arXiv subjects

John Cleveland

Publications and source records attributed to John Cleveland.

6 recordsLinked to original sources

arXiv2014

Evolutionary Dynamics and Lipschitz Maps

In \cite{ CLEVACKTHI, CLEVACK} an attempt is made to find a comprehensive mathematical framework in which to investigate the problems of well-posedness, asymptotic analysis and parameter estimation for fully nonlinear evolutionary game models. A theory is developed as a dynamical system on the state space of finite signed Borel measures under the weak star topology. Two drawbacks of the previous theory is that the techniques and machinery involved in establishing the results are awkward and have not shed light on the parameter estimation question. For example, in \cite{CLEVACK} the proof for the existence of the dynamical system is obtained via a fixed point argument using the total variation topology, however, the continuity of the model is established in the $weak^* $ topology. This has caused some confusion. I have remedied this by making all the vital rates Lipschitz and the dynamical system is defined on the dual of the bounded Lipschitz maps, a Banach space. I introduce a method of multiplying a functional by a family of functionals. This multiplication behaves nicely with respect to taking normed estimates. It allows us to form a semiflow that is locally Lipschitz, positive invariant, and covers all cases: discrete, continuous, pure selection, selection mutation and measure valued models. Under biologically motivated assumptions the model is uniformly eventually bounded. This remedies both the above problems as only one norm is used, this norm induces the $weak^*$ topology on the positive cone of measures and since we have a norm and local Lipschitzity we can form a theory of Parameter Estimation.

math.DS↗

arXiv2014

$\R \times BL^* $ Valued Consumer Resource Model

The ideas and techniques developed in \cite{CLEVACK, JC2} are applied to the basic pure selection (no mutation) parametric heterogeneous consumer resource model developed in \cite{SmithThieme} to derive a fully nonlinear resource dependent selection mutation $\R \times BL^*$ valued model. Where $BL^*$ is the dual of the Lipschitz maps, a Banach Space. By the appropriate choice of initial condition, and mutation kernel parameter this model unifies both discrete and continuous, pure selection and mutation selection, measure valued and density valued basic consumer resource models. In this paper well-posedness and uniform eventual boundedness under biologically sound assumptions is presented.

math.DS↗

arXiv2014

Selection Mutation Differential Equations: Long-time Behavior of Measure-Valued Solutions

We study the long-time behavior of solutions to a measure-valued selection-mutation model that we formulated in \cite{CLEVACK}. We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy (ASS). We also show that when the strategy space is discrete, the selection-mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy.

math.DS↗

arXiv2012

Evolutionary Game Theory on Measure Spaces: Well-Posedness

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.

math.DS↗

arXiv2012

Evolutionary Game Theory on Measure Spaces: Asymptotic Behavior of Solutions

In [12] we formulated an evolutionary game theory model as a dynamical system on the state space of finite signed Borel measures under the weak* topology. The focus of this paper is to extend the analysis to include the long-time behavior of solutions to the model. In particular, we show that M(Q), the finite signed Borel measures are asymptotically closed. This means that if the initial condition is a finite signed Borel measure and if the asymptotic limit of the model solution exists, then it will be a measure (note that function spaces such as L1(Q) and C(Q) do not have this property). We also establish permanence results for the full replicator mutator model. Then, we study the asymptotic analysis in the case where there is more than one strategy of a given fitness (a continuum of strategies of a given fitness), a case that often arises in applications. To study this case our mathematical structure must include the ability to demonstrate the convergence of the model solution to a measure supported on a continuum of strategies. For this purpose, we demonstrate how to perform completions of the space of measures and how to use these completions to formulate weak (generalized) asymptotic limits. In particular, we show that for the pure replicator dynamics the (weak) solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class; thus this Dirac measure is a globally attractive equilibrium point which is termed a continuously stable strategy (CSS). It is also shown that in the discrete case of the pure replicator dynamics and even for small perturbation of the pure replicator dynamics (i.e., selection with small mutation) there exists a globally asymptotically stable equilibrium.

math.DS↗

arXiv2004

Totally real origami and impossible paper folding

This paper gives one set of axioms for origami constructions, and describes the set of constructible points under these axioms. The determination of the set of cunstructible points for this particular set of axioms is related to Hilbert's 17 th problem.

math.HO↗