Invariant manifolds of models from population genetics

Many models in population genetics feature some form of convergence of the genetic state of the population, typically onto a globally attracting invariant manifold. This allows one to effectively reduce the dynamical system to a problem with fewer dimensions, making it easier to investigate the stability of the steady states in the model, as well as to predict the long-term evolution of the population. Moreover, along this manifold, there is a balance between multiple processes, such as selection and recombination. For some models, restrictive assumptions such as small selection coefficients or additivity of fertilities and mortalities has helped show global contraction of dynamics onto a manifold which is close to the well-known Hardy-Weinberg manifold, and on this `quasiequilibrium’ manifold the dynamics can be written in terms of allele frequencies (which is of more practical interest to geneticists than the genotype frequencies). This thesis focuses on proving the existence of an invariant manifold for two continuous-time models in population genetics: one is proposed by Nagylaki and Crow and features fertilities and mortalities (death rates), while the other is the selection-recombination model. Common themes in both proofs include a change of coordinates such that the dynamical system is monotone with respect to a certain cone. As a result, it is possible to construct an equicontinuous sequence of functions which has a convergent subsequence. We show this limiting function is indeed invariant. In fact, for the latter model, we show the manifold is globally attracting by proving the phase volume is contracting. The conditions obtained from the proofs are less restrictive than the use of parameters that are small or additive, hence our work is more widely applicable. For the former model, numerical examples are also provided in which the manifold need not be smooth, convex, unique or globally attracting

Dynamics of Macrosystems; Proceedings of a Workshop, September 3-7, 1984

There is an increasing awareness of the important and persuasive role that instability and random, chaotic motion play in the dynamics of macrosystems. Further research in the field should aim at providing useful tools, and therefore the motivation should come from important questions arising in specific macrosystems. Such systems include biochemical networks, genetic mechanisms, biological communities, neutral networks, cognitive processes and economic structures. This list may seem heterogeneous, but there are similarities between evolution in the different fields. It is not surprising that mathematical methods devised in one field can also be used to describe the dynamics of another. IIASA is attempting to make progress in this direction. With this aim in view this workshop was held at Laxenburg over the period 3-7 September 1984. These Proceedings cover a broad canvas, ranging from specific biological and economic problems to general aspects of dynamical systems and evolutionary theory

High Resolution Schemes for Conservation Laws With Source Terms.

This memoir is devoted to the study of the numerical treatment of source terms in hyperbolic conservation laws and systems. In particular, we study two types of situations that are particularly delicate from the point of view of their numerical approximation: The case of balance laws, with the shallow water system as the main example, and the case of hyperbolic equations with stiff source terms. In this work, we concentrate on the theoretical foundations of highresolution total variation diminishing (TVD) schemes for homogeneous scalar conservation laws, firmly established. We analyze the properties of a second order, flux-limited version of the Lax-Wendroff scheme which avoids oscillations around discontinuities, while preserving steady states. When applied to homogeneous conservation laws, TVD schemes prevent an increase in the total variation of the numerical solution, hence guaranteeing the absence of numerically generated oscillations. They are successfully implemented in the form of flux-limiters or slope limiters for scalar conservation laws and systems. Our technique is based on a flux limiting procedure applied only to those terms related to the physical flow derivative/Jacobian. We also extend the technique developed by Chiavassa and Donat to hyperbolic conservation laws with source terms and apply the multilevel technique to the shallow water system. With respect to the numerical treatment of stiff source terms, we take the simple model problem considered by LeVeque and Yee. We study the properties of the numerical solution obtained with different numerical techniques. We are able to identify the delay factor, which is responsible for the anomalous speed of propagation of the numerical solution on coarse grids. The delay is due to the introduction of non equilibrium values through numerical dissipation, and can only be controlled by adequately reducing the spatial resolution of the simulation. Explicit schemes suffer from the same numerical pathology, even after reducing the time step so that the stability requirements imposed by the fastest scales are satisfied. We study the behavior of Implicit-Explicit (IMEX) numerical techniques, as a tool to obtain high resolution simulations that incorporate the stiff source term in an implicit, systematic, manner