Parameterizing plankton functional type models: insights from a dynamical systems perspective

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Parameterizing plankton functional type models: insights from a dynamical

systems perspective

ROGER CROPP¹* AND JOHN NORBURY²

1CENTRE FOR ENVIRONMENTAL SYSTEMS RESEARCH,GRIFFITH SCHOOL OF ENVIRONMENT,GRIFFITH UNIVERSITY,NATHAN,QLD4111,AUSTRALIA AND 2MATHEMATICAL INSTITUTE,UNIVERSITY OF OXFORD,24-29ST GILES,OXFORD OX1 3LB,UK

*CORRESPONDING AUTHOR: r.cropp@griffith.edu.au

Received March 31, 2009; accepted in principle May 10, 2009; accepted for publication May 22, 2009; published online 21 June, 2009

Corresponding editor: Roger Harris

The spectre of anthropogenic global climate change has focused attention on biogeochemical cycling in the oceans as marine plankton ecosystems are involved in the cycling of several compounds thought to have significant implications for climate. To better understand these processes, modellers are developing plankton functional type (PFT) models that group plankton according to their biogeochemical properties. There is some debate as to whether our understanding of plankton ecosystems is sufficiently well developed for PFT models to be reliable and for their predictions to be treated with confidence. In this paper, we examine the dynamical properties of a generic predator – prey – prey PFT model, then apply these analysis techniques to a simple example PFT model with two phytoplankton and one zooplankton in order to explore its parameter space. We find that parameter combinations for which all PFTs stay extant for all time appear rare, but develop a simple heuristic that allows such parameter sets to be identified relatively easily for many PFT models. We observe that such systems often have phytoplankton with similar growth rates, but that differ in other properties such as differing nutrient utilization strategies or different susceptibilities to grazing. We also note that persistent PFT systems are more likely if neither phytoplankton have a low specific mortality rate or is a highly nutritious food for the grazer.

I N T RO D U C T I O N

The prospect of anthropogenic climate change has gen- erated substantial interest in the role of plankton in biogeochemical cycling in the ocean. Plankton may have a significant influence on climate by drawing down carbon dioxide from the atmosphere, sequestering it in the deep ocean, and by producing dimethylsulphide and other volatile compounds that may affect cloud for- mation over the oceans. Plankton models that include several plankton functional types (PFTs) are needed to resolve the role of plankton in biogeochemical cycling, as different plankton utilize different elements in different ways. PFTs are therefore often defined according to their biogeochemical role rather than their phylogeny (Falkowskiet al., 2003). The number of PFTs required is

dependent on the purpose of the model, with Hoodet al.

(Hood et al., 2006) noting that the performance of PFT models is more closely related to their tuning than their complexity. Le Que´re´ et al. (Le Que´re´ et al., 2005) suggested that 10 key PFTs were required to resolve cli- matically important biogeochemical cycling in the oceans. However, there has been significant debate over whether plankton ecosystems are sufficiently well understood to place any reliance on the results of models that include multiple PFTs (Anderson, 2005; Le Que´re´, 2006).

Pragmatically, the demand to resolve the roles of different PFTs in biogeochemical cycling in the oceans has meant that PFT models are already being developed and applied. The debate over the merits of includ- ing phytoplankton functional types in the generally

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successful nutrient – phytoplankton – zooplankton (NPZ) models suggests a pressing need for better understanding of the behaviour of these models. The emphasis of attempts to improve the understanding of, and build confidence in, PFT models is often focussed on more and more accurate measurements of PFT traits ( parameter values) and more and more accurate data to calibrate and validate the models (Le Que´re´et al., 2005;

Hood et al., 2006). Here, we investigate the dynamics of a simple, generic model with two phytoplankton functional types to analyse what insights might be gained into the attributes of more complex PFT models.

We choose the model and its parameters to ensure ecological realism under the conditions derived by Kolmogorov (Kolmogorov, 1936) and explicated by May (May, 1973).

Many climate change computer simulations include a model of plankton ecosystem processes. Many of these models of plankton dynamics, both simple and complex (Spitz et al., 2001; Franks, 2002; Cropp et al., 2004;

Vallina et al., 2008), may be classed as Kolmogorov systems, as they are of the general form:

_

ui¼fiðu1;u2;. . .;unÞui; i¼1;2;. . .;n; ð1Þ where u_¼du=dt for t.0, and the functions fi are bounded and continuously differentiable in their variables u1, u2,. . ., un. The fi describe the growth and mortality of each species, trophic guild or PFT, that is fi¼(growth2predation2mortality)i. These fi are often nonlinear (but smooth) functions of u1,u2,. . ., un

and include several parameters that describe the attributes (traits) of the plankton and how they interact.

There are many options for the fi, and much of the debate surrounding the application of PFT models centres on whether the forms of the fiand the values of the parameters that distinguish one PFT from another are sufficiently well understood. However, the fi commonly used in ecosystems models typically meet the above requirements.

In addition to defining these systems, Kolmogorov (Kolmogorov, 1936) derived conditions on the fi for predator– prey interactions in the form of equation (1) for n¼2 that ensured only ecologically realistic dynamics (stable spiral equilibriums or stable limit cycles that ensure continued co-existence of both predator and prey) were possible. Systems with stable spiral equilibrium points will exhibit blooms of ever- decreasing amplitude until they achieve a steady state, whereas systems with stable limit cycles will eventually settle to periodic repeating blooms of identical amplitude.

Ecological interpretations of Kolmogorov’s conditions are provided by May (May, 1973). A consequence of

Kolmogorov’s conditions is that they require the system to have both an autotroph-only and a predator– prey critical (equilibrium) point in the ecologically feasible region of the state space where all state variables are positive. The existence of these critical points is determined by the nature of the fi, but whether they lie within the ecologically feasible region of the state space is determined by the parameter values used in the model.

