head> Research Interests of Gail S. K. Wolkowicz

Research Interests of Gail S. K. Wolkowicz

keywords: differential equations, dynamical systems, bifurcation theory, mathematical biology, population dynamics, mathematical ecology and epidemiology, mathematical models of the chemostat.

My students and I have been formulating and analyzing models motivated by questions in ecology and epidemiology. One goal is to better understand basic population dynamics so that measurable criteria can be developed, enabling scientists to predict combinations of cultures of microorganisms, most effective and safest for use in such processes as water purification and biological waste decomposition. Other applications include pest control, prevention of species' extinction, and control or eradication of certain diseases. In order to elicit all the potential dynamics, a bifurcation theory approach is used so that the full spectrum of behaviour can be predicted for all appropriate parameter ranges and initial states. Computer simulations are used to elucidate complicated dynamics, to test conjectures and to reveal properties of the models that are useful in developing analytic proofs. Symbolic computation is used to carry out complicated calculations. The analyses often lead to interesting abstract mathematical problems in dynamical systems, including difference equations, ordinary, integro- and functional differential equations, and bifurcation theory.

  1. G.J. Butler and G.S.K. Wolkowicz, (1985) "A Mathematical Model of the Chemostat with a General Class of Functions Describing Nutrient Uptake," SIAM Journal on Applied Mathematics 45, 137-151. | postscript file| or | pdf file|
  2. G.J. Butler and G.S.K. Wolkowicz, (1986) "Predator-Mediated Competition in the Chemostat," Journal of Mathematical Biology 24, 167-191. | pdf file|
  3. H.I. Freedman and G.S.K. Wolkowicz, (1986) "Predator-Prey Systems with Group Defense: The Paradox of Enrichment Revisited," Bulletin of Mathematical Biology 48, 493-508. | pdf file|
  4. G.J. Butler and G.S.K. Wolkowicz, (1987) "Exploitative Competition in a Chemostat for Two Complementary and Possibly Inhibitory Resources," Mathematical Biosciences 83, 1-48. | pdf file|
  5. G.J. Butler and G.S.K. Wolkowicz, (1987) "Predator-Mediated Coexistence in a Chemostat: Coexistence and Competition Reversal," Mathematical Modelling in Science and Technology 8, 781-785. | pdf file|
  6. G.S.K. Wolkowicz, (1988) "Bifurcation Analysis of a Predator-Prey System Involving Group Defence," SIAM Journal on Applied Mathematics 48, 592-606. | pdf file |
  7. K. Mischaikow and G. S. K. Wolkowicz, (1988) A connection matrix approach illustrated by means of a predator-prey model involving group defense, Mathematical Ecology: Proc. of the Research Conf., Trieste, 1986 (ed. T. G. Hallam, L. J. Gross, and S. A. Levin), World Scienti c Publishing, 682-716. |pdf file|
  8. G.S.K. Wolkowicz, (1989) "Successful Invasion of a Food Web in a Chemostat," Mathematical Biosciences 93, 249-268. |pdf file|
  9. K. Mischaikow and G.S.K. Wolkowicz, (1990) "A Predator-Prey System Involving Group Defense: A Connection Matrix Approach," Nonlinear Analysis, Theory, Methods and its Applications 14, 955-969. |pdf file|
  10. G.S.K. Wolkowicz, (1990) "Invasion of a Uniformly Persistent System," Rocky Mountain Journal of Mathematics 20, 1-18. |pdf file|
  11. Gail S. K. Wolkowicz and Lu Zhiqi, (1992) "Global Dynamics of a Mathematical Model of Competition in the Chemostat: General Response Functions and Differential Death Rates," SIAM Journal on Applied Mathematics 52, 222--233. |pdf file|
  12. Betty Tang and Gail S.K. Wolkowicz, (1992) "Mathematical Models of Microbial Growth and Competition in the Chemostat Regulated by Cell-Bound Extracellular Enzymes," Journal of Mathematical Biology 31, 1-23. | pdf file|
