SPRAT A Spatially-Explicit Marine Ecosystem Model Based on Population Balance Equations

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SPRAT: A Spatially-Explicit Marine Ecosystem Model Based on

Population Balance Equations

Arne N. Johansona,b,∗, Andreas Oschliesa, Wilhelm Hasselbringb, Boris Wormc

aGEOMAR Helmholtz Centre for Ocean Research, Düsternbrooker Weg 20, 24105 Kiel, Germany

bKiel University, Department of Computer Science, 24098 Kiel, Germany

cDalhousie University, Biology Department, Halifax, NS, Canada B3H 4R2

Abstract

To successfully manage marine ﬁsheries using an ecosystem-based approach, long-term predictions of ﬁsh stock devel-

opment considering changing environmental conditions are necessary. Such predictions can be provided by end-to-end

ecosystem models, which couple existing physical and biogeochemical ocean models with newly developed spatially-

explicit ﬁsh stock models. Typically, Individual-Based Models (IBMs) and models based on Advection-Diﬀusion-Re-

action (ADR) equations are employed for the ﬁsh stock models. In this paper, we present a novel ﬁsh stock model

called SPRAT for end-to-end ecosystem modeling based on Population Balance Equations (PBEs) that combines the

advantages of IBMs and ADR models while avoiding their main drawbacks. SPRAT accomplishes this by describing the

modeled ecosystem processes from the perspective of individuals while still being based on partial diﬀerential equations.

We apply the SPRAT model to explore a well-documented regime shift observed on the eastern Scotian Shelf in the

1990s from a cod-dominated to a herring-dominated ecosystem. Model simulations are able to reconcile the observed

multitrophic dynamics with documented changes in both ﬁshing pressure and water temperature, followed by a predator-

prey reversal that may have impeded recovery of depleted cod stocks.

We conclude that our model can be used to generate new hypotheses and test ideas about spatially interacting ﬁsh

populations, and their joint responses to both environmental and ﬁsheries forcing.

Keywords: end-to-end modeling, population balance equation, ﬁsh stock prediction, ecosystem-based management

1. Introduction

Living marine resources and their exploitation by ﬁsh-

eries play an important role in sustaining global nutrition

but many of the world’s ﬁsh stocks are in poor condition

due to overharvesting (Worm et al., 2009; Costello et al.,

2016). This reduces the productivity of the stocks signiﬁ-

cantly and necessitates improved management in order to

achieve a sustainable use of global ﬁsheries resources.

Fishing, however, is not the only impact on the condi-

tion and productivity of ﬁsh stocks but long- and short-

term variability of environmental parameters due to cli-

mate change or other sources of variability (such as the

North Atlantic Oscillation (NAO)) imposes additional pres-

sures (Brander, 2007). The eﬀects of changes in the en-

vironment on ﬁsh can be direct (e.g., by altering individ-

ual growth rates) or indirect (by aﬀecting the net primary

productivity and, thus, the carrying capacity of the ecosys-

tem). Sometimes, these factors may interact with anthro-

pogenic inﬂuences in complex ways. For example, the ex-

∗Corresponding author

Email addresses: arj@informatik.uni-kiel.de (Arne N.

Johanson), aoschlies@geomar.de (Andreas Oschlies),

wha@informatik.uni-kiel.de (Wilhelm Hasselbring),

bworm@dal.ca (Boris Worm)

pansion of oxygen minimum zones in the tropical north-

east Atlantic Ocean due to climate change compresses the

suitable habitat of pelagic predator ﬁsh to a narrow sur-

face layer and, thus, increases their vulnerability to sur-

face ﬁshing gear (Stramma et al., 2012). The resulting

high catch rates in such areas can lead to overly optimistic

estimates of species abundance and, therefore, to exagger-

ated ﬁshing quotas that put the aﬀected stocks in danger

of overexploitation.

Another case illustrating the complexities of how ﬁsh-

ing and climate can interact in driven rapid ecosystem

change is the recent overﬁshing of Atlnatic cod (Gadus

morhua) stocks in the Gulf of Maine that occurred despite

stringent management practises. Here, retrospective anal-

ysis showed that this change can in large part be attributed

to rapid ocean warming that has led to an unrecognized

eﬀects on recruitment and mortality, and indirectly ren-

dered ﬁsheries exploitation rates unsustainable (Pershing

et al., 2015).

From such examples, it becomes apparent that ﬁsheries

management must address the eﬀects of ﬁshing and climate

variability and change in a joined framework, accounting

for the eﬀects of diﬀerent sources of mortality, including

changes in natural mortality, predation, and ﬁshing (Rose

et al., 2010). Thus a more holistic ecosystem-based ap-

Preprint submitted to Elsevier October 4, 2022

arXiv:2210.01100v1 [q-bio.PE] 30 Sep 2022

proach has been called for, which may focus on marine

ecosystems as a whole and takes into account the interde-

pendence of their components (Cury et al., 2008).

