NPZ-model with seasonal variability of plankton populationdynamicsCitation for published version (APA):Goorden, S. A., Korzilius, S. P., Thije Boonkkamp, ten, J. H. M., Anthonissen, M. J. H., & Rathish Kumar, B. V.(2011). NPZ-model with seasonal variability of plankton population dynamics. (CASA-report; Vol. 1130).Eindhoven: Technische Universiteit Eindhoven.

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computer Science

CASA-Report 11-30 April 2011

NPZ-model with seasonal variability of plankton population dynamics

by

S.A. Goorden, S. P. Korzilius, J.H.M. ten Thije Boonkkamp, M.J.H. Anthonissen, B.V. Rathish Kumar

Centre for Analysis, Scientific computing and Applications

Department of Mathematics and Computer Science

Eindhoven University of Technology

P.O. Box 513

5600 MB Eindhoven, The Netherlands

ISSN: 0926-4507

NPZ-model with seasonal variability of planktonpopulation dynamics

S.A. Goorden1, S.P. Korzilius1, J.H.M. ten Thije Boonkkamp1,M.J.H. Anthonissen1, B.V. Rathish Kumar2

1Department of Mathematics and Computer Science, Eindhoven University of TechnologyP.O. Box 513, 5600 MB Eindhoven, The Netherlands

2Department of Mathematics and Statistics, Indian Institute of Technology KanpurKanpur-208016, India

Abstract

In this paper we include seasonal variability in the NPZ-model of a population of phytoplankton.For the space discretization of the governing equations we employ the complete flux scheme. Astability analysis of the fully discrete scheme is presented. A new boundary condition, based on thecomplete flux scheme, is introduced. Numerical results show good agreement with previous resultsin literature.

Keywords. NPZ-model, seasonal variability, complete flux scheme, stability analysis,boundary treatment.

1 Introduction

The topic of phytoplankton has been widely studied for the last decades. Phytoplankton are small plantswhich inhabit the oceans and lakes of the earth. They are too small to be seen by the naked eye, but theycause a green discolouration of the water when present in large amounts. Phytoplankton are importantfor several reasons. First, they can extract carbon dioxide from the atmosphere and produce oxygen,through the process of photosynthesis. In fact, phytoplankton consumes as much carbon dioxide as allland vegetation does together. Therefore, it has a major influence on climate change [13]. Second,phytoplankton are also important because they are at the bottom of the food chain. Phytoplankton iseaten by marine life, ranging from microscopic organisms to whales. Small fish feed on phytoplanktonand they are in turn eaten by larger fish. Some species of phytoplankton can produce biotoxins, whichmight end up in fish caught for human consumption. Finally, after a bloom, large amounts of deadphytoplankton can sink to the bottom of the ocean or lake. Bacteria that decompose the phytoplanktoncan extract so much oxygen from the water that it suffocates marine life [10]. All these factors make itimportant to understand how a population of phytoplankton behaves under different circumstances.

For the concentration of phytoplankton we are only interested in the mixed layer, which stretchesfrom the water surface to a depth of approximately 100 meters. The reason for this is that the water belowthis layer does not receive enough sunlight for photosynthesis, which phytoplankton need to survive.Phytoplankton leave the mixed layer by sinking slowly. This sinking is descibed by downward advection.Besides sunlight phytoplankton also need nutrients, such as iron, to survive. These nutrients sojourn inthe deeper waters of the ocean and enter the mixed layer due to mixing. During summer, when the top

1

2 MATHEMATICAL MODEL 2

layer is warmer than the water below, there will be less mixing than in winter, when the temperaturedifference is smaller. Mixing of water is described by diffusion. Another effect of warmer water is thatphytoplankton reproduces faster in summer than in winter. Because nutrients are so light, and thereforehardly sink in comparison with phytoplankton, there is no advection term for nutrients. In addition tophytoplankton and nutrients we also include zooplankton in our model, which are microscopic organismsthat feed on phytoplankton. They do not need sunlight and do not sink because they have some capabilityof swimming, i.e., they are not advected downward.