  13. Mary M. Ballyk and Gail S. K. Wolkowicz, (1993) "Exploitative Competition in the Chemostat for Two Perfectly Substitutable Resources," Mathematical Biosciences 118, 127-180. |pdf file|
  14. S.-B. Hsu, Paul Waltman, and Gail S. K. Wolkowicz, (1994) "Global Analysis of a Model of Plasmid-Bearing, Plasmid-Free Competition in a Chemostat," Journal of Mathematical Biology 32, 731--742. | pdf file|
  15. Gail S. K. Wolkowicz, Mary M. Ballyk, and Spiro P. Daoussis, (1995) "Interaction in a Chemostat: Invasion by a Competitor can Promote Greater Diversity," Rocky Mountain Journal of Mathematics, 25, 515--543.|pdf file|
  16. Mary M. Ballyk and Gail S. K. Wolkowicz, (1995) "An Examination of the Thresholds of Enrichment: A Resource--Based Growth Model," Journal of Mathematical Biology, 33, 435--457. | pdf file|
  17. Shigui Ruan and Gail S. K. Wolkowicz, (1995) "Persistence in Plankton Models with Delayed Nutrient Recycling," Canadian Applied Mathematics Quarterly, 3, 219--235. | pdf file|
  18. Gail S. K. Wolkowicz, Mary M. Ballyk, and Zhiqi Lu, (1996) "Microbial Dynamics in a Chemostat: Competition, Growth, Implications of Enrichment," in Differential Equations and Control Theory, Lecture Notes in Pure and Applied Mathematics Vol. 176, Z. Deng, Z. Liang, G. Lu, and S. Ruan eds., Marcel Dekker, New York, 389--406. |pdf file|
  19. Shigui Ruan and Gail S. K. Wolkowicz, (1996) "Bifurcation Analysis of a Chemostat Model with a Distributed Delay," Journal of Mathematical Analysis and Applications, 204, 786--812. | pdf file|
  20. Gail S. K. Wolkowicz and Huaxing Xia, (1997) "Global Asymptotic Behavior of a Chemostat Model with Discrete Delays," SIAM Journal on Applied Mathematics, 57, 1019--1043. |pdf file| or |psfile|
  21. Gail S. K. Wolkowicz, Huaxing Xia, and Shigui Ruan, (1997) "Competition in the Chemostat: A Distributed Delay Model and its Global Asymptotic Behavior," SIAM Journal on Applied Mathematics, 57, 1281--1310. |pdf file| or | psfile |
  22. Gail S. K. Wolkowicz and Xiao-Qiang Zhao, (1998) "n-Species Competition in a Periodic Chemostat," Differential and Integral Equations: An International Journal for Theory and Applications, 11, 465--491. |pdf file|
  23. Gail S. K. Wolkowicz and Lu Zhiqi, (1998) "Direct Interference on Competition in a Chemostat," Journal of Biomathematics, 13, 282--291. |pdf file|
  24. Gail S. K. Wolkowicz, Huaxing Xia, and Jianhong Wu, (1999) "Global Dynamics of a Chemostat Competition Model with Distributed Delay," Journal of Mathematical Biology, 38, 285-316. |psfile| |pdf file|
  25. Shigui Ruan, Gail S. K. Wolkowicz, and Jianhong Wu, editors, (1999) Differential Equations with Applications to Biology, Fields Institute Communications, Vol. 21, 509 pages, American Mathematical Society.
  26. Bingtuan Li, Gail S. K. Wolkowicz, and Yang Kuang, (2000) "Global Asymptotic Behavior of a Chemostat Model with Two Perfectly Complementary Resources and Distributed Delay," SIAM J. Appl. Math, , 60, 2058-2086. |pdf file|
  27. Wu Jianhua and Gail S. K. Wolkowicz, (2001) "A system of resource-based growth models with two resources in the unstirred chemostat," Journal of Differential Equations 172, 300--332. |pdf file|
  28. Huaiping Zhu, Sue Ann Campbell, and Gail S. K. Wolkowicz, (2002) "Bifurcation Analysis of a Predator-Prey System with Nonmonotonic Functional Response," SIAM J. Appl. Math. 63, 636-682. |psfile| pdf file|
  29. Shigui Ruan, Gail S. K. Wolkowicz, and Jianhong Wu, (2003) editors, Dynamical Systems and Their Applications in Biology, Fields Institute Communications (FIC) series, Vol. 36, American Mathematical Society, 268 pages, 2003.