Ecological models that can supply this kind of infor-

mation are sometimes called end-to-end models because

they incorporate all ecosystem components from the dy-

namics of the abiotic environment to primary producers

to top predators (Travers et al., 2007). In such mod-

els, the diﬀerent elements of the ecosystem are linked to-

gether mainly through trophic interactions—i.e., by feed-

ing (Moloney et al., 2011). Ideally, all these links between

components are modeled bidirectionally (e.g., an increase

in ﬁsh biomass due to feeding on zooplankton is also re-

ﬂected in a decrease of zooplankton biomass). Such a

two-way coupling of model elements allows to explicitly

resolve at the same time both bottom-up and top-down

mechanisms of ecological control. It is the combination of

modeling these bidirectional links in the trophic structure

and considering the dynamics of the environment that en-

ables end-to-end models to provide long-term predictions

on the development of ﬁsheries ecosystems under environ-

mental change. In the context of ecosystem-based ﬁsheries

management, these predictive capabilities can be used to

evaluate diﬀerent management scenarios with regard to

their long-term eﬀectiveness (Stock et al., 2011).

In practice, end-to-end models are typically constructed

by using an existing physical and biogeochemical ocean

model (for the abiotic environment as well as for nutrient

and plankton dynamics) and creating a spatially-explicit

ﬁsh model that can be coupled with the ocean model (Shin

et al., 2010). In this context, ﬁsheries are usually included

in the model by assuming a mortality rate due to ﬁshing

(constant or changing with time), which applies homo-

geneously to the ﬁsh population beyond a certain lower

size limit. Implementing a complete end-to-end model

from scratch is discouraged by the amount of eﬀort that is

needed for developing sophisticated physical and biogeo-

chemical models.

The most widely used ﬁsh models for end-to-end mod-

eling are either Individual-Based Models (IBMs), such as

OSMOSE (Shin and Cury, 2001), or models based on Ad-

vection-Diﬀusion-Reaction (ADR) equations, such as SEA-

PODYM (Bertignac et al., 1998; Lehodey et al., 2013).

IBMs oﬀer the advantage that they are relatively easy to

parametrize as their parameters are typically observable in

individual ﬁsh. Additionally, these models can easily fea-

ture an emergent, dynamic food web structure. However,

since—at the ocean scale—it is not feasible to simulate

all individual ﬁsh of the study region, so-called super in-

dividual approximations of IBMs are employed (Scheﬀer

et al., 1995). With this approach, individuals that share

similar characteristics are replaced by a so-called super in-

dividual—i.e., an individual that has parameters similar to

those of the individuals it represents plus an additional pa-

rameter that describes the number of individuals it stands

for. This approximation technique is problematic because

there is no mathematical framework for IBMs that would

allow to formally study how many super-individuals are

necessary to simulate the inter-individual interactions with

suﬃcient accuracy.

Since ADR models are based on Partial Diﬀerential

Equations (PDEs), they integrate well with existing bio-

geochemical ocean models and feature a rigid mathemati-

cal framework with established approximation techniques

for which formal error bounds can be described. Further-

more, ADR equations are derived from the principle of

mass conservation and are, thus, well-suited for studying

mass ﬂuxes in marine ecosystems. However, ADR models

can be diﬃcult to parametrize because most of their pa-

rameters are usually not observable in individual ﬁsh and

the full life cycle of the ﬁsh species is not directly repre-

sented in their main equation (only discrete age classes can

be modeled).

In order to combine the advantages of IBMs and ADR

models and to prevent their main drawbacks, we propose

a ﬁsh model for end-to-end modeling that is based on

so-called Population Balance Equations (PBEs) (Ramkr-

ishna, 2000). Our PBE model—which is called SPRAT —

represents ﬁsh as density distributions on a combined con-

tinuous space-body size domain. Since PBEs are based on

diﬀerential equations, they share the advantages of ADR

models with regard to the integration with existing bio-

geochemical ocean models and to the existence of estab-

lished approximation techniques. At the same time, PBE

models share the distinct advantage of IBMs that most of

their parameters can directly be observed in individual ﬁsh

and that food web structure emerges dynamically from the

model.

Potential drawbacks introduced by our PBE-based mod-

el SPRAT in comparison to IBMs and ADR models in-

clude:

1. Since we represent ﬁsh as density distributions we

cannot track ﬁsh and their interactions down to the

level of single individuals (as it would be possible

with an IBM not using the super individual approx-

imation).

2. In comparison to ADR models, the SPRAT model

is associated with increased computational costs be-

cause PBE models represent the size of individuals

as an additional dimension of the domain of a PDE.

For a more detailed comparison of the PBE approach with

IBMs and ADR models refer to Johanson (2016, Chap. 10).

