arXiv subjects

Max O. Souza

Publications and source records attributed to Max O. Souza.

At least 19 recordsLinked to original sources

arXiv2026

Dynamical compatibility for finite and infinite population models used in genetics

Finite and infinite population models are frequently used in population dynamics. However, their interrelationship is rarely discussed. In this work, we examine the limits of large populations of the Moran process (a finite-population birth-death process) and the replicator equation (an ordinary differential equation) as paradigmatic examples of finite and infinite population models, respectively, both of which are extensively used in population genetics. Except for certain degenerate cases, we completely characterize when these models exhibit similar long-term dynamics, i.e., when there is a one-to-one relation between the stable attractors of the replicator equations and the metastable states of the Moran process. We will discuss our findings and the implications for an improved understanding of the famous Lewontin paradox. To achieve this goal, we first show that the asymptotic expression for the fixation probability in the Moran process, when the population size is large and individual interaction is almost arbitrary (including cases modeled through $d$-player game theory), is a convex combination of the asymptotic approximations obtained in the constant fitness case or 2-player game theory. We discuss several examples and the inverse problem, i.e., how to derive a Moran process that is compatible with a given replicator dynamics. In particular, we prove that modeling a Moran process with an inner metastable state may require the use of $d$-player game theory with possibly large $d$ values, depending on the precise location of the inner equilibrium. We finish by discussing possible connections between the newly introduced concepts and classical concepts of quasistationarity and metastability in the theory of stochastic processes.

q-bio.PE↗

arXiv2026

Optimal Trading in Automated Market Makers with Deep Learning

This article explores the optimisation of trading strategies in Constant Function Market Makers (CFMMs) and centralised exchanges. We develop a model that accounts for the interaction between these two markets, estimating the conditional dependence between variables using the concept of conditional elicitability. Furthermore, we pose an optimal execution problem where the agent hides their orders by controlling the rate at which they trade. We do so without approximating the market dynamics. The resulting dynamic programming equation is not analytically tractable, therefore, we employ the deep Galerkin method to solve it. Finally, we conduct numerical experiments and illustrate that the optimal strategy is not prone to price slippage and outperforms naïve strategies.

q-fin.TR↗

arXiv2025

Continuous approximations for the fixation probability of the Moran processes on star graphs

We consider a generalized version of the birth-death (BD) and death-birth (DB) processes introduced by Kaveh, Komarova, and Kohandel (2015), in which two constant fitnesses, one for birth and the other for death, describe the selection mechanism of the population. Rather than constant fitnesses, in this paper we consider more general frequency-dependent fitness functions (allowing any smooth functions) under the weak-selection regime. A particular case arises in evolutionary games on graphs, where the fitness functions are linear combinations of the frequencies of types. For a large population structured as a star graph, we provide approximations for the fixation probability which are solutions of certain ODEs (or systems of ODEs). For the DB case, we prove that our approximation has an error of order $1/N$, where $N$ is the size of the population. The general BD and DB processes contain, as special cases, the BD-* and DB-* (where * can be either B or D) processes described in Hadjichrysanthou, Broom, and Rychtář (2011) -- this class includes many examples of update rules used in the literature. Our analysis shows how the star graph may act as an amplifier, suppressor, or remains isothermal depending on the scaling of the initial mutant placement. We identify an analytical threshold for this transition and illustrate it through applications to evolutionary games, which further highlight asymmetric structural effects across different game types. Numerical examples show that our fixation probability approximations remain accurate even for moderate population sizes and across a wide range of frequency-dependent fitness functions, extending well beyond previously studied linear cases derived from evolutionary games, or constant fitness scenarios.

math.PR↗

arXiv2025

Insuperable strategies in two-player and reducible multi-player games

Real populations are seldom found at the Nash equilibrium strategy. The present work focuses on how population size can be a relevant evolutionary force diverting the population from its expected Nash equilibrium. We introduce the concept of insuperable strategy, a strategy that guarantees that no other player can have a larger payoff than the player that adopts it. We show that this concept is different from the rationality assumption frequently used in game theory and that for small populations the insuperable strategy is the most probable evolutionary outcome for any dynamics that equal game payoff and reproductive fitness. We support our ideas with several examples and numerical simulations. We finally discuss how to extend the concept to multiplayer games, introducing, in a limited way, the concept of game reduction.

