NPZ model structure, equations, and parameter

The NPZ model equations can be generalised as: dP dt = (LI×UN) P − GZP− ρP (1.1) dN dt = ρ P + ζ Z + (1 − λ )GZP−UNP (1.2) dZ dt = GzPZ− ζ Z (1.3)

Which describe the rate of change in phytoplankton (P), nutrient (N), and zooplankton (Z) with time. The general equations also shows how the changes in each compartment affect each other. Unlike a physical oceanography model which has the navier-stokes equations, the biogeochemical model does not have a known set of equations that describe different processes, which allows modellers to choose or

1.2. Early marine biogeochemical model: the NPZ model 5

derive their own functional forms to describe each biogeochemical process, as long as the model fits with experimental data (Flora et al., 2011). All of the governing process functional forms range from linear equations, that describe constant rate of process (density-independent), to nonlinear forms, where at first the rate increases with concentrations, and then saturates when it reaches the maximum rate (density- dependent) (Franks, 2002), as shown in Figure 1.2. Nonlinear forms have both a maximum rate (µ) and a shape-defining coefficient (usually a half saturation constant, k), and linear functions only have a constant rate. This makes the choice of functional form crucial as it can affect the dynamics of the model, and the choice of the parameter values that dictate the rate of grazing or nutrient uptake rates at a certain concentration. The next subsections will discuss the different functional forms and their parameters for each governing process within an NPZ model.

Figure 1.2: Common shapes of functional forms that describe the key processes in biogeochemical models. Saturating response is shown in hyperbolic and sigmoidal.

Phytoplankton Growth (LIand UN)

From equation 1.1, the phytoplankton growth is limited by light and nutrients. Phytoplankton growth rate in response to irradiance can be calculated using linear, saturating (using Hyperbolic, trigonometric, and exponential functions), or saturating with photoinhibiting response, summarised in Table 1.1, with a parameter to determine the irradiance at photosynthesis maximum, Io(Franks, 2002). In a more complex

NPZ model, where the mixed layer depth is taken into consideration (which assumes that the biological aspect of the model sits above the deep homogeneous layer with constant nutrient and no plankton (Anderson et al., 2015)), the total growth due to light has to be averaged over the mixed layer, described by: J(t, M) = Z 24h 0 Z M 0 LI(I(t, z))dzdt (1.4)

where J is the light limited phytoplankton growth, t is the time (days), M is the mixed layer depth, and 24h is the day length. The linear and saturating response functions can be chosen to match observed change in phytoplankton light limited growth rate over one day or how well they represent the nonlinear response of photosynthesis to irradiance (Franks, 2002). In order to make the functional forms described in Table 1.1 more similar to the laboratory experiment, instead of only using the term Io, it is possible

to split this term into two different parameters, such as in Smith (1936) function, which is similar to a hyperbolic saturating response. In Smith (1936) function, the term α/Vpmax is used to describe Io,

where α is the initial slope of the P-I curve, and maximum photosynthesis rate is described using Vpmax.

Similarly, the Ioterm in exponential function can also be split into α/Vpmax, and this have been used in

the recent NPZ model to described the light limited growth of phytoplankton (Anderson et al., 2015).

The most common function to represent the nutrient uptake is the logistic saturating rectangular hyper- bolic, also known as the Michaelis-Menten or Monod formulation, based on enzyme kinetics. As shown in Figure 1.2, this equation has a shape defining parameter, ku, and a maximum rate, in Vumax. This kind

of function is computationally cheap as it is not explicitly linked to resource availability (Flynn, 2018). However, phytoplankton have been shown to store nutrients in an internal pool before they are used for growth (Droop, 1973, 1983). The quota (Q) describes the amount of a substance in a unicellular organ- ism, and this controls the growth of phytoplankton. Q varies between the subsistence quota (Qmin), in

which it is impossible to use the nutrient for growing, and a growth maximum rate (Qmax) (Franks, 2002).

It should be noted that the Qmax here is not for the final phytoplankton growth as this is also limited by

light, as described in equation 1.1.

Zooplankton Grazing (Gz)

In a simple NPZ model, zooplankton would only graze one type of prey, the phytoplankton. Most of the formulations are based on laboratory experiment where populations of predators are acclimatised to different prey densities (Holling, 1959). From these experiments, there are four ‘Types’ of zooplankton grazing; (i) linear variations of grazing rate with prey, (ii) curved variation grazing rate with prey (Type II), which used rectangular hyperbolic function, and are often called Holling type II, (iii) another curved variations of grazing, which is similar to the sigmoidal curve, described in Figure 1.2 and is often referred to Holling type III, and (iv) grazing rate that reaches maximum at an intermediate density and decreases as the density gets higher, which mimics when the prey is toxic or the predator gets confused (Holling, 1959; Gentleman et al., 2003). The difference between type II and III is that the grazing rate in the former is higher at lower density prey concentration, compared to the latter, as shown in Figure 1.2. However, Flynn and Mitra (2016) argued that these hyperbolic functional forms do not explicitly represent the

1.2. Early marine biogeochemical model: the NPZ model 7

effect of physical oceanography processes (such as turbulence) when encountering prey, or how the motility of zooplankton changes when satiation or saturation occurred.