We consider the endogenous dynamics of a model with multiple phytoplankton functional types that is a three-dimensional (n¼3) Kolmogorov system.

u_1 ¼f1ðu₁;u2;u3Þu₁ _

u2 ¼f2ðu1;u2;u3Þu2

u_3 ¼f3ðu₁;u2;u3Þu₃

; ð2Þ

where u1 and u2 represent phytoplankton functional types andu3represents their zooplankton grazer. As the fi are generally nonlinear functions of u1, u2, and u3, analytic solutions to such systems are rare. We shall restrict our analysis tofithat comply with Kolmogorov’s (Kolmogorov, 1936) conditions, as do many ecosystem models in the contemporary literature (Huang and Zhu, 2005).

We look at the particular case of a model that con- serves the mass of limiting nutrient as many models applied in biological oceanography also have this property (Spitzet al., 2001; Franks, 2002; Vallinaet al., 2008).

Conservation of mass implies that the total mass of inorganic nutrient (N) present at any time is given by:

N ¼NTu1u2u3 ,

N_ ¼ u_1u_2u_3; ð3Þ where NT is a constant that gives the total effective nutrient in the system and theuiare the concentrations of the PFTs measured in this currency. We note that conservation of mass is required for a biogeochemical model to be written as a Kolmogorov system, as the nutrient equation in these models typically cannot be written in the Kolmogorov form. However, conservation of mass causes N to become a “virtual” variable; as shown in equation (3), it allows the N_ equation to be inferred from the other equations that are in Kolmogorov form.

We scale each state variable with respect to the total nutrient (i.e.ui/NT), which allows us to define an ecologically feasible “state space” where 0u1, u2, u31.

Conservation of mass then implies that the dynamics of the system are confined to the part of a multi- dimensional Cartesian co-ordinate space where each axis represents the concentration of one species or

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functional type (such as u1,u2, u3) and all the variables are positive and ,1. Mass conservation is said to apply when N_ .0 on N¼0, i.e. this gives the crucial phys- ically realistic constraint on ourfi, which then forces the dynamics to live in the feasible state space with u1þ u2þu31 for all time.

We consider the critical (equilibrium) points of this system, denoted by fu₁;u₂;u₃g where u__i¼f_iu_i ¼0 for all time. Implicit in the rationale for constructing plankton models with more than one functional type is the assumption that, in the absence of environmental factors, an interior critical point, with fu₁;u₂;u₃g=0, both exists and is an important determinant of the dynamics of the system. The theme of this work is an inquiry into the nature of these interior ( predator – prey – prey) critical points in systems with two autotrophs and a grazer, and how this might inform the development and calibration of more complex PFT models. We will show that most parameter sets do not produce useful PFT models; carefully tuned models are essential. While the range of parameter values may be narrowed down by laboratory studies, as Le Que´re´ (Le Que´re´, 2006) observes, our results suggest that laboratory studies need to include estimates of parameters for combinations of organisms, as suggested by Flynn (Flynn, 2006). We also note that, while parameter sets that produce realistic PFT models are rare, they are distributed throughout the parameter space, suggesting that in addition to laboratory measurements, model dynamics should also be rigorously validated against the observed data (Anderson, 2006). We derive biological heuristics for parameter combinations that assist in the tuning of parameter sets and yield useful PFT models.

M E T H O D

We first analyse the dynamics of a generic three state variable Kolmogorov system (equation (2)) where theu1

and u2 represent autotrophs and u3 represents their grazer. The analysis of this generic system provides general results that apply to all such three state variable Kolmogorov systems, irrespective of the process formulations (fi) chosen to represent the interactions between the state variables and independent of the parameter values used in the model.

We consider the critical points of the system, defined by u__i¼0 for all i. In Kolmogorov systems, critical points may be obtained from the isoclines in two ways for each equation, when fi¼0 or when ui¼0. Each critical point in a three state variable system has three eigenvalues and three associated eigenvectors. These eigenvalues and eigenvectors describe the local

(Lyapunov) stability of the system in the region of each of the critical points, and together form “signposts” that control the dynamics of the system. Negative eigenvalues attract nearby trajectories (states of the system) to the critical point along their eigenvectors, whereas positive eigenvalues repel nearby trajectories. Critical points that have all of their eigenvalues negative are stable, and produce systems that are attracted to, and then remain in, equilibrium. However, it only requires one eigenvalue of a critical point to be positive to render the point unstable, and the system will never be in the state described by the critical point for more than a brief period. The properties of critical points determined by their eigenvalues and eigenvectors will be discussed in more detail below.

The isoclines, critical points and eigenvectors for the whole system are shown in Fig. 1. Some eigenvectors have been omitted for clarity; none are shown for point E but these can be inferred from those shown for D and F. Eigenvectors that always have positive eigenvalues are shown pointing away from the critical point, those that always have negative eigenvalues are shown

Fig. 1. Generic diagram of the zero isoclines (f1,f2,f3¼0 andu1,u2, u3¼0), critical points (stars) and eigenvectors (bold arrow) for the Kolmogorov system described by equation (2). (a) The full system (u1, u2,u3), (b) the predator– prey subsystems (u1,u3andu2,u3) and (c) the competitive autotroph subsystem (u1,u2).

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pointing towards the critical point, and the others have double-ended arrows. Figure 1a shows the three- dimensional state space ð_u_i¼f_iðu₁;u₂;u₃Þu_i;i¼1;2;3Þ in which the isoclines are surfaces; where parts of the isoclines are hidden behind other isoclines they are shown by dotted lines. Figure 1b shows the predator–

prey state spaces (_u_i¼f_iðu₁;u₃Þu_i; i¼1;3 and _

u_i¼f_iðu₂;u₃Þu_i; i¼2;3) that form the left and right faces of the three-dimensional space in Fig. 1a, while Fig. 1c shows the competitive autotroph state space ð_u_i¼f_iðu₁;u₂Þu_i; i¼1;2Þ that forms the base of Fig. 1a. The dashed lines indicate the conservation of mass conditions for each face.

