Tiger mosquitoes (Aedes albopictus) are vectors of may diseases like dengue, Zika and chikungunya. They have been progressively invading many regions of the world, including Europe, due to global trade and global warming.
In this talk, I will present an overview of a series of recent works on mathematical models for mosquito population dynamics and on how they can be used to help control the mosquito population in order to reduce the risk of having epidemies of the diseases they transmit.
Certain biological processes - including neural dynamics - are characterized by transient dynamics that spend some time near one metastable state before a rapid transition to another state. Heteroclinic structures in phase space can provide an organizing center for such dynamics. We discuss heteroclinic dynamics in the context of transient synchronization and desynchronization of oscillators and their stability. This allows to gain insights how the dynamics bifurcate as the heteroclinic structures are broken. If time permits, we briefly touch upon on the possible complexity of dynamics near heteroclinic structures; this has implications for the computational ability of transient dynamics organized by heteroclinic structures.
One speaks of cross diffusion when in a system of equations, the diffusion rate of one unknown depends on the value (or the gradient) of another unknown. The appearance of cross diffusion in a system usually makes the mathematical analysis difficult and sometimes leads to a rich structure (for example complex bifurcations diagrams). Cross diffusion naturally appears in problems coming out physics (Maxwell-Stefan systems for mixtures of gases), or of biology (parabolic Keller-Segel equations in chemotaxis). We focus in this talk on systems coming out of population dynamics, and discuss the modeling and the mathematical structure of typical such systems.
Consider a nonlinear autonomous system of many ODE's describing N degrees of freedom randomly coupled via a smooth homogeneous Gaussian vector field with both gradient and divergence-free components. In the absence of coupling the system is assumed exponentially relaxing to a single equilibrium with a certain rate. We show that with increasing the ratio of the coupling strength to the relaxation rate such systems for N tending to infinity generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We also calculate the mean proportion of equilibria that have a fixed fraction of unstable directions. This picture provides a global view on the nature of the May−Wigner instability transition originally discovered by local linear stability analysis.
A popular model for suspensions of non-spherical particles in viscous fluids is the so-called Doi model, which is a nonlinear transport equation for the distribution of particles in space and orientation. The Doi model comes from a formal mean-field limit of a large system of particles interacting inside a Stokes flow. We will show in this talk that this formal limit is not accurate and will rigorously derive a correction to the model, under natural assumptions on the initial distribution of the particles. This is joint work with R. Höfer (Regensburg university).
Embryo development is a dynamic process governed by the regulation of timing and sequences of gene expression, which control the proper growth of the organism. Although many genetic programmes coordinating these sequences are common across species, the timescales of gene expression can vary significantly among different organisms. Currently, substantial experimental efforts are focused on identifying molecular mechanisms that control these temporal aspects. In contrast, the capacity of established mathematical models to incorporate tempo control while maintaining the same dynamical landscape remains less understood. In our recent work we have addressed this gap by developing a mathematical framework that links the functionality of developmental programmes to the corresponding gene expression orbits (or landscapes). We demonstrate that this framework allows for the prediction of molecular mechanisms governing tempo in synthetic networks such as the repressilator, and the gene networks governing brain development.
We study a branching particle system of diffusion processes on the real line interacting through their rank in the system. Namely, each particle follows an independent Brownian motion, but only K ≥ 1 particles on the far right are allowed to branch with constant rate, whilst the remaining particles have an additional positive drift of intensity χ > 0. This is the so called Go or Grow hypothesis, which serves as an elementary hypothesis to model cells in a capillary tube moving upwards a chemical gradient.
Despite the discontinuous character of the coefficients for the movement of particles and their demographic events, we first obtain the limit behavior of the population as K → ∞ by weighting the individuals by 1/K. Then, on the microscopic level when K is fixed, we investigate numerically the speed of propagation of the particles and recover a threshold behavior according to the parameter χ consistent with the already known behavior of the limit. Finally, by studying numerically the ancestral lineages we categorize the traveling fronts as pushed or pulled according to the critical parameter χ. This is a joint work with M. Demircigil (U. Arizona).