cs.GT↗

arXiv2023

On regularized optimal execution problems and their singular limits

We investigate the portfolio execution problem under a framework in which volatility and liquidity are both uncertain. In our model, we assume that a multidimensional Markovian stochastic factor drives both of them. Moreover, we model indirect liquidity costs as temporary price impact, stipulating a power law to relate it to the agent's turnover rate. We first analyze the regularized setting, in which the admissible strategies do not ensure complete execution of the initial inventory. We prove the existence and uniqueness of a continuous and bounded viscosity solution of the Hamilton-Jacobi-Bellman equation, whence we obtain a characterization of the optimal trading rate. As a byproduct of our proof, we obtain a numerical algorithm. Then, we analyze the constrained problem, in which admissible strategies must guarantee complete execution to the trader. We solve it through a monotonicity argument, obtaining the optimal strategy as a singular limit of the regularized counterparts.

q-fin.MF↗

arXiv2022

A mathematical framework for dynamical social interactions with dissimulation

Modeling social interactions is a challenging task that requires flexible frameworks. For instance, dissimulation and externalities are relevant features influencing such systems -- elements that are often neglected in popular models. This paper is devoted to investigating general mathematical frameworks for understanding social situations where agents dissimulate, and may be sensitive to exogenous objective information. Our model comprises a population where the participants can be honest, persuasive, or conforming. Firstly, we consider a non-cooperative setting, where we establish existence, uniqueness and some properties of the Nash equilibria of the game. Secondly, we analyze a cooperative setting, identifying optimal strategies within the Pareto front. In both cases, we develop numerical algorithms allowing us to computationally assess the behavior of our models under various settings.

math.OC↗

arXiv2022

Entropy and the arrow of time in population dynamics

The concept of entropy in statistical physics is related to the existence of irreversible macroscopic processes. In this work, we explore a recently introduced entropy formula for a class of stochastic processes with more than one absorbing state that is extensively used in population genetics models. We will consider the Moran process as a paradigm for this class, and will extend our discussion to other models outside this class. We will also discuss the relation between non-extensive entropies in physics and epistasis (i.e., when the effects of different alleles are not independent) and the role of symmetries in population genetic models.

q-bio.PE↗

arXiv2020

Gradient flow formulations of discrete and continuous evolutionary models: a unifying perspective

We consider three classical models of biological evolution: (i) the Moran process, an example of a reducible Markov Chain; (ii) the Kimura Equation, a particular case of a degenerated Fokker-Planck Diffusion; (iii) the Replicator Equation, a paradigm in Evolutionary Game Theory. While these approaches are not completely equivalent, they are intimately connected, since (ii) is the diffusion approximation of (i), and (iii) is obtained from (ii) in an appropriate limit. It is well known that the Replicator Dynamics for two strategies is a gradient flow with respect to the celebrated Shahshahani distance. We reformulate the Moran process and the Kimura Equation as gradient flows and in the sequel we discuss conditions such that the associated gradient structures converge: (i) to (ii) and (ii) to (iii). This provides a geometric characterisation of these evolutionary processes and provides a reformulation of the above examples as time minimization of free energy functionals.

q-bio.PE↗

arXiv2018

From fixation probabilities to d-player games: an inverse problem in evolutionary dynamics

The probability that the frequency of a particular trait will eventually become unity, the so-called fixation probability, is a central issue in the study of population evolution. Its computation, once we are given a stochastic finite population model without mutations and a (possibly frequency dependent) fitness function, is straightforward and it can be done in several ways. Nevertheless, despite the fact that the fixation probability is an important macroscopic property of the population, its precise knowledge does not give any clear information about the interaction patterns among individuals in the population. Here we address the inverse problem: From a given fixation pattern and population size, we want to infer what is the game being played by the population. This is done by first exploiting the framework developed in FACC Chalub and MO Souza, J. Math. Biol. 75: 1735, 2017., which yields a fitness function that realises this fixation pattern in the Wright-Fisher model. This fitness function always exists, but it is not necessarily unique. Subsequently, we show that any such fitness function can be approximated, with arbitrary precision, using $d$-player game theory, provided $d$ is large enough. The pay-off matrix that emerges naturally from the approximating game will provide useful information about the individual interaction structure that is not itself apparent in the fixation pattern. We present extensive numerical support for our conclusions.