All governing equations are of (advection-)diffusion-reaction type. Previous research focussed onvarious aspects, such as stability of the governing PDEs [19], inclusion of seasonal variations [17], one-dimensional models [14] or multi-dimensional models [13]. In this paper we adopt a one-dimensionalmodel, which includes several features of other models reported in literature, and in particular seasonalvariability. The numerical scheme we use is the finite volume-complete flux scheme, described in [15,16]. The complete flux scheme is a flux approximation scheme for the advection-diffusion-reactionequation. The basic idea is to compute the numerical flux from a local BVP for the entire equation,including the source term. As a result, the numerical flux consists of two part, i.e., the homogeneous andinhomogeneous flux, corresponding to the advection-diffusion operator and the source term, respectively.The (constant coefficient) homogeneous flux is used in many applications as numerical flux; see e.g.[11]. The inhomogeneous flux is an extension, and is particulary of importance for dominant advectionin combination with a strong source term. It is therefore anticipated that the scheme will work well forturbulence models of phytoplanton, where the (horizontal) advection is dominant over diffusion [7].

In Section 2 we describe our mathematical model, and subsequently in Section 3, we outline thediscretisation scheme. A stability analysis of the scheme is presented in Section 4. In Section 5 weintroduce the boundary conditions. Numerical results are given in Section 6, and finally, concludingremarks are formulated in Section 7.

2 Mathematical model

In our model we incorporate phytoplankton, zooplankton and nutrients. Phytoplankton consumes nutri-ents and zooplankton feeds on phytoplankton. Phytoplankton needs light for photosynthesis and there-fore it can only live near the surface of the water. Because of gravity, phytoplankton tends to sinkslowly, however, near the surface it can be mixed upwards with the water due to wind stress. Plank-ton mostly lives in this mixed layer, which is generally not deeper than 100 meters. We assume thatthe (hydro)dynamics of the mixed layer does not depend on the horizontal directions, but only on thevertical space coordinate x. The key assumption in this article is that the dynamics of the mixed layerchanges with the seasons. In summers the water is warmer, which has two effects. First, phytoplanktonreproduces faster than in winter, see e.g. [4], and second, because of higher temperature gradients, thereis less mixing of the water, see e.g. [1] and [14].

The governing equations for the concentrations of phytoplankton P [mmol N2 m−3], zooplankton Z[mmol N2 m−3] and nutrients N [mmol N2 m−3] are given by:

∂P

∂t+ w

∂P

∂x=

∂

∂x

(K(x, t)

∂P

∂x

)+(p(x, t,N)− `P

)P − g(P )Z, (2.1a)

∂Z

∂t=

∂

∂x

(K(x, t)

∂Z

∂x

)− `ZZ + γg(P )Z, (2.1b)

∂N

∂t=

∂

∂x

(K(x, t)

∂N

∂x

)− αp(x, t,N)P + rα

(`PP + `ZZ

), (2.1c)

2 MATHEMATICAL MODEL 3

where w > 0 [m s−1] is the constant sinking velocity of phytoplankton, K(x, t) [m2 s−1] is the eddydiffusivity coefficient, p(x, t,N) [s−1] is the production rate of phytoplankton and g(P ) [s−1] is thegrazing rate, i.e., the rate at which zooplankton eats phytoplankton. Moreover, system (2.1) containsthe following coefficients: the mortality rates `P [s−1] and `Z [s−1] of phytoplankton and zooplankton,respectively, the assimilation coefficient γ [-], which is the fraction of phytoplankton that is convertedinto new zooplankton, the conversion factor α [-], which is the amount of nutrients that is needed toproduce phytoplankton and the regeneration rate r [-], i.e., the fraction of phytoplankton and zooplanktonthat is converted into nutrients. The advection term for the zooplankton is absent, because it has somecapability of staying near the surface. There is also no advection term for nutrients, because they do nothave enough mass to sink with a significant velocity. The system (2.1) is referred to as the NPZ-model.

Next, we introduce models for K, p and g. Mixing is generally described by a diffusion term, witheddy diffusivity coefficient K given by

K(x, t) = Ks(x)(1 + µK sin

(2πtT

)), (2.2a)

Ks(x) =Kmax

1 + C2

(exp

(−(x−x0C1

)2)+ C2

), (2.2b)

whereKs(x) [m2 s−1] is the stationary eddy diffusivity. Parameters in (2.2) are: the diffusivity variabilitycoefficient µK [-], the period of seasonal variation T , corresponding to one year, the maximum diffusivityKmax [m2 s−1] occuring at x = x0, C1 [m], which determines the thickness of the mixed layer and C2

[m], which determines the diffusivity at the bottom of the mixed layer. The diffusivity depends onmany things, such as wind stress and temperature of the water. We are not modelling any particulargeographical location and therefore we have chosen a representative profile for Ks(x), see Figure 1,which is based on profiles found in literature, see e.g. [6] and [14]. The sinusoidal term in (2.2a) isadded to take into account that there is less mixing in summers than in winters. Assuming that weconsider the Northern Hemisphere, the water temperature is maximal around August 1 and it is minimalaround February 1. This means that, with our definition of K, the initial time t = 0 corresponds toNovember 1.