The eigenvalues and eigenvectors of a system reflect the essential system-level characteristics of the community, or Jacobian, matrix that describes how each variable changes in response to changes in itself and other variables. Technically, the elements of the community matrix (J) are given by the partial derivatives of the equations with respect to the state variables, that is Ji,j¼@(fiui)/@ujfori,j¼1,2,. . .,n.

The critical points and their associated eigenvalues and eigenvectors control the dynamics of the system by influencing the trajectories that the system takes through the state space. The state space of a three state variable Kolmogorov system is a three-dimensional Cartesian coordinate space with u1, u2 and u3 along the axes (Fig. 1). A trajectory is defined by an initial condition (i.e. the initial values ascribed to each of u1, u2 and u3) and the subsequent behaviour of the system over time as defined by the equations

_

u_i¼f_iðu₁;u₂;u₃Þu_i; i¼1;2;3. A negative eigenvalue at a critical point means that trajectories near the eigenvector associated with the eigenvalue are attracted to the critical point along the eigenvector. Conversely, a positive eigenvector at a critical point indicates that trajectories near the associated eigenvector will be repelled from the critical point along the direction of the eigenvector. Each critical point has as many eigenvalues as the dimension of the system (in this case three). Critical points that have all negative real parts are locally stable, that is, all nearby trajectories will be attracted towards them and the system will display an equilibrium state, but it only requires one positive eigenvalue for a point to be unstable.

Eigenvalues that are real numbers indicate that nearby trajectories will approach/retreat from the critical point asymptotically (i.e. continuously increasing or continuously decreasing). However, many predator–

prey systems have eigenvalues that are complex numbers, with a real part and an imaginary part. The real part of the eigenvalue determines whether nearby trajectories are attracted to, or repelled from, the critical

point, while the complex part indicates that nearby trajectories will spiral around the critical point (i.e. the system will oscillate). We shall use the following terms to describe the stability properties of the critical points:

† Stable: the point has three negative real eigenvalues

† Unstable or saddle: the point has at least one positive real eigenvalue

† Stable spiral: the point has a pair of complex eigenvalues that have negative real parts

† Unstable spiral: the point has a pair of complex eigenvalues that have positive real parts.

When we discuss the stability properties of the predator– prey – prey critical point, which typically has a pair of complex eigenvalues and a single real eigenvalue, we shall differentiate its stability properties into those of the complex eigenvalues and those of the real eigenvalue.

We then consider a specific example of an NP1P2Z Kolmogorov system in order to examine the parameteri- zations of the system that result in realistic PFT dynamics.

The study system has conventional phytoplankton (P1,P2) growth on inorganic nutrient (N) balanced by zooplankton (Z) grazing and linear mortality. We note that there is currently significant debate over the appropriate form of the process representations in PFT models (Flynn, 2003;

Mitra, 2009). We choose “simple” fi that are commonly used to allow, as far as possible, for closed form analytic expressions to be found for the critical points and their eigenvalues. Even in this simple model, explicit analytic evaluation is not always possible, and we are forced to develop numerical estimates of one interior value. We justify this choice because

† we will demonstrate in the analysis of the generic Kolmogorov system that many of our results are independent of the form of thefi, and

† at the heart of our approach is the great value that analytic expressions provide over numerical estimates in understanding why PFT models behave the way they do.

Analytic expressions for the key properties of PFT systems allow us to understand and predict PFT dynamics rather than merely observe them. We shall present the results of the analysis of theNP1P2Zmodel, equivalent to our analysis of the generic Kolmogorov system, and use these results to explore the parameter space and associated dynamical properties of this simple system.

The example system is written in a currency of inorganic nutrient, with all state variables expressed as concentrations of nutrient, as described by equations

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(4) – (6):

P_₁ ¼f_P₁P₁¼ mN

Nþk wZ

1þ1P₁s

P₁; ð4Þ

P_2¼fP2P2 ¼ m^N

Nþk^ w^Z 1þ^1P₂s^

P2; ð5Þ

Z_ ¼fZZ¼ wð1cÞP₁

1þ1P₁ þw^ð1c^ÞP₂ 1þ^1P₂ s_Z

" #

Z: ð6Þ

We check our conservation of mass criterion as per equation (3) and see that

N_ ¼ s_Zþ wcP1

1þ1P₁þ w^c^P2

1þ^1P₂

" #

Z

mN N þks

P1 m^N Nþk^s^

P2; ð7Þ when we put N¼0 in the right-hand side thenN_ .0 at N;0¼NT2P12P22Z. This forces N to be positive, so that P1þP2þZ,NT for all time in our system. The useful PFT dynamics, where nothing goes extinct, occur in 0,P1, P2, Z with P1þP2þZ,NT. We shall define several parameter sets for this model to demonstrate qualitatively different dynamics. These parameter sets are derived from measured values that are described in Table I. As we will be considering the effect of varying parameter values on the model dynamics, it is essential that we scale the parameters so that the influence of each parameter is revealed unequi- vocally. We have therefore non-dimensionalized the parameter values in Table I from their measured values by scaling time by the maximum growth rate ofP1(m) and concentrations by the total nutrient (NT).

We have chosen the form of theZ grazing terms so that the parameter values of 1and^1in equations (4) – (7) can be used to control the dynamics of the predator– prey sub-systems (NP1ZandNP2Z). Using values of 1 or^1equal to zero results in these terms behaving as Lotka – Volterra grazing formulations, while non-zero values cause them to behave as Michalis – Menten grazing formulations. This provides a simple mechan- ism to determine the dynamics of the predator– prey subsystems as it is easy to find parameter sets that result in asymptotically stable spiral dynamics in this system with Lotka – Volterra grazing, and similarly easy to find parameter sets that result in stable limit cycle dynamics in this system with Michalis – Menten grazing. In general, in what follows we will set 010.1 to generate stable spirals in theNP1Zsubsystem and^11 to generate stable limit cycles in theNP2Zsubsystem. This system can therefore reproduce all of the dynamical states that Kolmogorov (Kolmogorov, 1936), May (May, 1973) and most subsequent ecologists have considered ecologically realistic for predator– prey systems.

