A replicator model with transport dynamics on networks for species evolution

This paper proposes a network-based framework to model and analyze the evolution and dynamics of a marine ecosystem. The model involves two different length scales: the evolution of species in local reserves and the exchange of species between reserves. At the inter-reserve level, species evolution is ruled by the replicator equation, while a transport function accounts for the transport at the network level. This multi-scale approach allows for capturing both local dynamics within individual reserves and the broader connectivity and interactions across the network. We study how equilibria are modified due to the exchange between connected nodes and prove that evolutionarily stable states are asymptotically stable if the velocity transfer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ν</mml:mi></mml:math> is contained within a condition involving the maximum degree of the network. A fourth-order P-(EC) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mmultiscripts><mml:mrow></mml:mrow> <mml:mrow></mml:mrow> <mml:mi>k</mml:mi></mml:mmultiscripts> </mml:math> formulation of the Gauss-Legendre Runge Kutta scheme is adopted. This numerical procedure is challenged against a suitable numerical experiment involving three species on a single node for validating the robustness of the scheme in terms of accuracy for a large observation time. Several numerical experiments are provided for characterizing the abilities and limitations of the model. Three prototypical networks are considered for the case of two- and three-agent games with both linear and nonlinear transport terms. Moreover, the ability of the proposed model to reproduce synchronization phenomena on networks is discussed. This approach has been demonstrated to have the potential to uncover insights into the stability, resilience, and long-term behavior of these ecosystems, offering valuable tools for their conservation and management.