In this talk, we present a discrete-time population dynamics model for snow crab in the southern Gulf of St. Lawrence that incorporates sex-specific developmental stages: immature, adolescent, and adult males, and immature, prepubescent, and adult females. The model includes density-dependent interactions through intercohort cannibalism, while excluding groundfish predation due to its limited influence in this ecosystem.

Using this framework, we analyze how biological interactions and demographic processes influence snow crab population dynamics. In particular, we investigate the effects of cannibalism, recruitment variability, natural mortality, fertility, and fishing mortality on long-term population behaviour. We also examine the conditions under which the model exhibits stable equilibria or cyclic dynamics through the use of dynamical systems tools such as bifurcation analysis and periodograms.

This work aims to improve understanding of the mechanisms driving variability in snow crab populations and to contribute to the development of stock assessment models that better account for nonlinear biological processes in a changing environment.

Being too small to move on their own, snow crab larvae are transported by water currents across the Gulf of St. Lawrence. Their small size also causes them to behave similarly to suspended particles in water, making diffusion an important factor in their movement.

The transport and dispersion of snow crab larvae can therefore be modeled using the advection-diffusion equation :

\begin{equation*} \frac{\partial C}{\partial t} + u\frac{\partial C}{\partial x} + v\frac{\partial C}{\partial y} = D \left(\frac{\partial^2 C}{\partial x^2} + \frac{\partial^2 C}{\partial y^2}\right) \end{equation*}

In this equation, the advection velocities $u$ and $v$ represent ocean current data provided by Fisheries and Oceans Canada. Because the available data was not sufficiently detailed for direct mathematical modeling, it was interpolated using the Kriging method. The model also includes a diffusion coefficient $D$ and the larval concentration field $C$, which must be solved for numerically.

Since no realistic analytical solution exists for this system, numerical approximation techniques are required. A finite element method was therefore applied to solve the advection–diffusion equation on a computational mesh designed to simulate the conditions of the Gulf of St. Lawrence.

Snow crab (Chionoecetes opilio) is a major component of the maritime economy in the Gulf of St.~Lawrence, with landings generating over \$300 million in 2021 according to Fisheries and Oceans Canada (DFO). Beyond its commercial importance, this species plays a key ecological role, but its management is complicated by high sensitivity to climate change and a life cycle characterized by discontinuous growth and cannibalistic behavior.

In this work, we investigate the relevance of continuous-time modeling for describing the dynamics of this resource using a stage-structured framework distinguishing immature and mature crabs. We compare two approaches: a model incorporating cannibalism as a trophic regulation mechanism, and a delay differential equation model accounting for maturation time.

We conduct a mathematical analysis of both models, focusing on equilibrium points and the occurrence of Hopf bifurcations that may explain observed population oscillations. Analytical results are complemented by numerical simulations to assess the ability of each model to reproduce stock dynamics. These findings provide insight into the suitability of stage-structured continuous-time models for understanding snow crab population cycles and supporting sustainable fisheries management under changing environmental conditions.

Continuation methods, such as the Moore-Penrose continuation, provide an effective framework for tracking equilibrium solutions as parameters vary and can successfully handle many types of bifurcations. However, in more extreme cases of bifurcations, these methods may exhibit convergence difficulties. Detecting such problematic regions in advance would therefore significantly improve the reliability of numerical simulations.

In this work, we propose a deep learning approach for the early detection and classification of bifurcation points from time-series data. The model is trained to recognize characteristic patterns associated with different bifurcation types. Once validated, the model will be integrated with the Moore-Penrose continuation to improve the quality of its simulations. The proposed framework will then be tested on a variety of dynamical systems arising in applied mathematics and population dynamics.