We consider the dynamics of the NP1P2Z system under six parameter sets that provide six representative dynamical behaviours, and examine the influence of the various eigenvalues and eigenvectors on the dynamics.

In each case, typical trajectories are integrated from two different initial conditions with a fourth – fifth-order adaptive step size Runge – Kutta routine implemented with relative and absolute step size tolerances set to machine epsilon. The dynamics of the system for the six parameter sets explicate the dynamics of three classes of PFT systems: bona fide PFT systems where all functional types remain extant forever; pseudo-PFT systems where the initial population sizes determine which type will go extinct; and non-PFT systems where the parameter values determine which type will go extinct irrespective of the initial conditions.

Table I: Measured parameter values for the NP₁P₂Z model

Par Process Value Reference

m Max rate ofNuptake byP1 1.00 Gabricet al. (1999)

m^ Max rate ofNuptake byP2 1.15 Muller-Niklas and Herndl (1996)

x Half-sat const forP1uptake ofN 0.25 Slagstad and Stole-Hansen (1991)

k^ Half-sat const forP2uptake ofN 0.07 Billen and Becquevort (1991)

F Zgrazing rate (per ind) onP1 6.18 Hansenet al. (1996)

w^ Zgrazing rate (per ind) onP2 1.85 Gabricet al. (1999)

1 Half-sat const forZuptake ofP1 5.50 Fenchel (1982)

^

1 Half-sat const forZuptake ofP2 5.50 Fenchel (1982)

S P1specific mortality rate 0.00 Gabricet al. (2001)

s^ P2specific mortality rate 0.26 Moloneyet al. (1986)

sZ Zspecific mortality rate 0.19 Moloneyet al. (1986)

C Prop ofP1uptake excreted byZ 0.40 Moloneyet al. (1986)

c^ Prop ofP2uptake excreted byZ 0.40 Moloneyet al. (1986)

These values have been non-dimensionalised by scaling by the maximum rate ofNuptake byP1(m) and the total nutrient (NT). These values form the basis for the values used in the numerical experiments.

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We then investigate the sensitivity of the interior predator– prey – prey critical point of a bona fide PFT system to variations in parameter values. This analysis is not intended to be a comprehensive sensitivity analysis of the model, but a demonstration of the finely balanced nature ofbona fidePFT systems, and an explication of the important parameter sensitivities of the example model at one point in the parameter space (column 1 of Table II). The parameters were varied one at a time and the location of the predator– prey – prey critical point was followed using a numerical continuation algorithm (Dhooge et al., 2003), which follows the critical point through the state space, until either the parameter or the critical point achieved ecologically infeasible values. The maximum and minimum values of the parameter that resulted in the predator– prey – prey critical point being in the feasible region of the state space were recorded, and a sensitivity metric calculated by scaling the range of valid parameter values by the mean parameter value.

The parameter set used resulted in a stable spiral on one face of the state space, characterized by eigenvalues with negative real parts, and a limit cycle on the other, characterized by eigenvalues with positive real parts. We therefore also recorded the parameter value at which the real parts of the eigenvalues passed through zero and the dynamics changed from a stable spiral to a limit cycle or vice versa as the predator– prey – prey critical point traversed the state space.

Finally, we consider the ubiquity of parameter sets that result in bona fidePFT systems. The biological parameter spaces that marine biogeochemical models occupy are notoriously poorly constrained, so we defined a parameter space ranging from 50% of the smallest to 150% of the largest values of each parameter listed in Table II. We randomly sampled 510⁶

parameter sets from uniform distributions within this parameter space. We established criteria for the validity of each parameter set based on the application of Kolmogorov’s (Kolmogorov, 1936) criteria to the analysis of our exemplar model. If a parameter set was valid, we calculated the eigenvalues of the inward-pointing eigenvectors of the predator – prey critical points on the faces and classified each parameter set according to the signs of the eigenvalues. We tested the effect of extend- ing the parameter space beyond the limits specified above, but this did not significantly alter the number or proportion of valid parameter sets.

R E S U LT S

We initially present the results of the analysis of the generic Kolmogorov system (equation (2)), where we define u1 and u2 to be autotrophs and u3 to be their grazer. We present analytic expressions for the critical points and eigenvalues where possible, and present the generic dynamics of the state space described by the critical points, eigenvalues and eigenvectors graphically (Fig. 1). We have labelled the critical points in Fig. 1 and shall use these labels to identify the critical points and their eigenvalues in the following analysis. The values of the state variables at the critical points are denoted by the asterisk superscripts, and the critical points differentiated by their subscripts. The signs of the eigenvalues are shown where these are defined by the ecological properties of the system and always hold.

Origin critical point (O)

Every Kolmogorov system has a critical point at the origin (O in Fig. 1) where u_i ¼0 (i.e. no life) and (usually)fi=0 for alli. The eigenvalues at the origin are lO1¼f1 Oj .0; ð8Þ l_O2¼f2 Oj .0; ð9Þ lO3¼f3 Oj ,0; ð10Þ where f_ij_O¼0 means that the expressionfiis evaluated at the critical point O (i.e. using the values of the state variables at the critical point). Analytic expressions for the eigenvalues that control the stability characteristics of this point are therefore easily obtained from inspection of the equations as they are simply thefiof each PFT, and their magnitudes given by the value of thefiat the origin.