0 20 40 60 80 1000

0.5

1

1.5x 10

−3

x [m]

Ks(x

) [m

2 s−

1 ]

Figure 1: Stationary eddy diffusivity Ks(x).

2 MATHEMATICAL MODEL 4

Analogous to the expression for K, the production rate p exhibits a seasonal variation given by

p(x, t,N) = ps(x,N)(1− µp sin

(2πtT

)), (2.3a)

ps(x,N) = pmax min( Line−νx

LH + Line−νx,

N

NH +N

), (2.3b)

where ps(x,N) [s−1] is the stationary production rate. Furthermore, µp [-] is the production variabilitycoefficient, pmax [s−1] is the maximum production rate, Lin [mol photons m−2 s−1] is the incident lightintensity, ν [m−1] is the light absorption coefficient and LH [mol photons m−2 s−1] and NH [mmolN2 m−3] are the half saturation constants of light and nutrients, respectively. It is assumed that theproduction rate of phytoplankton is limited by the amount of light and nutrients available. Both aredescribed according to the Holling type II functional response. For the nutrients this is widely used, seee.g. [1, 14]. For the dependence on light there are several options, and we choose the same Holling typeII functional response as in [12]. The light intensity is assumed to decrease exponentially with depth.Some limiting profiles of the stationary production rate are presented in Figure 2. The sinusoidal term in(2.3a) is included to account for the higher production rate in summers.

0 0.02 0.04 0.06 0.08 0.10

0.5

1

1.5

2x 10

−5

N [mmol N2 m−3]

p s [s−

1 ]

(a) unlimited light

0 20 40 60 80 1000

0.5

1

1.5

2x 10

−5

x [m]

p s [s−

1 ]

(b) unlimited nutrients

Figure 2: Profiles of the stationary production rate ps(x,N) in case of unlimited light and nutrients,respectively.

Finally, the grazing rate is defined as

g(P ) = Gmax(1− exp

(− kiv(P − Pmin)

), 0), (2.4)

whereG [s−1] is the phytoplankton predation rate, kiv

[(mmol N2

)−1 m3]

is Ivlev’s coefficient and Pmin

[mmol N2 m−3] is the treshold concentration of phytoplankton. This term has been taken from [14]. Asone might expect, the grazing rate approaches G when there is sufficient phytoplankton and reduces to0 when there is very little phytoplankton. A typical profile is given in Figure 3. All parameter values inthis section are listed in Table 1.

3 THE COMPLETE FLUX SCHEME 5

0 0.1 0.2 0.3 0.4 0.50

1

2

3

x 10−6

P [mmol N2 m−3]

g(P

) [s

−1 ]

Figure 3: Profile of the grazing rate g(P ).

3 The complete flux scheme

In this section we outline the derivation of the complete flux scheme for (2.1); for a detailed accountsee [15, 16]. We first present the stationary flux approximation and subsequently its extension to time-dependent problems.

Consider the stationary model equation

ddx

(wϕ−K(x)

dϕdx

)= s, (3.1)

for ϕ = P,Z or N and where s is the corresponding source term from (2.1). Introducing the flux

f = wϕ−K(x)dϕdx, (3.2)

we can write df/dx = s. We cover the domain with a finite number of disjunct cells (intervals) Ij =(xj−1/2, xj+1/2

)and define grid points xj in the centre of each interval where the variable ϕ has to

be approximated. Integrating equation (3.1) over the interval Ij and adopting the midpoint rule for theintegral of the source term, we obtain the discrete equation

Fj+1/2 − Fj−1/2 = sj∆x, (3.3)

where Fj+1/2 is the numerical flux approximating f at the cell boundary x = xj+1/2 and where sj =s(xj). We require that Fj+1/2 depends on both ϕ and s in the adjacent grid points xj and xj+1, i.e., weare looking for an expression of the form

Fj+1/2 = αj+1/2ϕj − βj+1/2ϕj+1 + ∆x(γj+1/2sj + δj+1/2sj+1

), (3.4)

where the coefficients αj+1/2 etc. depend on w and K. A similar expression holds for Fj−1/2. Re-lation (3.4) is the numerical approximation of the integral representation of f(xj+1/2), which can bederived from the local BVP on (xj , xj+1) for the entire equation (3.1), including the source term. Con-sequently, the numerical flux Fj+1/2 is the superposition of the homogeneous flux F h

j+1/2, dependingon the advection-diffusion operator, and the inhomogeneous flux F i

j+1/2, taking into account the effect

3 THE COMPLETE FLUX SCHEME 6

of the source term. Introducing the function R(x) = w∆x/K(x) as generalisation of the cell-Reynoldsnumber, we obtain the following expressions