  30. Robert Smith and Gail S. K. Wolkowicz, (2003) "Growth and Competition in the Nutrient Driven Self-Cycling Fermentation Process," Canadian Applied Mathematics Quarterly, Vol 10, No. 1, 171-177. |pdf format|
  31. R.J. Smith and G.S.K. Wolkowicz, (2003) "A Size-Structured Model for the Nutrient-Driven Self-Cycling Fermentation Process," Dynamics of Discrete, Continuous, and Impulsive Systems, Series B: Applications and Algorithms, Vol 10, 207-220. | psfile | pdf file|
  32. Julien Arino, Sergei Pilyugin, and Gail S. K. Wolkowicz, (2003) "Considerations on yield, nutrient uptake, cellular growth, and competition in chemostat models," Canadian Applied Mathematics Quarterly, Vol 11, 107-142. |pdf file|
  33. R.J. Smith and G.S.K. Wolkowicz, (2004) "Analysis of a Model of the Nutrient Driven Self-cycling Fermentation Process," Dynamics of Continuous, Discrete and Impulsive Systems, Series B: Applications and Algorithms 11, 239-265. | pdf file |
  34. Jianhua Wu, Hua Nie, and Gail S. K. Wolkowicz, (2004) "A mathematical model of competition for two essential resources in the unstirred chemostat," SIAM J. Appl. Math., 65, 209-229. |pdf file|
  35. Huaxing Xia, Gail S.K. Wolkowicz, and Lin Wang, (2005) "Transient Oscillation Induced by Delayed Growth Response in the Chemostat," Journal of Mathematical Biology, 50, 489-530. |pdf file|
  36. Mary M. Ballyk, C. Connell McCluskey, and Gail S. K. Wolkowicz, (2005) "Global Analysis of Competition for Perfectly Substituable Resources with Linear Response," Journal of Mathematical Biology 51, 458-490. | pdf file|
  37. Lin Wang and Gail S. K. Wolkowicz, (2006) "A delayed chemostat model with general nonmonotone response functions and differential removal rates," Journal of Mathematical Analysis and Applications, 321, 452-468.| pdf file|
  38. Julien Arino, Lin Wang, and Gail S. K. Wolkowicz, (2006) "An alternative formulation for a delayed logistic equation," Journal of Theoretical Biology, Vol. 241, Issue 1, 109-119. | pdf file|
  39. Gail S. K. Wolkowicz, (2006) "Interpretation of the generalized asymmetric May-Leonard model of three species competition as a food web in a chemostat," Fields Institute Communications, 48, 279-289. | pdf file|
  40. Guihong Fan and Gail S. K. Wolkowicz, (2007) "Analysis of a model of nutrient driven self-cycling fermentation allowing unimodal response functions," DCDS-B Vol. 8, No. 4, 801-831. | pdf file|
  41. Jianhua Wu, Hua Nie, and Gail S. K. Wolkowicz, (2007) "The Effect of Inhibitor on the Plasmid-Bearing and Plasmid-Free Model in the Unstirred Chemostat," SIAM J. Math. Anal. Volume 38, Issue 6, pp. 1860-1885. | pdf file|
  42. Guihong Fan, Maung Min-oo, and Gail S. K. Wolkowicz, (2009) "Hopf bifurcation of delay differential equations with delay dependent parameters," Canadian Applied Mathematics Quarterly, Vo1 17, No. 1, 37-60. | pdf file|
  43. Guihong Fan and Gail S. K. Wolkowicz, (2010) "A predator-prey model in the chemostat with time delay," International Journal of Differential Equations, Article ID 287969, 41 pages doi:10.1155/2010/287969, | html link |
  44. Mary M. Ballyk and Gail S. K. Wolkowicz, (2011) "Classical and resource-based competition: A unifying graphical approach," Journal of Mathematical Biology, 62:81-109, DOI 10.1007/s00285-010-0328-x. |downloadable author's version .pdf| Published on-line first at | on-line first| "The final publication is available at www.springerlink.com".
  45. D. L. DeAngelis, G. S. K. Wolkowicz, Y. Lou, Y. Jiang, M. Novak, R. Svanback, M. Araujo, Y. S. Jo, and E. Cleary, (2011) ``The effect of travel loss on evolutionarily stable distributions of populations in space,'' The American Naturalist, Vol. 178, No. 1, 15--29, http://www.jstor.org/stable/10.1086/660280 (on-line Appendix A (7 pages) and Appendix B (2 pages)). http://www.jstor.org/stable/10.1086/660280 on-line Appendix A (7 pages) and Appendix B (2 pages)). DOI: 10.1086/660280 |pdf file|
  46. K. Northcott, M. Imran, G. S. K. Wolkowicz, (2012) "Competition in the presence of a virus in an aquatic system," Journal of Mathematical Biology, 64, 1043-1086, DOI: 10.1007/s00285-011-2439-z |downloadable author's version .pdf|
  47. Marion Weedermann, Gunog Seo, Gail S.K. Wolkowicz, (2013) "Mathematical model of anaerobic digestion in a chemostat: effects of syntrophy and inhibition," Journal of Biological Dynamics, 7:1, 59-85 To link to this Open Access article
  48. Guihong Fan, Sue Ann Campbell, G. S. K. Wolkowicz, Huaiping Zhu (2013) "The bifurcation study of 1:2 resonance in a delayed system of two coupled neurons," JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS Volume: 25 Issue: 1 Pages: 193-216 DOI: 10.1007/s10884-012-9279-9
  49. Marion Weedermann, Gail S. K. Wolkowicz, and Joanna Sasara, (2015) "Optimal biogas production in a model for anaerobic digestion," Nonlinear Dynamics, Volume 81, Issue 3, Pages 1097-1112, DOI 10.1007/s11071-015-2051-z
  50. Gunog Seo and Gail S. K. Wolkowicz, (2015) "Existence of multiple limit cycles in a predator-prey model with arctan(ax) as functional response," Communications in Mathematical Analysis Volume 18, Issue 1, Pages 64-68, ISSN 1938-9787.