The origin represents the state of the system where only inorganic nutrient exists. Near this point, Table II: Parameter values used in the

numerical simulations

Par

Parameter values

Fig. 2 Fig. 3 Fig. 4 Fig. 5 Fig. 6 Fig. 7

m 0.73 0.68 1.00 1.00 1.00 0.70

m^ 1.14 1.14 0.50 0.40 1.14 0.50

x 0.71 0.71 0.25 0.25 0.25 0.71

k^ 0.10 0.10 0.01 0.01 0.10 0.01

w 2.25 2.25 2.00 2.00 2.00 2.25

w^ 6.11 6.11 1.50 1.50 6.11 2.50

1 0.00 0.00 0.01 0.01 0.01 0.00

^

1 1.24 1.24 0.01 0.90 1.24 0.10

s 0.00 0.00 0.01 0.01 0.01 0.00

s^ 0.19 0.19 0.01 0.01 0.19 0.01

sZ 0.37 0.37 0.20 0.20 0.20 0.37

c 0.78 0.78 0.40 0.40 0.40 0.78

c^ 0.36 0.36 0.40 0.40 0.36 0.40

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autotrophs must grow (or there could never be any life) and predators must die (as they have insufficient prey to survive on). This point will therefore always be a saddle (i.e. have some positive and some negative eigenvalues) and its eigenvectors (shown by the bold arrows in the Fig. 1) will always lie along the axes of the state space.

The unstable directions (lO21 and lO22) are a result of the autotrophs growing by consuming nutrient in iso- lation along their respective axes, while the stable direction (lO23) is that of the predator dying in the absence of prey along its axis. Note that each eigenvalue is com- prised only of parameters related to each PFT’s own growth and/or mortality, and that grazing interactions are not relevant to this critical point. Further, ecological realism requires that it should be possible to make the origin a stable node. The fi of the autotrophs should therefore all include an explicit mortality term such that it is possible to increase the mortality rates of the autotrophs so that they exceed the maximum growth rates possible and extinction of all life ensues.

Prey-only critical points (A, C)

Every model that complies with Kolmogorov’s (Kolmogorov, 1936) criteria will have a prey-only (autotroph) critical point in each of the predator– prey subsystems. These are defined by u₁=0;u₂¼0;u₃¼0 (A) and u₁¼0;u₂=0;u₃ ¼0 (C) and are located where the f1¼0 isocline in the (u1,u3) plane intersects theu1axis and where the f2¼0 isocline in the (u2, u3) plane intersects theu2axis, respectively. The eigenvalues of these points are given by, for A:

lA1¼@f1

@u₁u1jA,0; ð11Þ

lA2¼f₂j_A; ð12Þ

lA3¼f3jA.0; ð13Þ and for C:

lC1¼f₁j_C; ð14Þ

lC2¼@f2

@u2

u₂j_C ,0; ð15Þ

lC3¼f3jC .0: ð16Þ Here again the eigenvalues have neat ecological interpretations. The stable eigenvalues (lA21for A and

lC22for C) are given by the response of the blooming autotroph to increases in its own biomass (@fi/@ui) and are always negative, as autotroph growth rates reduce as nutrient becomes less available. The eigenvectors associated with these eigenvalues always lie along theu1andu2

axes, respectively, as shown by the bold arrows in Fig. 1.

The other eigenvalues are obtained by evaluating the growth functions (fi) of the other PFTs at the critical point. These reflect the grazing pressure applied by the predator (lA23 for A and lC23 for C) or the competition from the other autotroph for nutrient (lA22for A and lC21 for C). Systems that comply with Kolomogorov’s criteria will always have lA23 and lC23, the eigenvalues associated with grazing pressure, positive (destabilizing) at these points. The directions of these eigenvalues will vary according to the nature of the fi, but will always point into the interior of, and lie in the plane of, the (u1, u3) plane for A or the (u2, u3) plane for C. The eigenvalues associated with the competing autotroph (lA22 for A and lC23 for C) may be positive if the blooming autotroph leaves enough nutrient so that the competing autotroph’s growth from nutrient uptake exceeds its specific mortality rate, or negative otherwise. The eigenvectors for these critical points are always orthogonal to the (u1, u3) plane for A or the (u2,u3) plane for C as shown in Fig. 1.

Dual prey-only critical point (B)

Figure 1 shows a third prey-only critical point (B), where both competing autotrophs coexist. The existence of this point is not assured by Kolomogorov’s criteria, as the (u1, u2) plane does not constitute a predator–prey system. This point is defined byu₁=0;u₂ =0;u₃ ¼0 and is located where thef1¼0 isocline intersects thef2¼0 isocline in the (u1,u2) plane. The eigenvalues of this point are given by:

2lB1;2 ¼@f1

@u1

u1þ@f2

@u2

u2

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

@f1

@u1

u1þ@f2

@u2

u2

2

4u1u2

@f1

@u1

@f2

@u2

@f1

@u2

@f2

@u1

s

jB; ð17Þ

lB3 ¼f3jB .0: ð18Þ The eigenvalue associated with grazing pressure (lB23) is always positive, whereas the eigenvalues associated with the autotroph growth (lB21,2) will have one negative eigenvalue representing autotroph growth on available nutrient. The other eigenvalue may be positive or negative, either repelling or attracting nearby trajectories, but is not critical to the co-existence of u1 and u2.

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The directions of the eigenvectors associated with these eigenvalues will vary according to the fi and the parameter values used but will generally lie as shown in Fig. 1 for parameter values that result in B being located in the feasible region of the state space.

Predator– prey critical points (D, F)

Every sub-system that complies with Kolmogorov’s criteria will have a predator– prey critical point.