q-bio.PE↗

arXiv2018

Fitness potentials and qualitative properties of the Wright-Fisher dynamics

We present a mechanistic formalism for the study of evolutionary dynamics models based on the diffusion approximation described by the Kimura Equation. In this formalism, the central component is the fitness potential, from which we obtain an expression for the amount of work necessary for a given type to reach fixation. In particular, within this interpretation, we develop a graphical analysis --- similar to the one used in classical mechanics --- providing the basic tool for a simple heuristic that describes both the short and long term dynamics. As a by-product, we provide a new definition of an evolutionary stable state in finite populations that includes the case of mixed populations. We finish by showing that our theory -- rigorous for two types evolution without mutations-- is also consistent with the multi-type case, and with the inclusion of rare mutations.

q-bio.PE↗

arXiv2017

On the stochastic evolution of finite populations

This work is a systematic study of discrete Markov chains that are used to describe the evolution of a two-types population. Motivated by results valid for the well-known Moran (M) and Wright-Fisher (WF) processes, we define a general class of Markov chains models which we term the Kimura class. It comprises the majority of the models used in population genetics, and we show that many well-known results valid for M and WF processes are still valid in this class. In all Kimura processes, a mutant gene will either fixate or become extinct, and we present a necessary and sufficient condition for such processes to have the probability of fixation strictly increasing in the initial frequency of mutants. This condition implies that there are WF processes with decreasing fixation probability --- in contradistinction to M processes which always have strictly increasing fixation probability. As a by-product, we show that an increasing fixation probability defines uniquely an M or WF process which realises it, and that any fixation probability with no state having trivial fixation can be realised by at least some WF process. These results are extended to a subclass of processes that are suitable for describing time-inhomogeneous dynamics. We also discuss the traditional identification of frequency dependent fitnesses and pay-offs, extensively used in evolutionary game theory, the role of weak selection when the population is finite, and the relations between jumps in evolutionary processes and frequency dependent fitnesses.

q-bio.PE↗

arXiv2016

Conservative parabolic problems: non-degenerated theory and degenerated examples from population dynamics

We consider partial differential equations (PDE) of drift-diffusion type in the unit interval, supplemented by either two conservation laws or by a conservation law and a further boundary condition. We treat two different cases: (i) uniform parabolic problems; (ii) degenerated problems at the boundaries. The former can be treated in a very general and complete way, much as the traditional boundary value problems. The latter, however, bring new issues, and we restrict our study to a class of forward Kolmogorov equations that arise naturally when the corresponding stochastic process has either one or two absorbing boundaries. These equations are treated by means of a uniform parabolic regularisation, which then yields a measure solution in the vanishing regularisation limit that is unique. Two prototypical problems from population dynamics are treated in detail.

math.AP↗

arXiv2015

A Hedged Monte Carlo Approach to Real Option Pricing

In this work we are concerned with valuing optionalities associated to invest or to delay investment in a project when the available information provided to the manager comes from simulated data of cash flows under historical (or subjective) measure in a possibly incomplete market. Our approach is suitable also to incorporating subjective views from management or market experts and to stochastic investment costs. It is based on the Hedged Monte Carlo strategy proposed by Potters et al (2001) where options are priced simultaneously with the determination of the corresponding hedging. The approach is particularly well-suited to the evaluation of commodity related projects whereby the availability of pricing formulae is very rare, the scenario simulations are usually available only in the historical measure, and the cash flows can be highly nonlinear functions of the prices.