The first type of grazing has no satiation, and is formulated using a linear function, but there is no theoretical basis to this equation. The second type is described using a Disk Equation, which is based on predator-prey theory, and defined by handling and attack rate. The Michaelis-Menten (rectangular hyperbolic equation), is mathematically equivalent to the Disk equation, however instead of defining the equation using attack, a, and handling time, h, rectangular hyperbolic has maximum rate (gmax) and half

saturation constant (kg), where in this case, gmaxbecomes 1_h, and kgbecomes_ah1 (Gentleman et al., 2003).

The Michaelis-Menten type function has been used in early NPZ models (Fasham et al., 1990; Steele and Henderson, 1992) and even a more complex model with various types of phytoplankton (Follows et al., 2007; Prowe et al., 2012), rather than the actual Disk equation. Similar response curves can also be formulated using Ivlev functional forms (Ivlev, 1961), however the rate of change in Ivlev is different to the Disk equation, and has a shape defining functions λ instead of kgor a and h (Gentleman, 2002). The

third type can also be described using a Disk equation, however, the attack rate would vary linearly with prey density, according to a constant (a= constant × prey). Using a Michaelis-Menten type equation, this could be described using a sigmoidal (Holling type III) equation, with gmax= 1/h and kg= 1/

√ ch. The Holling type III function is more commonly used in later biogeochemical models (Fasham et al., 1993; Edwards and Brindley, 1996; Edwards and Yool, 2000; Palmer and Totterdell, 2001; Anderson et al., 2015), and is referred as Holling type III. The fourth type can be described using Holling type II, but with an additional term in the denominator that results in maximum and half saturation constants depending on complicated functions of other parameters (Gentleman et al., 2003). These equations are described in Table 1.1.

Plankton Mortality (ρ and ζ )

The mortality terms in the NPZ model allow the recycling of nutrients from phytoplankton and zoo- plankton. This is usually modelled using either linear, quadratic, hyperbolic, or sigmoidal functions. The linear mortality has a constant rate independent of plankton concentrations, which may represent higher predators when this is used for zooplankton, whose biomass does not fluctuate (Edwards and Yool, 2000). In an NPZ model, the phytoplankton mortality term is usually modelled using a linear or a quadratic function (Steele and Henderson, 1981, 1992; Fasham et al., 1990; Fasham, 1995; Edwards and Brindley, 1996), although it is possible to represent phytoplankton mortality using a Michaelis-Menten type equation (Fasham et al., 1993). However in a more recent NPZ model, such as that of Anderson et al. (2015), both linear and quadratic phytoplankton mortality is used. This is because the linear mor-

tality account for metabolic losses or natural mortality, and the density dependent loss (non-linear) may represent mortality due to infection or viruses (Palmer and Totterdell, 2001; Yool et al., 2011, 2013; Anderson et al., 2015).

Zooplankton mortality terms have been shown to alter model dynamic more than altering the parameter values when the equations are altered. Similar to phytoplankton mortality, in an NPZ model, this process is usually represented by linear mortality, however instead of producing a steady state, it produces os- cillations over a wider parameter range (Edwards and Yool, 2000; Franks, 2002). The range of different functional forms for zooplankton mortality may be due to the various feeding strategies of zooplankton. When cannibalism occurs within the zooplankton compartment, or if the population of higher predators changes proportionally with zooplankton, the mortality term may be represented using quadratic mor- tality. This functional form will have a mortality rate that depends on the zooplankton biomass. The hyperbolic and sigmoidal forms mimic the higher predator grazing strategy (Edwards and Yool, 2000), similar to the zooplankton grazing. Non-linear mortality functions do not produce limit cycles (Steele and Henderson, 1992), and therefore are more preferred. When compared to in situ data, using quadratic mortality produces the largest deviations compared to sigmoidal and hyperbolic (Mitra, 2009). Although it is suggested that using only one mortality term is inappropriate and necessary in a planktonic food web model as the description of zooplankton loss processes is biologically inaccurate (Mitra, 2009), more recent simple NPZD model (e.g. Anderson et al. (2015)), and even more complex biogeochemical model such as Palmer and Totterdell (2001); Dutkiewicz et al. (2009); Halloran et al. (2010) still use these density dependent mortality functions, whose equations are summarised on Table 1.1.