In the system defined by equation (2) these are defined by u₁=0;u₂¼0;u₃=0 for D and u₁¼0;u₂=0;u₃=0 for F and are located where the f1¼0 isocline intersects the f3¼0 isocline in the (u1, u3) plane and where the f2¼0 isocline intersects the f3¼0 isocline in the (u2, u3) plane, respectively. The eigenvalues of these critical points are given by:

2lD1;3¼@f1

@u1

u₁þ@f3

@u3

u₃

+

@f1

@u₁u1þ@f3

@u₃u3

2

4u1u3

@f1

@u₁

@f3

@u₃@f1

@u₃

@f3

@u₁

s

jD;

ð19Þ lD2¼f2jD; ð20Þ for D, and by:

lF1¼f₁j_F; ð21Þ 2lF2;3¼@f2

@u2

u2þ@f3

@u3

u3

+

@f2

@u2

u2þ@f3

@u3

u3

2

4u2u3

@f2

@u2

@f3

@u3

@f2

@u3

@f3

@u2

s

jF; ð22Þ for F. In almost all cases lD21,3 and lF22,3 will be complex numbers, with positive or negative real parts, indicating that trajectories will either spiral into or away from the critical point. An example of each case is shown in Fig. 1, with D shown with negative real parts and F shown with positive real parts. Spiral curves then lie in the (u1,u3) or (u2,u3) planes and start or end at D or F. lD21,3and lF22,3 therefore control the dynamics in the (u1,u3) and (u2,u3) planes, respectively.

The dynamics of the system in the direction orthogonal to these planes is of critical importance to PFT modellers as the eigenvalues in this direction determine whether a system will maintain all plankton extant during simulations. The eigenvalues in this direction are given bylD22for D and bylF21for F. The eigenvalues are associated with eigenvectors that are orthogonal to

the (u1,u3) and (u2,u3) planes, respectively, with magnitudes given by the fiof the competing autotroph evaluated at the critical point. These eigenvalues are always real numbers, their direction is known, and analytic expressions for them are easily obtained from inspection of the model equations. The signs of these eigenvalues determine whether a predator– prey – prey critical point (E), fundamental to the construction of a bona fidePFT model, exists and is stable.

If lD22 and lF21 are both positive, E exists in the feasible region of state space and is stable in the direction orthogonal to the predator– prey planes (u1,u3) and (u2,u3). This means that the system will eventually come to some balance whereu1andu2co-exist, and are both grazed on byu3. This may be a stable equilibrium point or a stable limit cycle.

If lD22 and lF21are both negative, E exists in the feasible region of state space but is unstable in the direction orthogonal to the predator– prey planes (u1,u3) and (u2,u3). This means thatu1andu2cannot co-exist, and one must always out-compete the other. The winner and loser of the competition in this case are determined by their starting values (initial conditions). We shall refer to these cases as pseudo-PFT systems. WhenlD22

and lF21 have opposite signs, E does not exist in the feasible region of state space, and again u1 and u2

cannot co-exist. In this case, the winner and loser are determined by the eigenvalues, and the initial conditions have no influence on the outcome of the competition. We shall refer to these cases as non-PFT systems.

The ecological interpretation of eitherlD22or lF21

being positive is that if the equilibrium state (or equiva- lently the long term average if a limit cycle) of the competing autotroph and its grazer leaves enough nutrient available for the autotroph growth rate to exceed its losses due to grazing and mortality, then the autotroph has the capacity to grow. Technically, this destabilizes the critical point allowing the trajectory to explore the interior of the state space.

Predator– prey – prey critical point (E)

As noted above, the system may have a critical point E defined by u₁=0;u₂=0;u₃=0 located where the f1¼0,f2¼0 andf3¼0 isoclines all intersect in the (u1, u2,u3) volume. In this case the isoclines are surfaces, as shown in Fig. 1. The existence of this point is not assured by Kolmogorov’s criteria as it is a predator–

prey – prey system rather than a predator– prey system.

However, systems that have critical points at D and F will, for some parameter sets, have a predator– prey – prey critical point E that lies in the interior of the state space.

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The eigenvalues of the critical point E are generally intricate in analytic form, and difficult to interpret, as they involve the roots of a cubic equation derived from the community matrix, and hence it is usually simpler to obtain them numerically. It appears that the spiral dynamics enforced on the (u1, u3) and (u2,u3) planes by the Kolmogorov criteria are reflected throughout the interior of the state space. We therefore expect that the critical point E will have one pair of complex conjugate eigenvalues that control its spiral behaviour in the (j1P1þ j2P2,Z) “plane”, wherej1andj2are constants defining a ray on the (P1,P2) face. The third eigenvalue must be a real number, and its sign will control the outcome of the competition between the two autotrophs. The real eigenvalue will therefore either repel trajectories away from E toward the predator–prey critical points on the faces (if positive) or attract them away from the faces toward the interior predator–prey–prey critical point E (if negative).

The global dynamics of the state space must be consistent with the above local information. Therefore, if the interior critical point E has positive real eigenvalues, the critical points on the faces D and F must both have negative real eigenvalues associated with the eigenvectors that are orthogonal to the faces. Similarly, if E has negative real eigenvalues, D and F must both have positive real eigenvalues. In the case where D and F have real eigenvalues of opposite signs, E cannot exist in the interior of the state space. In this circumstance, the point is located exterior to the ecologically feasible state space, and it and its eigenvalues do not influence the dynamics of the system. These cases will be demon- strated in our analysis of the example NP1P2Z Kolmogorov system to follow.

Critical points of theNP₁P₂Z system

Although relatively simple, theNP1P2Z system lies at the boundary of complexity for systems that are wholly analytically tractable. We are able to obtain analytic expressions for all but one of the critical points and eigenvalues; the location of the interior predator–prey – prey critical point E and its eigenvalues and eigenvectors is easily obtained numerically. The analytic expressions describing the locations of the critical points, and their eigenvalues where possible, analogous to and labelled identically to that presented above for the generic model are presented in the Appendix. The numerical evalu- ations of these expressions are presented in Table III, and the locations of the critical points that these values specify are shown in Figs 2– 7. Note that critical point B is a special case of a critical line in this model (see Appendix for details) and is therefore not shown in Figs 2–7 that show the dynamics of the system.

We now review the dynamics possible in NP1P2Z systems. This is not a comprehensive review, but rather a pragmatic survey of the dynamics; other dynamics can be inferred from the sample presented. We formulated six parameter sets (Table II) derived from the measured values listed in Table I. These parameter sets were chosen to produce dynamics representative of the six qualitatively different cases, and always result in a complex conjugate pair of eigenvalues in the (P1,Z) or (P2,Z) vertical planes, whereas the eigenvalues in the (P1, P2) horizontal plane are always real. The dynamics shown in Figs 2–7 demonstrate the various possibilities of the signs of the real parts of these eigenvalues. The figures each show two trajectories that start from different initial conditions to demonstrate the different trajectories possible. The arrows on the faces represent the vector fields of the sub-systems (i.e.

