Abstract. We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used.
We investigate the linear stability of a neural network with distributed delay, where the neurons are identical. We examine the stability of a symmetrical equilibrium point via the analysis of the characteristic equation both when the connection matrix is symmetric and when it is not. We determine a mean delay and distribution independent stability region. We then illustrate a way of improving on this conservative result by approximating the true region of stability when the actual distribution is not known, but some moments or cumulants of the distribution are. Finally, we compare the approximate stability regions with the stability regions in the case of the uniform and gamma distributions. We show that the approximations improve as more moments or cumulants are used, and that the approximations using cumulants give better results than the ones using moments.
Abstract. We discuss how distributed delays arise in biological models and review the literature on such models. We indicate why it is important to keep the distributions in a model as general as possible. We then demonstrate, through the analysis of a particular example, what kind of information can be gained with only minimal information about the exact distribution of delays. In particular we show that a distribution independent stability region may be obtained in a similar way that delay independent results are obtained for systems with discrete delays. Further, we show how approximations to the boundary of the stability region of an equilibrium point may be obtained with knowledge of one, two or three moments of the distribution. We compare the approximations with the true boundary for the case of uniform and gamma distributions and show that the approximations improve as more moments are used.
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