Fj+1/2 = F hj+1/2 + F i

j+1/2, (3.5a)

F hj+1/2 = −

Rj+1/2

Rj+1/2

Kj+1/2

∆x

(B(Rj+1/2

)ϕj+1 −B

(− Rj+1/2

)ϕj

), (3.5b)

F ij+1/2 =

(12−W

(Rj+1/2

))∆x sj , (3.5c)

where B(z) := z/(ez − 1

), W (z) :=

(ez − 1 − z

)/(z(ez − 1

)), see Figure 4, and where the average

vj+1/2 and weighted average vj+1/2 (v = R,K) are defined as follows

vj+1/2 = 12

(vj + vj+1

), vj+1/2 = W

(− Rj+1/2

)vj +W

(Rj+1/2

)vj+1. (3.6)

Note that the weighted average vj+1/2 reduces to the arithmetic average vj+1/2 when w = 0 (no advec-tion) and to the upwind value vj when K(x) = 0 (no diffusion). We refer to the flux approximation (3.5)as the complete flux scheme.

Combining (3.3) and (3.5) we obtain the following discretisation scheme

−aW,jϕj−1 + aC,jϕj − aE,jϕj+1 = bW,jsj−1 + bC,jsj , (3.7a)

with coefficients aW,j etc. given by

aW,j =Kj−1/2

∆xRj−1/2

Rj−1/2

B(− Rj−1/2

), aE,j =

Kj+1/2

∆xRj+1/2

Rj+1/2

B(Rj+1/2

),

aC,j = aW,j+1 + aE,j−1, bW,j = ∆x(

12 −W

(Rj−1/2

)), bC,j = ∆x

(12 +W

(Rj+1/2

)).

(3.7b)

The discretisation scheme in (3.7a) gives rise to a linear system of the form

Aϕ = Bs + b, (3.8)

−10 −8 −6 −4 −2 0 2 4 6 8 10

2

4

6

8

10

−10 −8 −6 −4 −2 0 2 4 6 8 10

0.2

0.4

0.6

0.8

1

Figure 4: The functions B (left) and the function W (right).

4 STABILITY ANALYSIS 7

with A = tridiag(aW,j, aC,j, aE,j

)and B = tridiag

(bW,j,bC,j, 0

). The matrix A is an M-matrix for

the special case K(x) = Const.Next, consider the time-dependent extension of (3.1), i.e.,

∂ϕ

∂t+

∂

∂x

(wϕ−K(x, t)

∂ϕ

∂x

)= s, (3.9)

which can be written as ∂ϕ/∂t + ∂f/∂x = s with f defined in (3.2), however with a time dependentdiffusivity coefficient K(x, t). Analogous to the derivation of (3.3), we integrate equation (3.9) over thecell Ij to obtain the semidiscretisation

∆xϕj + Fj+1/2 − Fj−1/2 = sj∆x, (3.10)

where ϕj(t) denotes the approximation of ∂ϕ/∂t(xj , t). The procedure to derive the numerical flux iscompletely analogous to the stationary case, except that we include the time derivative in the source term,i.e., we replace s by s = s − ∂ϕ/∂t and determine the numerical flux from a local BVP on (xj , xj+1)for the quasi-steady equation ∂x

(wϕ−K(x, t)∂xϕ

)= s. This way we obtain

Fj+1/2 = F hj+1/2 +

(12 −W

(Rj+1/2

))∆x(sj − ϕj

), (3.11)

with F hj+1/2 defined as in (3.5b). Thus, the time derivative is included in the inhomogeneous flux. The

resulting semidiscretisation reads

bW,jϕj−1 + bC,jϕj − aW,jϕj−1 + aC,jϕj − aE,jϕj+1 = bW,jsj−1 + bC,jsj , (3.12)

with coefficients aW,j etc. defined in (3.7b). The extension of (3.8) is therefore an implicit ODE systemof the form

Bϕ+ Aϕ = Bs + b. (3.13)

To conclude, we have to apply a suitable time integration method to (3.13), for which we employ theϑ-method [9].