NP1Z,NP2ZandNP1P2). The vector fields are determined by the eigenvalues and eigenvectors on the faces and allow generalized dynamics to be inferred from the indi- vidual trajectories shown. Colour versions of these figures are provided online in the Supplementary material.

Parameter sets for which both real eigenvalues of the predator– prey critical points on the faces are positive are the only cases in which the original set of functional types is always maintained in the model. In all other cases (i.e. both negative or one positive and one negative eigenvalue), one species or other of the phytoplankton will dominate. The system dynamics will then no longer inhabit the interior of the state space, but will reside on either the (P1,Z) or (P2,Z) face.

We observe that linear constant coefficient ordinary differential equation models have exponential solutions.

For ecological models, this means that, in the absence of growth, state variables with linear mortality terms exponentially decay. They cannot therefore ever become exactly equal to zero (although when solved numerically they may equal zero, or even become negative, if the numerical integration routine used is not sufficiently accurate). However, we note that when the dynamics of our PFT system are attracted to one of the faces then the other PFT, even though non-zero, can never recover, and is for all biogeochemical purposes extinct. For ease of discussion, we will therefore refer to this situation as the extinction of that PFT.

Bona fide PFT systems

We define bona fidePFT systems to be those in which no PFT goes extinct. There is therefore a consistency between the model equations that describe multiple PFTs and the numerical simulations that maintain those PFTs for all time. Figure 2 presents typical dynamics for abona fidePFT system with a stable spiral internal critical point.

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The origin and autotroph critical points are saddles and the predator– prey critical points on the faces both have positive real eigenvalues (equations (A.26) and (A.27)).

The complex eigenvalues of the predator–prey critical point on the (P1, Z) face have negative real parts, whereas the complex eigenvalues of the predator–prey critical point on the (P2, Z) face have positive real parts.

The (P1, Z) face therefore has stable spiral dynamics, whereas the (P2,Z) face has stable limit cycle dynamics.

Every initial condition for the parameterization of the model in Fig. 2 will end up at the spirally stable predator– prey – prey critical point in the interior of the feasible state space. This is perhaps the ideal dynamics for a PFT model, as a balance between all PFT is achieved, and no PFT is ever lost to the system.

Figure 3 demonstrates the dynamics of perhaps the next most desirable system for PFT modelling, that of a

stable limit cycle predator– prey – prey critical point.

Again, the (P1,Z) face has stable spiral dynamics while the (P2,Z) face has stable limit cycle dynamics, and both have positive real eigenvalues. Here, the system regularly cycles on a plane in the interior of the state space, in this case with large blooms ofP2and relatively small blooms ofP1. Again, these dynamics are robust and no PFT is lost to the system as the real eigenvalues of the predator–

prey critical points on the faces direct all orbits away from the faces and towards the interior point.

Pseudo-PFT systems

We define pseudo-PFT systems to be one class of models for which there is an inconsistency between the model that is ostensibly described by the system of differential equations and the results of the numerical solution of the Table III: Summary of critical point locations and their eigenvalues for the parameter values used in the numerical simulations

Figure CP NN* P₁ P₂ ZZ* ll1 ll2 ll3

2 O 1 0 0 0 0.43 0.85 20.37

A 0 1 0 0 21.03 20.19 0.12

C 0.02 0 0.98 0 0.02 27.63 1.36

D 0.18 0.75 0 0.07 20.24þ0.11i 0.14 20.24 – 0.11i F 0.74 0 0.11 0.15 0.03 20.04þ0.52i 20.04 – 0.52i E 0.49 0.32 0.06 0.13 20.03þ0.45i 20.03 20.03 – 0.45i

3 O 1 0 0 0 0.40 0.84 20.37

A 0 1 0 0 20.95 20.19 0.12

C 0.02 0 0.98 0 0.02 27.63 1.36

D 0.19 0.75 0 0.06 20.22þ0.13i 0.16 20.22 – 0.13i

F 0.74 0 0.11 0.15 0.01 0.04þ0.52i 0.04 – 0.52i

E 0.67 0.09 0.09 0.15 0.02þ0.50i 20.01 0.02þ0.50i

4 O 1 0 0 0 0.79 0.49 20.20

A 0.01 0.99 0 0 23.91 0.09 0.99

C 0.01 0 0.99 0 20.01 248 0.69

D 0.50 0.17 0 0.33 20.04þ0.40i 20.01 20.04 – 0.40i F 0.46 0 0.22 0.32 20.01 20.01þ0.31i 20.01 – 0.31i E 0.47 0.02 0.19 0.32 20.01þ0.33i 0.01 20.01 – 0.33i

5 O 1 0 0 0 0.79 0.37 20.20

A 0.01 0.99 0 0 23.91 0.07 0.99

C 0.02 0 0.98 0 20.01 238 0.27

D 0.50 0.17 0 0.33 20.04þ0.40i 20.11 20.04 – 0.40i F 0.40 0 0.28 0.32 20.03 0.03þ0.25i 0.03 – 0.25i

E 0.42 0.02 0.25 0.31 0.02þ0.26i 0.02 0.02 – 0.26i

6 O 1 0 0 0 0.79 0.85 20.20

A 0.01 0.99 0 0 23.91 20.16 0.99

C 0.02 0 0.98 0 0.07 27.63 1.53

D 0.50 0.17 0 0.33 20.04þ0.40i 21.26 20.04 – 0.40i

F 0.80 0 0.06 0.14 0.47 0.02þ0.40i 0.02 – 0.40i

E – – – – – – –

7 O 1 0 0 0 0.41 0.49 20.37

A 0 1 0 0 20.99 20.01 0.12

C 0.01 0 0.99 0 0.01 248 1.00

D 0.19 0.75 0 0.06 20.23þ0.12i 0.30 20.23 – 0.12i F 0.55 0 0.25 0.20 20.14 0.01þ0.41i 0.01 – 0.41i

