# Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters

@article{Beretta2002GeometricSS, title={Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters}, author={Edoardo Beretta and Yang Kuang}, journal={SIAM J. Math. Anal.}, year={2002}, volume={33}, pages={1144-1165} }

In most applications of delay differential equations in population dynamics,the need of incorporation of time delays is often the result of the existence of some stage structure. Since the through-stage survival rate is often a function of time delays,it is easy to conceive that these models may involve some delay dependent parameters. The presence of such parameters often greatly complicates the task of an analytical study of such models. The main objective of this paper is to provide… Expand

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#### References

SHOWING 1-10 OF 39 REFERENCES

Discrete delay, distributed delay and stability switches

- Mathematics
- 1982

In modelling in the biological, physical and social sciences, it is sometimes necessary to take account of time delays inherent in the phenomena. The inclusion of delays explicitly in the equations… Expand

Interaction of maturation delay and nonlinear birth in population and epidemic models

- Biology
- 1999

Numerical simulations indicate that oscillations can also be induced by disease related death in a model with maturation delay in a population with birth rate function B(N) N and linear death rate for the adult stage. Expand

Global attractivity and periodic solutions in delay-differential equations related to models in physiology and population biology

- Mathematics
- 1992

AbstractWith appropriate assumptions, the following two general first-order nonlinear differential delay equations may be employed to describe some physiological control systems as well as some… Expand

Introduction to Functional Differential Equations

- Mathematics, Computer Science
- Applied Mathematical Sciences
- 1993

The present book builds upon the earlier work of J. Hale, "Theory of Functional Differential Equations" published in 1977. The authors have attempted to maintain the spirit of that book and have… Expand

Analysis of an SEIRS epidemic model with two delays

- Mathematics, Medicine
- Journal of mathematical biology
- 1996

Stability of the disease free proportion equilibrium, and existence, uniqueness, and stability of an endemic proportionilibrium, are investigated and results are stated in terms of a key threshold parameter. Expand

Analysis of a Delayed Two-Stage Population Model with Space-Limited Recruitment

- Mathematics, Computer Science
- SIAM J. Appl. Math.
- 1995

A global qualitative analysis of a delayed two-stage population model proposed by Bence and Nisbet to study the dynamic behavior of open systems where older or larger individuals can inhibit the recruitment of juveniles or smaller ones into the population, such as an open marine population with space-limited recruitment. Expand

Space-limited recruitment in open systems: the importance of time delays

- Biology
- 1989

This work found, as did Roughgarden et al., that two qualitatively different dynamic outcomes were possible: a stable steady state, and cyclic fluctuation in population density and space occupied. Expand

Analysis of a model representing stage-structured population growth with state-dependent time delay

- Mathematics
- 1992

A stage-structured model of population growth is proposed, where the time to ma- turity is itself state dependent. It is shown that under appropriate assumptions, all solutions are positive and… Expand

Stage Structure Models Applied in Evolutionary Ecology

- Biology
- 1989

This work states that practical applications of structured models demand large quantities of biological information, and it is seldom easy to formulate models that only require parameters which can be calculated from existing data. Expand

Global Behavior of Nonlinear Difference Equations of Higher Order with Applications

- Mathematics
- 1993

Preface. 1. Introduction and Preliminaries. 2. Global Stability Results. 3. Rational Recursive Sequences. 4. Applications. 5. Periodic Cycles. 6. Open Problems and Conjectures. Appendix: A. The… Expand