Spatially heterogeneous discrete waves in predator-prey communities over a patchy environment

Mathematical Biosciences

Volumn 131, Issue 2, 1996, Pages 135-155

  Koh, D ^a   Wei, J ^a   Wu, J ^a  

^a YORK UNIVERSITY   (Canada)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; CALCULATION; ENVIRONMENT; MATHEMATICAL ANALYSIS; MATHEMATICAL MODEL; PREDATOR; THEORY;

PATCHY ENVIRONMENT; PREDATOR-PREY COMMUNITY; SPATIAL HETEROGENEITY; TIME LAG;

EID: 0029668372     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/0025-5564(95)00035-6     Document Type: Article

References (36)

1. J. D. Murray, Spatial structures in predator-prey communities - a nonlinear time delay diffusional model, Math. Biosci. 30:73-85 (1976).
2. E. Beretta, F. Solimano, and Y. Takeuchi, Global stability and periodic orbits for two-patch predator-prey diffusion-delay models, Math, Biosci. 85:153-183 (1987).
3. E. Beretta and Y. Takeuchi, Global stability of single-species diffusion models with continuous time delays, Bull. Math. Biol. 49:431-448 (1987).
4. E. Beretta and Y. Takeuchi, Global asymptotic stability of Lotka-Volterra diffusion models with continuous time delay, SIAM J. Appl. Math. 48:627-651 (1988).
5. H. I. Freedman, B. Rai, and P. Waltman, Mathematical models of population interactions with disperal. II: Differential survival in a charge of habitat, J. Math. Anal. Appl. 115:140-154 (1986).
6. H. I. Freedman, J. B. Shukla, and Y. Takeuchi, Population diffusion in a two-patch environment, Math. Biosci. 95:111-123 (1989).
7. H. I. Freedman and Y. Takeuchi, Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear Anal. 13:993-1002 (1989).
8. H. I. Freedman and P. Waltman, Mathematical models of population interaction with dispersal. I: Stability of two habitats with and without a predator, SIAM J. Appl. Math. 32:631-648 (1977).
9. H. I. Freedman and J. Wu, Persistence and global asymptotic stability of single species dispersal models with stage structure, Quart. Appl. Math. 49:351-371 (1991).
10. H. Hastings, Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates, J. Math. Biol. 16:49-55 (1982).
11. Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Differ. Integral Equations 4:117-128 (1991).
12. S. A. Levin, Dispersion and population interactions. Am. Nat. 108:207-228 (1974).
13. S. L. Levin, Mathematical population biology, Proc. Symp. Appl. Math. 30:1-8 (1984).
14. S. A. Levin and L. A. Segel, An hypothesis to explain the origin of planktonic patchiness, Nature 259:659 (1976).
15. J. Lin and P. B. Kahn, Random effects in population models with hereditary effects, J. Math. Biol. 10: 101-112 (1980).
16. J. Lin and P. B. Kahn, Phase and amplitude instability in delay-diffusion population models, J. Math. Biol. 13:383-393 (1982).
17. R. McMurtie, Persistence and stability of single-species and prey-predator systems in spatially heterogeneous environments, Math. Biosci. 39:11-51 (1978).
18. S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Popul. Biol. 21:92-113 (1982).
19. N. Shigesada and J. Roughgarden, The role of rapid dispersal in the population dynamics of competition, Theor. Popul. Biol. 21:353-372 (1982).
20. J. G. Skellam, Random dispersal in theoretical populations, Biometrika 38:196-218 (1951).
21. D. Stirzaker, On a population model, Math. Biosci. 23:329-336 (1976).
22. Y. Takeuchi, Diffusion effect in stability of Lotka-Volterra models, Bull. Math. Biol. 48:585-601 (1986).
23. R. R. Vance, The effect of dispersal on population stability in one-species, discrete-space population growth models, Am. Nat. 123:230-254 (1984).
24. J. Wu and W. Krawcewicz, Discrete waves and phase-locked oscillations in the growth of a single-species population over a patchy environment, Open Syst. Inf. Dynam. 1:127-147 (1992).
25. R. D. Braddock and P. van den Driessche, On a two lag differential delay equation, J. Austral. Math. Soc. 248:292-317 (1983).
26. S. A. Campbell and J. Belair, Stability and bifurcations of equilibrium in a multiple-delayed differential equation, SIAM J. Appl. Math. to appear.
27. J. Hale and W. Huang, Global geometry of the stable regions for two delay differential equations, Tech. Rep. CDSNS90-41, Georgia Tech., 1990.
28. J. M. Mahaffy, P. J. Zak, and K. M. Joiner, A geometric analysis of the stability regions for a linear differential equation with two delays, preprint.
29. R. D. Nussbaum, Differential delay equations with two time lags, Mem. Amer. Math. Soc. 16:1-62 (1975).
30. K. Yoshida and K. Kishimoto, Effect of two time delays on partial differential equations, Kumamoto J. Sci. Math. 15:91-109 (1983).
31. M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. 2, Springer-Verlag, New York, 1988.
32. J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
33. J. Wu, The effect of delay and diffusion on spontaneous symmetry breaking in functional differential equations, Rocky Mt. J. Math., to appear.
34. J. Wu, Delay-induced discrete waves of large amplitudes in neural networks with circulant connection matrices, Preprint, Fields Institute, 1993.
35. B. Hassard, N. Kazarinoff, and Y. Wan, Theory and Applications of Hopf Bifurcation, Cambridge Univ. Press, Cambridge, 1981.
36. J. C. Alexander and G. Auchmuty, Global bifurcations of phase-locked oscillations, Arch. Ration. Mech. Anal. 93:253-270 (1986).