In my talk I will discuss several simple food web modules that combine population dynamics with evolutionary trait dynamics. Classical food web modules (e.g., trophic chain, exploitative competition, apparent competition, intraguild predation etc.) have proved to be instrumental in our understanding of mechanisms regulating biodiversity. These simple food web modules assume that the interaction strength between species is fixed. In other words, they do not consider various individual adaptations to changing environment. Because decisions an individual has to make often depend on what the others are doing, the optimal strategies require game theoretical framework. The two times scales involved in these models (demographic and evolutionary) lead to two possible scalings that simplify resulting models. The approach that assumes trait dynamics operate on a slower time scale when compared with population dynamics led to the so-called "adaptive dynamics". When trait dynamics operate on much faster time scale, the resulting models are described by control system (where controls describe traits) with state dependent feedbacks. As these feedbacks are often multivalued, the population dynamics are described by a differential inclusion. In my talk I will show how models that combine population with evolutionary dynamics change our understanding of mechanisms of biodiversity in simple food web modules.
In subdivided populations, adaptation to a local environment may be hampered by maladaptive gene flow from other subpopulations. We study a continent-island model in which an ancestral population sends migrants to a colony exposed to a different environment. At an isolated locus, i.e., unlinked to other loci under selection, a locally beneficial mutation can be established and maintained only if its selective advantage exceeds the immigration rate of alternative allelic types. We show that, if a beneficial mutation arises in linkage to a locus at which a locally adapted allele is already segregating in migration-selection balance, the new mutant can invade and be maintained under much higher immigration rates than predicted by one-locus theory. We deduce the maximum amount of gene flow that admits the preservation of the locally adapted haplotype on the strength of recombination and selection. We calculate the selective advantage of recombination-reducing mechanisms, such as chromosome inversions, which often seem to play a role in speciation. Our analysis provides conditions for the evolution of clusters of locally adaptive genes, or islands of divergence, as found by some empirical studies. For an extended model that allows for epistasis, we discuss how much gene flow is needed to inhibit speciation by the accumulation of Dobzhansky-Muller incompatibilities.
Mechanistic mathematical models are important tools for understanding the processes that shape ecological systems. Models have been used to describe life cycles of individuals, population dynamics, behavior, and more. However, in order for these models to reach their full potential as both tools for understanding and for prediction we must be able to link modeled quantities to data and infer model parameters.
However, general methods of parameter inference for many of these models are not available, and we must think carefully about how to link sophisticated models with robust inferential techniques. Here I discuss three examples of ecological models of these types. First is a model of the temperature dependence of malaria transmission, which shows the power of even simple models combined with data. The second example uses an example of an individual-based model (IBM) developed to describe the spread of Chytridiomycosis in a population of frogs.
This case study shows how one can perform inference for IBMs that exhibit certain characteristics with a traditional likelihood-based approach. Third, I present a bioenergetic model of individual growth and reproduction in a dynamic environment. This example highlights how input mis-specification can affect inference, and the consequences for prediction in novel environments.
October 15, 2013; 10:20-11:15am
Sam Karlin was one of the great mathematicians to contribute to evolutionary theory, both in the forward direction (dynamics) and the backward direction (bioinformatics). Karlin introduced two theorems in 1976 to analyze the effect of population subdivision on the protection of genetic diversity. Both could be summarized as the phenomenon that mixing reduces growth, with the consequence that greater dispersal in heterogenous environments reduces the survival of rare alleles.
They provide the basis to prove very generally that populations can always be invaded by genetic variants for information transmission (mutation and recombination) that better preserve information during reproduction --- the Reduction Principle initially discovered by Marc Feldman. The Reduction Principle has made an appearance in recent work on reaction diffusion models of dispersal in continuous space. Could a continuous-space version of Karlin's theorem be at work here? I will describe my recent extension of Karlin's theorem to infinite dimensional Banach spaces. This result unifies the reaction diffusion models showing that the slower disperser wins. It also applies to the generators of strongly continuous semigroups, elliptic operators, Schrivdinger operators, and local and nonlocal diffusions. The phenomenon that mixing reduces growth and hastens decay could be described as a universal phenomenon.
Cholera is a severe water-borne disease that remains a global threat to public health. In this talk, we review some recent studies on cholera dynamics from the viewpoint of mathematical epidemiology. We then present a generalized cholera model which explicitly incorporates both human-to-human and environment-to-human transmission pathways, and which unifies several existing deterministic models. We conduct a stability analysis on the local and global dynamics, and demonstrate the application of this framework with realistic case study. Some results from optimal control simulation are also discussed. In addition, We extend the model to periodic environments to investigate cholera transmission with seasonal variation.
Regulation or management to a constant set–point is fundamental across science and engineering. In conservation management or pest control, population managers aim to regulate the population to a desired density. In order to be useful in applications, set–point regulation should be robust to parametric uncertainty and measurement errors. We address how set–point regulation can be achieved in a robust way. We describe the control theory concept of Integral Control. Integral control is a simple yet powerful technique developed by control engineers, which is ubiquitous in engineering but has not yet (to our knowledge) received attention in population dynamics. One striking feature of integral controllers is that they can be implemented on the basis of both minimal knowledge of the system to be managed or regulated, and in the presence of considerable system uncertainty. This that makes them appealing for population management/conservation, where uncertainty and incomplete measurements are expected. In this talk we discuss the theory of integral controllers and give hypothetical examples.
November 19, 2013; 10:20-11:15am
I will introduce mathematical models describing the influence of external factors
in the temporal dynamics of populations. One model incorporates climate
factors into the dynamics of seasonal influenza through three ecology-based response
functions: response of influenza virus survival and human susceptibility
to air temperature as well as influenza virus transmission response to specific
humidity. I will discuss numerical simulation results obtained when the model
are driven by temperature and specific humidity data. Interestingly, the models
reproduce not only the reported double peaks of influenza A cases in subtropicalregion,
but also the observed temporal pattern of flu in temperate regions (one
Two other models incorporating the effect of insect outbreaks either as a single disturbance in the forest population dynamics or coupled with wildfire disturbances. The results show that 1) the beetle-tree system parameterized model exhibits the well known temporal dynamics of beetle-tree interaction described by the dual equilibria theory. 2) The beetle-tree-fire model reveals the existence of positive feedback between wildfire and insect outbreak disturbances in certain region of fire strength. This result agrees with one of the current theories in the field.
Not only carbon (C) but also nutrient elements such as nitrogen (N) and phosphorous (P) are pivotal for organismal growth, reproduction, and maintenance. Newly emerging mathematical models linking population dynamics with these key elements greatly improve historic trophic interaction models and resolve many existing paradoxes. Most of these models assume strict homeostasis in heterotroph and non-homeostasis in autotroph due to the fact that the stoichiometric variability of heterotroph is much less than that of autotroph. Via bifurcations we study when the "strict homeostasis" assumption is sound and when not. Incorporating light dependence on the growth of autotrophs, the resulting dynamics reveal a series of homoclinic and heteroclinic bifurcations in low light conditions giving the explanation for why microcosm experiments have had unreliable results in low light conditions.