نظریه بازی تکاملی در فضاهای اندازه گیری: خوب مطرح بودن

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.

1. Introduction Evolutionary game theory (EGT) is the creation and study of mathematical models that describe how the strategy profile in games change over time due to mutation and selection (replication). In this paper we address the problem of finding a comprehensive mathematical framework suitable for studying the problems of well-posedness and long-term solution behavior for fully nonlinear evolutionary game theoretic models. We form a unified theory for evolutionary game theory as a dynamical system on the state space of finite signed Borel measures under the weak star topology. In this theory, we unify the discrete and continuous strategy (trait) spaces and the pure replicator and replicator–mutator dynamics under one model. A natural question to ask is why the formulation of a dynamical system on the state space of finite signed Borel measures under the weak star topology? Why isn’t the existing mathematical machinery adequate? The next two examples will illustrate the need for such a formulation. First, we consider the following EGT model of generalized logistic growth with pure selection (i.e., strategies replicate themselves exactly and no mutation occurs) which was developed and analyzed in [1]: equation(1) View the MathML sourceddtx(t,q)=x(t,q)(q1−q2X(t)), Turn MathJax on where X(t)=∫Qx(t,q)dqX(t)=∫Qx(t,q)dq is the total population, View the MathML sourceQ⊂int(R+2) is compact and the state space is the set of continuous real valued functions C(Q)C(Q). Each q=(q1,q2)∈Qq=(q1,q2)∈Q is a two tuple where q1q1 is an intrinsic replication rate and q2q2 is an intrinsic mortality rate. The solution to this model converges to a Dirac mass centered at the fittest qq-class. This is the class with the highest birth to death ratio View the MathML sourceq1q2, and this convergence is in a topology called weak∗weak∗ (point wise convergence of functions) [1]. However, this Dirac limit is not in the state space as it is not a continuous function. It is a measure. Thus, under this formulation one cannot treat this Dirac mass as an equilibrium (a constant) solution and hence the study of linear stability analysis is not possible. Other examples for models developed on classical state spaces such as L1(X,μ)L1(X,μ) that demonstrate the emergence of Dirac measures in the asymptotic limit from smooth initial densities are given in [2], [1], [3], [4], [5], [6], [7] and [8]. In particular, how the measures arise naturally in a biological and adaptive dynamics environment is illustrated quite well in [7, Chapter 2]. These examples show that the chosen state space for formulating such selection–mutation models must contain densities and Dirac masses and the topology used must contain the ability to demonstrate convergence of densities to Dirac masses. The first example above assumes a continuous strategy space QQ and hence the model solution is sought among density functions denoted by x(t,q)x(t,q). Our second example, is the classic discrete EGT model known as the replicator–mutator equation (in this model the strategy space is assumed to be discrete). In [9, p. 273] it is given as equation(2) View the MathML sourceddtxi(t)=∑j=1nxj(t)fj(x⃗(t))Qij−ϕ(x⃗(t))xi(t) Turn MathJax on where View the MathML sourcex→(t)=(x1(t),x2(t),…,xn(t)) is a vector consisting of nn classes each of size xi(t)xi(t), and QijQij is the payoff kernel, i.e., QijQij is the proportion of the jj-class that mutates into the ii-class and ϕϕ is a weighted (average) fitness. The author states that the language equation (replicator–mutator equation) is a unifying description of deterministic evolutionary dynamics. He further states that the replicator–mutator equation is used to describe the dynamics of complex adaptive systems in population dynamics, biochemistry and models of language acquisition. Equation (1) and a version of (2) are special cases of a more general measure-valued model that we present in this paper. In particular, with the discrete model (2) if we assume that fjfj and ϕϕ are functions of the total population View the MathML sourceX=∑j=1nxj then this model can be obtained by choosing the proper initial condition composed of a linear combination of Dirac masses and the proper replication–mutation kernel which is also composed of a linear combination of Dirac masses. The example of the pure selection density model given in (1) can be realized from the measure-valued model by choosing an absolutely continuous initial measure and a continuous family of Dirac measures for the selection–mutation kernel (which represents the pure replication case). Thus, these density and discrete models can be unified under this formulation. Furthermore, our new theory combines both the pure replicator and replicator–mutator dynamics in a continuous manner. By this we mean that as the mutations get smaller and smaller the replicator–mutator model will approach the pure replicator model. This is possible because our mutation kernels are allowed to be (family of) measures as well. This presents a serious difficulty in the analysis which requires the development of some technical tools in studying the well-posedness of the new model. Many researchers have recently devoted their attention to the study of such EGT models (e.g. [2], [1], [3], [4], [10], [11], [12] and [13]). To date almost all EGT models are formulated as density models [1], [3], [4], [12] and [13] with linear mutation term. There are several formulations of pure selection or replicator equation dynamics on measure spaces [2], [14] and [15]. The recent formulations of selection–mutation balance equations on the probability measures by [16] and [17] are novel constructions. These models describe the aging of an infinite population as a process of accumulation of mutations in a genotype. The dynamical equation which describes the system is of Kimura–Maruyama type. Thus far in selection–mutation studies the mutation process has been modeled using two different approaches: (1) a diffusion type operator [5] and [13]; (2) an integral type operator that makes use of a mutation kernel [2], [3], [4], [16] and [17]. Here we focus on the second approach for modeling mutation. Perhaps the work most related to the one presented here is that in [2]. In that paper, the authors considered a pure selection model with density dependent birth and mortality function and a 2-dimensional trait space on the space of finite signed measures. They discussed existence–uniqueness of solutions and studied the long term behavior of the model. Here, we generalize the results in that paper in several directions. Most salient is the fact that the present paper is one in evolutionary game theory; hence the applications are possibly other than population biology. In particular, in the present paper we construct a (measure valued) EGT model. This is an ordered triple (Q,μ,F)(Q,μ,F) subject to equation(3) View the MathML sourceddtμ(t)(E)=F(μ(t)(Q))(E),for every E∈B(Q). Turn MathJax on Here QQ is the strategy (metric) space, B(Q)B(Q) are the Borel sets on Q,μ(t)Q,μ(t) is a time dependent family of finite signed Borel measures on QQ and FF is a density dependent vector field such that μμ and FF satisfy Eq. (3). The main contributions of the present work are as follows: (1) we establish well-posedness of the new measure-valued dynamical system; (2) we are able to combine models that consider both discrete and continuous parameter spaces under this formulation; no separate machinery is needed for each; (3) we are able to include both selection and mutation in one model because our setup allows for choosing the mutation to be a family of measures; (4) unlike the linear mutation term commonly used in the literature, we allow for nonlinear (density dependent) mutation term that contains all classical nonlinearities, e.g., Ricker, Beverton–Holt, Logistic; (5) unlike the one or two dimensional strategy spaces used in the literature, we allow for a strategy space QQ that is possibly infinite dimensional. In particular, we assume that QQ is a compact complete separable metric space, i.e., a compact Polish space. This paper is organized as follows. In Section 2 we demonstrate how to proceed from a density model to a measure valued one and we formulate the model on the (natural) space of measures. In Section 3 we establish the well-posedness of this model. In Section 4 we demonstrate how this model encompasses the discrete, continuous replicator–mutator and species and quasi-species models. In Section 5 we provide concluding remarks.

