A discrete-delayed model with plasmid-bearing, plasmid-free competition in a chemostat
Sze-Bi Hsu Cheng-Che Li
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
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
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
Heteroclinic foliation, global oscillations for the Nicholson-Bailey model and delay of stability loss
Sze-Bi Hsu Ming-Chia Li Weishi Liu Mikhail Malkin
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
Relaxation oscillation profile of limit cycle in predator-prey system
Sze-Bi Hsu Junping Shi
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
Dynamics of two phytoplankton species competing for light and nutrient with internal storage
Sze-Bi Hsu Chiu-Ju Lin
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
Special issue dedicated to the memory of Paul Waltman
Sze-Bi Hsu Hal L. Smith Xiaoqiang Zhao
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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Global dynamics of a Predator-Prey model with Hassell-Varley Type functional response
Sze-Bi Hsu Tzy-Wei Hwang Yang Kuang
Predator-prey models with Hassell-Varley type functional response are appropriate for interactions where predators form groups and have applications in biological control. Here we present a systematic global qualitative analysis to a general predator-prey model with Hassell-Varley type functional response. We show that the predator free equilibrium is a global attractor only when the predator death rate is greater than its growth ability. The positive equilibrium exists if the above relation reverses. In cases of practical interest, we show that the local stability of the positive steady state implies its global stability with respect to positive solutions. For terrestrial predators that form a fixed number of tight groups, we show that the existence of an unstable positive equilibrium in the predator-prey model implies the existence of an unique nontrivial positive limit cycle.
keywords: limit cycles extinction. Functional response predator-prey model global stability
Analysis of a model of two parallel food chains
Sze-Bi Hsu Christopher A. Klausmeier Chiu-Ju Lin
In this paper we study a mathematical model of two parallel food chains in a chemostat. Each food chain consists of a prey species $x$ and a predator species $y$. Two food chains are symmetric in the sense that the prey species are identical and so are the specialized predator species. We assume that both of the prey species in the parallel food chains share the same nutrient $R$. In this paper we show that as the input concentration $R^{(0)}$ of the nutrient varies, there are several possible outcomes: (1) all species go extinct; (2) only the two prey species survive; (3) all species coexist at equilibrium; (4) all species coexist in the form of oscillations. We analyze cases (1)-(3) rigorously; for case (4) we do extensive numerical studies to present all possible phenomena, which include limit cycles, heteroclinic cycles, and chaos.
keywords: chaos. heteroclinic orbits food chains limit cycles chemostat
Growth of single phytoplankton species with internal storage in a water column
Linfeng Mei Sze-Bi Hsu Feng-Bin Wang
In this paper, we analyze a system modeling the growth of single phytoplankton populations in a water column, where population growth increases monotonically with the nutrient quota stored within individuals. We establish a threshold result on the global extinction and persistence of phytoplankton. Condition for persistence is shown to depend on the principal eigenvalue of a boundary value problem, which is related to the physical transport properties of the water column (i.e. the diffusivity and the sinking speed), nutrient uptake rate, and growth rate.
keywords: threshold dynamics spatial variations internal storage Steady states a water column.
Avner Friedman Sze-Bi Hsu Yuan Lou
Recent years have seen dramatic increase in the number and variety of new mathematical models describing biological processes. Many of these models are formulated in terms of systems of partial differential equations. Relevant biological questions give rise to interesting questions regarding properties of the solutions of these equations. The present volume includes eleven articles, each describing a set of problems and results in PDEs inspired by biology. Although in many instances the mathematical analysis may help to better understand the underlying biological processes, the emphasis here is on new mathematical ideas and new mathematical results. The goal is to demonstrate the broad spectrum of new PDE theories that are emerging on the border of two fields: biology and mathematics.

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