Delay Differential Equations

Instability regions in delay models with delay dependent parameters: an application in the biosciences
Margherita Carletti
(Istituto di Biomatematica, Università di Urbino,Italy)

In this work we describe a method to determine the instability regions of the positive equilibria in delay models involving delay dependent parameters. The method relies on a geometric stability switch criterion introduced by Beretta and Kuang in $[2]$ and it is here applied to a $2D$ model of phage-bacteria interaction in an open marine environment $[1]$. As for similar delay models, the numerical experiments show that, as time delay increases, stability changes from stable to unstable to stable, implying that large delays can be stabilizing, thus contradicting the common scenario provided by delay models with delay independent parameters $[3]$.

References

[1]

E. Beretta, M. Carletti, F. Solimano, On the effects of environmental fluctuations in a simple model of bacteria-bacteriophage interaction, to appear in The Canadian Applied Mathematics Quarterly.

[2]

E. Beretta, Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, to appear in SIAM J. Math. Anal.

[3]

M. Carletti, Numerical determination of the instability region for a delay model of phage-bacteria interaction, to appear in Numerical Algorithms.

[4]

A. Campbell, Conditions for the existence of bacteriophage, Evolution 15 (1961) 153-165.

[5]

L. M. Proctor, A. Okubo, J. A. Fuhrman, Calibrating of phage-induced mortality in marine bacteria. Ultrastructural studies of marine bacteriophage development from one-step growth experiments, Microb. Ecol. 25 (1993) 161-182.

Radau IIA methods for the numerical integration of stiff delay differential equations
Nicola Guglielmi
(Università dell'Aquila, Italy)

We consider initial value problems for delay differential equations

$$\begin{array}{rcl} M y'(t) &\! = \!& f\Bigl( t,y(t),y\bigl(... ...qquad y(t) = g(t) \quad \mbox{for}   t< t_0 , \end{array}$$ (1)

where $M$ is a constant $d\times d$ matrix and $\alpha_i (t,y(t))\le t$ for all $t\ge t_0$ and for all $i$. The value $g(t_0)$ may be different from $y_0$, allowing for a discontinuity at $t_0$.

Allowing the matrix $M$ to be singular, the above formulation includes all kinds of differential-algebraic delay equations as well as problems of neutral type.

We discuss how collocation methods based on Radau nodes can be applied to solve problems of type (1). We consider both theoretical properties of such methods and practical implementation aspects [1,2].

In particular we focus attention on the following issues:

(i)

choice of the continuous extension;

(ii)

stability and accuracy of the numerical process;

(iii)

error control and stepsize selection strategies;

(iv)

efficient solution of the algebraic equations.

Finally we show the practical behaviour of such methods on real-life examples [3,4].

References

[1]

N. Guglielmi, E. Hairer, Implementing Radau IIA methods for stiff delay differential equations, Computing, to appear (2001).

[2]

N. Guglielmi, Asymptotic stability barriers for natural Runge-Kutta processes for delay equations, SIAM J. Numer. Anal., to appear (2001).

[3]

P. Waltman, A threshold model of antigen-stimulated antibody production, Theoretical Immunology (Immunology Ser. 8), Dekker, New York, 437-453 (1978).

[4]

G.A. Bocharov, G.I. Marchuk, A.A. Romanyukha, Numerical solution by LMMs of stiff delay differential systems modelling an immune response, Numer. Math. 73, 131-148 (1996).

An asymptotically stable approach in the numerical solution of Delay Differential Equations
Stefano Maset
(Department of mathematical sciences, University of Trieste, Italy)

It is known (see [1]) that there do not exist Runge-Kutta (RK) methods unconditionally preserving the asymptotic stability when they are applied to linear systems of Delay Differential Equations (DDEs)

$$y'(t)=Ly(t)+My(t-\tau).$$ (1)

In other words for any RK method there exist problems (complex system of dimension 2 or real system of dimension 4) where the method has stability restriction on the stepsize. In view of such a very negative result we now present an approach for solving DDE which unconditionally preserves the asymptotic stability on all systems of DDE with $L,M$ matrices of arbitrary dimension. We restate the DDE as an abstract Cauchy problem (equivalently as an hyperbolic PDE) and then we use a classic RK method on the abstract problem. At every step of the method we have to solve a Boundary Value Problem which can be solved exactly.

References

[1]

S. Maset, Instability of Runge-Kutta methods when they are applied to linear systems of delay differential equations, Numer. Math., in press.

RK methods for Functional Differential Equations
Lucio Torelli & Stefano Maset & Rossana Vermiglio
(Department of mathematical sciences - University of Trieste,Italy)

The class of functional differential equations comprises discrete delay and distributed delay equations. The approach for numerically solving them is different in these two cases.
In this talk we show how RK-like schemes can be applied to general functional differential equations

$$y'(t)=f(t,y_t)$$ (1)

This approach requires only the computation of the functional f and gives new methods which works for both discrete and distributed delays. At every step of the RK method a functional algebraic equation arises which is solved by using interpolatory projection on finite dimensional spaces.