  51. Gail S. K. Wolkowicz and Yuan Yuan, (2016) "Effect of light on the growth of non-nitrogen-fixing and nitrogen-fixing phytoplankton in an aquatic system," J. Math. Biol. 72:1663-1692. DOI 10.1007/s00285-015-0924-x
  52. F. Barraquand et al. (2017) "Moving forward in circles: challenges and opportunities in modeling population cycles," Ecology Letters DOI: 10.1111/ele.12789
  53. Liang Wang, Daqing Jiang, Gail S. K. Wolkowicz and Donal O'Regan, (2017) "Dynamics of the stochastic chemostat with Monod-Haldane response function," Scientific Reports, | 7: 13641 | DOI:10.1038/s41598-017-13294-3.
  54. Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz and Huaiping Zhu, (2018) "HOW SEASONAL FORCING INFLUENCES THE COMPLEXITY OF A PREDATOR-PREY SYSTEM," Discrete & Continuous Dynamical Systems - B, 23(2): 785-807. doi: 10.3934/dcdsb.2018043
  55. Gunog Seo and Gail S. K. Wolkowicz, (2018) "Sensitivity of the dynamics of the general Rosenzweig–MacArthur model to the mathematical form of the functional response: a bifurcation theory approach," Journal of Mathematical Biology, 76:1873-1906. doi.org/10.1007/s00285-017-1201-y, ( https://doi.org/10.1007/s00285-017-1201-y and link to openly accessible read only version: http://rdcu.be/Eef3)
  56. Chiu-Ju Lin, Lin Wang, Gail S. K. Wolkowicz, (2018) “An alternative formulation of a distributed delayed logistic equation,” Bulletin of Mathematical Biology, 80(7):1713-1735. doi.org/10.1007/s11538-018-0432-4
  57. Ting-Hao Hsu, Tyler Meadows, Lin Wang, Gail S. K. Wolkowicz (2019) “Growth on two limiting essential resources in a self-cycling fermentor,” Mathematical Biosciences and Engineering. 16(1):78-100. DOI: 10.3934/mbe.2019005i, (https://www.aimspress.com/fileOther/PDF/MBE/mbe-16-01-004.pdf)
  58. Liang Wang, Daqing Jiang, and Gail S. K. Wolkowicz, (2019) "Global asymptotic behavior of a multi-species stochastic chemostat model with discrete delays," Journal of Dynamics and Differential Equations, On-line first https://doi.org/10.1007/s10884-019-09741-6.
  59. Tyler Meadows, Marion Weederman, and Gail S. K. Wolkowicz, (2019) ``Global analysis of a simplified model of anaerobic digestion and a new result for the chemostat,'' SIAM Journal on Applied Mathematics, 79:2, 668--689. (http://arxiv.org/abs/1903.01257 )
  60. Ting-Hao Hsu and Gail S. K. Wolkowicz, (2020) ``A criterion for the existence of relaxation oscillations with applications to predator-prey systems and an epidemic model,'' Discrete and Continuous Dynamical Systems, Series-B, 25(4):1257-1277 doi: 10.3934/dcdsb.2019219. (https://arxiv.org/abs/1811.08307)
  61. Gunog Seo and Gail S. K. Wolkowicz, (2020) ``Pest control by generalist parasitoids: a bifurcation theory approach,'' Discrete and Continuous Dynamical Systems, Series-S, 13(11):3157--3187 doi: 10.3934/dcdss.2020163.
