The spread in time of a mutation through a population is studied analytically and computationally in fully connected networks and on spatial lattices. The time t* for a favorable mutation to dominate scales with the population size N as N(D+1)/D in D-dimensional hypercubic lattices and as NlnN in fully-connected graphs. It is shown that the surface of the interface between mutants and nonmutants is crucial in predicting the dynamics of the system. Network topology has a significant effect on the equilibrium fitness of a simple population model incorporating multiple mutations and sexual reproduction.

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The existence of sex is one of the major unsolved problems in biology. We use computer simulations to model conditions in which sex may first become established. We develop an individual-based population model and show that a hypothetical facultative sex gene can fix, provided that the initial cost is low. It is demonstrated that the equilibrium fitness in the population increases with increasing population size and decreasing mutation rate. The probability of the establishment of the sex gene is found not to be directly related to the fitness difference between the asexual and sexual populations. This change in fitness on changing the parameters of the model is investigated.

The level-spacing distribution in the tails of the eigenvalue bands of the power-law random banded matrix (PRBM) ensemble have been investigated numerically. The change of level-spacing statistics across the band is examined for different coupling strengths and compared to the density of states for the different systems. It is confirmed that, by varying the eigenvalue region, the same level-spacing statistics can be reached as by varying the coupling strength.PACS numbers: 71.30.+h, 72.15.Rn, 71.55.Jv The Anderson metal-insulator transition (MIT) 1 is a phenomenon of major physical importance that continues to attract a substantial research effort 2,3 . With just short-range interactions, localization-delocalization (LD) transitions are only found in systems with dimensionality, D, greater than two 4 . However, with the addition of long-range interactions, or correlations between the short-range interactions, it is possible to study the LD transition in systems with dimensionality less than two 5 . In this respect, power-law random-banded matrices (PRBMs), that exhibit this transition, have recently attracted much attention 6,7,8,9,10 . The PRBM ensemble was introduced by Mirlin et. al.11 and, in the real case, is defined as the ensemble of N × N random symmetric matrices,Ĥ. The PRBM elements, H ij , are randomly drawn from a Gaussian distribution, centred around zero, with a variance governed by a power-law decay:where α and b ∈ (0, ∞) are parameters. RegardingĤ as a Hamiltonian, the eigenvalues, E, are energies. For α = 1, it has been shown 11 that all the eigenstates of these matrices are critical (i.e. at the LD transition). The parameter b is inversely related to the coupling strength between the nodes. In the limit b ≫ 1 and α = 1, the PRBM critical states are analogous to the critical states at the Anderson transition with D = 2 + ǫ and ǫ ≪ 1, and for b ≪ 1 to those found in the Anderson model with D ≫ 1 12 . By varying b, it is possible to access a set of different critical theories parameterized by dimension (2 < D < ∞) in the conventional Anderson transition. This ability to examine Anderson transitions in different dimensions, in the same effectively one-dimensional model, makes the study of PRBMs a powerful method to make progress in this rich field. As well as being an analogue for the study of important transitions elsewhere, the PRBM is physically important in its own right and has been applied to the study of the finite-temperature Luttinger liquid 13 , the coherent propagation of two interacting particles in a 1D weak random potential 14 and other problems 7,15 . Recently, it has also been realized that, with the addition of chirality, the PRBM also describes the LD transition of quark zero modes in QCD 8 . In the QCD vacuum, the quark zeromode wavefunction decay has a power-law dependence and long-range hopping between sites is possible. In this model, the eigenvalues away from the centre of the spectral band are not affected by the chiral structure; this makes it relevant to the analysis ...

The influence of spatial structure on the equilibrium properties of a sexual population model defined on networks is studied numerically. Using a small-world-like topology of the networks as an investigative tool, the contributions to the fitness of assortative mating and of global mutant spread properties are considered. Simple measures of nearest-neighbor correlations and speed of spread of mutants through the system have been used to confirm that both of these dynamics are important contributory factors to the fitness. It is found that assortative mating increases the fitness of populations. Quick global spread of favorable mutations is shown to be a key factor increasing the equilibrium fitness of populations.