A parameterized shift-splitting preconditioner for saddle point problems.

Recently, Chen and Ma [A generalized shift-splitting preconditioner for saddle point problems, Applied Mathematics Letters, 43 (2015) 49-55] introduced a generalized shift-splitting preconditioner for saddle point problems with symmetric positive definite (1,1)-block. In this paper, I establish a parameterized shift-splitting preconditioner for solving the large sparse augmented systems of linear equations. Furthermore, the preconditioner is based on the parameterized shift-splitting of the saddle point matrix, resulting in an unconditional convergent fixed-point iteration, which has the intersection with the generalized shift-splitting preconditioner. In final, one example is provided to confirm the effectiveness.

Permutation transformations of tensors with an application.

The permutation transformation of tensors is introduced and its basic properties are discussed. The invariance under permutation transformations is studied for some important structure tensors such as symmetric tensors, positive definite (positive semidefinite) tensors, Z-tensors, M-tensors, Hankel tensors, P-tensors, B-tensors and H-tensors. Finally, as an application of permutation transformations of tensors, the canonical form theorem of tensors is given. The theorem shows that some problems of higher dimension tensors can be translated into the corresponding problems of lower dimension weakly irreducible tensors so as to handle easily.

The evolution of optimal resource allocation and mating systems in hermaphroditic perennial plants.

By incorporating the effects of inbreeding depression (ID) on both juveniles and adults survivorship, we developed a new theoretical model for hermaphroditic perennial plants. Our model showed that the effect of the selfing rate on the evolutionarily stable strategy (ESS) reproductive allocation depends on three parameters: (1) the self-fertilized juvenile relative survivorship (SFJRS), (2) the self-fertilized adult relative survivorship (SFARS) and (3) the growth rate of self-fertilized adult, where the SFJRS is the survivorship of self-fertilized juveniles divided by the survivorship of outcrossed juveniles, and likewise for the SFARS. However, the ESS sex allocation decreases as the selfing rate increases. This relationship seems independent of the SFJRS, the SFARS, and the growth rate of self-fertilized adults. Additionally, our model showed that the complete outcrossing is an ESS when the fraction of juvenile inbreeding depression (FJID) is less than 1/2 - τ, where τ is the self-fertilized adults mortality rate caused by ID. In contrast, the complete selfing also acts as an ESS when the FJID is greater than 1/2 - τ. These results could explain the diversity of mating strategies and related resource allocations for plants.

Collective punishment is more effective than collective reward for promoting cooperation.

Collective punishment and reward are usually regarded as two potential mechanisms to explain the evolution of cooperation. Both scenarios, however, seem problematic to understand cooperative behavior, because they can raise the second-order free-rider problem and many organisms are not able to discriminate less cooperating individuals. Even though they have been proved to increase cooperation, there has been a debate about which one being more effective. To address this issue, we resort to the N-player evolutionary snowdrift game (NESG), where a collective punishment/reward mechanism is added by allowing some players to display punishment/reward towards all remaining players. By means of numerous simulations and analyses, we find that collective punishment is more effective in promoting cooperation for a relatively high initial frequency of cooperation or for a relatively small group. When the intensity of punishment exceeds a certain threshold, a stable state of full cooperation emerges for both small and large groups. In contrast, such state does not appear for large groups playing a NESG with reward mechanism. In the case of mutualistic interactions, finally, our results show the new payoff with collective punishment/reward can lead to the coexistence of cooperators and defectors when discrimination between these two is not possible.

The shift between the Red Queen and the Red King effects in mutualisms.

Interspecific mutualisms consist of partners trading services that yield common benefits to both species. Until now, understanding how the payoffs from mutualistic cooperation are allocated among the participants has been problematic. Two hypotheses have been proposed to resolve this problem. The Red Queen effect argues that faster-evolving species are favoured in co-evolutionary processes because they are able to obtain a larger share of benefits. Conversely, the Red King effect argues that the slower-evolving species gains a larger share of benefits. The model we propose shows that the allocations for a common benefit vary when the effect of a reward mechanism is included in the model. The outcome is a shift from the Red Queen effect to the Red King effect and vice versa. In addition, our model shows that either an asymmetry in payoff or an asymmetry in the number of cooperative partners causes a shift between the Red Queen effect and the Red King effect. Even in situations where the evolutionary rates are equal between the two species, asymmetries in rewards and in participant number lead to an uneven allocation of benefits among the partners.

Asymmetric interaction paired with a super-rational strategy might resolve the tragedy of the commons without requiring recognition or negotiation.

