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, ordinary,
integro and functional differential equations and bifurcation theory.
 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, 137151.

postscript file or 
pdf file

G.J. Butler and G.S.K. Wolkowicz, (1986) "PredatorMediated Competition
in the Chemostat," Journal of Mathematical Biology 24, 167191.
 pdf file
 H.I. Freedman and G.S.K. Wolkowicz, (1986) "PredatorPrey Systems with Group
Defense: The Paradox of Enrichment Revisited,"
Bulletin of Mathematical Biology 48, 493508.
 pdf file

G.J. Butler and G.S.K. Wolkowicz, (1987) "Exploitative Competition in a
Chemostat for Two Complementary and Possibly Inhibitory Resources,"
Mathematical Biosciences 83, 148.  pdf file

G.J. Butler and G.S.K. Wolkowicz, (1987) "PredatorMediated Coexistence in a
Chemostat: Coexistence and Competition Reversal,"
Mathematical Modelling in Science and Technology 8, 781785.
 pdf file

G.S.K. Wolkowicz, (1988) "Bifurcation Analysis of a PredatorPrey System
Involving Group Defence," SIAM Journal on Applied Mathematics 48,
592606.  pdf file 
 K. Mischaikow and G. S. K. Wolkowicz, (1988) A connection matrix approach
illustrated by means of a predatorprey 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 Scientic Publishing,
682716. pdf file

G.S.K. Wolkowicz, (1989) "Successful Invasion of a Food Web in a Chemostat,"
Mathematical Biosciences 93, 249268.
pdf file

K. Mischaikow and G.S.K. Wolkowicz, (1990) "A PredatorPrey System Involving
Group Defense: A Connection Matrix Approach," Nonlinear Analysis, Theory,
Methods and its Applications 14, 955969. pdf file

G.S.K. Wolkowicz, (1990) "Invasion of a Uniformly
Persistent System," Rocky Mountain Journal of Mathematics 20, 118.
pdf file

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, 222233. pdf file

Betty Tang and Gail S.K. Wolkowicz, (1992)
"Mathematical Models of Microbial Growth and
Competition in the Chemostat Regulated by CellBound Extracellular Enzymes,"
Journal of Mathematical Biology 31, 123.
 pdf file

Mary M. Ballyk and Gail S. K. Wolkowicz, (1993)
"Exploitative Competition in the Chemostat for Two Perfectly
Substitutable Resources," Mathematical Biosciences
118, 127180.
pdf file

S.B. Hsu, Paul Waltman, and Gail S. K. Wolkowicz, (1994)
"Global Analysis of a Model of PlasmidBearing, PlasmidFree
Competition in a Chemostat," Journal of Mathematical Biology 32, 731742.  pdf file

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, 515543.pdf file

Mary M. Ballyk and Gail S. K. Wolkowicz, (1995) "An Examination of the
Thresholds of Enrichment: A ResourceBased Growth Model,"
Journal of Mathematical Biology, 33, 435457.
 pdf file

Shigui Ruan and Gail S. K. Wolkowicz, (1995) "Persistence in Plankton
Models with
Delayed Nutrient Recycling," Canadian Applied Mathematics Quarterly,
3, 219235.  pdf file

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, 389406. pdf file
 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, 786812.  pdf file
 Gail S. K. Wolkowicz and Huaxing Xia, (1997) "Global Asymptotic Behavior of a
Chemostat Model with Discrete Delays,"
SIAM Journal on Applied Mathematics, 57, 10191043.
pdf file
or psfile
 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, 12811310.
pdf file or  psfile 
 Gail S. K. Wolkowicz and XiaoQiang Zhao, (1998)
"nSpecies Competition in a Periodic
Chemostat," Differential and Integral Equations:
An International Journal for Theory and Applications,
11, 465491.
pdf file
 Gail S. K. Wolkowicz and Lu Zhiqi, (1998) "Direct Interference on Competition in a Chemostat," Journal of Biomathematics, 13, 282291. pdf file
 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, 285316.
psfile pdf file
 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.
 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, 20582086. pdf file
 Wu Jianhua and Gail S. K. Wolkowicz, (2001) "A
system of resourcebased growth models with two resources
in the unstirred chemostat,"
Journal of
Differential Equations 172, 300332. pdf file

Huaiping Zhu, Sue Ann Campbell, and Gail S. K. Wolkowicz, (2002) "Bifurcation
Analysis of a PredatorPrey System with Nonmonotonic Functional
Response," SIAM J. Appl. Math. 63, 636682. psfile pdf file
 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.
 Robert Smith and Gail S. K. Wolkowicz, (2003) "Growth and Competition
in the Nutrient Driven SelfCycling Fermentation Process,"
Canadian Applied Mathematics Quarterly, Vol 10, No. 1, 171177.
pdf format

R.J. Smith and G.S.K. Wolkowicz, (2003) "A SizeStructured Model for the
NutrientDriven SelfCycling Fermentation Process,"
Dynamics of Discrete, Continuous, and Impulsive Systems, Series B:
Applications and Algorithms,
Vol 10, 207220.  psfile  pdf file
 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, 107142.
pdf file

