The risk of extinction - the mutational meltdown or the overpopulation

The phase diagrams survival-extinction for the Penna model with parameters: (mutations rate)-(birth rate), (mutation rate)-(harmful mutations threshold), (harmful mutation threshold)-(minimal reproduction age) are presented. The extinction phase may be caused by either mutational meltdown or overpopulation. When the Verhulst factor is responsible for removing only newly born babies and does not act on adults the overpopulation is avoided and only genetic factors may lead to species extinction.Comment: RevTex4, 5 pages, 4 figures (8 eps files

Ferromagnetic Ising spin systems on the growing random tree

We analyze the ferromagnetic Ising model on a scale-free tree; the growing random network model with the linear attachment kernel $A_k=k+\alpha$ introduced by [Krapivsky et al.: Phys. Rev. Lett. {\bf 85} (2000) 4629-4632]. We derive an estimate of the divergent temperature $T_s$ below which the zero-field susceptibility of the system diverges. Our result shows that $T_s$ is related to $\alpha$ as $\tanh(J/T_s)=\alpha/[2(\alpha+1)]$, where $J$ is the ferromagnetic interaction. An analysis of exactly solvable limit for the model and numerical calculation support the validity of this estimate.Comment: 15 pages, 5 figure

Memory effect in growing trees

We show that the structure of a growing tree preserves an information on the shape of an initial graph. For the exponential trees, evidence of this kind of memory is provided by means of the iterative equations, derived for the moments of the node-node distance distribution. Numerical calculations confirm the result and allow to extend the conclusion to the Barabasi--Albert scale-free trees. The memory effect almost disappears, if subsequent nodes are connected to the network with more than one link.Comment: 9 pages, 9 figure

Dependence of the average to-node distance on the node degree for random graphs and growing networks

In a graph, nodes can be characterized locally (with their degree $k$) or globally (e.g. with their average length path $\xi$ to other nodes). Here we investigate how $\xi$ depends on $k$. Our earlier algorithm of the construction of the distance matrix is applied to the random graphs. Numerical calculations are performed for the random graphs and the growing networks: the scale-free ones and the exponential ones. The results are relevant for search strategies in different networks.Comment: 7 pages, 2 figure