Free Access
Math. Model. Nat. Phenom.
Volume 11, Number 5, 2016
Bifurcations and Pattern Formation in Biological Applications
Page(s) 137 - 157
Published online 07 December 2016
  1. R. M. Anderson, R. M. May. The invasion, persistence and spread of infectious diseases within animal and plant communities. Philosophical Transactions of the Royal Society of London B, 314 (1986), 533–570. [Google Scholar]
  2. S. Bedhomme, P. Agnew, Y. Vital, C. Sidobre, Y. Michalakis. Prevalence-dependent costs of parasite virulence. PLoS Biology, 3 (2005), e262. [CrossRef] [PubMed] [Google Scholar]
  3. G. E. P. Box, M. E. Muller. A note on the generation of random normal deviates. Annals of Mathematical Statistics, 29 (1958), no. 2, 610–611. [Google Scholar]
  4. J.-B. Burie, A. Calonnec, M. Langlais. Modeling of the invasion of a fungal disease over a vineyard. In A. Deutsch, R. B. de la Parra, R. J. de Boer, O. Diekmann, P. Jagers, E. Kisdi, M. Kretzschmar, P. Lansky, H. Metz (Eds.), Mathematical Modeling of Biological Systems, Volume II. Epidemiology, Evolution and Ecology, Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods, Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, 2008, pages 11–21. [Google Scholar]
  5. E. M. Coombs, J. K. Clark, G. L. Piper, A. F. Cofrancesco Jr. (Eds.). Biological control of invasive plants in the United States. Oregon State University Press, Corvallis OR, 2004. [Google Scholar]
  6. A. d’Onofrio (Ed.). Bounded noises in physics, biology, and engineering. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser–Springer, New York, 2013. [CrossRef] [Google Scholar]
  7. J. A. Drake, H. A. Mooney (Eds.). Biological invasions: a global perspective, vol. 27 of SCOPE. Wiley, Chichester, 1989. [Google Scholar]
  8. H. I. Freedman. A model of predator-prey dynamics as modified by the action of a parasite. Mathematical Biosciences, 99 (1990), 143–155. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  9. J. S. Fulda. The logistic equation and population decline. Journal of Theoretical Biology, 91 (1981), 255–259. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  10. L. Q. Gao, H. W. Hethcote. Disease transmission models with density dependent demographics. Journal of Mathematical Biology, 30 (1992), 717–731. [MathSciNet] [PubMed] [Google Scholar]
  11. D. García-Álvarez. A comparison of a few numerical schemes for the integration of stochastic differential equations in the Stratonovich interpretation. arXiv:1102.4401v1 [physics.comp-ph], (2011). [Google Scholar]
  12. J. García-Ojalvo, J. M. Sancho. Noise in spatially extended systems. Institute for Nonlinear Science. Springer, New York, 1999. [CrossRef] [Google Scholar]
  13. J. García-Ojalvo, J. M. Sancho, L. Ramírez-Piscina. Generation of spatiotemporal colored noise. Physical Review E, 46 (1992), no. 8, 4670–4675. [Google Scholar]
  14. K. P. Hadeler, H. I. Freedman. Predator-prey populations with parasitic infection. Journal of Mathematical Biology, 27 (1989), 609–631. [Google Scholar]
  15. J. M. Halley. Ecology, evolution and l/f-noise. Trends in Ecology & Evolution, 11 (1996), 33–37. [CrossRef] [PubMed] [Google Scholar]
  16. J. M. Halley, P. Inchausti. The increasing importance of 1=f-noises as models of ecological variability. Fluctuations and Noise Letters, 4 (2004), no. 2, R1–R26. [CrossRef] [Google Scholar]
  17. K. Harley, I. W. Forno. Biological control of weeds: a handbook for practitioners and students. Inkata Press, Melbourne, 1992. [Google Scholar]
  18. R. Hengeveld (Ed.). Dynamics of biological invasions. Chapman and Hall, London, 1989. [Google Scholar]
  19. F. M. Hilker, H. Malchow. Strange periodic attractors in a prey-predator system with infected prey. Mathematical Population Studies, 13 (2006), no. 3, 119–134. [Google Scholar]
  20. W. H. Hundsdorfer, J. G. Verwer. Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems. Mathematics of Computation, 53 (1989), no. 187, 81–101. [Google Scholar]
  21. K. Itó. On stochastic differential equations. Memoirs of the American Mathematical Society, 4 (1951), 1–51. [MathSciNet] [Google Scholar]
  22. R. Jarrow, P. Protter. A short history of stochastic integration and mathematical finance: the early years, 1880–1970. In Anirban DasGupta (Ed.), A Festschrift for Herman Rubin, vol. 45 of Lecture Notes – Monograph Series. Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2004, pages 75–91. [CrossRef] [Google Scholar]
