The existing classification of evolutionarily singular strategies in Adaptive Dynamics (Geritz et al. in Evol Ecol 12:35-57, 1998; Metz et al. in Stochastic and spatial structures of dynamical systems, pp 183-231, 1996) assumes an invasion exponent that is differentiable twice as a function of both the resident and the invading trait. Motivated by nested models for studying the evolution of infectious diseases, we consider an extended framework in which the selection gradient exists (so the definition of evolutionary singularities extends verbatim), but where the invasion fitness may lack the smoothness necessary for the classification à la Geritz et al. We derive the classification of singular strategies with respect to convergence stability and invadability and determine the condition for the existence of nearby dimorphisms. In addition to ESSs and invadable strategies, we observe what we call one-sided ESSs: singular strategies that are invadable from one side of the singularity but uninvadable from the other. Studying the regions of mutual invadability in the vicinity of a one-sided ESS, we discover that two isoclines spring in a tangent manner from the singular point at the diagonal of the mutual invadability plot. The way in which the isoclines unfold determines whether these one-sided ESSs act as ESSs or as branching points. We present a computable condition that allows one to determine the relative position of the isoclines (and thus dimorphic dynamics) from the dimorphic as well as from the monomorphic invasion exponent and illustrate our findings with an example from evolutionary epidemiology.
COBISS.SI-ID: 1024534868
Cvetković et al. (2010) [3] presented an algorithm for dynamic load balancing in discrete load model based on the eigenvectors of the adjacency matrix of the processor network. In case of integral graphs, whose eigenvalues are integers and whose eigenvector basis may be chosen to consist of integers only, this algorithm executes in integer arithmetic, providing an argument for the use of integral graphs as suitable candidates for processor networks. It is observed here that this algorithm, however, is not capable of balancing all integer load distributions. After specifying a necessary and sufficient condition for properly managing all integer load distributions, it becomes apparent that the proposed algorithm does not actually depend on the integrality of eigenvectors, but on employing a set of balancing flows carrying a unit load from a fixed vertex to each of the remaining vertices in the graph. Such sets exist for every connected graph and, thus, yield a simple load balancing algorithm that can be executed in integer arithmetic for every processor network.
COBISS.SI-ID: 1536723908