4 Stability analysis

In this section we investigate stability of the ϑ-method applied to the implicit ODE system (3.13). Forthat purpose, we assume that K(x) = Const and ignore the source term and boundary conditions.An analysis of (3.13) for a constant coefficient advection-diffusion equation in terms of dissipation anddispersion is presented in [16].

For s = b = 0, the ϑ-method applied to (3.13) reads

1∆t

B(ϕn+1 −ϕn

)+ (1− ϑ)Aϕn + ϑAϕn+1 = 0, (4.1)

where 0 ≤ ϑ ≤ 1. Stability of (4.1) is estabished in the following theorem.

Theorem.The ϑ-method (4.1) is stable if the following conditions hold{

0 ≤ ϑ < 12 and d ≤ 1

2(1−2ϑ)C(R),12 ≤ ϑ ≤ 1,

4 STABILITY ANALYSIS 8

with d = ∆t/∆x2, R = w∆x/K and where C is defined by

C(z) =1−

(1− 2W (z)

)(1− zW (z)

)(1 + z

(12 −W (z)

))2 ,

see Figure 5.Proof. Written componentwise, the scheme (4.1) reads

∆x(

12 −W (R)

) 1∆t(ϕn+1j−1 − ϕ

nj−1

)+ ∆x

(12 +W (R)

) 1∆t(ϕn+1j − ϕnj

)+

(1− ϑ)K

∆x(−B−ϕnj−1 +

(B− +B+

)ϕnj −B+ϕnj+1

)+

ϑK

∆x(−B−ϕn+1

j−1 +(B− +B+

)ϕn+1j −B+ϕn+1

j+1

)= 0,

(4.2)

with B± := B(±R). Substituting the discrete Fourier mode

ϕnj = gneiκxj ,

we obtain the following expressions for the amplification factor g, i.e.,

g =a1 − (1− ϑ)db1 + i(a2 − (1− ϑ)db2)

a1 + ϑdb1 + i(a2 + ϑdb2)(4.3a)

a1 =(

12 −W (R)

)cosψ + 1

2 +W (R), a2 = −(

12 −W (R)

)sinψ (4.3b)

b1 = 2(B+ +B−) sin2

(12ψ), b2 = R sinψ, ψ = κ∆x. (4.3c)

From the stability requirement |g|2 ≤ 1 we can deduce

a1b1 + a2b2 + (ϑ− 12)d(b21 + b22

)≥ 0,

or equivalently, in terms of the vectors a = (a1 a2)T and b = (b1 b2)T,

a·b +(ϑ− 1

2

)d|b|2 ≥ 0. (∗)

From (∗) we conclude that if the scheme is stable for ϑ = 12 , then it is certainly stable for 1

2 < ϑ ≤ 1. So,consider first θ = 1

2 , for which (∗) reduces to a·b ≥ 0. Expressing a and b in terms of 12ψ, substituting

the resulting formulae and using the relation 12

(B(z) + B(−z)

)= 1 + z

(12 − W (z)

)we obtain the

inequality

1− sin2(

12ψ)(

1− 2W (R))(1−RW (R)

)≥ 0,

which should hold for 0 ≤ ψ < π. Clearly, a sufficient condition reads(1− 2W (R)

)(1−RW (R)

)≤ 1,

which is identically satisfied, implying stability for 12 ≤ ϑ ≤ 1. Next, consider the case 0 ≤ ϑ < 1

2 . Inthis case, a sufficient condition for (∗) to hold is

1−(1− 2W (R)

)(1−RW (R)

)− (2− 4ϑ)d

(1 +R

(12 −W (R)

))2 ≥ 0,

from which we can derived the desired stability condition. QED

5 BOUNDARY TREATMENT 9

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

z

C(z)

Figure 5: The function C.

5 Boundary treatment

In this section we outline the boundary conditions for the semidiscretisation (3.12).At the surface, plankton nor nutrients can leave or enter the domain, therefore the corresponding

fluxes must be 0, i.e.,

wP (0, t)−K(0, t)∂P

∂x(0, t) = 0,

∂Z

∂x(0, t) =

∂N

∂x(0, t) = 0, t > 0. (5.1)

