Research

Gaining a mechanistic understanding of spatial biodiversity patterns

Biodiversity results from differentiation mechanisms developing within biological populations. Such mechanisms are influenced by the properties of the landscape over which individuals interact, disperse and evolve. The documentation of high levels of species diversity in complex mountain region or riverine systems suggest that some peculiar landscape properties foster diversity.

Hengduan mountains in China

Hengduan mountains in China. Montane regions are more biodiverse than other environments on Earth. Why so?

I use mathematical models to investigate how connectivity and habitat heterogeneity foster diversity. My modeling approach complements empirical work on biodiversity by connecting documented patterns to the processes that have generated it.

Hengduan mountains in China

Effect of habitat heterogeneity on differentiation in a mathematical model where the vertices of the graph, corresponding to geographical areas, are assigned different environmental conditions favoring different niche traits. The model consists of individuals possessing traits that are neutral ($u$-traits) and adaptive ($s$-trait). Offsprings can migrate to neighboring vertices and experience mutations. On the adaptive trait dimension $s$, two clusters emerge around the optimal values $\theta$. The configuration of the environmental conditions $\theta$ determine the differentiation of populations across the vertices.

Preprint available here

Developping numerical schemes for solving highly dimensional non-local nonlinear PDEs

Partial Differential Equations (PDEs) are equations that arise in a variety of models in physics, engineering, finance and biology. I develop numerical schemes based on machine learning techniques to solve for a special class of PDEs (cf below) in high dimension.

$$ \begin{equation} \begin{aligned} (\tfrac{\partial}{\partial t}u)(t,x) &= \int_{D} f\big(t,x,{\bf x}, u(t,x),u(t,{\bf x}), ( \nabla_x u )(t,x ),( \nabla_x u )(t,{\bf x} ) \big) \, \nu_x(d{\bf x}) \\ & \quad + \big\langle \mu(t,x), ( \nabla_x u )( t,x ) \big\rangle + \tfrac{ 1 }{ 2 } \text{Trace}\!\big( \sigma(t,x) [ \sigma(t,x) ]^* ( \text{Hess}_x u)( t,x ) \big). \end{aligned} \end{equation} $$


The equation appearing above defines the class of PDEs I am interested in. Such PDEs are also referred in the literature as non-local reaction diffusion equations.

Such PDEs permit to e.g. model the evolution of biological populations in a realistic manner, by describing the dynamics of several traits characterising individuals (for a population of birds, think of traits as the beak length, the body size, or the color of the birds).

Neural network architectures

Neural network architectures used to approximate solutions to Eq. (1). The output neuron $\mathbb{V}_n(\theta ,x)$ approximates $u(t_n,x)$.

I have implemented those schemes in HighDimPDE.jl, a Julia package that should allow scientists to develop models that better capture the complexity of life. These techniques extend beyond biology and are also relevant for other fields, such as finance.

Check out my blog post to read more about it!

Papers in prep.

What can Economics learn from Biology?

The fields of Evolutionary Biology and Economics have mutually exchanged ideas that lead to breakthrough for the past two centuries, starting with Darwin who got inspired from the economist Malthus for developing his theory on Evolution. Indeed, it is more and more acknowledged that companies, which are structured around market niches within which they develop competitive and mutualistic interactions, interact in similar ways as biological species. I rely on the scientific literature, data science and mechanistic models to better understand how evolutionary processes arising in economic systems connect to those observed in biological systems.

Glärnishfirn

As glaciers are melting in mountains, ecological succession proceeds. Mosses and grasses settle on the bare rock after the melting and transform it into soil, slowly allowing vegetation to colonize the new environment until the "climax" forest is reached. I am demonstrating that ecological succession dynamics reflect in many aspects how economies develop.

My work highlights the potential of cross fertilization of ideas and methods developed in apparently unrelated disciplines. It also suggests new directions for economic policies. We could surely get insights from ecological systems, that have survived major environmental crisis for more that 3.5 billion years.

Paper in prep.