q-fin.CP↗

arXiv2015

On the dynamics of a class of multi-group models for vector-borne diseases

The resurgence of vector-borne diseases is an increasing public health concern, and there is a need for a better understanding of their dynamics. For a number of diseases, e.g. dengue and chikungunya, this resurgence occurs mostly in urban environments, which are naturally very heterogeneous, particularly due to population circulation. In this scenario, there is an increasing interest in both multi-patch and multi-group models for such diseases. In this work, we study the dynamics of a vector borne disease within a class of multi-group models that extends the classical Bailey-Dietz model. This class includes many of the proposed models in the literature, and it can accommodate various functional forms of the infection force. For such models, the vector-host/host-vector contact network topology gives rise to a bipartite graph which has different properties from the ones usually found in directly transmitted diseases. Under the assumption that the contact network is strongly connected, we can define the basic reproductive number $\mathcal{R}_0$ and show that this system has only two equilibria: the so called disease free equilibrium (DFE); and a unique interior equilibrium---usually termed the endemic equilibrium (EE)---that exists if, and only if, $\mathcal{R}_0>1$. We also show that, if $\mathcal{R}_0\leq1$, then the DFE equilibrium is globally asymptotically stable, while when $\mathcal{R}_0>1$, we have that the EE is globally asymptotically stable.

q-bio.PE↗

arXiv2015

Fixation in large populations: a continuous view of a discrete problem

We study fixation in large, but finite, populations with two types, and dynamics governed by birth-death processes. By considering a restricted class of such processes, we derive a continuous approximation for the probability of fixation that is valid beyond the weak-selection (WS) limit. From the continuous approximations, we then obtain asymptotic approximations for evolutionary dynamics with at most one equilibrium, in the selection-driven regime, that does not preclude a weak-selection regime. As an application, we study the fixation pattern when the infinite population limit has an interior Evolutionary Stable Strategy (ESS): (i) we show that the fixation pattern for the Hawk and Dove game satisfies what we term the one-half law: if the Evolutionary Stable Strategy (ESS) is outside a small interval around $\sfrac{1}{2}$, the fixation is of dominance type; (ii) we also show that, outside of the weak-selection regime, the long-term dynamics of large populations can have very little resemblance to the infinite population case; in addition, we also present results for the case of two equilibria. Finally, we present continuous restatements valid for large populations of two classical concepts naturally defined in the discrete case: (i) the definition of an $\textsc{ESS}_N$ strategy; (ii) the definition of a risk-dominant strategy. We then present two applications of these restatements: (i) we obtain an asymptotic definition valid in the quasi-neutral regime that recovers both the one-third law under linear fitness and the generalised one-third law for $d$-player games; (ii) we extend the ideas behind the (generalised) one-third law outside the quasi-neutral regime and, as a generalisation, we introduce the concept of critical-frequency; (iii) we recover the classification of risk-dominant strategies for $d$-player games.

q-bio.PE↗

arXiv2013

Multiscale Analysis for a Vector-Borne Epidemic Model

Traditional studies about disease dynamics have focused on global stability issues, due to their epidemiological importance. We study a classical SIR-SI model for arboviruses in two different directions: we begin by describing an alternative proof of previously known global stability results by using only a Lyapunov approach. In the sequel, we take a different view and we argue that vectors and hosts can have very distinctive intrinsic time-scales, and that such distinctiveness extends to the disease dynamics. Under these hypothesis, we show that two asymptotic regimes naturally appear: the fast host dynamics and the fast vector dynamics. The former regime yields, at leading order, a SIR model for the hosts, but with a rational incidence rate. In this case, the vector disappears from the model, and the dynamics is similar to a directly contagious disease. The latter yields a SI model for the vectors, with the hosts disappearing from the model. Numerical results show the performance of the approximation, and a rigorous proof validates the reduced models.

q-bio.PE↗

arXiv2013

The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations (PDE) for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented.

q-bio.PE↗

arXiv2009

The SIR epidemic model from a PDE point of view

We present a derivation of the classical SIR model through a mean-field approximation from a discrete version of SIR. We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR model. Moreover, we show that the long time limit of the evolution will be a Dirac measure. The exact position will depend on the well-know $R_0$ parameter, and it will be supported on the corresponding stable SIR equilibrium.

q-bio.PE↗