Discrete Competitive and Cooperative Models of Lotka-Volterra Type

Journal of Computational Analysis and Applications

Volumn 3, Issue 1, 2001, Pages 53-73

  Liu, Pingzhou ^a   Elaydi, Saber N ^b  

^a Dept of Mathematics and Statistics ^*  (Australia)

^b TRINITY UNIVERSITY   (United States)

Author keywords

Competitive; Cooperative; Global stability; Lotka Volterra systems

Indexed keywords

EID: 0345734844     PISSN: 15211398     EISSN: None     Source Type: Journal    
DOI: 10.1023/a:1011539901001     Document Type: Article

References (41)

1. R. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, New York, 1992.
2. M. Araki and B. Kondo, Stability and transient behaviour of composite systems, IEEE Trans. Auto. Contr. AC-17, 537-541 (1972).
3. D. G. Aronson, M. A. Chory, G. R. Hall, and R. P. McGehee, Bifurcation from an invariant circle for two-parameter families of maps of the plane: A computer-assisted study, Commun. Math. Phys. 83, 303-354 (1982).
4. J. M. Gushing, Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Spring-Verlag, Berlin, New York, 1977.
5. A. Dohtani, Occurrence of chaos in higher-dimensional discrete-time systems, SIAM J. Appl. Math. 52, 1702-1721 (1992).
6. L. Edelstein-Keshet, Mathematical Models in Biology, Random House, New York, 1978.
7. S. N. Elaydi, An Introduction to Difference Equations, Springer-Verlag, Berlin, 1996.
8. S. N. Elaydi, V. L. Kocic, and J. Li, Global stability of nonlinear delay difference equations, J. Difference Equations Appl. 2, 87-96 (1996).
9. M. E. Fisher, Analysis of Difference Equation Models in Population Dynamics, Ph.D. thesis, University of Western Australia (1982).
10. G. Gandolfo, Economics Dynamics, 3rd edn, Springer-Verlag, Berlin, 1997.
11. B. S. Goh, Stability in models of mutualism, Amer. Natural 113, 261-275 (1979).
12. K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic, Dordrecht, 1992.
13. J. L. Gouzé and K. P. Hadeler, Monotone flows and order intervals, Nonlinear World 1, 23-34 (1994).
14. K. P. Hadeler and I. Gerstmann, The discrete Rosenzweig model, Math. Biosci. 98, 49-72 (1990).
15. J. K. Hale and A. S. Somolinos, Competition for fluctuating nutrient, J. Math. Biol. 18, 255-280 (1983).
16. F. B. Hildebrand, Finite-Difference Equations and Simulations, Prentice-Hall, Englewood Cliffs, NJ, 1968.
17. M. W. Hirsch, Systems of differential equations which are competitive or cooperative I, Limit set, SIAM J. Math. Anal 13, 167-179 (1982).
18. M. W. Hirsch, Systems of differential equations which are competitive or cooperative II, Convergence almost everywhere, SIAM J. Math. Anal. 16, 432-439 (1985).
19. M. W. Hirsch, Systems of differential equations which are competitive or cooperative IV, Structural stability in three dimensional systems, SIAM J. Math. Anal. 21, 125-1244 (1990).
20. J. Hofbauer, V. Hutson, and W. Jansen, Coexistence for systems governed by difference equations of Lotka-Volterra type, Bull. Math. Biol. 24, 553-579 (1978).
21. V. Hutson and G. Vickers, A criterion for permanent coexistence of species, with an application to a two-prey, one-predator system, Math. Biosci. 63, 253-269 (1983).
22. H. Jiang and T. D. Rogers, The discrete dynamics of symmetric competition in the plane, J. Math. Biol. 25, 573-596 (1987).
23. E. I. Jury, The inners approach to some problems of system theory, IEEE Trans. Automatic Contr. AC16, 233-240 (1971).
24. W. Kelley and A. Peterson, (Different Equations: An Introduction with Applications, Academic Press, New York, 1991.
25. W. Krawcewicz and T. D. Rogers, Perfect harmony: The discrete dynamics of cooperation, J. Math. Biol. 28, 383-410 (1990).
26. V. L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order, Kluwer Academic, Dordrecht, 1993.
27. Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993.
28. G. Ladas, G. Tzanetopoulos, and A. Tovbis, On May's host parasitoid model, J. Difference Equations Appl. 2, 195-204 (1996).
29. P. H. Leslie, A stochastic model for studying the properties of certain biological systems by numerical methods, Biometrika 45, 16-31 (1958).
30. J. J. Levin, The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. Math. Soc. 14, 533-541 (1963).
31. R .M. May, Biological populations obeying difference equations: Stable cycles and chaos, J. Theoret. Biol. 51, 511-524 (1975).
32. J. Maynard Smith, Models in Ecology, Cambridge Univ. Press (Cambridge, 1974).
33. J. Mazumdar, An Introduction to Mathematical Physiology and Biology, Cambridge Univ. Press, Cambridge, 1989, pp. 63-65.
34. R. E. Mickens, Exact solutions to a finite-difference model for a nonlinear reaction-diffusion equation: Implications for numerical analysis, Numer. Methods Partial Diff. Eq. 5, 313-325 (1989).
35. R. E. Mickens, Genesis of elementary numerical instabilities in finite-difference models of ordinary differential equations, Proceedings of Dynamics Systems and Applications (G. S. Ladde and M. Sambandham eds.), Dynamics Publisher, 1, 1994, pp. 251-258.
36. R. E. Mickens and A. Smith, Finite-difference models of ordinary differential equations: Influence of denominator functions, J. Franklin Inst. 327, 143-149 (1990).
37. R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.
38. P. de Mottoni and A. Schiaffino, Competition systems with periodic coefficients: A geometric approach, J. Math. Biol. 11, 319-335 (1981).
39. E. C. Pielou, Mathematical Ecology, Wiley (Interscience), New York, 1977.
40. H. L. Smith, Monotone Dynamical Systems, An Introduction to the Theory of Competitive and Cooperative Systems, Proc. Amer. Math. Soc., Rhode Island (1995).
41. S. Ushiki, Central difference scheme and chaos, Physica D 4, 407-424 (1982).