Phytoplankton-zooplankton model

Edwards, A. M. (2001). Adding detritus to a nutrient-phytoplankton—zooplankton model a dynamical-systems approach. J. Plankton Res. 23, 389-413. 

We have seen that chemical and biological interactions lead to mathematical models displaying a variety of linear and nonlinear behavior relaxation to fixed points, multistability, excitability, oscillations, chaos, etc. Despite the different origin of the models, and the diverse nature of the variables they represent (chemical concentrations, population numbers, or even membrane electric potentials) the mathematical structures are quite similar, and it is possible to understand some aspects of the dynamics in one field (e.g. the chemical oscillations in the BZ reaction) with the help of models from other fields (for example the FN model of neurophysiology, or a phytoplankton-zooplankton model). This possibility of common mathematical description will be used in the rest of the book to highlight the similarities and relationships between chemical and biological dynamics when occurring in fluid flows. 

Fig. 2.5 Timeseries of daily phytoplankton, zooplankton, dissolved organic carbon, detritus, and phosphorus concentration, and photosyntesis over one model year at two location the shelf seas of the Pacific Ocean, 170 E 65 N and 140 E 10 S.

The predominance of phytoplankton-derived carbon in diets of many fish species, despite its small contribution to floodplain production, can be explained by the selective consumption of algae. However, Bayley (1989) argued that phytoplankton production was too low to contribute significantly to regional fish production. He based his argument on a food chain with three trophic levels (phytoplankton—zooplankton— fish), and assumed a 10% transfer efficiency between trophic levels. This model is inappropriate for detritivorous and herbivorous fish which consume plant materials directly. A model with two trophic levels would be more appropriate and would indicate a higher potential contribution to... 

The models proposed by Riley et al. and Steele are basically similar. Each consider the primary dependent variables to be the phytoplankton, zooplankton, and nutrient concentration. A conservation of mass equa-... 

Other models have been proposed which follow the outlines of the equations already discussed. Equations with parameters that vary as a function of temperature, sunlight, and nutrient concentration have been presented by Davidson and Clymer (9) and simulated by Cole (10). A set of equations which model the population of phytoplankton, zooplankton, and a species of fish in a large lake have been presented by Parker (II). The application of the techniques of phytoplankton modeling to the problem of eutrophication in rivers and estuaries has been proposed by Chen (12). The interrelations between the nitrogen cycle and the phytoplankton population in the Potomac Estuary has been investigated using a feed-forward-feed-back model of the dependent variables, which interact linearly following first order kinetics (13). 

A model of the dynamics of phytoplankton populations based on the principle of conservation of mass has been presented. The growth and death kinetic formulations of the phytoplankton and zooplankton have been empirically determined by an analysis of existing experimental data. Mathematical expressions which are approximations to the biological mechanisms controlling the population are added to the mass transport terms of the conservation equation for phytoplankton, zooplankton, and nutrient mass in order to obtain the equations for the phytoplankton model. The resulting equations are compared with two years data from the tidal portion of the San Joaquin River, California. Similar comparisons have been made for the lower portion of Delta and are reported elsewhere (62). 

Oscillations can also arise from the nonlinear interactions present in population dynamics (e.g. predator-prey systems). Mixing in this context is relevant for oceanic plankton populations. Phytoplankton-zooplankton (PZ) and other more complicated plankton population models often exhibit oscillatory solutions (see e.g. Edwards and Yool (2000)). Huisman and Weissing (1999) have shown that oscillations and chaotic fluctuations generated by the plankton population dynamics can provide a mechanism for the coexistence of the huge number of plankton species competing for only a few key resources (the plankton paradox ). In this chapter we review theoretical, numerical and experimental work on unsteady (mainly oscillatory) systems in the presence of mixing and stirring. 

A simple dynamical box-model was constructed in order to test the seasonal variation features of phytoplankton, zooplankton, DIN, DIP, DOC, POC, as well as dissolved oxygen (DO) in the northern part of Jiaozhou Bay in 1995. The annual variations of the phytoplankton production show two high value periods (Mar. to Apr. and July to Aug.) and two low value periods (May to July and after Oct ). DOC shows the common features it is high in summer and low in winter (Wu and Yu, 1999). Their model equations describing the cycles of phytoplankton and OC in the ocean are as follows ... 

Fig. 32.14. Idealized nutrients cycling in Tokyo Bay s ecosystem in the model of Koibuchi et aX. Cycling between the 13 state variables phytoplanktons, zooplanktons, nutrients (nitrogen, phosphorus, and silicate), labile detritus, and refractory detritus for each nutrients and dissolved oxygen as well as sedimentation processes of particulate organic material.

Fig. 4. Compartmental model describing the cycling of nitrogen in a planktonic community in the mixed layer of a water column. Flow pathways are represented by arrows and numbers which correspond to mathematical expressions described in Table 2. The nitrogen pool represents all abiotic nitrogen (nitrate, ammonia and urea), and other compartments represent bacteria, zooflagellates, larger protozoa, and micro-mesozooplankton, giving off waste products (F+U). Arrows (13) and (14) depict sedimentation of zooplankton faeces and phytoplankton cells, respectively (After Moloney et al., 1985).

The simulation model depicts the flows of nitrogen between the compartments, and in particular is used to investigate the effect of sedimentation of phytoplankton and zooplankton faeces out of the euphotic zone which is assumed to be 60 m deep however the depth does not affect the conclusions drawn from the results. 

Like algae, phytoplankton, and macrophytes, an equilibrium partitioning model commonly is used to estimate the chemical concentration in zooplankton (Cz, g chemical/kg organism). Hence, Cz is the product of the freely dissolved chemical concentration in the water (CVVI), g chemical/L water), the lipid content of zooplankton (Lz, kg lipid/kg organism), and the lipid-water partition coefficient (KL, L water/kg lipid), which the octanol-water partition coefficient approximates (Clayton et al. 1977) ... 

Gobas (1993) published a foodweb bioaccumulation model to predict chemical concentrations in phytoplankton, macrophytes, zooplankton, benthic invertebrates, and fish, based... 

The model applies equilibrium partitioning to estimate chemical concentrations in phytoplankton, macrophytes, zooplankton, and benthic invertebrates. Chemical concentrations in sediment and water, along with environmental and trophodynamic information, are used to quantify chemical concentrations in all aquatic biota. This model can be applied to many aquatic food webs and relies on a relatively small set of input parameters which are readily accessible. 

Ray S, Berec L, Straskraba M, Jorgensen SE. 2001. Optimization of exergy and implications of body sizes of phytoplankton and zooplankton in an aquatic ecosystem model. Ecol Model 140 219-234. 

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