A Physical–biological Coupled Model for Algal Dynamics in Lakes

Abstract

A coupled model is presented for simulating physical and biological dynamics in fresh water lakes. The physical model rests upon the assumption that the turbulent kinetic energy in a water column of the lake is fully contained in a mixed layer of variable depth. Below this layer the mechanical energy content is assumed to vanish. Additionally, the horizontal currents are ignored. This one-dimensional two-layered model describes the internal conversion of the mechanical and thermal energy input from the atmosphere into an evolution of the mixed layer depth by entrainment and detrainment mechanisms. It is supposed to form the physical domain in which the simulation of the biological processes takes place. The biological model describes mathematically the typical properties of phyto- and zooplankton, their interactions and their response to the physical environment. This description then allows the study of the behaviour of Lagrangian clusters of virtual plankton that are subjected to such environments. The essence of the model is the dynamical simulation of an arbitrary number of nutrient limited phytoplankton species and one species of zooplankton. The members of the food web above and below affect the model only statically. The model is able to reproduce the typical progression of a predator-prey interaction between phyto- and zooplankton as well as the exploitative competition for nutrients between two phytoplankton species under grazing pressure of Daphnia. It suggests that the influence of the biological system on the physical system results in a weak increase of the surface temperature for coupled simulations, but a considerably higher seasonal thermocline in spring and a lower one in autumn.

Mathematical Modelling of Plankton–Oxygen Dynamics Under the Climate Change

“…Whilst function f (c) describes the rate of increase in the concentration of the dissolved oxygen because of its transport from phytoplankton cells to the surrounding water (see the details below), factor A takes into account the effect of the environmental factors (such as the water temperature) on the rate of oxygen production inside the cells. Function g(c, u) describes the phytoplankton growth rate which is known to be correlated to the rate of photosynthesis [26]; we therefore assume it to depend on the amount of available oxygen. Terms u r and v r in Eq.…”

“…As for the positive equilibria (if any), it does not seem possible to solve Eqs. (26)(27) explicitly. Instead, since system's equilibria are the intersection points of the two isoclines, important conclusions can be made by analyzing the mutual position of the corresponding curves.…”

“…Instead, since system's equilibria are the intersection points of the two isoclines, important conclusions can be made by analyzing the mutual position of the corresponding curves. Figure 2a shows the isoclines (26) and (27) obtained for some hypothetical parameter values. Therefore, system (24)(25) can have two positive equilibria (c 1 ,ũ 1 ) and (c 2 ,ũ 2 ).…”

“…In a more general context, understanding of plankton dynamics is important for a reliable estimation of marine ecosystems productivity [5,38]. We mention here that mathematical modelling has proved to be an efficient research tool to reveal the properties of the plankton dynamics using both simple, conceptual models and more complicated 'realistic' ones [3,5,26,57,64,66,77].…”

Global Warming Can Lead to Depletion of Oxygen by Disrupting Phytoplankton Photosynthesis: A Mathematical Modelling Approach

“…Function M(c, u) describes the rate of oxygen decrease due to different processes such as its consumption by phytoplankton during the night-time, breathing of marine fauna, decay in the oxygen concentration due to (bio)chemical reactions in the water (e.g., decay and remineralization of detritus) [22,[30][31][32][33][34], etc. The phytoplankton growth rate g(c, u) is known to be correlated with the rate of photosynthesis [35]. The mechanisms resulting in this correlation are poorly understood; here, we assume that the growth rate depends on the amount of available oxygen.…”