Ergodicity and regularity of invariant measure for branching Markov processes with immigration

In this thesis we consider systems of finitely many particles moving on paths given by a strong Markov process and undergoing branching and reproduction at random times. The branching rate of a particle, its number of offspring and their spatial distribution are allowed to depend on the particle's position and possibly on the configuration of coexisting particles. In addition there is immigration of new particles, with the rate of immigration and the distribution of immigrants possibly depending on the configuration of pre-existing particles as well. In the first two chapters of this work, we concentrate on the case that the joint motion of particles is governed by a diffusion with interacting components. The resulting process of particle configurations was studied by E. Löcherbach (2002, 2004) and is known as a branching diffusion with immigration (BDI). Chapter 1 contains a detailed introduction of the basic model assumptions, in particular an assumption of ergodicity which guarantees that the BDI process is positive Harris recurrent with finite invariant measure on the configuration space. This object and a closely related quantity, namely the invariant occupation measure on the single-particle space, are investigated in Chapter 2 where we study the problem of the existence of Lebesgue-densities with nice regularity properties. For example, it turns out that the existence of a continuous density for the invariant measure depends on the mechanism by which newborn particles are distributed in space, namely whether branching particles reproduce at their death position or their offspring are distributed according to an absolutely continuous transition kernel. In Chapter 3, we assume that the quantities defining the model depend only on the spatial position but not on the configuration of coexisting particles. In this framework (which was considered by Höpfner and Löcherbach (2005) in the special case that branching particles reproduce at their death position), the particle motions are independent, and we can allow for more general Markov processes instead of diffusions. The resulting configuration process is a branching Markov process in the sense introduced by Ikeda, Nagasawa and Watanabe (1968), complemented by an immigration mechanism. Generalizing results obtained by Höpfner and Löcherbach (2005), we give sufficient conditions for ergodicity in the sense of positive recurrence of the configuration process and finiteness of the invariant occupation measure in the case of general particle motions and offspring distributions.

A simple and novel method to monitor breathing and heart rate in awake and urethane-anesthetized newborn rodents

Rodents are most useful models to study physiological and pathophysiological processes in early development, because they are born in a relatively immature state. However, only few techniques are available to monitor non-invasively heart frequency and respiratory rate in neonatal rodents without restraining or hindering access to the animal. Here we describe experimental procedures that allow monitoring of heart frequency by electrocardiography (ECG) and breathing rate with a piezoelectric transducer (PZT) element without hindering access to the animal. These techniques can be easily installed and are used in the present study in unrestrained awake and anesthetized neonatal C57/Bl6 mice and Wistar rats between postnatal day 0 and 7. In line with previous reports from awake rodents we demonstrate that heart rate in rats and mice increases during the first postnatal week. Respiratory frequency did not differ between both species, but heart rate was significantly higher in mice than in rats. Further our data indicate that urethane, an agent that is widely used for anesthesia, induces a hypoventilation in neonates whilst heart rate remains unaffected at a dose of 1 g per kg body weight. Of note, hypoventilation induced by urethane was not detected in rats at postnatal 0/1. To verify the detected hypoventilation we performed blood gas analyses. We detected a respiratory acidosis reflected by a lower pH and elevated level in CO2 tension (pCO2) in both species upon urethane treatment. Furthermore we found that metabolism of urethane is different in P0/1 mice and rats and between P0/1 and P6/7 in both species. Our findings underline the usefulness of monitoring basic cardio-respiratory parameters in neonates during anesthesia. In addition our study gives information on developmental changes in heart and breathing frequency in newborn mice and rats and the effects of urethane in both species during the first postnatal week.