Automated techniques for the qualitative analysis of ecological models: Continuous models

Ecology and Society

Volumn 1, Issue 1, 1997, Pages

  Van Coller, Lynn ^a,b  

^a UNIVERSITY OF BRITISH COLUMBIA   (Canada)

^b NONE   (South Africa)

Author keywords

Bifurcation; Computer analysis; Dynamical systems; Ecological models; Qualitative analysis; Ratio dependent model; XPPAUT

Indexed keywords

EID: 54249129435     PISSN: 17083087     EISSN: 17083087     Source Type: Journal    
DOI: 10.5751/es-00017-010105     Document Type: Article

References (75)

1. Abrams, P.A. 1994. The fallacies of "ratio-dependent" predation. Ecology 75(6): 1842-1850.
2. Akçakaya, H.R., R. Arditi, and L.R. Ginzburg. 1995. Ratio-dependent predation: an abstraction that works. Ecology 76(3): 995-1004.
3. Allgower, E.L., and K. Georg. 1992. Continuation and path following. Acta Numerica 1-64.
4. Arditi, R., and H. Saïah. 1992. Empirical evidence of the role of heterogeneity in ratio-dependent consumption. Ecology 73(5):1544-1551.
5. Arnold, V.I. 1983. Geometrical methods in the theory of ordinary differential equations. Springer-Verlag, New York, New York, USA.
6. Aronson, D.G., M.A. Chory, G.R. Hall, and R.P. McGehee. 1982. Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study. Communications in Mathematical Physics 83: 303-354.
7. Back, A., J. Guckenheimer, M. Myers, F. Wicklin, and P. Worfolk. 1992. dstool: computer-assisted exploration of dynamical systems. Notices of the American Mathematical Society 39: 303-309.
8. Bazykin, A.D. 1974. Volterra's system and the Michaelis-Menten equation. Pages 103-142 in V.A. Ratner, editor. Problems in mathematical genetics. U.S.S.R. Academy of Science, Novosibirsk, U.S.S.R.
9. Beddington, J.R., C.A. Free, and J.H. Lawton. 1975. Dynamic complexity in predator-prey models framed in difference equations. Nature 255: 58-60.
10. Beddington, J.R., C.A. Free, and J.H. Lawton. 1976. Concepts of stability and resilience in predator-prey models. Journal of Animal Ecology 45: 791-816.
11. Berlinski, D. 1976. On systems analysis: an essay concerning the limitations of some mathematical methods in the social, political, and biological sciences. Massachusetts Institute of Technology Press, Cambridge, Massachusetts, USA.
12. Berryman, A.A. 1992. The origins and evolution of predator-prey theory. Ecology 73(5): 1530-1535.
13. Brylinski, M. 1972. Steady-state sensitivity analysis of energy flow in a marine ecosystem Pages xxx-xxx in B.C. Patten, editor. Systems analysis and simulation in ecology, II. Academic Press, New York, New York, USA.
14. Collings, J.B., and D.J. Wollkind. 1990. A global analysis of a temperature-dependent model system for a mite predator-prey interaction. SIAM Journal of Applied Mathematics 50(5): 1348-1372.
15. Collings, J.B., D.J. Wollkind, and M.E. Moody. 1990. Outbreaks and oscillations in a temperature-dependent model for a mite predator-prey interaction. Theoretical Population Biology 38: 159-191.
16. Deuflhard, P., B. Fiedler, and P. Kunkel. 1987. Efficient numerical pathfollowing beyond critical points. SIAM Journal of Numerical Analysis 24: 912-927.
17. Doedel, E.J. 1981. AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congressus Numerantium 30: 265-284.
18. Dowden, P.B., H.A. Jaynes, and V.M. Carolin. 1953. The role of birds in a spruce budworm outbreak in Maine. Journal of Economic Entomology 46: 307-312.
19. Edelstein-Keshet, L. 1988. Mathematical models in biology. Random House, New York, New York, USA.
20. Ermentrout, G.B. 1995. XPPAUT 1.63: the differential equations tool.
21. Fairen, V., and M.G. Velarde. 1979. Time-periodic oscillations in a model for respiratory process of a bacterial culture. Journal of Mathematical Biology 8: 147-157.