R.J. Smith and G.S.K. Wolkowicz, (2004) "Analysis of a
Model of the Nutrient Driven Selfcycling Fermentation Process,"
Dynamics of Continuous, Discrete and Impulsive Systems, Series B:
Applications and Algorithms 11, 239265.  pdf file 
 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, 209229. pdf file
 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, 489530. pdf file
 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, 458490.
 pdf file
 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, 452468. pdf file
 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, 109119.  pdf file
 Gail S. K. Wolkowicz, (2006) "Interpretation of the generalized asymmetric MayLeonard
model of three species competition as a food web in a chemostat," Fields Institute Communications, 48, 279289.  pdf file
 Guihong Fan and Gail S. K. Wolkowicz, (2007) "Analysis of a model of nutrient driven selfcycling fermentation allowing unimodal response functions," DCDSB Vol. 8, No. 4, 801831.  pdf file
 Jianhua Wu, Hua Nie, and Gail S. K. Wolkowicz, (2007) "The Effect of Inhibitor on the PlasmidBearing and PlasmidFree Model in the Unstirred Chemostat," SIAM J. Math. Anal. Volume 38, Issue 6, pp. 18601885.  pdf file
 Guihong Fan, Maung Minoo, and Gail S. K. Wolkowicz, (2009) "Hopf bifurcation of delay differential equations with delay dependent parameters," Canadian Applied Mathematics Quarterly, Vo1 17, No. 1, 3760.  pdf file
 Guihong Fan and Gail S. K. Wolkowicz, (2010) "A predatorprey model in the chemostat with time delay," International Journal of Differential Equations, Article ID 287969, 41 pages doi:10.1155/2010/287969,  html link 
 Mary M. Ballyk and Gail S. K. Wolkowicz, (2011) "Classical and resourcebased competition: A unifying graphical approach," Journal of Mathematical Biology, 62:81109, DOI 10.1007/s002850100328x. downloadable author's version .pdf
Published online first at  online first "The final publication is available at www.springerlink.com".
 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, 1529, http://www.jstor.org/stable/10.1086/660280 (online Appendix A (7 pages) and Appendix B (2 pages)). http://www.jstor.org/stable/10.1086/660280 online Appendix A (7 pages) and Appendix B (2 pages)). DOI: 10.1086/660280 pdf file
 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, 10431086, DOI: 10.1007/s002850112439z downloadable author's version .pdf
 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, 5985
To link to this Open Access article
 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: 193216 DOI: 10.1007/s1088401292799
 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 10971112, DOI 10.1007/s110710152051z
 Gunog Seo and Gail S. K. Wolkowicz, (2015) "Existence of multiple limit cycles in a predatorprey model with arctan(ax) as functional response," Communications in Mathematical Analysis Volume 18, Issue 1, Pages 6468, ISSN 19389787.
 Gail S. K. Wolkowicz and Yuan Yuan, (2016) "Effect of light on the growth of nonnitrogenﬁxing and nitrogenﬁxing phytoplankton in an aquatic system," J. Math. Biol. 72:16631692. DOI 10.1007/s002850150924x
 F. Barraquand et al. (2017) "Moving forward in circles: challenges and opportunities in modeling population cycles," Ecology Letters DOI: 10.1111/ele.12789
 Liang Wang, Daqing Jiang, Gail S. K. Wolkowicz and Donal O'Regan, (2017) "Dynamics of the stochastic chemostat with MonodHaldane response function," Scientific Reports,  7: 13641  DOI:10.1038/s41598017132943.
 Xueping Li, Jingli Ren, Sue Ann Campbell, Gail S. K. Wolkowicz and Huaiping Zhu, (2018) "HOW SEASONAL FORCING INFLUENCES THE COMPLEXITY
OF A PREDATORPREY SYSTEM," Discrete & Continuous Dynamical Systems  B, 23(2): 785807. doi: 10.3934/dcdsb.2018043
 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, 76:18731906. doi.org/10.1007/s002850171201y,
( https://doi.org/10.1007/s002850171201y and link to openly accessible read only version: http://rdcu.be/Eef3)
 ChiuJu Lin, Lin Wang, Gail S. K. Wolkowicz, (2018) “An alternative formulation of a distributed delayed logistic equation,” Bulletin of Mathematical Biology, 80(7):17131735. doi.org/10.1007/s1153801804324
 TingHao Hsu, Tyler Meadows, Lin Wang, Gail S. K. Wolkowicz (2018) “Growth on two limiting essential resources in a selfcycling fermentor,” Mathematical Biosciences and Engineering. 16(1):78100. DOI: 10.3934/mbe.2019005
 Liang Wang, Daqing Jiang, and Gail S. K. Wolkowicz, (2019) "Global asymptotic behavior of a multispecies stochastic chemostat model with discrete delays," Journal of Dynamics and Differential Equations, Online first https://doi.org/10.1007/s10884019097416.
 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, 668689.q
 TingHao Hsu and Gail S. K. Wolkowicz, (2019) ``A criterion for the existence of relaxation oscillations with applications to predatorprey systems and an epidemic model,'' Discrete and Continuous Dynamical Systems Series B
 Gunog Seo and Gail S. K. Wolkowicz, (2019) ``Pest control by generalist parasitoids: a bifurcation theory approach,'' accepted Discrete and Continuous Dynamical Systems Series.
Gail Wolkowicz