  23. M. Julien, G. White (Eds.). Biological control of weeds: theory and practical application. No. 49 in ACIAR Monograph Series. Australian Centre for International Agricultural Research, Bruce ACT, 1997. [Google Scholar]
  24. N. Keiding. Extinction and exponential growth in random environments. Theoretical Population Biology, 8 (1975), no. 1, 49 – 63. [CrossRef] [PubMed] [Google Scholar]
  25. P. E. Kloeden, E. Platen. Numerical solution of stochastic differential equations, vol. 23 of Applications of Mathematics. Springer, Berlin, 1999. [Google Scholar]
  26. E. Kuno. Some strange properties of the logistic equation defined with r and K: Inherent effects or artefacts? Researches on Population Ecology, 33 (1991), 33–39. [Google Scholar]
  27. T. G. Kurtz. Diffusion approximations for branching processes. In K. B. Arthreya, P. E. Ney (Eds.), Branching processes, vol. 5 of Conférence Saint Hippolyte, Québec, 1976. New York, 1978, pages 269–292. [Google Scholar]
  28. V. Lehmann. Invasion, Konkurrenz und Kontrolle einer fremden Art. Diplomarbeit, Institut für Umweltsystemforschung, Fachbereich Mathematik/Informatik, Universität Osnabrück (2011). [Google Scholar]
  29. H. Malchow, A. James, R. Brown. Competitive and diffusive invasion in a noisy environment. Mathematical Medicine and Biology, 28 (2011), 153–163. [CrossRef] [MathSciNet] [Google Scholar]
  30. H. Malchow, A. James, R. Brown. Control of competitive bioinvasion. In M. E. Lewis, P. K. Maini, S. V. Petrovskii (Eds.), Dispersal, individual movement and spatial ecology: A mathematical perspective, vol. 2071 of Lecture Notes in Mathematics. Springer, Berlin, 2013, pages 293–305. [Google Scholar]
  31. H. Malchow, L. Schimansky-Geier. Noise and diffusion in bistable nonequilibrium systems, vol. 5 of Teubner-Texte zur Physik. Teubner-Verlag, Leipzig, 1985. [Google Scholar]
  32. J. Mallet. The struggle for existence: how the notion of carrying capacity, K, obscures the links between demography, Darwinian evolution, and speciation. Evolutionary Ecology Research, 14 (2012), 627–665. [Google Scholar]
  33. M. Matsumoto, T. Nishimura. Mersenne Twister: a 623-dimensionally equidistributed uniform pseudorandom number generator. ACM Transactions on Modeling and Computer Simulation, 8 (1998), no. 1, 3–30. [CrossRef] [Google Scholar]
  34. R. M. May. Stability and complexity in model ecosystems, vol. 6 of Monographs in Population Biology. Princeton University Press, Princeton, 1973. [Google Scholar]
  35. R. M. May. Stability in randomly uctuating versus deterministic environments. The American Naturalist, 107 (1973), no. 957, 621–650. [Google Scholar]
  36. H. McCallum, N. Barlow, J. Hone. How should pathogen transmission be modelled? Trends in Ecology & Evolution, 16 (2001), no. 6, 295–300. [CrossRef] [PubMed] [Google Scholar]
  37. P. B. McEvoy, E. M. Coombs. Biological control of plant invaders: Regional patterns, field experiments, and structured population models. Ecological Applications, 9 (1999), no. 2, 387–401. [CrossRef] [Google Scholar]
  38. P.-A. Meyer. Stochastic processes from 1950 to the present. Electronic Journal for History of Probability and Statistics, 5 (2009), no. 1, 1–42. [Google Scholar]
  39. G. N. Milstein. Chislennoe integrirovanie stokhasticheskikh differentsial’nykh uravnenii. Izdatel’stvo Ural’skogo Universiteta, Sverdlovsk, 1988. [Google Scholar]
  40. G. N. Milstein. Numerical integration of stochastic differential equations, vol. 313 of Mathematics and Its Applications. Kluwer Academic Publishers, Dordrecht, 1995. [Google Scholar]
  41. D. Mollison. Modelling biological invasions: chance, explanation, prediction. Philosophical Transactions of the Royal Society of London B, 314 (1986), 675–693. [CrossRef] [Google Scholar]
  42. C. Mueller. Some tools and results for parabolic stochastic partial differential equations. In D. Khoshnevisan, F. Rassoul-Agha (Eds.), A minicourse on stochastic partial differential equations, vol. 1962 of Lecture Notes in Mathematics, chap. 4. Springer, Berlin, Heidelberg, 2009, pages 111–144. [Google Scholar]
  43. A. Nitzan, P. Ortoleva, J. Ross. Nucleation in systems with multiple stationary states. Faraday Symposia of the Chemical Society, 9 (1974), 241–253. [CrossRef] [Google Scholar]