22. Fitzhugh, R. 1961. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal 1: 445-466.
23. Forrester, J.W. 1961. Industrial dynamics. Massachusetts Institute of Technology Press, Cambridge, Massachusetts, USA.
24. Frank, P.M. 1978. Introduction to system sensitivity theory. Academic Press, New York, New York, USA.
25. Freedman, H.I., and R.M. Mathsen. 1993. Persistence in predator-prey systems with ratio-dependent predator influence. Bulletin of Mathematical Biology 55(4): 817-827.
26. Gerald, C.F., and P.O. Wheatley. 1989. Applied numerical analysis. Addison-Wesley, Reading, Massachusetts, USA.
27. Gilpin, M.E. 1974. A model of the predator-prey relationship. Theoretical Population Biology 5: 333-344.
28. Ginzburg, L.R., and H.R. Akçakaya. 1992. Consequences of ratio-dependent predation for steady-state properties of ecosystems. Ecology 73(5): 1536-1543.
29. Gleeson, S.K. 1994. Density dependence is better than ratio dependence. Ecology 75(6): 1834-1835.
30. Gragnani, A., and S. Rinaldi. 1995. A universal bifurcation diagram for seasonally perturbed predator-prey models. Bulletin of Mathematical Biology 57(5): 701-712.
31. Guckenheimer, J., and P. Holmes. 1983. Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer-Verlag, New York, New York, USA.
32. Gutierrez, A.P. 1992. Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm. Ecology 73(5): 1552-1563.
33. Gutierrez, A.P., N.J. Mills, S.J. Schreiber, and C.K. Ellis. 1994. A physiologically based tritrophic perspective on bottom up-top down regulation of populations. Ecology 75: 2227-2242.
34. Hadamard, J. 1952. Lectures on Cauchy's problem in linear partial differential equations. Dover, New York, New York, USA.
35. Hale, J.K., and H. Kogak. 1989. Differential and difference equations through computer experiments. Springer-Verlag, New York, New York, USA.
36. Hanski, I. 1991. The functional response of predators: worries about scale. Trends in Ecology and Evolution 6(5): 141-142.
37. Hearne, J.W. 1987. An approach to resolving the parameter sensitivity problem in system dynamics methodology. Applied Mathematical Modelling 11: 315-318.
38. Hethcote, H.W. 1976. Qualitative analyses of communicable disease models. Mathematical Biosciences 28: 335-356.
39. Holling, C.S. 1965. The functional response of predators to prey density and its role in mimicry and population regulation. Memoirs of the Entomological Society of Canada 45: 1-60.
40. _. 1973. Resilience and stability of ecological systems. Annual Review of Ecology and Systematics 4: 1-23.
41. Hultquist, P.F. 1988. Numerical methods for engineers and computer scientists. Benjamin Cummings, Menlo Park, California, USA.
42. Jeffers, J.N.R. 1978. An introduction to systems analysis with ecological applications. University Park Press, Baltimore, Maryland, USA.
43. Kaas-Petersen, C. 1989. PATH: user's guide. University of Leeds, Leeds, UK.
44. Khibnik, A., Y. Kuznetsov, V. Levitin, and E. Nikolaev. 1993. Continuation techniques and interactive software for bifurcation analysis of ordinary differential equations and iterated maps. Physica D 62: 360-371.
45. Kowal, N.E. 1971. A rationale for modeling dynamic ecological systems. Pages xxx-xxx in B.C. Patten, editor. Systems analysis and simulation in ecology, I. Academic Press, New York, New York, USA.
46. Ludwig, D., D.D. Jones, and C.S. Holling. 1978. Qualitative analysis of insect outbreak systems: the spruce budworm and forest. Journal of Animal Ecology 47: 315-332.
47. Lundberg, P., and J.M. Fryxell. 1995. Expected population density vs. productivity in ratio-dependent and prey-dependent models. American Naturalist 146(1): 153-161.
48. MacDonald, N. 1978. Time lags in biological models. Springer-Verlag, Berlin, Germany.
49. _. 1989. Biological delay systems. Cambridge University Press, Cambridge, Massachusetts, USA.
