Sze-Bi Hsu

A Mathematical Theory for Single-Nutrient Competition in Continuous Cultures of Micro-Organisms

The continuous culture of micro-organisms using the chemostat is an important research technique in microbiology and population biology. It offers advantages in the form of economical production of micro-organisms for the industrial microbiologist and is a laboratory idealization of nature for population studies. The paper studies a mathematical model, based on Michaelis-Menten kinetics, for one substrate and n competing species. Given the parameters of the system, we answer the basic question as to which species survive and which do not, and determine the limiting behaviors. The primary conclusion is that the species will survive whose Michaelis-Menten constant is smallest in comparison with its intrinsic rate of natural increase.

Global stability for a class of predator-prey systems

This paper deals with the question of global stability of the positive locally asymptotically stable equilibrium in a class of predator-prey systems. The Dulac’s criterion is applied and Liapunov functions are constructed to establish the global stability.

Limiting Behavior for Competing Species

Two competition models concerning n species consuming a single, limited resource are discussed. One is based on the Holling-type functional response and the other on the Lotka–Volterra-type. The focus of the paper is on the asymptotic behavior of solutions. LaSalle’s extension theorem of Lyapunov stability theory is the main tool.

On Global Stability of a Predator-Prey System

A two-dimensional model for predator-prey interaction is proposed. Two criteria for global stability of the locally stable equilibrium are presented. These make the graphical method of Rosenzweig and MacArthur more significant.

A ratio-dependent food chain model and its applications to biological control

While biological controls have been successfully and frequently implemented by nature and human, plausible mathematical models are yet to be found to explain the often observed deterministic extinctions of both pest and control agent in such processes. In this paper we study a three trophic level food chain model with ratio-dependent Michaelis-Menten type functional responses. We shall show that this model is rich in boundary dynamics and is capable of generating such extinction dynamics. Two trophic level Michaelis-Menten type ratio-dependent predator-prey system was globally and systematically analyzed in details recently. A distinct and realistic feature of ratio-dependence is its capability of producing the extinction of prey species, and hence the collapse of the system. Another distinctive feature of this model is that its dynamical outcomes may depend on initial populations levels. Theses features, if preserved in a three trophic food chain model, make it appealing for modelling certain biological control processes (where prey is a plant species, middle predator as a pest, and top predator as a biological control agent) where the simultaneous extinctions of pest and control agent is the hallmark of their successes and are usually dependent on the amount of control agent. Our results indicate that this extinction dynamics and sensitivity to initial population levels are not only preserved, but also enriched in the three trophic level food chain model. Specifically, we provide partial answers to questions such as: under what scenarios a potential biological control may be successful, and when it may fail. We also study the questions such as what conditions ensure the coexistence of all the three species in the forms of a stable steady state and limit cycle, respectively. A multiple attractor scenario is found.

A Mathematical Model of the Chemostat with Periodic Washout Rate

In its simplest form, the chemostat consists of several populations of microorganisms competing for a single limiting nutrient. If the input concentration of nutrient and the washout rate are constant, theory predicts and experiment confirms that at most one of the populations will survive. In nature, however, one may expect the input concentration and washout rate to vary with time. In this paper we consider a model for the chemostat with periodic washout rate. Conditions are found for competitive exclusion to hold, and bifurcation techniques are employed to show that under suitable circumstances there will be coexistence of the competing populations in the form of positive periodic solutions.

COMPETITIVE EXCLUSION AND COEXISTENCE FOR COMPETITIVE SYSTEMS ON ORDERED BANACH SPACES

The dynamics of competitive maps and semiflows defined on the product of two cones in respective Banach spaces is studied. It is shown that exactly one of three outcomes is possible for two viable competitors. Either one or the other population becomes extinct while the surviving population approaches a steady state, or there exists a positive steady state representing the coexistence of both populations.

A Competition Model for a Seasonally Fluctuating Nutrient

A model of two species consuming a single, limited, periodically added resource is discussed. The model is based on chemostat-type equations, which differ from the classical models of Lotka and Volterra. The model incorporates nonlinear ‘functional response’ curves of the Holling or Michaelis-Menten type to describe the dependence of the resource-exploitation rate on the amount of resource. Coexistence of two species due to seasonal variation is indicated by numerical studies.

Lectures on Chaotic Dynamical Systems

Basic concepts Zero-dimensional dynamics One-dimensional dynamics Two-dimensional dynamics Systems with 1.5 degrees of freedom Systems generated by three-dimensional vector fields Lyapunov exponents Appendix Bibliography Index.

Some results on global stability of a predator-prey system

In this paper we derive some results to ensure the global stability of a predator-prey system. The results cover most of the models which have been proposed in the ecological literature for predator-prey systems. The first result is very geometric and it is very easy to check from the graph of prey and predator isoclines. The second one is purely algebraic, however, it covers the defects of the first one especially in dealing with Hollings type-3 functional response in some sense. We also discuss the global stability of Kolmogorovs model. Some examples are presented in the discussion section.