Avoiding the tragedy of the commons requires that one or more individuals in a group or partnership "volunteer", benefiting the group at a cost to themselves. Recognition and negotiation with social partners can maintain cooperation, but are often not possible. If recognition and negotiation are not always the mechanism by which cooperative partnerships avoid collective tragedies, what might explain the diverse social cooperation observed in nature? Assuming that individuals interact asymmetrically and that both "weak" and "strong" players employ a super-rational strategy, we find that tragedy of the commons can be avoided without requiring either recognition or negotiation. Whereas in the volunteer's dilemma game a rational "strong" player is less likely to volunteer to provide a common good in larger groups, we show that under a wide range of conditions a super-rational "strong" player is more likely to provide a common good. These results imply that the integration of super-rationality and asymmetric interaction might have the potential to resolve the tragedy of the commons. By illuminating the conditions under which players are likely to volunteer, we shed light on the patterns of volunteerism observed in variety of well-studied cooperative social systems, and explore how societies might avert social tragedies.

Evolutionary stability in the asymmetric volunteer's dilemma.

It is often assumed that in public goods games, contributors are either strong or weak players and each individual has an equal probability of exhibiting cooperation. It is difficult to explain why the public good is produced by strong individuals in some cooperation systems, and by weak individuals in others. Viewing the asymmetric volunteer's dilemma game as an evolutionary game, we find that whether the strong or the weak players produce the public good depends on the initial condition (i.e., phenotype or initial strategy of individuals). These different evolutionarily stable strategies (ESS) associated with different initial conditions, can be interpreted as the production modes of public goods of different cooperation systems. A further analysis revealed that the strong player adopts a pure strategy but mixed strategies for the weak players to produce the public good, and that the probability of volunteering by weak players decreases with increasing group size or decreasing cost-benefit ratio. Our model shows that the defection probability of a "strong" player is greater than the "weak" players in the model of Diekmann (1993). This contradicts Selten's (1980) model that public goods can only be produced by a strong player, is not an evolutionarily stable strategy, and will therefore disappear over evolutionary time. Our public good model with ESS has thus extended previous interpretations that the public good can only be produced by strong players in an asymmetric game.

Resource elasticity of offspring survival and the optimal evolution of sex ratios.

The fitness of any organisms includes the survival and reproductive rate of adults and the survival of their offspring. Environmental selection pressures might not affect these two aspects of an organism equally. Assuming that an organism first allocates its limited resources to maintain its survival under environmental selection pressure, our model, based on the evolutionarily stable strategy theory, surprisingly shows that the sex ratio is greatly affected by the environmental pressure intensity and by the reproductive resource elasticity of offspring survival. Moreover, the concept of the resource elasticity of offspring survival intrinsically integrates the ecological concepts of K selection and r selection. The model shows that in a species with reproductive strategy K, increased environmental selection pressure will reduce resource allocation to the male function. By contrast, in a species with reproductive strategy r, harsher environmental selection pressure will increase allocation to the male function. The elasticity of offspring survival might vary not only across species, but also across many other factors affecting the same species (e.g., age structure, spatial heterogeneity), which explains sex ratio differences across species or age structures and spatial heterogeneity in the same species.

[Evolutionary stability analysis of asymmetric hawk-dove game considering the impact of common resource].

Explaining the evolution of cooperation remains one of the important problems in both biology and social science. Classical theories mainly based on an assumption that cooperative players are symmetrically interacted. However, almost all the well-studied systems showed that cooperative players are in fact asymmetrically interacted and that asymmetric interaction might greatly affect cooperation behavior of the involved players. Considering the asymmetric interaction and the selection pressure of resources, we present a model that possesses four strategies: strength- cooperation (SC), strength-defection (SD), weakness-cooperation (WC) and weakness-defection (WD). Combining evolutionary game theory with dynamical stability theory, we find that the evolutionary results closely depend on the asymmetric interaction and selection pressure of resources as well as cost-to-benefit ratio of conflict. When the common resources are plentiful, the cost-to-benefit ratio of conflict is negatively correlated with the probability of SC, while it is positively correlated with the probability of SD and WD. With increasing the strength ratio between the strong and weak players, the proportion of SC and SD will increase, while the proportion of WD will reduce. The model developed here has intrinsically integrated Boxed Pigs game and Hawk-Dove game. When the common resource is at shortage, the Boxed Pigs game will transform into Hawk-Dove game under the increase of the strength ratio between the strong and weak players.