Differential equations is a central area of mathematics, and one of its recent and most important applications is in mathematical biology and medicine. In today's integrative and interdisciplinary research climate, mathematical biosciences and engineering is quickly evolving from a mainstream area of study into a core and hot domain of scientific activities. The newly launched journal of Mathematical Biosciences and Engineering underlines this exciting scenario.

My current and past research interests cover almost the full spectrum of differential equations, mathematical and computational biosciences.

**MATHEMATICAL BIOLOGY**

My recent interests in this area are focused on
__ stoichiometry__
based population models and their implications. These
models carefully imbed the powerful and natural chemical heterogeneity
that innate to all biomasses. This is a solid platform for
understanding most of the complex dynamics found in biosciences,
including the current issues on biocomplexity and biodiversity. It
naturally connects gene to individual growth through the so-called
growth rate hypothesis, and this in turn connects all the way to
complex ecosystem dynamics. The ratio-dependent population models
provide natural mechanisms for the deterministic total extinction of
all species. These painfully formulated and systematically studied
mathematical models can often reveal deep biological insights that are
important for our efforts in protect and managing our fragile
ecosystems and vast but diminishing biodiversity. Questions we ask
include such as what mechanisms allow species to coexist, why and how
species evolves and what causes species go extinct.

All my recent papers are the results of close collaborations involving a group of extremely competent and leading researchers in biomathematics and theoretical biology. Below are some of my relevant papers on these topics.

80. M. Fan, Y. Kuang and Z. Feng: Cats Protecting Birds Revisited, Bulletin of Mathematical Biology, (pdf 785k), in press.

79. M. Fan, I. Loladze, Y. Kuang and J. J. Elser: Dynamics of a stoichiometric discrete producer-grazer model, (pdf3.8M, ps 1.9M), J. Difference Equations and Applications, in press. zip file

78. Yang Kuang, Jef Huisman and James J. Elser: Stoichiometric plant-herbivore models and their interpretation, Math. Biosc. and Eng., 1, 215-222(2004) (pdf 228K)

73. C. R. Miller, Y. Kuang, W. F. Fagan and J. J. Elser: Modeling and analysis of stoichiometric two-patch consumer-resource systems. (31 pages), (pdf 2067k ), Mathematical Biosciences, in review.

71. I. Loladze, Y. Kuang, J. J. Elser and W. F. Fagan: Coexistence of two predators on one prey mediated by stoichiometry. (pdf 406k ), Theoretical Population Biology, 65, 1-15, 2004.

68. Y. Kuang, W. Fagan and I. Loladze: Biodiversity, habitat area, resource growth rate and interference competition, (22 pages), Bulletin of Mathematical Biology, 65(2003), 497-518 ( pdf 266k ).

67. S.-B. Hsu, T.-W. Hwang and Y. Kuang: A Ratio-Dependent Food Chain Model and Its Applications to Biological Control, Math. Biosc., 181(2003), 55-83. ( pdf 795k )

66. T.-W. Hwang and Y. Kuang: Deterministic extinction effect of parasites on host populations, J. Math. Biol., 46(2003), 17-30.( pdf 215k )

55. I. Loladze, Y. Kuang and J. Elser: Stoichiometry in producer-grazer systems: linking energy flow and element cycling, Bull. Math. Biol., 62, 1137-1162 (2000).( pdf, 425k )

46. Y. Kuang and E. Beretta: Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol. 36, 389--406 (1998) (pdf 259k )

3. Y. Kuang and H. I. Freedman: Uniqueness of limit cycles in Gause-type Models of predator-prey systems, Math. Biosci. 88, 67-84 (1988) (745k)

**
MATHEMATICAL MEDICINE**

My interests in mathematical medicine is
growing rapidly. The current focus is on
modeling various aspects of tumor growth and management. This line of
research involves several biological faculty members and students. I
also maintain an active interest in modeling glucose-insulin
interactions and currently working with several colleagues/Ph.D
students on this topic. Below are some of my relevant papers on these
topics.

81.
A. Makroglou, J. Li and Y. Kuang: Mathematical models and software
tools for the glucose-insulin

regulatory system and diabetes: an overview,
Applied Numerical Mathematics, (pdf 264K) in press.

72. Y. Kuang, J. Nagy
and J. Elser:
Biological
stoichiometry of tumor dynamics:
mathematical models and analysis, Disc.
Cont. Dyn. Sys., series B, 4, 221-240.
2004. (
pdf 243k )

70. J. Elser, J. Nagy and Y. Kuang: Biological stoichiometry: an ecological perspective on tumor dynamics, ( pdf 339k ), BioScience, 53(2003), 1112-1120.

57. J. Li, Yang Kuang and B. Li: Analyses of IVGTT glucose-insulin interaction models with time delay, Discrete Contin. Dynam. Systems, B. 1, 103--124 (2001). ( pdf 298k )

56. E. Beretta and Y.
Kuang: Modeling and analysis of a marine bacteriophage infection
with
latency period,
Nonlinear Analysis: Real World Applications, 2, 35-74(2001). (496K**)**

48. K. Cooke, Y. Kuang and Bingtuan Li: Analysis of an antiviral immune response model with time delays, Proceedings of the third Butler conference, Canadian Applied Math. Quarterly , 6, 321--354 (1998).

47. E. Beretta and Y. Kuang: Modeling and analysis of a marine bacteriophage infection. Math. Biosc. 149, 57--76(1998). ( pdf 207k )

**
DELAY DIFFERENTIAL EQUATIONS**

My
interests in this area are deep and far
reaching. My well received monograph on
**Delay Differential Equations with Applications
in Population Dynamics **is sold out in 2001. Time permitting, I
will develop a second edition or a sequel. Due to the intrinsic time
delays in all life processes and the complexity of the observed
dynamics in such processes, delay differential equation becomes a
popular choice for modeling most life science problems. Recent
theoretical and computational progresses make this choice a highly
fruitful one. My interest in this deep rooted field of mathematics will
continue in earnest and will always be motivated by important issues of
biology and medicine. Below are some of my relevant papers in
this area.

77. S. A.
Gourley and Y. Kuang: A
Delay Reaction-Diffusion Model of the Spread of Bacteriophage
Infection, (18 pages),
SIAM J. Appl. Math., in
press. (
pdf 482k)

76. S. A. Gourley and Y. Kuang: A Stage Structured Predator-Prey model and Its Dependence on Maturation Delay and Death Rate, J. Math. Biol., 49, 188-200 (2004). ( pdf 396k)

69. S. A. Gourley and Y. Kuang: Wavefronts and global stability in a time-delayed population model with stage structure, Proc Roy Soc Lond Ser A. , 459, 1563-1579 (2003). ( pdf 334k )

63. E. Beretta and Y. Kuang: Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165. ( pdf 236k)

56. E. Beretta and Y. Kuang: Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Analysis: Real World Applications, 2, 35-74 (2001). (496K).

42. T. Zhao, Y. Kuang and H.L. Smith: Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Analysis, TMA, 28, 1373--1394(1997).

35. Y. Kuang: Global stability in delay differential systems without dominating instantaneous negative feedbacks. J. Diff. Eqns., 119, 503--532 (1995).( jde95.pdf, 721k )

30. Y. Kuang, **Delay
Differential Equations with Applications in Population Dynamics,**
volume 191 in the series of Mathematics in
Science and Engineering, Academic Press.
1993. (398 pages) (
where to buy, if any )

28. Y. Kuang and H. L. Smith: Global stability for infinite delay Lotka-Volterra type systems, J. Differential Equations, 103, 221--246(1993). (jde93.pdf, 721k )