  62. Tyler Meadows and Gail S. K. Wolkowicz, (2020) “Growth on Multiple Interactive-Essential Resources in a Self-Cycling Fermentor: ``An Impulsive Differential Equations Approach,’’ Nonlinear Analysis: Real World Applications 56 103157. (https://doi.org/10.1016/j.nonrwa.2020.103157 ) (17 pages final form online) (https://arxiv.org/abs/2004.12242)
  63. Tedra Bolger, Brydon Eastman, Madeleine Hill, and Gail S. K. Wolkowicz, (2020) ``A Predator-Prey Model in the Chemostat with Holling Type II Response Functions,’’ Mathematics in Applied Sciences and Engineering, Online first, pp. 1-22, (https://ojs.lib.uwo.ca/mase)
  64. Szymon Sobieszek, Matthew J. Wade, and Gail S. K. Wolkowicz, (2020) ``Rich Dynamics of a Three-Tiered Anaerobic Food-Web in a Chemostat with Multiple Substrate Inflow,’’ Mathematical Biosciences and Engineering, 17(6): 7045-7073.
  65. Matthew J. Wade and Gail S. K. Wolkowicz, (2021) ``Bifurcation Analysis of a Impulsive System Describing Nitritation and Anammox in a Hybrid Reactor,’’ Environmental Science & Technology Article ASAP ( https://doi.org/10.1021/acs.est.0c06275 ), American Chemical Society (11 pages in final form online).
  66. Guihong Fan and Gail S. K. Wolkowicz, (2021) ``Chaotic Dynamics in a Simple Predator-Prey Model with Discrete Delay,'' Discrete and Continuous Dynamical Systems, Series-B 26(1): 191-216. (doi: 10.3934/dcdsb.2020263 ) ( https://arxiv.org/abs/2007.16140 )
  67. Hayriye Gulbudak, Paul L. Salceanu and Gail S. K. Wolkowicz, (2021) ``A Delay Model for Persistent Viral Infection in Replicating Cells,'' Journal of Mathematical Biology, 82(7) 1-52 https://rdcu.be/ckGLZ or https://doi.org/10.1007/s00285-021-01612-3
  68. Sabrina H. Streipert and Gail S. K. Wolkowicz, (2021) ``An Alternative Delayed Population Growth Difference Equation Model,’’ Journal of Mathematical Biology, 83:25 https://doi.org/10.1007/s00285-021-01652-9
  69. Chiu-Ju Lin, Ting-Hao Hsu, and Gail S. K. Wolkowicz (2022) ``Population Growth and Competition Models with Decay and Competition Consistent Decay,'' Journal of Mathematical Biology 84:39 https://doi.org/10.1007/s00285-022-01741-3
  70. Sabrina H. Streipert, Martin Bohner, and Gail S. K. Wolkowicz, (2022) ``Derivation and Analysis of a Discrete Predator-Prey Model,'' Bulletin of Mathematical Biology 84:67 https://doi.org/10.1007/s11538-022-01016-4
  71. Sabrina H. Streipert and Gail S. K. Wolkowicz, (2023) “ An augmented phase plane approach for discrete planar maps: Introducing next-iterate operators ,“ Mathematical Biosciences, 355, 108924, 19 https://doi.org/10.1016/j.mbs.2022.108924
  72. S. H. Streipert and G. S. K. Wolkowicz. ``A method to derive discrete population models,'' In: Advances in Discrete Dynamical Systems, Difference Equations and Applications, eds: S. Elaydi, M.R.S. Kulanovic, & S. Kalabusic, Springer Proceedings in Mathematics & Statistics 416,473--494, Springer, (2023). https://doi.org/10.1007/978-3-031-25225-9_22
  73. Alexandra Teslya and Gail S. K. Wolkowicz, (2023) ``Dynamics of a predator-prey model with distributed delay to represent the conversion process or maturation,'' Differential Equations and Dynamical Systems, (3)(613-649) doi.org/10.1007/s12591-020-00546-4, ( https://rdcu.be/b6gOy )
  74. S. H. Streipert and G. S. K. Wolkowicz. (2023) ``A technique to derive discrete population models wth delayed growth,'' Journal of Biologial Dynamics, 17:1, 2244987, DOI: 10.1080/17513758.2023.2244987, https://doi.org/10.1080/17513758.2023.2244987
  75. S. H. Streipert and G. S. K. Wolkowicz. (2024) ``Derivation and dynamics of discrete population models with distributed delay in reproduction,'' Mathematical Biosciences, 376, https://doi.org/10.1016/j.mbs.2024.109279
  76. S. R. Smith?, T. Meadows, G. S. K. Wolkowicz, (2024) ``Competition in the nutrient-driven self-cyclcing fermentation process,'' Nonlinear Analysis: Hybrid Systems, 54: 101519 https://doi.org/10.1016/j.nahs.2024.101519


    Gail Wolkowicz