Optimal Switching for Hybrid Semilinear Evolutions

We consider the optimization of a dynamical system by switching at discrete time points between abstract evolution equations composed by nonlinearly perturbed strongly continuous semigroups, nonlinear state reset maps at mode transition times and Lagrange-type cost functions including switching costs. In particular, for a fixed sequence of modes, we derive necessary optimality conditions using an adjoint equation based representation for the gradient of the costs with respect to the switching times. For optimization with respect to the mode sequence, we discuss a mode-insertion gradient. The theory unifies and generalizes similar approaches for evolutions governed by ordinary and delay differential equations. More importantly, it also applies to systems governed by semilinear partial differential equations including switching the principle part. Examples from each of these system classes are discussed

Mutual invadability near evolutionarily singular strategies for multivariate traits, with special reference to the strongly convergence stable case

Over the last two decades evolutionary branching has emerged as a possible mathematical paradigm for explaining the origination of phenotypic diversity. Although branching is well understood for one-dimensional trait spaces, a similarly detailed understanding for higher dimensional trait spaces is sadly lacking. This note aims at getting a research program off the ground leading to such an understanding. In particular, we show that, as long as the evolutionary trajectory stays within the reign of the local quadratic approximation of the fitness function, any initial small scale polymorphism around an attracting invadable evolutionarily singular strategy (ess) will evolve towards a dimorphism. That is, provided the trajectory does not pass the boundary of the domain of dimorphic coexistence and falls back to monomorphism (after which it moves again towards the singular strategy and from there on to a small scale polymorphism, etc.). To reach these results we analyze in some detail the behavior of the solutions of the coupled Lande-equations purportedly satisfied by the phenotypic clusters of a quasi-n-morphism, and give a precise characterisation of the local geometry of the set (Formula presented.) in trait space squared harbouring protected dimorphism. Intriguingly, in higher dimensional trait spaces an attracting invadable ess needs not connect to (Formula presented.). However, for the practically important subset of strongly attracting ess-es (i.e., ess-es that robustly locally attract the monomorphic evoltionary dynamics for all possible non-degenerate mutational or genetic covariance matrices) invadability implies that the ess does connect to (Formula presented.), just as in 1-dimensional trait spaces. Another matter is that in principle there exists the possibiliy that the dimorphic evolutionary trajectory reverts to monomorphism still within the reign of the local quadratic approximation for the invasion fitnesses. Such locally unsustainable branching cannot occur in 1- and 2-dimensional trait spaces, but can do so in higher dimensional ones. For the latter trait spaces we give a condition excluding locally unsustainable branching which is far stricter than the one of strong convergence, yet holds good for a relevant collection of published models. It remains an open problem whether locally unsustainable branching can occur around general strongly attracting invadable ess-es

The Evolution of Simple Life-Histories: Step Towards Classification

We present a classification of the evolutionary dynamics for a class of simple life-history models. The model class considered is characterised by discrete time population dynamics, density-dependent population growth, by the assumption that individuals can occur in two states, and that two evolving traits are coupled by a trade-off. Individual models differ in the choice of traits that are presumed to evolve and in the way population regulation is incorporated. The classification is based on a fitness measure that is sign equivalent to invasion fitness but algebraically simpler. We classify models according to curvature properties of the fitness landscape and whether the evolutionary dynamics can be analysed by means of an optimisation criterion. The first classification allows us to infer whether trait combinations that are characterised by a zero fitness gradient are susceptible to invasion by similar trait combinations. The second classification distinguishes models where evolutionary change is frequency-independent from models that give rise to frequency dependence. Given certain symmetry assumptions we can extend the classification in the latter case by splitting selection into a density-dependent and a frequency-dependent component. We apply our approach to several simple life-history models and demonstrate how our classification facilitates an analytical analysis. We conclude by discussing some general patterns that emerge from our analysis and by hinting at several possible extensions

The Evolution of Resource Specialization through Frequency-Dependent and Frequency-Independent Mechanisms

Levin's fitness set approach has shaped the intuition of many evolutionary ecologists about resource specialization: if the set of possible phenotypes is convex, a generalist is favored, while either of the two specialists is predicted for concave phenotype sets. An important aspect of Levins approach is that it explicitly excludes frequency-dependent selection. Frequency-dependence emerged in a series of models that studied the degree of character displacement of two consumers coexisting on two resources. Surprisingly, the evolutionary dynamics of a single consumer type under frequency-dependence has not been studied in detail yet. We analyze a model of one evolving consumer feeding on two resources and show that, depending on the trait considered to be subject to evolutionary change, selection is either frequency-independent or frequency-dependent. This difference is explained by the effects different foraging traits have on the consumer-resource interactions. If selection is frequency-dependent, then the population can become dimorphic through evolutionary branching at the trait value of the generalist. Those traits with frequency-independent selection, however, do indeed follow the predictions based on Levin's fitness set approach. This dichotomy in the evolutionary dynamics of traits involved in the same foraging process was not previously recognized

