The paradox of plankton

Abstract

The principle of competitive exclusion states that, in equilibrium, the amount of coexisting species is limited by the amount of limiting resource types in an ecosystem. However, in phytoplankton communities the paradox of plankton appears, amounts of plankton species can coexist that far exceed this
upper limit. A resource competition model is formulated and it is shown that the paradox arises for several systems, which indicates that the bloom in biodiversity is a result of the resource competition and not of any external factors. A proof is given that the principle of competitive exclusion only holds in equilibrium solutions. Therefore, as long as a system does not intersect with an equilibrium solution the biodiversity is not restricted by the amount of limiting resource types. It is concluded that intersecting with an equilibrium solution is avoided when there are only unstable equilibrium solutions present in the system. When a plankton species allows an asymptotically stable equilibrium solution, with a region of convergence equal to the domain of the system, to appear it will be called dominant. It is proven that an asymptotically stable equilibrium solution always exists in a simplified system with less than three limiting resource types. Furthermore, an algorithm is constructed that determines all the new equilibrium solutions, and their respective stabilities, when a new plankton species is introduced to a system.
By applying this algorithm it can be determined whether a species is suitable for an ecosystem, when the goal is to maintain biodiversity. The resource competition model is expanded to include light as an additional resource for all plankton species. It is observed that the coexistence of the plankton species and the total biomass is limited
if there is too little light for the plankton species to consume, or if one plankton species becomes dominant due to it being significantly better at consuming light than the other species.
Additionally, the physical context of a flowing river is introduced, with dispersive and advective mass transfer and finite length. It is observed that while the spatial distribution of the plankton species along the river is strongly influenced by the spatial parameters, the biodiversity of the ecosystem is still primarily determined by the original parameters from the resource competition model, as long as the dispersive mass transfer is the dominant type of mass transfer not too large in comparison to the length of the river.