models in population ecology , 1982 .

[19]  J M Cushing,et al.  A predator prey model with age structure , 1982, Journal of mathematical biology.

[20]  K. Gopalsamy.  Harmless delays in model systems , 1983 .

[21]  A Hastings,et al.  Delays in recruitment at different trophic levels: Effects on stability , 1984, Journal of mathematical biology.

[22]  K. Gopalsamy,et al.  Delayed responses and stability in two-species systems , 1984, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[23]  L. Nunney,et al.  Absolute stability in predator-prey models , 1985 .

[24]  Mary R. Myerscough,et al.  An analysis of an ordinary differential equation model for a two-species predator-prey system with harvesting and stocking , 1992 .

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

[26]  Yang Kuang,et al.  Convergence Results in a Well-Known Delayed Predator-Prey System , 1996 .

[27]  Moxun Tang,et al.  Coexistence Region and Global Dynamics of a Harvested Predator-Prey System , 1998, SIAM J. Appl. Math..

[28]  S. Ruan.  Absolute stability, conditional stability and bifurcation in Kolmogrov-type predator-prey systems with discrete delays , 2001 .