Traits traded off

The course of evolution is restricted by constraints. A special type of constraint is a trade-off where different traits are negatively correlated. In this situation a mutant type that shows an improvement in one trait suffers from a decreased performance through another trait. In a fixed fitness landscape evolution is expected to come up with a compromise of the competing fitness components that is optimal in the sense that no other realized compromise can be more successful. However, in most ecological settings the resident community will form a vital part of the selective environment experienced by a mutant. In this case each component in a fitness trade-off can be affected by the phenotype of the conspecifics, which causes the fitness landscape to change as evolution proceeds. We refer to selection in a changing fitness landscape as frequency dependent. With frequency dependence the concept of optimality cannot be applied anymore. This thesis explores, by means of mathematical models, how frequency dependence can be detected and how it alters the evolutionary dynamics of traits that are coupled by a trade-off. Special attention is paid to the phenomenon of evolutionary branching where directional selection drives a population's trait distribution into a region of the trait space where selection turns disruptive.Supported by the Research Council for Earth and Life Sciences (ALW), which is subsidized by the Netherlands Organisation for Scientific Research (NWO)UBL - phd migration 201

Adaptive Walks on Changing Landscapes: Levins' Approach Extended

The assumption that trade-offs exist is fundamental in evolutionary theory. Levins (Am. Nat. 96 (1962) 361-372) introduced a widely adopted graphical method for analyzing evolution towards an optimal combination of two quantitative traits, which are traded off. His approach explicitly excluded the possibility of density- and frequency-dependent selection. Here we extend Levins method towards models, which include these selection regimes and where therefore fitness landscapes change with population state. We employ the same kind of curves Levins used: trade-off curves and fitness contours. However, fitness contours are not fixed but a function of the resident traits and we only consider those that divide the trait space into potentially successful mutants and mutants which are not able to invade ('invasion boundaries'). The developed approach allows to make a priori predictions about evolutionary endpoints and about their bifurcations. This is illustrated by applying the approach to several examples from the recent literature

The components of directional and disruptive selection in heterogeneous group-structured populations.

We derive how directional and disruptive selection operate on scalar traits in a heterogeneous group-structured population for a general class of models. In particular, we assume that each group in the population can be in one of a finite number of states, where states can affect group size and/or other environmental variables, at a given time. Using up to second-order perturbation expansions of the invasion fitness of a mutant allele, we derive expressions for the directional and disruptive selection coefficients, which are sufficient to classify the singular strategies of adaptive dynamics. These expressions include first- and second-order perturbations of individual fitness (expected number of settled offspring produced by an individual, possibly including self through survival); the first-order perturbation of the stationary distribution of mutants (derived here explicitly for the first time); the first-order perturbation of pairwise relatedness; and reproductive values, pairwise and three-way relatedness, and stationary distribution of mutants, each evaluated under neutrality. We introduce the concept of individual k-fitness (defined as the expected number of settled offspring of an individual for which k-1 randomly chosen neighbors are lineage members) and show its usefulness for calculating relatedness and its perturbation. We then demonstrate that the directional and disruptive selection coefficients can be expressed in terms individual k-fitnesses with k=1,2,3 only. This representation has two important benefits. First, it allows for a significant reduction in the dimensions of the system of equations describing the mutant dynamics that needs to be solved to evaluate explicitly the two selection coefficients. Second, it leads to a biologically meaningful interpretation of their components. As an application of our methodology, we analyze directional and disruptive selection in a lottery model with either hard or soft selection and show that many previous results about selection in group-structured populations can be reproduced as special cases of our model

Necessary and sufficient conditions for R0 to be a sum of contributions of fertility loops

Recently, de-Camino-Beck and Lewis (Bull Math Biol 69:1341.1354, 2007) have presented a method that under certain restricted conditions allows computing the basic reproduction ratio R0 in a simple manner from life cycle graphs, without, however, giving an explicit indication of these conditions. In this paper, we give various sets of sufficient and generically necessary conditions. To this end, we develop a fully algebraic counterpart of their graph-reduction method which we actually found more useful in concrete applications. Both methods, if they work, give a simple algebraic formula that can be interpreted as the sum of contributions of all fertility loops. This formula can be used in e.g. pest control and conservation biology, where it can complement sensitivity and elasticity analyses. The simplest of the necessary and sufficient conditions is that, for irreducible projection matrices, all paths from birth to reproduction have to pass through a common state. This state may be visible in the state representation for the chosen sampling time, but the passing may also occur in between sampling times, like a seed stage in the case of sampling just before flowering. Note that there may be more than one birth state, like when plants in their first year can already have different sizes at the sampling time. Also the common state may occur only later in life. However, in all cases R0 allows a simple interpretation as the expected number of new individuals that in the next generation enter the common state deriving from a single individual in this state. We end with pointing to some alternative algebraically simple quantities with properties similar to those of R0 that may sometimes be used to good effect in cases where no simple formula for R0 exists