Epidemiology and Disease-Control Under Gene-for-Gene Plant-Pathogen Interaction

An introduction of disease-resistant variety of a crop plant often leads to the development of a virulent race in pathogen species that restores the pathogenicity to the resistant crop. This often makes disease control of crop plants extremely difficult. In this paper, we theoretically explore the optimal multiline control, which makes use of several different resistant varieties, that minimizes the expected degree of crop damages caused by epidemic outbreaks of the pathogen. We examine both single-locus and two-locus gene-for-gene (GFG) systems for the compatibility relationship between host genotypes and pathogen genotypes, in which host haplotype has either susceptible or resistant allele in each resistance locus, and the pathogen haplotype has either avirulent or virulent allele in the corresponding virulence locus. We then study the optimal planting strategy of host resistant genotypes based on standard epidemiological dynamics with pathogen spore stages. The most striking result of our single locus GFG model is that there exists an intermediate optimum mixing ratio for the susceptible and resistant crops that maximizes the final yield, in spite of the fact that the susceptible crop has no use to fight against either avirulent or virulent race of the pathogen. The intermediate mixture is optimum except when the initial pathogen spore population in the season consists exclusively of the virulent race. The optimal proportion of resistant crops is approximately 1/R0, where R0 is the basic reproductive ratio of pathogen the rest (the vast majority if R0 is large) of crops should be the susceptible genotype. By mixing susceptible and resistant crops, we can force the pathogen races to compete with each other for their available hosts. This competition between avirulent and virulent races prevents the fatal outbreak of the virulent race (the super-race) that can infect all the host genotypes. In the two-locus GFG control, there again exists the optimal mixing ratio for the fraction of universally susceptible genotype and the total fraction of various resistant genotypes, with the ratio close to 1/R0

Evolutionary Stability on Graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree $k > 2$. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth–death (BD), death–birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.MathematicsOrganismic and Evolutionary Biolog

Anderson transitions in three-dimensional disordered systems with randomly varying magnetic flux

The Anderson transition in three dimensions in a randomly varying magnetic flux is investigated in detail by means of the transfer matrix method with high accuracy. Both, systems with and without an additional random scalar potential are considered. We find a critical exponent of $\nu=1.45\pm0.09$ with random scalar potential. Without it, $\nu$ is smaller but increases with the system size and extrapolates within the error bars to a value close to the above. The present results support the conventional classification of universality classes due to symmetry.Comment: 4 pages, 2 figures, to appear in Phys. Rev.

Direct Reciprocity on Graphs

Direct reciprocity is a mechanism for the evolution of cooperation based on the idea of repeated encounters between the same two individuals. Here we examine direct reciprocity in structured populations, where individuals occupy the vertices of a graph. The edges denote who interacts with whom. The graph represents spatial structure or a social network. For birth–death or pairwise comparison updating, we find that evolutionary stability of direct reciprocity is more restrictive on a graph than in a well-mixed population, but the condition for reciprocators to be advantageous is less restrictive on a graph. For death–birth and imitation updating, in contrast, both conditions are easier to fulfill on a graph. Moreover, for all four update mechanisms, reciprocators can dominate defectors on a graph, which is never possible in a well-mixed population. We also study the effect of an error rate, which increases with the number of links per individual; interacting with more people simultaneously enhances the probability of making mistakes. We provide analytic derivations for all results.MathematicsOrganismic and Evolutionary Biolog

The Replicator Equation on Graphs

We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called ‘birth-death’, ‘death-birth’ and ‘imitation’. A fourth update rule, ‘pairwise comparison’, is shown to be equivalent to birth-death updating in our model. We use pair-approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner’s Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors-Paper game.MathematicsOrganismic and Evolutionary Biolog

The Anderson transition: time reversal symmetry and universality

We report a finite size scaling study of the Anderson transition. Different scaling functions and different values for the critical exponent have been found, consistent with the existence of the orthogonal and unitary universality classes which occur in the field theory description of the transition. The critical conductance distribution at the Anderson transition has also been investigated and different distributions for the orthogonal and unitary classes obtained.Comment: To appear in Physical Review Letters. Latex 4 pages with 4 figure

The One-Third Law of Evolutionary Dynamics

Evolutionary game dynamics in finite populations provide a new framework for studying selection of traits with frequency-dependent fitness. Recently, a “one-third law” of evolutionary dynamics has been described, which states that strategy A fixates in a B-population with selective advantage if the fitness of A is greater than that of B when A has a frequency View the MathML source. This relationship holds for all evolutionary processes examined so far, from the Moran process to games on graphs. However, the origin of the “number” View the MathML source is not understood. In this paper we provide an intuitive explanation by studying the underlying stochastic processes. We find that in one invasion attempt, an individual interacts on average with B-players twice as often as with A-players, which yields the one-third law. We also show that the one-third law implies that the average Malthusian fitness of A is positive.MathematicsOrganismic and Evolutionary Biolog

Evolutionary stability on graphs

Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k > 2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs

The replicator equation on graphs

We study evolutionary games on graphs. Each player is represented by a vertex of the graph. The edges denote who meets whom. A player can use any one of n strategies. Players obtain a payoff from interaction with all their immediate neighbors. We consider three different update rules, called ‘birth-death’, ‘death-birth’ and ‘imitation’. A fourth update rule, ‘pairwise comparison’, is shown to be equivalent to birth-death updating in our model. We use pair-approximation to describe the evolutionary game dynamics on regular graphs of degree k. In the limit of weak selection, we can derive a differential equation which describes how the average frequency of each strategy on the graph changes over time. Remarkably, this equation is a replicator equation with a transformed payoff matrix. Therefore, moving a game from a well-mixed population (the complete graph) onto a regular graph simply results in a transformation of the payoff matrix. The new payoff matrix is the sum of the original payoff matrix plus another matrix, which describes the local competition of strategies. We discuss the application of our theory to four particular examples, the Prisoner’s Dilemma, the Snow-Drift game, a coordination game and the Rock-Scissors-Paper game

Breaking the Symmetry Between Interaction and Replacement in Evolutionary Dynamics on Graphs

We study the evolution of cooperation modeled as symmetric 2×2 games in a population whose structure is split into an interaction graph defining who plays with whom and a replacement graph specifying evolutionary competition. We find it is always harder for cooperators to evolve whenever the two graphs do not coincide. In the thermodynamic limit, the dynamics on both graphs is given by a replicator equation with a rescaled payoff matrix in a rescaled time. Analytical results are obtained in the pair approximation and for weak selection. Their validity is confirmed by computer simulations.MathematicsOrganismic and Evolutionary Biolog