E – – – – – – –

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model. In our pseudo-PFT systems, one or other of the phytoplankton functional types will always go extinct in simulations, with the PFT destined for extinction being determined by the initial conditions selected to com- mence the integration from. Figure 4 shows the first of the cases in which extinction of one Poccurs. Both (P1, Z) and (P2, Z) faces have stable spiral dynamics in this case, and also have negative real eigenvalues attracting nearby orbits “horizontally” onto the face. This is perhaps the worst scenario for the dynamics of a PFT model, as the initial dynamics may give the impression of a bona fide PFT system. However, the system will always collapse to one or the other of the stable states on the faces, and which P survives is determined solely by the initial conditions. We note, however, that for different parameter sets that result in this dynamical situation, the survivorship of the P will be determined by different

initial conditions, that is, the basins of attraction of the predator–prey critical points on the faces will be different for different parameter sets. We further observe that it would be possible to start simulations with phytoplankton functional type populations very close to the separating surface dividing dominance by P1 from dominance by P2. In such an event, the system would give the initial impression that it was maintaining both types, but long simulations would eventually result in the extinction of one functional type.

Figure 5 shows a similar situation to Fig. 4, as both predator– prey critical points have negative real eigenvalues, except in this case the (P2, Z) face has stable limit cycle dynamics. It is interesting to note that although the initial conditions again solely determine which PFT will survive, it is not necessarily immediately obvious which PFT that will be. The left trajectory, for

Fig. 2. Critical points and dynamics for theNP1P2Zsystem showing attraction of both trajectories to an internal stable spiral critical point.

Parameter set used is listed in column 1 of Table II. Initial conditions for red trajectory areN¼0.898,P1¼0.100,P2¼0.001,Z¼0.001.

Initial conditions for green trajectory areN¼0.799,P1¼0.100,P2¼0.100,Z¼0.001.

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example, reveals that P1 dominates the initial bloom before the trajectory sweeps over to near the (P2, Z) face, where subsequent smaller blooms are dominated by P2. As subsequent blooms wax and wane, however, P1 gradually re-asserts its dominance, resulting in the

“extinction” ofP2in this case.

Non-PFT systems

We define non-PFT systems to be those in which one phytoplankton functional type will always go extinct, and in which the PFT destined for extinction is pre- determined by the parameterization and is independent of the initial conditions. In these cases, there is a fundamental dichotomy between what the model equations suggest is the case (in this case a system with two competing phytoplankton) and how the model actually functions (as a single phytoplankton – zooplankton

predator– prey system). Two non-PFT systems, in which the real eigenvector of one of the predator– prey critical points on the faces is positive and the other one is negative, are shown in Figs 6 and 7. In these cases, there is no predator– prey – prey critical point in the feasible region of the state space and the phytoplankton functional type model is illusory, in that more than one PFT exists only in the initial transient dynamics; ultimately one PFT will always dominate and the other PFT will always go extinct. In contrast to the pseudo-PFT systems, extinction in these situations is determined not by the initial populations, but by an intrinsic property of the parameter set chosen, and there are no circumstances under which the loser can survive. In Fig. 6, for example, the predator–prey critical point on the (P1,Z) face has a negative real eigenvalue while the predator–prey critical point on the (P2, Z) face has a positive real eigenvalue.

All initial conditions are therefore eventually attracted to

Fig. 3. Critical points and dynamics for theNP1P2Zsystem showing attraction of both trajectories to an internal stable limit cycle. Parameter set used is listed in column 2 of Table II. Initial conditions for red trajectory are N¼0.898, P1¼0.100,P2¼0.001, Z¼0.001. Initial conditions for green trajectory areN¼0.655,P1¼0.092,P2¼0.092,Z¼0.161.

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the stable spiral critical point on the (P1,Z) face (“extinction” of P2), even though the initial dynamics of the rightmost trajectory suggest thatP2may dominate.

Figure 7 shows the reverse situation, when the stable spiral on the (P1, Z) face has a positive real eigenvalue and the stable limit cycle on the (P2,Z) face has a negative real eigenvalue. In this case, all initial conditions result in the “extinction” ofP1. The dynamics in Figs 6 and 7 are consistent with the “Competitive Exclusion Principle” first articulated by Gause (Gause, 1934).

Sensitivity of the predator– prey – prey critical point

The results of the numerical investigation of the sensitivity of the predator– prey – prey critical point for the bona fide PFT system of Fig. 2 are summarized in Table V. The minimum and maximum values in Table V refer to the values of each parameter for which

the predator– prey – prey critical point (E) leaves the ecologically feasible region of the state space. These are defined analytically by equation (A.29). The effect of parameter variations on the predator– prey – prey critical point is consistent across all parameters and in accord with the critical point and eigenvalue analysis above. The sensitivity analysis identifies the range of parameter values for which E exists in the interior of the parameter space.

Variations in the parameter values move the predator–

prey –prey critical point (E) across the state space between the predator–prey critical points (D and F). The parameter values for which E is located within the feasible region of the state space are listed in Table V, with the sensitivity metric where available and a ranking of relative sensitivities to one-at-a-time parameter variations.

We observe that because we have used a parameter set that results in a stable spiral on one face and a limit cycle on the other, the sensitivity of the critical point E

Fig. 4. Critical points and dynamics for theNP1P2Zsystem showing attraction of both trajectories to a stable spiral on each face. Parameter set used is listed in column 3 of Table II. Initial conditions for red trajectory areN¼0.849,P1¼0.100,P2¼0.050,Z¼0.001. Initial conditions for green trajectory areN¼0.849,P1¼0.050,P2¼0.100,Z¼0.001.