The bottom of the mixed layer is an artificial boundary where conditions are less obvious. So we neednumerical boundary conditions. First, consider the semidiscretisation (3.12) for ϕ = P at the boundarypoint xM = 100, i.e.,

bW,M PM−1 + bC,M PM −aW,MPM−1 +aC,MPM −aE,MPM+1 = bW,MsM−1 + bC,MsM , (5.2)

where the coefficients aC,M , bC,M etc. are defined in (3.7b). Note that this equation contains the un-known PM+1 in the virtual grid point xM+1. In order to eliminate PM+1 we assume that there is no dif-fusion in the virtual gridpoint, i.e., we set KM+1 = 0. Taking into account the definitions of (weighted)average in (3.6) we conclude that aE,M = 0, and consequently, PM+1 disappears naturally from (5.2).On the other hand, substituting KM+1 = 0 in (3.11) we obtain the equivalent condition

FP,M+1/2 = wPM + 12∆x

(sM − ϕM

), (5.3)

for the numerical flux at the virtual interface xM+1/2, where the subscript P is added to denote the fluxcorresponding to the plankton concentration, thus the numerical flux is determined by the balance ofadvection and net production at x = xM+1/2. Next, for zooplankton, we derive a similar condition.In the absence of advection the semidiscretisation (3.12) for ϕ = Z at x = xM reduces to the centraldifference discretisation

∆xZM −1

∆x

(KM−1/2ZM−1 − (KM−1/2 + KM+1/2)ZM + KM+1/2ZM+1

)= ∆xsM . (5.4)

In this case the conditionKM+1 = 0 is not sufficient to eliminate ZM+1 and therefore we put KM+1/2 =0. This is equivalent to the zero-flux condition FZ,M+1/2 = 0. Finally, the concentration of nutrientsbelow the mixed layer is assumed to be constant, which means that the following boundary conditionholds

N(100, t) = NB, t > 0. (5.5)

6 NUMERICAL EXAMPLES 10

6 Numerical examples

We have computed numerical solutions in the mixed layer for a period of 4 years using 100 grid pointsand 6000 time steps. We used ϑ = 1

2 for the time integration of (3.13), which proved to be uncondi-tionally stable. The parameter values used in the simulations can be found in Table 1 and are based on[2, 4, 12, 14]. The behavior after some time, say 1 or 2 years, is independent of the initial conditions.We have chosen P (x, 0) = Z(x, 0) = 1 and N(x, 0) = 0.01x. This means that N varies from 0 at thesurface to 1 at the bottom of the mixed layer.

First, we investigate the case without seasonal variability: µK = µp = 0. The solution is shown inFigure 6(a). We see that it converges to the stationary solution, which is shown in Figure 7. The profilesare similar to the ones in [7]. The concentration of phytoplankton is maximal at a depth of around 20meters and is also quite large at the surface and at 50 meters depth. The concentration of nutrients issmall near the surface and starts to increase significantly from 40 meters downward. The main differenceis a smaller concentration of zooplankton near the surface in [7], which is due to a sinking term forzooplankton.

Making the diffusion seasonally variable, µK = 0.5, we obtain the results in Figure 6(b). In this casewe see that P,Z and N slowly vary with the diffusion. This is expected, because higher diffusion meansthat nutrients diffuse towards the surface more rapidly. More nutrients results in more phytoplankton andmore phytoplankton results in more zooplankton.

When the phytoplankton production rate is seasonally variable, µp = 0.4, we obtain the results inFigure 6(c) and 6(d). Here we observe an important effect: a very steep phytoplankton bloom occursin March. Levels are back to normal approximately one month after the bloom starts, so it lasts only a

Parameters ValueH 100 mtmax 4 yearsKmax 1.2773× 10−3 m2s−1

w 10−4 m s−1

pmax 1.8× 10−5 s−1

ν 0.2 m−1

Lin 6× 10−4mol photons m−2s−1

LH 2× 10−5mol photons m−2s−1

`P 10−6 s−1

`Z 5× 10−7 s−1

G 3× 10−6 s−1

NH 10−2mmol N2 m−3

kiv 16.67(mmol N2

)−1 m3

Pmin 1.2× 10−2mmol N2 m−3

γ 0.5α 0.0175r 0.3µK 0, 0.5µp 0, 0.4

Table 1: Parameter values in the NPZ-model.

6 NUMERICAL EXAMPLES 11

(a) µK = 0, µp = 0

(b) µK = 0.5, µp = 0

(c) µK = 0, µp = 0.4

(d) µK = 0.5, µp = 0.4

Figure 6: Numerical solutions for different values of µK and µp.

6 NUMERICAL EXAMPLES 12

0 20 40 60 80 1000

0.05

0.1

0.15

x [m]

C [m

mol

N2

m−

3 ]

PZN

Figure 7: Stationary solution in case of no seasonal variability.