  44. D. W. Peaceman, H. H. Rachford Jr. The numerical solution of parabolic and elliptic differential equations. Journal of the Society for Industrial and Applied Mathematics, 3 (1955), 28–41. [CrossRef] [MathSciNet] [Google Scholar]
  45. D. Pimentel (Ed.). Biological invasions. Economic and environmental costs of alien plant, animal, and microbe species. CRC Press, Boca Raton, 2002. [Google Scholar]
  46. J. Ripa, P. Lundberg, V. Kaitala. A general theory of environmental noise in ecological food webs. American Naturalist, 151 (1998), 256–263. [CrossRef] [Google Scholar]
  47. L. Ruokolainen, A. Lindén, V. Kaitala, M. S. Fowler. Ecological and evolutionary dynamics under coloured environmental variation. Trends in Ecology and Evolution, 24 (2009), no. 10, 555–563. [Google Scholar]
  48. D. F. Sax, J. J. Stachowicz, S. D. Gaines (Eds.). Species invasions. Insights into ecology, evolution, and biogeography. Sinauer, Sunderland, 2005. [Google Scholar]
  49. T. Schaffter. Numerical integration of SDEs: a short tutorial (2010). Laboratory of Intelligent Systems, Swiss Federal Institute of Technology at Lausanne (EPFL), Switzerland. [Google Scholar]
  50. N. Shigesada, K. Kawasaki. Biological invasions: Theory and practice. Oxford University Press, Oxford, 1997. [Google Scholar]
  51. M. Sieber, H. Malchow, F. M. Hilker. Disease-induced modification of prey competition in eco-epidemiological models. Ecological Complexity, 18 (2014), 74–82. [CrossRef] [Google Scholar]
  52. I. Siekmann. Mathematical modelling of pathogen-prey-predator interactions. Verlag Dr. Hut, München, 2009. [Google Scholar]
  53. I. Siekmann, H. Malchow. Local collapses in the Truscott-Brindley model. Mathematical Modelling of Natural Phenomena, 3 (2008), no. 4, 114–130. [Google Scholar]
  54. W. L. Smith. Necessary conditions for almost sure extinction of a branching process with random environment. Annals of Mathematical Statistics, 39 (1968), no. 6, 2136–2140. [CrossRef] [MathSciNet] [Google Scholar]
  55. W. L. Smith, W. E. Wilkinson. On branching processes in random environments. Annals of Mathematical Statistics, 40 (1969), no. 3, 814–827. [CrossRef] [MathSciNet] [Google Scholar]
  56. J. H. Steele. A comparison of terrestrial and marine ecological systems. Nature, 313 (1985), 355–358. [Google Scholar]
  57. R. L. Stratonovich. Topics in the theory of random noise, vol. 3 (1&2) of Mathematics and Its Applications. Gordon and Breach, New York, 1963&1967. [Google Scholar]
  58. J. W. Thomas. Numerical partial differential equations: Finite difference methods, vol. 22 of Texts in Applied Mathematics. Springer, New York, 1995. [CrossRef] [Google Scholar]
  59. P. van den Driessche, M. L. Zeeman. Disease induced oscillations between two competing species. SIAM Journal of Applied Dynamical Systems, 3 (2004), no. 4, 601–619. [CrossRef] [Google Scholar]
  60. J. E. van der Plank. Host-pathogen interactions in plant disease. Academic Press, New York, 1982. [Google Scholar]
  61. D. A. Vasseur, P. Yodzis. The color of environmental noise. Ecology, 85 (2004), no. 4, 1146–1152. [Google Scholar]
  62. E. Venturino. The influence of diseases on Lotka-Volterra systems. IMA Preprint Series 913, Institute of Mathematics and its Applications, University of Minnesota, Minneapolis (1992). [Google Scholar]
  63. E. Venturino. The influence of diseases on Lotka-Volterra systems. Rocky Mountain Journal of Mathematics, 24 (1994), 381–402. [Google Scholar]
  64. E. Venturino. The effect of diseases on competing species. Mathematical Biosciences, 174 (2001), 111–131. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  65. J. B. Walsh. An introduction to stochastic partial differential equations. In R. Carmona, H. Kesten, J. B. Walsh (Eds.), École d’été de probabilités de Saint-Flour XIV - 1984, vol. 1180 of Lecture Notes in Mathematics. Springer, Berlin, 1986, pages 265–437. [Google Scholar]
  66. M. Williamson. Biological invasions, vol. 15 of Population and Community Biology Series. Chapman & Hall, London, 1996. [Google Scholar]
  67. T. Woyzichovski. Der Einuss von Rauschen und Infektion in einem Konkurrenzmodell fremder und indigener Spezies. Diplomarbeit, Institut für Umweltsystemforschung, Fachbereich Mathematik/Informatik, Universität Osnabrück (2013). [Google Scholar]

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