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My primary research interest is understanding the behavior of complex systems and their models. This is at the intersection of mathematics (methods and theorems from PDEs, dynamical systems, linear algebra, numerical techniques), physics (modeling approach, ideas from thermodynamics, statistical physics, synergetics), and biology or ecology (a lot of objects for modeling). The style of research may be very different -- from rigorous proofs (ideally) to numerical and non-rigorous approximate studies.
My primary research interest is understanding the behavior of complex systems and their models. This problem is at the intersection of mathematics (methods and theorems from PDEs, dynamical systems, linear algebra, numerical techniques), physics (modeling approach, ideas from thermodynamics, statistical physics, synergetics), and biology or ecology (a lot of objects for modeling). The style of research may be very different -- from rigorous proofs (ideally) to numerical and non-rigorous approximate studies. The models may be both mathematical (dynamical systems) and statistical.
My major most recent projects, which I did at the Centre for Mathematical Biology, include:
- Modeling of chronic wasting disease in Alberta deer. This is the only prion disease known in wild animals. A lot of information about the disease transmission is still unknown, so building a model includes developing and testing many submodels for specific transmission paths. Seasonal changes in deer social behavior additionally complicate the picture. From practical point of view most interesting is the possibility to control the disease spread, and one of the modeling tasks was to test various management scenarios. One of interesting findings is that for the disease management important are not only the disease transmission paths, but also density-dependent mechanisms of population self-regulation.
- Building a statistical model of invasion risk for an aquatic invader spiny waterflee in Ontario lakes. This project included two parts. First, we built a stochastic gravity model, giving an approximation of intensity of boater traffic between two lakes as a function of the lakes properties (mainly area and the distance between the lakes). We did a model fit to data collected by J. Muirhead earlier and model selection to find the best functional form and the best set of lake parameters for the model. The second part was to build a model for waterflee invasion risk using the results of the gravity model and the data obtained by N. Yan's group about 300 Ontario lakes: presence/absence of the invader and 16 parameters of lake water. Again we did model selection to find the best predictor of the invasion. One of the outcomes of the work was a conclusion that BIC criterion works better than AIC for selecting the best set of model parameters. This work was within CAISN network project.
Major topics of my past research are:
- Spatial models of invasion control. We considered a networks of lakes where invasion spreads, and the problem to optimize the control efforts at each lake to slow down or prevent the invasion. For invasion stopping the problem appears not too complicated since analysis of a steady state is enough. However, optimization of control in space in time requires stochastic dynamic programming, which fails to be solvable for more than ~15 lakes. I developed a method of finding an approximate solution basing on the ideas of neurodynamic programming. In model calculations it provided a reasonably good control policy for 50 lakes, which is close to true ecological problems.
Modeling and analysis of complex behavior have many applications in ecology, biology, neuroscience as well as in practical management. I hope for new interesting challenges in my future work
- Application of PCA and cluster analysis for analysis of molecular dynamics trajectories for proteins to detect functional groups of atoms.
- Influence of climate change on species competition. We considered competition of two species in space in time in a patch that moves due to isotherm shift. We proved several theorem about species coexistence within a patch, and found numerically two interesting effects: increased invasibility at one patch end and change of the competition outcome with the increase of patch speed.
- Complex spatio-temporal behavior in the models of chemotaxis. We studied metastable structures with extremely long lifetime and developed approximate models for this type of structures.
- Bioeconomic models of invasion and optimal control for Great Lakes (as a part of the ISIS project and David Lodge lab at the University of Notre Dame). We try to unify model of invasion, control, and accounting for the corresponding costs. This brings an optimal control problem, for which we obtained some new results, partially analytical, partially numerical.
- Dissipative structures and their spatial organization;
- Dynamical chaos, properties of nonlinear models;
- Nonlinear time series analysis by methods of nonlinear dynamics and corresponding numerical techniques -- attractor dimension, Lyapunov exponents, etc.;
- Complex behavior in information processing neural networks.