(a) (b)

Figure 8: Concentration of phytoplankton versus nutrients, and zooplankton versus phytoplankton, ata depth of 20 meters, for the case µK = 0.5 and µp = 0.4. In the NP figure, a darker shade of redrepresents a higher concentration of zooplankton. In the PZ figure, darker shade of blue represents ahigher concentration of nutrients.

very short period. During the phytoplankton bloom, also a zooplankton bloom starts. The zooplanktonconcentration does not increase as rapidly and as much as the phytoplankton concentration, but thezooplankton bloom does last longer. These phenomena correspond to observations in [17], [14] and [4].

The ways in which phytoplankton, zooplankton and nutrients interact, in case µK = 0.5 and µp =0.4, can be seen more clearly in Figure 8. Both of the periodic orbits are traversed in the counter-

7 CONCLUDING REMARKS 13

clockwise direction. This corresponds to our expectations, because in both cases the species on thevertical axis (predators) feeds on the species on the horizontal axis (prey). This means that we get Lotka-Volterra type behaviour. Generally, when there is a lot of food, i.e., when the system is relatively far tothe right in the figures, the concentration of predators will increase. This means that when the system isat the right, it will tend to move upward. Meanwhile, because the concentration of predators increases,the concentration of prey will decrease, which causes the system to move to the left. When the system isat the top left there are a lot of predators but not enough prey, and therefore the number of predators willdecrease and the systems moves downward again. After enough predators have died, the concentrationof prey can increase again, which means that the system moves to the right and the cycle is closed.

The phytoplankton bloom can easily be recognized in Figure 8. In Figure 8(a) it is the vertical peakand in Figure 8(b) it is the horizontal extreme. In both cases the bloom makes up a very large part ofthe curve. However, the trajectory moves much more rapidly in this part of the curve than in the restof it. This must be the case, because the phytoplankton bloom lasts only about one month whereas thecomplete orbit corresponds to one year.

We see that two conditions are satisfied at the moment the phytoplankton bloom starts. First, theconcentration of zooplankton is low and second, the concentration of nutrients is high. It is clear thatthe concentration of phytoplankton should increase in this case. However, the question remains why itincreases so suddenly and why the bloom always occurs in March, regardless of the time of the yearat which the simulation is started. The main reason appears to be that in March the water temperatureincreases and therefore the growth rate of the phytoplankton increases. The consequence of this increasedgrowth rate is that the zooplankton, which is at that time only present in relatively small numbers, cannot eat the phytoplankton quickly enough to keep the growth under control. This uncontrolled growth isapproximately exponential, typical for an exploding population. The growth lasts until the concentrationof nutrients becomes too small to support the growth of the phytoplankton and the concentration ofzooplankton has become large enough.

7 Concluding remarks

The model that we propose describes the basic features of the ecosystem. One thing that makes thismodel different from most others, is the way we deal with the boundary at the bottom of the domain: weassume that the diffusion of phytoplankton and zooplankton is 0 at the lower boundary. This providesa good approximation of the real behavior. Another important aspect of the model is that we take intoaccount seasonal variability. The seasonal effects of a higher phytoplankton production rate when thewater is warmer and a higher turbulent diffusivity when the water is colder are included into the model.

The numerical scheme which we use to solve the system is the finite volume-complete flux scheme.The complete flux scheme includes the source term and time derivative already in the flux approximationand has proven to be a very accurate scheme.

We have applied the model to a (hypothetical) representative real life problem. Our results correspondto those in some of the literature. The results in other publications can be quite different, which is notsurprising because (very) different parameter values are used. In order to validate the model properly,a geographical location should be chosen for which the parameter values can be obtained. The realbehaviour at that location should be measured and the results of our model should be compared to that.

A few extensions of our model are suggested. It would be useful to have a separate model to calculatethe hydrodynamical behaviour. This way it is possible to get more accurate profiles for, e.g., the tem-perature and the diffusion coefficient. Another extension would be to include transport in the horizontal

REFERENCES 14

direction. Doing this, advection will play a bigger role, but this should not be a problem consideringthe numerical scheme that is used. Another possibility is to include more species, such as more kinds ofplankton.

References

[1] O. Arino, K. Boushaba and A. Boussouar (2000). A mathematical model of the dynamics of thephytoplankton-nutrient system. Nonlinear analysis: real world applications 1, 69-87.

[2] A. Dube and G. Jayaraman (2008). Mathematical modelling of the seasonal variability of planktonin a shallow lagoon. Nonlinear Analysis 69, 850-865.

[3] H.I. Freedman (1980). Deterministic Mathematical Models in Population Ecology, Marcel Dekker,New York.

