DCDS-S
Dynamics of two phytoplankton species competing for light and nutrient with internal storage
Sze-Bi Hsu Chiu-Ju Lin
Discrete & Continuous Dynamical Systems - S 2014, 7(6): 1259-1285 doi: 10.3934/dcdss.2014.7.1259
We analyze a competition model of two phytoplankton species for a single nutrient with internal storage and light in a well mixed aquatic environment. We apply the theory of monotone dynamical system to determine the outcomes of competition: extinction of two species, competitive exclusion, stable coexistence and bistability of two species. We also present the graphical presentation to classify the competition outcomes and to compare outcome of models with and without internal storage.
keywords: Two species competition competitive exclusion well-mixed water column Droop model Tilman's graph presentation. bistability stable coexistence
DCDS-B
A discrete-delayed model with plasmid-bearing, plasmid-free competition in a chemostat
Sze-Bi Hsu Cheng-Che Li
Discrete & Continuous Dynamical Systems - B 2005, 5(3): 699-718 doi: 10.3934/dcdsb.2005.5.699
A discrete-delayed model of plasmid-bearing, plasmid-free organisms competing for a single-limited nutrient in a chemostat is established. Rigorous mathematical analysis of the asymptotic behavior of this model is presented. An interesting method to analyze the local stability of interior equilibrium is developed. The argument is also applicable to a model of plasmid-bearing, plasmid-free organisms competing for two complementary nutrients in a chemostat.
keywords: chemostat plasmid-bearing global attractivity. plasmid-free Delayed growth response perturbation
CPAA
On a mathematical model arising from competition of Phytoplankton species for a single nutrient with internal storage: steady state analysis
Sze-Bi Hsu Feng-Bin Wang
Communications on Pure & Applied Analysis 2011, 10(5): 1479-1501 doi: 10.3934/cpaa.2011.10.1479
In this paper we construct a mathematical model of two microbial populations competing for a single-limited nutrient with internal storage in an unstirred chemostat. First we establish the existence and uniqueness of steady-state solutions for the single population. The conditions for the coexistence of steady states are determined. Techniques include the maximum principle, theory of bifurcation and degree theory in cones.
keywords: degree theory. Maximum principle coexistence global bifurcation Chemostat
DCDS
Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss
Sze-Bi Hsu Ming-Chia Li Weishi Liu Mikhail Malkin
Discrete & Continuous Dynamical Systems - A 2003, 9(6): 1465-1492 doi: 10.3934/dcds.2003.9.1465
This paper is concerned with the classical Nicholson-Bailey model [15] defined by $f_\lambda(x,y)=(y(1-e^{-x}), \lambda y e^{-x})$. We show that for $\lambda=1$ a heteroclinic foliation exists and for $\lambda>1$ global strict oscillations take place. The important phenomenon of delay of stability loss is established for a general class of discrete dynamical systems, and it is applied to the study of nonexistence of periodic orbits for the Nicholson-Bailey model.
keywords: singular perturbation. global oscillation Nicholson-Bailey model heteroclinic foliation delay of stability loss
DCDS-B
Relaxation oscillation profile of limit cycle in predator-prey system
Sze-Bi Hsu Junping Shi
Discrete & Continuous Dynamical Systems - B 2009, 11(4): 893-911 doi: 10.3934/dcdsb.2009.11.893
It is known that some predator-prey system can possess a unique limit cycle which is globally asymptotically stable. For a prototypical predator-prey system, we show that the solution curve of the limit cycle exhibits temporal patterns of a relaxation oscillator, or a Heaviside function, when certain parameter is small.
keywords: predator-prey model. limit cycle Relaxation oscillator
DCDS-B
Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage
Sze-Bi Hsu Junping Shi Feng-Bin Wang
Discrete & Continuous Dynamical Systems - B 2014, 19(10): 3169-3189 doi: 10.3934/dcdsb.2014.19.3169
The dynamics of a reaction-diffusion system for two species of microorganism in an unstirred chemostat with internal storage is studied. It is shown that the diffusion coefficient is a key parameter of determining the asymptotic dynamics, and there exists a threshold diffusion coefficient above which both species become extinct. On the other hand, for diffusion coefficient below the threshold, either one species or both species persist, and in the asymptotic limit, a steady state showing competition exclusion or coexistence is reached.
keywords: Unstirred chemostat internal storage reaction-diffusion system steady state.