50. May, R.M. 1973. Time delay vs. stability in population models with two and three trophic levels. Ecology 54: 315-325.
51. _. 1981. Models for two interacting populations. Pages 78-104 in R.M. May, editor. Theoretical ecology: principles and applications. Blackwell, Oxford, UK.
52. Maynard Smith, J. 1974. Models in ecology. Cambridge University Press, Cambridge, UK.
53. McCarthy, M.A., L.R. Ginzburg, and H.R. Akçakaya. 1995. Predator interference across trophic chains. Ecology 76(4): 1310-1319.
54. Murray, J.D. 1981. A pre-pattern formation mechanism for animal coat markings. Journal of Theoretical Biology 88: 161-199.
55. Nusse, H.E., and J.A. Yorke. 1994. Dynamics: numerical explorations. Applied Mathematical Sciences 101. Springer-Verlag, New York, New York, USA.
56. Patten, B.C. 1971. A primer for ecological modeling and simulation with analog and digital computers. Pages xxx-xxx in B.C. Patten, editor. Systems analysis and simulation in ecology, I. Academic Press, New York, New York, USA.
57. Rheinboldt, W.C., and J.V. Burkardt. 1983a. A locally parameterized continuation process. ACM Transactions on Mathematical Software 9: 215-235.
58. Rheinboldt, W.C., and J.V. Burkardt. 1983b. Algorithm 596: a program for a locally parametrized continuation process. ACM Transactions on Mathematical Software 9: 236-241.
59. Rinaldi, S., S. Muratori, and Y. Kuznetsov. 1993. Multiple attractors, catastrophes, and chaos in seasonally perturbed predator-prey communities. Bulletin of Mathematical Biology 55(1): 15-35.
60. Rosenzweig, M.L., and R.H. MacArthur. 1963. Graphic representation and stability conditions of predator-prey interactions. American Naturalist 97: 209-223.
61. Royama, T. 1984. Population dynamics of the spruce budworm Choristoneura fumiferana. Ecological Monographs 54(4): 429-462.
62. Sanders, C.J., R.W. Stark, E.J. Mullins, and J. Murphy, editors. 1985. Recent advances in spruce budworm research. Proceedings of CANUSA Spruce Budworm Research Symposium, 16-20 September 1984, Bangor, Maine, Canadian Forestry Service, Ottawa, Ontario, Canada.
63. Sarnelle, O. 1994. Inferring process from pattern: trophic-level abundances and imbedded interactions. Ecology 75(6): 1835-1841.
64. Seydel, R. 1988. From equilibrium to chaos: practical bifurcation and stability analysis. Elsevier, New York, New York, USA.
65. _. 1991. BIFPACK: a program package for continuation, bifurcation, and stability analysis. Version 2.3+. University of Ulm, Ulm, Germany.
66. Strogatz, S.H. 1994. Nonlinear dynamics and chaos. Addison-Wesley, Reading, Massachusetts, USA.
67. Swart, J. 1987. Sensitivity of a hyrax-lynx mathematical model to parameter uncertainty. South African Journal of Science 83: 545-547.
68. Swart, J., and J.W. Hearne. 1989. A mathematical model to analyze predation and competition problems in a sheep-farming region. System Dynamics Review 5: 35-50.
69. Tomovic, R. 1963. Sensitivity analysis of dynamic systems. Translated by D. Tornquist. McGraw-Hill, London, UK.
70. Waterloo Maple Software. 1990-1992. The maple roots report. Waterloo Maple Software, Waterloo, Canada.
71. Watt, K.E.F. 1966. The nature of systems analysis. Pages xxx-xxx in K.E.F. Watt, editor. Systems analysis in ecology. Academic Press, New York, New York, USA.
72. Wiggins, S. 1990. Introduction to applied nonlinear dynamical systems and chaos. Springer-Verlag, New York, New York, USA.
73. Williams, T., and C. Kelley. 1986. GNUPLOT: an interactive plotting program: user manual. http://www.cs.dartmouth.edu/gnuplot/gnuplot.html
74. Wollkind, D.J., J.B. Collings, and J.A. Logan. 1988. Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees. Bulletin of Mathematical Biology 50(4): 379-409.
75. Yodzis, P. 1989. Introduction to theoretical ecology. Harper and Row, New York, New York, USA.