Overview
We use mathematics and computational approaches to address questions in evolutionary biology and ecology. We primarily work on adaptive evolution, dynamics, and biodiversity from theoretical perspectives. Our approaches inlude but are not limited to:
- dynamical systems,
- game theory,
- applied linear algebra,
- stochastic process / probability theory,
- graph theory / group theory,
- information theory,
- computer simulations.
But these appraoches are not mutually exclusive, but highly interdependent.
General Theory
Inclusive fitness theory
Darwin’s truely breakthrough idea of adaptive evolution, based on individuals' ‘fitness’, has provided an explanation for the ‘design’ principle of organisms. But what remained puzzling for Darwin was the exception of sterile workers in ants, wasps, termites, beetles, shellfish, and even some mammals. William Donald Hamilton’s idea of ‘inclusive fitness’ provides a game-changing explanation for such systems, where cooperation of workers can increase the fitness of queens’ who are genetically related to the workers. In other words, helping sibs can, despite the null fecundity of the actors' own, contribute to the spread of the gene responsible for the act. The key idea is that spatial, temporal, or mechanistic assortment between individuals can modify the course of evolutionary dynamics, by extending the individual-level fitness to gene-level fitness, which is the inclusive fitness.
The inclusive fitness theory is now recognized the most powerful framework to study the evolutionary dynamics of behavioral interactions, including cooperation, between genetically similar individuals. We primarily apply the inclusive fitness theory to address the questions in life history evolution such as cooperation, sex allocation, and other behavioral traits in a range of systems.
Lifecycle Graph Theory
Modeling adaptive consequences in a single, homogeneous population is relatively straightforward: we pick one representative individual (usually with ‘mutant’ phenotype) and assess its life-time reproductive success, which is a product of birth rate times longevity. But if the populations are stuctured (into ‘classes’), assessing the lifetime reproductive success would become a challenging task, often rendering the problem analytically intractable.
We are generally interested in developing analytical frameworks to study evolutionary dynamics in class-structured populations. We apply graph theoretical approaches to circumvent the intractability problem, and assess analyitical formula of fitness, within the framework of adaptive dynamics. We are also interested in general, dynamical properties of class-structured populations, e.g., symmetry, feedback dimensions, and geometric singular perturbation.
The developed theory is applicable to make predictions of adaptation in highly heterogeneous, class-structured populations, such as sex-structured epidemiological models, spatially heterogeneous populations, populations with group-size variations, and temporally structured populations (niche-construction).
Diversity Measurement
Diversity is a ‘notorious’ concept, because of its conceptual and practical difficulty. The problem is the lack of conceptualization of diversity. We study diversity measurement from mathematical perspective: what is diversity, and how does it change with parameters? We apply the Schur-convexity theory and majorization theory to diversity measurement, to reveal patterns of the effects of biotic and abiotic factors on diversity. We are also interested in how diversity dynamically changes with ecological processes.