DCDS-B
Coexistence solutions of a competition model with two species in a water column
Hua Nie Sze-Bi Hsu Jianhua Wu
Discrete & Continuous Dynamical Systems - B 2015, 20(8): 2691-2714 doi: 10.3934/dcdsb.2015.20.2691
Competition between species for resources is a fundamental ecological process, which can be modeled by the mathematical models in the chemostat culture or in the water column. The chemostat-type models for resource competition have been extensively analyzed. However, the study on the competition for resources in the water column has been relatively neglected as a result of some technical difficulties. We consider a resource competition model with two species in the water column. Firstly, the global existence and $L^\infty$ boundedness of solutions to the model are established by inequality estimates. Secondly, the uniqueness of positive steady state solutions and some dynamical behavior of the single population model are attained by degree theory and uniform persistence theory. Finally, the structure of the coexistence solutions of the two-species system is investigated by the global bifurcation theory.
keywords: degree theory coexistence solution global bifurcation. Water column uniqueness
DCDS-B
A qualitative study of the damped duffing equation and applications
Zhaosheng Feng Goong Chen Sze-Bi Hsu
Discrete & Continuous Dynamical Systems - B 2006, 6(5): 1097-1112 doi: 10.3934/dcdsb.2006.6.1097
In this paper, we analyze the damped Duffing equation by means of qualitative theory of planar systems. Under certain parametric choices, the global structure in the Poincaré phase plane of an equivalent two-dimensional autonomous system is plotted. Exact solutions are obtained by using the Lie symmetry and the coordinate transformation method, respectively. Applications of the second approach to some nonlinear evolution equations such as the two-dimensional dissipative Klein-Gordon equation are illustrated.
keywords: equilibrium point Duffing's equation global structure autonomous system phase plane Lie symmetry.
DCDS-B
Classification of potential flows under renormalization group transformation
Sze-Bi Hsu Bernold Fiedler Hsiu-Hau Lin
Discrete & Continuous Dynamical Systems - B 2016, 21(2): 437-446 doi: 10.3934/dcdsb.2016.21.437
Competitions between different interactions in strongly correlated electron systems often lead to exotic phases. Renormalization group is one of the powerful techniques to analyze the competing interactions without presumed bias. It was recently shown that the renormalization group transformations to the one-loop order in many correlated electron systems are described by potential flows. Here we prove several rigorous theorems in the presence of renormalization-group potential and find the complete classification for the potential flows. In addition, we show that the relevant interactions blow up at the maximal scaling exponent of unity, explaining the puzzling power-law Ansatz found in previous studies. The above findings are of great importance in building up the hierarchy for relevant couplings and the complete classification for correlated ground states in the presence of generic interactions.
keywords: polar equations blow-up in finite time RG flow gradient flow Strongly correlated electron system LaSalles' invariance principle.
DCDS-B
Special issue dedicated to the memory of Paul Waltman
Sze-Bi Hsu Hal L. Smith Xiaoqiang Zhao
Discrete & Continuous Dynamical Systems - B 2016, 21(2): i-ii doi: 10.3934/dcdsb.2016.21.2i
This volume is dedicated to the memory of Paul Waltman. Many of the authors of articles contained here were participants at the NCTS International Conference on Nonlinear Dynamics with Applications to Biology held May 28-30, 2014 at National Tsing-Hua University, Hsinchu, Taiwan. The purpose of the conference was to survey new developments in nonlinear dynamics and its applications to biology and to honor the memory of Professor Paul Waltman for his influence on the development of Mathematical Biology and Dynamical Systems. Attendees at the conference included Paul's sons Fred and Dennis, many of Paul's former doctoral and post-doctoral students, many others who, although not students of Paul, nevertheless were recipients of Paul's valuable advice and council, and many colleagues from all over the world who were influenced by Paul's mathematics and by his personality. We thank the NCTS for its financial support of the conference and Dr. J.S.W. Wong for supporting the conference banquet.

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