Most evolutionary thinking is based on the notion of fitness and related ideas such as fitness landscapes and evolutionary optima. Nevertheless, it is often unclear what fitness actually is, and its meaning often depends on the context. Here we argue that fitness should not be a basal ingredient in verbal or mathematical descriptions of evolution. Instead, we propose that evolutionary birth-death processes, in which individuals give birth and die at ever-changing rates, should be the basis of evolutionary theory, because such processes capture the fundamental events that generate evolutionary dynamics. In evolutionary birth-death processes, fitness is at best a derived quantity, and owing to the potential complexity of such processes, there is no guarantee that there is a simple scalar, such as fitness, that would describe long-term evolutionary outcomes. We discuss how evolutionary birth-death processes can provide useful perspectives on a number of central issues in evolution.

The longstanding debate about the importance of group (multilevel) selection suffers from a lack of formal models that describe explicit selection events at multiple levels. Here, we describe a general class of models for two-level evolutionary processes which include birth and death events at both levels. The models incorporate the state-dependent rates at which these events occur. The models come in two closely related forms: (1) a continuous-time Markov chain, and (2) a partial differential equation (PDE)derived from (1) by taking a limit. We argue that the mathematical structure of this PDE is the same for all models of two-level population processes, regardless of the kinds of events featured in the model. The mathematical structure of the PDE allows for a simple and unambiguous way to distinguish between individual-and group-level events in any two-level population model. This distinction, in turn, suggests a new and intuitively appealing way to define group selection in terms of the effects of group-level events. We illustrate our theory of group selection by applying it to models of the evolution of cooperation and the evolution of simple multicellular organisms, and then demonstrate that this kind of group selection is not mathematically equivalent to individual-level (kin) selection. K E Y W O R D S :Evolutionary dynamics, evolution of cooperation, evolution of multicellular organisms, multilevel selection.

Many quantities of interest in open queueing systems are expected values which can be viewed as functions of the arrival rate to the system. We are thus led to consider f(λ), where λ is the arrival rate, and f represents some quantity of interest. The aim of this paper is to investigate the behavior of f(λ), for λ near zero. This ‘light traffic’ information is obtained in the form of f(0) and its derivatives, f(n)(0), n ≥ 1. Focusing initially on Poisson arrival processes, we provide a method to calculate f(n)(0) for any ‘admissible’ function f. We describe a large class of queueing networks for which we show several standard quantities of interest to be admissible. The proof that the method yields the correct values for the derivatives involves an interchange of limits, whose justification requires a great deal of effort. The determination of f(n)(0) involves consideration of sample paths with n + 1 or fewer arrivals over all of time. These calculations are illustrated via several simple examples. These results can be extended to arrival processes which, although not Poisson, are ‘driven’ by a Poisson process. We carry out the details for phase type renewal processes and nonstationary Poisson processes.

A network is defined as an undirected graph and a routing which consists of a collection of simple paths connecting every pair of vertices in the graph. The forwarding index of a network is the maximum number of paths passing through any vertex in the graph. Thus it corresponds to the maximum amount of forwarding done by any node in a communication network with a fixed routing. For a given number of vertices, each having a given degree constraint, we consider the problem of finding networks that minimize the forwarding index. Forwarding indexes are calculated' for cube networks and generalized de Bruijn networks. General bounds are derived which show that de Bruijn networks are asymptotically optimal. Finally, efficient techniques for building large networks with small forwarding indexes out of given component networks are presented and analyzed.

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