[4] J.A. Freund, S. Mieruch, B. Scholze, K. Wiltshire and U. Feudel (2006). Bloom dynamics ina seasonally forced phytoplankton-zooplankton model: trigger mechanisms and timing effects.Ecological Complexity 3, 129-139.

[5] L. Grieco, L.B. Tremblay and E. Zambianchi (2005). A hybrid approach to transport processesin the Gulf of Naples: an application to phytoplankton and zooplankton population dynamics.Continental Shelf Research 25, 711-728.

[6] A. Larsson (2004). The role of mixing in an Antarctic ocean ecosystem: observations and modelcomputations of vertical distributions of related parameters. Deep Sea Research Part II: TopicalStudies in Oceanography 51, 2807-2825.

[7] D.M. Lewis (2005). A simple model of plankton population dynamics coupled with a LES of thesurface mixed layer. Journal of Theoretical Biology 234, 565-591.

[8] J.F. Lopes, A.C. Cardoso, M.T. Moita, A.C. Rocha and J.A. Ferreira (2009). Modelling the tem-perature and the phytoplankton distributions at the Aveiro near coastal zone, Portugal. EcologicalModelling 220, 940-961.

[9] R.M.M. Mattheij, S.W. Rienstra and J.H.M. ten Thije Boonkkamp (2005). Partial DifferentialEquations, Modeling, Analysis, Computation. SIAM, Philadelphia.

[10] http://earthobservatory.nasa.gov/Features/Phytoplankton.

[11] S.V. Patankar (1980). Numerical Heat Transfer and Fluid Flow. Series in Computational Methodsin Mechanics and Thermal Sciences, Hemisphere Publishing Corporation, New York.

[12] N.N. Pham Thi (2005). On positive solutions in a phytoplankton-nutrient model. Journal of Com-putational and Applied Mathematics 117, 467-473.

[13] N.N. Pham Thi, J. Huisman and B.P. Sommeijer (2005). Simulation of three-dimensional phy-toplankton dynamics: competition in light-limited environments. Journal of Computational andApplied Mathematics 174, 57-77.

REFERENCES 15

[14] N. Skliris, K. Elkalay, A. Goffart, C. Frangoulis and J.H. Hecq (2001). One-dimensional modellingof the plankton ecosystem of the north-western Corsican coastal area in relation to meteorologicalconstraints. Journal of Marine Systems 27, 337-362.

[15] J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen (2011). The finite volume-complete fluxscheme for advection-difussion-reaction equations. J Sci Comput 46, 47-70.

[16] J.H.M. ten Thije Boonkkamp and M.J.H. Anthonissen (2010). Extension of the complete fluxscheme to time-dependent conservation laws. In: G. Kreiss et al. (eds.) Numerical Mathematicsand Advanced Applications 2009, Proceedings ENUMATH 2009, Springer-Verlag, Berlin, 865-873.

[17] C. Troupin (2010). Seasonal variability of the oceanic upper layer and its modulation of biologicalcycles in the Canary Island region. Journal of Marine Systems 80, 172-183.

[18] A. Verma, S. Goorden, E. Alemayehu and P.A. Marin Zapata (2010). Modeling of the residencetime of phytoplankton in the surface mixed layer of the ocean. private communication.

[19] J. Woods (2005). Stability and predictability of a virtual plankton ecosystem created with anindividual-based model. Progress in Oceanography 67, 43-83.

PREVIOUS PUBLICATIONS IN THIS SERIES:

Number Author(s) Title Month

11-26 11-27 11-28 11-29 11-30

M. Rudnaya R.M.M. Mattheij J.M.L. Maubach H.G. ter Morsche J.A.W.M. Groot C.G. Giannopapa R.M.M. Mattheij C.G. Giannopapa J.A.W.M. Groot A. Muntean T.L. van Noorden S.A. Goorden S. P. Korzilius J.H.M. ten Thije Boonkkamp M.J.H. Anthonissen B.V. Rathish Kumar

Gradient-based sharpness function Modelling stretch blow moulding of polymer containers using level set methods Modeling the blow-blow forming process in glass container manufacturing: A comparison between computations and experiments Corrector estimates for the homogenization of a locally-periodic medium with areas of low and high diffusivity NPZ-model with seasonal variability of plankton population dynamics

Apr. ‘11 Apr. ‘11 Apr. ‘11 Apr. ‘11 Apr. ‘11

Ontwerp: de Tantes,

Tobias Baanders, CWI