We have formulated a density dependent EGT (selection–mutation) model on the space of measures and provided a framework which is rich enough to allow pure selection, selection–mutation, and discrete and continuous strategy spaces, all under one setting. We also established the well-posedness of this EGT model. There are several future paths to take from this point. We will mention one application and one mathematical future pathway. Towards the application, tumor heterogeneity is one main cause of tumor robustness. Tumors are robust in the sense that tumors are systems that tend to maintain stable functioning despite various perturbations. Tumor heterogeneity describes the existence of distinct subpopulations of tumor cells with specific characteristics within a single neoplasm. The mutation between the subpopulations is one major factor that makes the tumor robust. To date there is no unifying framework in mathematical modeling of carcinogenesis that would account for parametric heterogeneity [25]. To introduce distributed parameters (heterogeneity) and mutation is essential as we know that cancer recurrence, tumor dormancy and other dynamics can appear in heterogeneous settings and not in homogeneous settings. Increasing technological sophistication has led to a resurgence of using oncolytic viruses in cancer therapy. So in formulating a cancer therapy it is useful to know that in principle a heterogeneous oncolytic virus must be used to eradicate a tumor cell. One mathematical future path is to perform asymptotic analysis on the model. There are two essential things that need to be addressed if we wish to be able to perform asymptotic analysis of our model. First, we need a state space with the property that if the measure valued dynamical system has an initial condition as a finite signed Borel measure then the asymptotic limits will also be in this space. The second problem is that often there will be more than one strategy of a given fitness. In (1), a Dirac mass emerged as it is assumed that only a unique fittest class exists. In reality, this may not be the case and more than one fittest class can exist. In particular, it is possible that a continuum of fittest strategies exist (see Fig. 1 for an example). So our mathematical structure must include the ability to demonstrate the convergence of the model solution to a measure supported on a continuum of strategies.These two difficulties coupled with our desire to study the problem of parameter estimation in these models imply that some form of a “weak” or “generalized” asymptotic limit must be formulated. These weak limits need to live in a certain “completion” of the space of finite signed measures. We will explore these topics in a forthcoming study.