From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step

Baar, Martina; Bovier, Anton; Champagnat, Nicolas

doi:10.48550/arxiv.1505.02421

Cited by 2 publications

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“…[27], which describes the continuous evolution of a monomorphic population in a fitness landscape. The convergence of the individual-based model to the CEAD in a single step has recently be shown by Baar et al [26], i.e. the limit (K, µ, σ) → (∞, 0, 0) of large populations, rare mutations and small mutation steps are taken simultaneously, provided 1/(Kµ) log(K)/σ and 1/ √ K σ 1.…”

Section: Arrival Of a Mutantmentioning

confidence: 91%

A stochastic individual-based model for immunotherapy of cancer

Baar,

Coquille,

Mayer

et al. 2015

Preprint

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We propose an extension of a standard stochastic individual-based model in population dynamics which broadens the range of biological applications. Our primary motivation is modelling of immunotherapy of malignant tumours. In this context the different actors, T-cells, cytokines or cancer cells, are modelled as single particles (individuals) in the stochastic system. The main expansions of the model are distinguishing cancer cells by phenotype and genotype, including environment-dependent phenotypic plasticity that does not affect the genotype, taking into account the effects of therapy and introducing a competition term which lowers the reproduction rate of an individual in addition to the usual term that increases its death rate. We illustrate the new setup by using it to model various phenomena arising in immunotherapy. Our aim is twofold: on the one hand, we show that the interplay of genetic mutations and phenotypic switches on different timescales as well as the occurrence of metastability phenomena raise new mathematical challenges. On the other hand, we argue why understanding purely stochastic events (which cannot be obtained with deterministic models) may help to understand the resistance of tumours to therapeutic approaches and may have non-trivial consequences on tumour treatment protocols. This is supported through numerical simulations.

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Section: Arrival Of a Mutantmentioning

confidence: 91%

A stochastic individual-based model for immunotherapy of cancer

Baar,

Coquille,

Mayer

et al. 2015

Preprint

Self Cite

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“…Stochastic individual based models are often used to describe the evolution of individuals undergoing birth, death and mutations. Relations with other approaches can also been made at the large population limit, see [2,12] and the references therein.…”

Section: Evolution Of Dispersionmentioning

confidence: 99%

Rare Mutations Limit of a Steady State Dispersal Evolution Model

Perthame

Souganidis

2016

Math. Model. Nat. Phenom.

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The evolution of a dispersal trait is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fittest dispersal rate for a population in a bounded domain, and, more recently, for traveling waves in the full space.In the present study, we reformulate the problem in the context of adaptive evolution. We consider a population structured by space and a genetic trait acting directly on the dispersal (diffusion) rate under the effect of rare mutations on the genetic trait. We show that, as in simpler models, in the limit of vanishing mutations, the population concentrates on a single trait associated to the lowest dispersal rate. We also explain how to compute the evolution speed towards this evolutionary stable distribution.The mathematical interest stems from the asymptotic analysis which requires a completely different treatment for each variable. For the space variable, the ellipticity leads to the use the maximum principle and Sobolev-type regularity results. For the trait variable, the concentration to a Dirac mass requires a different treatment. This is based on the WKB method and viscosity solutions leading to an effective Hamiltonian (effective fitness of the population) and a constrained Hamilton-Jacobi equation.

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From stochastic, individual-based models to the canonical equation of adaptive dynamics - In one step

Cited by 2 publications

References 0 publications

A stochastic individual-based model for immunotherapy of cancer

A stochastic individual-based model for immunotherapy of cancer

Rare Mutations Limit of a Steady State Dispersal Evolution Model

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