zbMATH — the first resource for mathematics

The computer-aided bifurcation analysis of predator-prey models. (English) Zbl 0547.92015
Summary: Mathematical analysis of dynamical systems can often benefit from companying numerical computations. This is particularly true if one has software capable of providing an automatic bifurcation analysis of such systems. Computer programs of this type now exist.
We describe the application of such software to a predator-prey model. Phenomena that arise in this analysis include stationary bifurcations, limit points, Hopf bifurcations and secondary periodic bifurcations. A two-parameter numerical analysis leads quite naturally to the detection of higher order singularities.

92D25 Population dynamics (general)
65L10 Numerical solution of boundary value problems involving ordinary differential equations
65-04 Software, source code, etc. for problems pertaining to numerical analysis
37-XX Dynamical systems and ergodic theory
92-04 Software, source code, etc. for problems pertaining to biology
Full Text: DOI
[1] Aronson, D., Doedel, E. J., Othmer, H. G.: An analytical and numerical study of the bifurcations in a system of linearly-coupled oscillators. Technical Report, Department of Mathematics, University of Utah, 1984 · Zbl 0624.34029
[2] Ascher, U., Christiansen, J., Russell, R. D.: Collocation software for boundary value ODE’s. ACM Trans. Math. Software 7, (No. 2) 209-222 (1981) · Zbl 0455.65067 · doi:10.1145/355945.355950
[3] Beyn, W.-J., Doedel, E. J.: Stability and multiplicity of solutions to discretizations of nonlinear ordinary differential equations. SIAM J. Sci. Stat. Comput. 2, (No. 1) 107-120 (1981) · Zbl 0466.65049 · doi:10.1137/0902009
[4] Clark, C. W.: Mathematical Bioeconomics. New York: Wiley, 1976 · Zbl 0364.90002
[5] Crandall, M. G., Rabinowitz, P. H.: The Hopf bifurcation theorem. MRC Report 1604, University of Wisconsin 1976 · Zbl 0385.34020
[6] Doedel, E. J.: AUTO: A program for the automatic bifurcation analysis of autonomous systems. Cong. Num. 30, 265-284 (1981) (Proc. 10th Manitoba Conf. on Num. Math. and Computing, Winnipeg Canada, October 1980)
[7] Doedel, E. J.: Continuation techniques in the study of chemical reaction schemes, to appear in: Proc. Special Year in Energy Math., Univ. of Wyoming, K. I. Gross, ed., SIAM Publ.
[8] Doedel, E. J., Heinemann, R. F.: Numerical computation of periodic solution branches and oscillatory dynamics of the stirred tank reactor with A ? B ? C reactions. Chem. Engrg Sci. 38, (No. 9) 1493-1499 (1983) · doi:10.1016/0009-2509(83)80084-0
[9] Doedel, E. J., Jepson, A. D., Keller, H. B.: Numerical methods for Hopf bifurcation and continuation of periodic solution paths, to appear in: Computing Methods in Applied Sciences and Engineering VI, Glowinsky, R., Lions, J. L., (eds.). North Holland 1984 · Zbl 0566.65041
[10] Doedel, E. J., Leung, P. C.: Numerical techniques for bifurcation problems in delay equations. Cong. Num. 34, 225-237 (1982) (Proc. 11th Manitoba Conf. on Num. Math. and Computing, Winnipeg Canada, October 1981)
[11] Jepson, A. D.: Numerical Hopf bifurcation. Thesis, Part II, California Institute of Technology, Pasadena, Ca., 1981
[12] Jepson, A., Spence, A.: Folds in solutions of two parameter systems and their calculation: Part I, Numerical Analysis Project, Manuscript NA-82-02, Computer Science Department, Stanford University, March 1982 · Zbl 0576.65052
[13] Jepson, A., Spence, A.: The numerical solution of nonlinear equations having several parameters, Part I: Scalar equations, Technical Report 168/83, Department of Computer Science, The University of Toronto, 1983 · Zbl 0597.65051
[14] Keller, H. B.: Numerical solution of bifurcation and nonlinear eigenvalue problems. In: Applications of Bifurcation Theory, Rabinowitz, P. H., (ed.). New York: Academic Press, 1977, pp. 359-384 · Zbl 0581.65043
[15] Keller, H. B.: Continuation methods in Computational Fluid Dynamics. In: Numerical and Physical Aspects of Aerodynamic Flows, Cebeci, T., (ed.). Berlin-Heidelberg-New York: Springer 1981
[16] Kernevez, J. P.: Enzyme Mathematics. North-Holland Press 1980
[17] Langford, W. F.: Periodic and steady state mode interactions lead to tori. SIAM J. Appl. Math. 37, (No. 1), 22-48 (1979) · Zbl 0417.34030 · doi:10.1137/0137003
[18] Golubitsky, M., Langford, W. F.: Classification and unfoldings of degenerate Hopf bifurcations. J. Differential Equations 41, (No. 3) 375-415 (1981) · Zbl 0462.58023 · doi:10.1016/0022-0396(81)90045-0
[19] Griewank, A., Reddien, G. W.: Characterization and computation of generalized turning points. SIAM J. Numer. Anal. 21, (No. 1) 176-185 (1984) · Zbl 0536.65031 · doi:10.1137/0721012
[20] Griewank, A., Reddien, G. W.: The calculation of Hopf points by a direct method. IMA J. Numer. Anal. 3, 295-303 (1983) · Zbl 0521.65070 · doi:10.1093/imanum/3.3.295
[21] Hassard, B. D., Kazarinoff, N. D., Wan, Y.-H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge, 1981 · Zbl 0474.34002
[22] Poore, A. B.: On the theory and application of the Hopf-Friedrichs bifurcation theory. Arch. Rat. Mech. Anal. 60, 371-393 (1976) · Zbl 0358.34005 · doi:10.1007/BF00248886
[23] Rheinboldt, W. C.: Numerical methods for a class of finite dimensional bifurcation problems. SIAM J. Numer. Anal. 15, 1-11 (1978) · Zbl 0389.65024 · doi:10.1137/0715001
[24] Rheinboldt, W. C.: Computation of critical boundaries on equilibrium manifolds. SIAM J. Numer. Anal. 19, (No. 3) 653-669 (1982) · Zbl 0489.65033 · doi:10.1137/0719046
[25] Rinzel, J., Miller, R. N.: Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations. Math. Biosc. 49, 27-59 (1980) · Zbl 0429.92014 · doi:10.1016/0025-5564(80)90109-1
[26] Russell, R. D., Christiansen, J.: Adaptive mesh selection strategies for solving boundary value problems. SIAM J. Numer. Anal. 15, 59-80 (1978) · Zbl 0384.65038 · doi:10.1137/0715004
[27] Saupe, D.: Beschleunigte PL-Kontinuitätsmethoden und periodische Lösungen parametrisierter Differentialgleichungen mit Zeitverzögerung. Dissertation, Universität Bremen, 1982 · Zbl 0506.65032
[28] Szeto, R.: The flow between rotating coaxial disks. Ph.D. Thesis, California Institute of Technology, Pasadena, Ca., 1977
[29] Weber, H.: Numerical solution of Hopf bifurcation problems. Math. Meth. in the Appl. Sc. 2, 178-190 (1980) · Zbl 0448.65055 · doi:10.1002/mma.1670020205
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.