On the Gause predator-prey model with a refuge: A fresh look at the history

Journal of Theoretical Biology

Volumn 274, Issue 1, 2011, Pages 67-73

  Křivan, Vlastimil ^a,b  

^a INSTITUTE OF ENTOMOLOGY   (Czech Republic)

^b University of Tennessee and National Institute for Mathematical and Biological Synthesis   (United States)

Author keywords

Discontinuous differential equations; Filippov solution; Holling functional response; Limit cycle; Lotka Volterra model

Indexed keywords

EQUATION; LOTKA-VOLTERRA MODEL; NUMERICAL MODEL; POPULATION DENSITY; POPULATION ECOLOGY; PREDATOR-PREY INTERACTION;

ARTICLE; ECOLOGY; POPULATION DENSITY; POPULATION DYNAMICS; PREDATION; PREDATOR PREY INTERACTION; PRIORITY JOURNAL; STATISTICAL MODEL;

ANIMALS; MODELS, BIOLOGICAL; PREDATORY BEHAVIOR; TIME FACTORS;

EID: 79251592857     PISSN: 00225193     EISSN: 10958541     Source Type: Journal    
DOI: 10.1016/j.jtbi.2011.01.016     Document Type: Article

References (39)

1. Aubin J.-P., Cellina A. Differential Inclusions 1984, Springer-Verlag, Berlin, Germany.
2. Casey G., de Jong H., Gouzé J.-L. Piecewise-linear models of genetic regulatory networks: equilibria and their stability. Journal of Mathematical Biology 2006, 52:27-56.
3. Colombo R., Křivan V. Selective strategies in food webs. IMA Journal of Mathematics Applied in Medicine and Biology 1993, 10:281-291.
4. Crowley P.H. Dispersal and the stability of predator-prey interactions. The American Naturalist 1981, 118:673-701.
5. de Jong H., Gouz'e J.-L., Hernandez C., Page M., Sari T., Geiselmann J. Qualitative simulation of genetic regulatory networks using piecewise-linear models. Bulletin of Mathematical Biology 2004, 66:301-340.
6. Dercole F., Gragnani A., Rinaldi S. Bifurcation analysis of piecewise smooth ecological models. Theoretical Population Biology 2007, 72(2):197-213.
7. Dercole, F., Kuznetsov, Y.A., 2004. User guide to SlideCont 2.0. Department of Mathematics, Utrecht University, The Netherlands http://www.math.uu.nl/people/kuznet/cm/slidecont.pdf.
8. Dercole F., Kuznetsov Y.A. SlideCont: an Auto97 driver for bifurcation analysis of Filippov systems. ACM Transactions on Mathematical Software 2005, 31:95-119.
9. Dulac H. Recherche des cycles limites. Comptes Rendus Hebdomadaires des Séances de L'Académie des Sciences 1937, 204:1703-1706.
10. Edwards R. Analysis of continuous-time switching networks. Physica D 2000, 146:165-199.
11. Filippov A.F. Differential equations with discontinuous right-hand side. Matematicheskii sbornik 1960, 51:99-128. (in Russian. English translation published in American Mathematical Society Translations, Series 2, 1964, pp. 199-231).
12. Filippov A.F. Differential Equations with Discontinuous Righthand Sides 1988, Kluwer Academic Publishers, Dordrecht.
13. Gause G.F. The Struggle for Existence 1934, Williams and Wilkins, Baltimore.
14. Gause G.F., Smaragdova N.P., Witt A.A. Further studies of interaction between predators and prey. The Journal of Animal Ecology 1936, 5:1-18.
15. Gouzé J.L., Sari T. A class of piecewise linear differential equations arising in biological models. Dynamical Systems-an International Journal 2003, 17:299-316.
16. Hassell M.P. The Dynamics of Arthropod Predator-prey Systems 1978, Princeton University Press, Princeton, NJ, USA.
17. Hassell M.P., May R.M. Stability in insect host-parasite models. Journal of Animal Ecology 1973, 42:693-736.
18. Hochberg M.E., Holt R. Refuge evolution and the population dynamics of coupled host-parasitoid associations. Evolutionary Ecology 1995, 9:633-661.
19. Hofbauer J., Sigmund K. Evolutionary Games and Population Dynamics 1998, Cambridge University Press, Cambridge.
20. Holling C.S. Some characteristics of simple types of predation and parasitism. The Canadian Entomologist 1959, 91:385-398.
21. Huang X.C., Merrill S.J. Conditions for uniqueness of limit cycles in general predator-prey systems. Mathematical Biosciences 1989, 96:47-60.
22. Ives A.R., Dobson A.P. Antipredator behaviour and the population dynamics of simple predator-prey systems. The American Naturalist 1987, 130:431-447.
23. Kuznetsov Y.A., Rinaldi S., Gragnani A. One-parameter bifurcations in planar Filippov systems. International Journal of Bifurcation and Chaos 2003, 13:2157-2188.
24. Křivan V. Optimal foraging and predator-prey dynamics. Theoretical Population Biology 1996, 49:265-290.
25. Křivan V. Effects of optimal antipredator behavior of prey on predator-prey dynamics: role of refuges. Theoretical Population Biology 1998, 53:131-142.
26. Křivan V. The Lotka-Volterra predator-prey model with foraging-predation risk trade-offs. American Naturalist 2007, 170:771-782.
27. Křivan V. Prey-predator models. Encyclopedia of Ecology 2008, vol. 4:2929-2940. Elsevier, Oxford.
28. Lotka A.J. Elements of Physical Biology 1926, Williams and Wilkins, Baltimore.
29. Maynard Smith J. Models in Ecology 1974, Cambridge University Press, Cambridge, UK.
30. Meza M.E.M., Bhaya A., Kaszkurewicz E., Costa M.I.S. Threshold policies control for predator-prey systems using a control Liapunov function approach. Theoretical Population Biology 2005, 67:273-284.
31. Murdoch W.W., Oaten A. Predation and population stability. Advances in Ecological Research 1975, 1-131. Academic Press. A. MacFadyen (Ed.).
32. Piiroinen P.T., Kuznetsov Y.A. An event-driven method to simulate Filippov systems with accurate computing of sliding motions. ACM Transactions on Mathematical Software 2008, 34:13:1-13:24.
33. Rosenzweig M.L., MacArthur R.H. Graphical representation and stability conditions of predator-prey interactions. The American Naturalist 1963, 97:209-223.
34. Ruxton G.D. Short term refuge use and stability of predator-prey models. Theoretical Population Biology 1995, 47:1-17.
35. Sih A. Prey refuges and predator-prey stability. Theoretical Population Biology 1987, 31:1-12.
36. Svirezhev Y.M., Logofet D.O. Stability of Biological Communities 1983, Mir Publishers, Moscow, USSR.
37. Thota P., Dankowicz H. Tc-hat (TC): a novel toolbox for the continuation of periodic trajectories in hybrid dynamical systems. SIAM Journal on Applied Dynamical Systems 2008, 7(4):1283-1322.
38. van Baalen M., Křivan V., van Rijn P.C.J., Sabelis M. Alternative food, switching predators, and the persistence of predator-prey systems. American Naturalist 2001, 157:512-524.
39. Volterra V. Fluctuations in the abundance of a species considered mathematically. Nature 1926, 118:558-560.