Conformally symplectic systems send a symplectic form into a multiple of itself. They appear in mecanical systems with friction proportional to the velocity and as Euler-Lagrange equations of the time discounted actions common in economics. The conformally symplectic structure provides identities that we use to prove "a-posteriori" theorems that show that if we have an approximate solution which satisfies some non-degeneracy conditions, we can obtain a true solution close to the approximate one. The identities used to prove the theorem, also lead to very efficient algorithms and computer assisted proofs.
Ixodes scapularis is a tick vector of several zoonosis (such as Lyme disease) in northeastern North America. Many field studies have been shown that both climate change and migratory birds have crucial effects on range expansion of I. scapularis in Canada. In this study, we use mathematical model to examine the effects of migration timing, migration speed, bird abundance, tick load and landscape barriers to migration on the ability of birds to connect the source region and destination region of the tick population. Firstly, we propose a within- patch model to describe the local dynamics of I. scapularis population, and we establish the threshold condition for the tick population to survive. Then we move on to consider a patch model with delay to study the process of long distance dispersal of the tick vector of Lyme disease carried by bird migration. This is a jointly ongoing work with P. Leighton, J. Wu and X. Zou.
Microbial communities are commonly observed in nature. It has been observed that such communities seem to be more efficient in use of resources as well as more robust, than monocultures. We study a simple synthetic microbial community, where two metabolic pathways of E.coli are split into two different strains of E.coli. Thorough analysis of a model shows that the total biomass of the consortium must be smaller than that of monoculture under very general conditions on the shape of the growth curves. This contradicts the experimental observations. To explain this observation, we conclude that some adaptation must have occurred in the members of the consortium.
Active regulation in gene networks poses mathematical challenges regarding flow in transition regions that have led to conflicting approaches to analysis. That is, competing regulation that keeps concentrations of some transcription factors at or near threshold values leads to so-called singular dynamics when steeply sigmoidal interactions are approximated by step functions. We have argued that the spurious solutions of the Filippov approach could be avoided, and an extension, due to Artstein and coauthors, of the classical singular perturbation approach is an appropriate way to handle the most complex situation, where non-trivial dynamics of fast variables occurs in singular domains. Now we show that even in this context, it is possible for non-uniqueness to arise in such a system in the case of limiting step-function interactions, and thus cannot be avoided completely, even if we avoid the overly inclusive set-valued Filippov definition of solutions. Real gene networks have sigmoidal interactions, however, and in the examples considered here, it is shown that the corresponding behaviour in smooth systems is a sensitivity to initial conditions that leads in the limit to densely interwoven basins of attraction of different attractors.