Abstract:
An attempt is made to find a comprehensive mathematical framework in which to investigate the problems of well-posedness and asymptotic analysis for fully nonlinear evolutionary game theoretic models. The model should be rich enough to include all classical nonlinearities, e.g., Beverton-Holt or Ricker type. For several such models formulated on the space of integrable functions, it is known that as the variance of the payoff kernel becomes small the solution converges in the long term to a Dirac measure centered at the fittest strategy; thus the limit of the solution is not in the state space of integrable functions. Starting with the replicator-mutator equation and a generalized logistic equation as bases, a general model is formulated as a dynamical system on the state space of finite signed measures. Well-posedness is established, and then it is shown that by choosing appropriate payoff kernels this model includes all classical density models, both selection and mutation, and discrete and continuous strategy (trait) spaces.

Abstract:
In the first part of this paper, we show that the Cauchy problem for wave propagation in some static spacetimes presenting a singular time-like boundary is well posed, if we only demand the waves to have finite energy, although no boundary condition is required. This feature does not come from essential self-adjointness, which is false in these cases, but from a different phenomenon that we call the alternative well-posedness property, whose origin is due to the degeneracy of the metric components near the boundary. Beyond these examples, in the second part, we characterize the type of degeneracy which leads to this phenomenon.

Abstract:
A class of non-autonomous differential inclusions in a Hilbert space setting is considered. The well-posedness for this class is shown by establishing the mappings involved as maximal monotone relations. Moreover, the causality of the so established solution operator is addressed. The results are exemplified by the equations of thermoplasticity with time dependent coefficients and by a non-autonomous version of the equations of viscoplasticity with internal variables.

Abstract:
Ownership and trade emerge from anarchy as evolutionary stable strategies. In these evolutionary game models, ownership status provides an endogenous asymmetrizing criterion enabling cheaper resolution of property conflicts.

Abstract:
In this paper, we prove that the periodic higher-order KdV-type equation \[\left\{\begin{array}{ll} \partial_t u + (-1)^{j+1} \partial_x^{2j+1}u + \frac12 \partial_x(u^2)=0, \hspace{1em} &(t,x) \in \mathbb{R} \times \mathbb{T}, \\ u(0,x) = u_0(x), &u_0 \in H^s(\mathbb{T}). \end{array} \right.\] is globally well-posed in $H^s$ for $s \ge -\frac{j}{2}$, $j \ge 3$. The proof of the global well-posedness is based on "I-method" introduced by Colliander et al. \cite{CKSTT1}. To apply "I-method", we factorize the resonant functions by using the different ways from Hirayama \cite{Hirayama}. Furthermore, we prove the nonsqueezing property of the periodic higher-order KdV-type equation as well. The proof relies on Gromov's nonsqueezing theorem for the finite dimensional Hamiltonian system and approximation for the solution flow. More precisely, after taking the frequency truncation to the solution flow, we applied the nonsqueezing theorem and then the result is transferred to the infinite dimensional original flow. This argument was introduced by Kuksin \cite{Kuksin:1995ue} and made concretely by Bourgain \cite{Bourgain:1994tr} for 1D cubic NLS flow, and Colliander et. al. \cite{CKSTT3} for the KdV flow. One of our observation is that the higher-order KdV-type equation has the better modulation effect from the non-resonant interaction than KdV equation. Hence, unlike the work of Colliander et. al. \cite{CKSTT3}, we can get the nonsqueezing property for the solution flow without the Miura transform.

Abstract:
In this article we present local well-posedness results in the classical Sobolev space H^s(R) with s > 1/4 for the Cauchy problem of the Gardner equation, overcoming the problem of the loss of the scaling property of this equation. We also cover the energy space H^1(R) where global well-posedness follows from the conservation laws of the system. Moreover, we construct solitons of the Gardner equation explicitly and prove that, under certain conditions, this family is orbitally stable in the energy space.

Abstract:
Having the ill-posedness in the range $s<-3/4$ of the Cauchy problem for the Benjamin equation with an initial $H^{s}({\mathbb R})$ data, we prove that the already-established local well-posedness in the range $s>-3/4$ of this initial value problem is extendable to $s=-3/4$ but also that such a well-posed property is globally valid for $s\in [-3/4,\infty)$.

Abstract:
Evolutionary branching points are a paradigmatic feature of adaptive dynamics, because they are potential starting points for adaptive diversification. The antithesis to evolutionary branching points are Continuously stable strategies (CSS's), which are convergent stable and evolutionarily stable equilibrium points of the adaptive dynamics and hence are thought to represent endpoints of adaptive processes. However, this assessment is based on situations in which the invasion fitness function determining the adaptive dynamics have non-zero second derivatives at a CSS. Here we show that the scope of evolutionary branching can increase if the invasion fitness function vanishes to higher than first order at a CSS. Using a class of classical models for frequency-dependent competition, we show that if the invasion fitness vanishes to higher orders, a CSS may be the starting point for evolutionary branching, with the only additional requirement that mutant types need to reach a certain threshold frequency, which can happen e.g. due to demographic stochasticity. Thus, when invasion fitness functions vanish to higher than first order at equilibrium points of the adaptive dynamics, evolutionary diversification can occur even after convergence to an evolutionarily stable strategy.

Abstract:
Human language is still an embarrassment for evolutionary theory, as the speaker's benefit remains unclear. The willingness to communicate information is shown here to be an evolutionary stable strategy (ESS), even if acquiring original information from the environment involves significant cost and communicating it provides no material benefit to addressees. In this study, communication is used to advertise the emitter's ability to obtain novel information. We found that communication strategies can take two forms, competitive and uniform, that these two strategies are stable and that they necessarily coexist.

Abstract:
Evolutionary game theory is used to model the evolution of competing strategies in a population of players. Evolutionary stability of a strategy is a dynamic equilibrium, in which any competing mutated strategy would be wiped out from a population. If a strategy is weak evolutionarily stable, the competing strategy may manage to survive within the network. Understanding the network-related factors that affect the evolutionary stability of a strategy would be critical in making accurate predictions about the behaviour of a strategy in a real-world strategic decision making environment. In this work, we evaluate the effect of network topology on the evolutionary stability of a strategy. We focus on two well-known strategies known as the Zero-determinant strategy and the Pavlov strategy. Zero-determinant strategies have been shown to be evolutionarily unstable in a well-mixed population of players. We identify that the Zero-determinant strategy may survive, and may even dominate in a population of players connected through a non-homogeneous network. We introduce the concept of `topological stability' to denote this phenomenon. We argue that not only the network topology, but also the evolutionary process applied and the initial distribution of strategies are critical in determining the evolutionary stability of strategies. Further, we observe that topological stability could affect other well-known strategies as well, such as the general cooperator strategy and the cooperator strategy. Our observations suggest that the variation of evolutionary stability due to topological stability of strategies may be more prevalent in the social context of strategic evolution, in comparison to the biological context.