The PBE approach to ﬁsh stock modeling is similar to

so-called size spectra models, which also describe ﬁsh via

distribution functions on a continuous body size domain

(see, e.g., Carozza et al., 2016; Andersen and Beyer, 2006;

Maury and Poggiale, 2013). In the context of size spec-

tra models, however, space is typically not resolved in the

deduction of the models and is only introduced later on

by assigning an instance of the respective model to each

box or grid point of a discretized spatial grid (hence these

models could be characterized as replicated one-dimen-

sional or univariate PBE models). An exception to this

is the APECOSM model by ?, which is designed to study

apex predators (namely tuna). APECOSM is a spatially-

continuous, mass-balanced size spectrum model that, like

SPRAT, oﬀers a uniﬁed continuous description of ﬁsh in

both space and body size via a single distribution function.

Hence, SPRAT could also be described as a spatially-con-

tinuous size spectrum model. Despite the strong similar-

ities between these approaches, we prefer to call SPRAT

a PBE model to highlight that SPRAT is derived from

a model type which is widely applied in engineering and

has a large body of research associated with it (especially

regarding fast discretization techniques; see, e.g., ?).

In this paper, we apply SPRAT to simulate and mech-

anistically explore the complex interactions between the

diﬀerent components of the eastern Scotian Shelf ecosys-

tem, speciﬁcally plankton and ﬁsh populatipons, ﬁsheries

and climate. The SPRAT model was implemented using

a software engineering approach of the same name, which

we presented earlier (Johanson et al., 2016; Johanson and

Hasselbring, 2014a,b).

2. Material and Methods

2.1. Model Description

Before describing the SPRAT model in detail, we ﬁrst

give a conceptual overview of the model, loosely following

the ODD (Overview, Design concepts, Details) protocol

by Grimm et al. (2006), which was designed as a standard

protocol for describing IBMs.

Overview.

Purpose. The main goal in the development of SPRAT is

to construct a ﬁsh stock prediction model for end-to-end

modeling which can represent ﬁsh in a completely contin-

uous, rigorous mathematical framework and at the same

time is still formulated from the perspective of the indi-

vidual ﬁsh to allow for a dynamic food web structure. The

aim of this speciﬁc study is to employ SPRAT to simulate

and explore a well-doumented regime shift on the eastern

Scotian Shelf in the 1990s from a mainly cod-dominated

to a herring-dominated ecosystem (Section 2.2).

State variables and scales. In our baseline simulation, we

include ﬁve state variables: nutrient (N), phytoplankton

(P), and zooplankton (Z) concentrations as well as mass

distributions of two ﬁsh species complexes (predator and

prey species representing mainly cod and herring). All

state variables vary in time and space (two spatial dimen-

sions representing the vertically-integrated ocean). The

two ﬁsh mass distributions furthermore vary in an addi-

tional body size dimension. The N,P, and Zvariables

are modeled discretely in space and continuously in time.

The ﬁsh mass distributions are modeled continuously in

all dimensions.

Biogeochemical Ocean Model

Currents Temperature

Zooplankton

SPRAT

Movement

Passive

Active

Predictive

Reactive

Reproduction

Background Mortality

Metabolic Costs

Net Swimming

Costs

Resting

Metabolic Rate

Fishing

Predation (Opportunistic)

Intake Losses

Growth

Controlled by

Time/Temp.

Controlled by

Biomass Uptake

Figure 1: Conceptual diagram of the SPRAT model.

The vertically-integrated ocean region of the eastern Sco-

tian Shelf is assumed to be a homogeneous 450 by 450 km

square. For approximating the solution of our PDE-based

model, this space is discretized into 48 by 48 equally-sized

rectangular cells. The size dimension is divided into 32

cells using logarithmically distributed division points. The

model is integrated in time from the year 1970 till 2010

with a time step of about 1 hour.

Process overview. An overview of the processes resolved

by our model is given in Figure 1. SPRAT is loosely based

on dynamic energy budget models, which model the ﬂow

of energy or biomass through the ecosystem (see, e.g.,

Maury and Poggiale, 2013). The most important path-

ways in such models are predation, metabolic costs (rest-

ing metabolic rate and locomotion), structural growth, re-

production, and mortality (esp. due to ﬁshing). Where

applicable, we model these processes to be dependent on

temperature because of the major signiﬁcance of this vari-

able for controlling metabolic rates (Clarke and Johnston,

1999). In general, we include ocean currents into SPRAT

but ignore them for the Scotian Shelf scenario because we

make the simplifying assumption of space being homoge-

neous (see Section 2.3).

Our ﬁsh stock model is coupled bi-directionally with a

simple, spatially-explicit NPZ model. The NPZ model re-

solves only one class of phytoplankton and zooplankton,

each. SPRAT and the NPZ model are linked via the Z

state variable: when ﬁsh feed on zooplankton, the corre-

sponding amount of biomass is transferred from the zoo-

plankton state variable to the ﬁsh mass distribution.

Design Concepts.

Mass conservation. In the absence of sources and sinks,

such as zooplankton grazing and ﬁshing, the SPRAT model

is mass-conserving: the overall biomass in the system stays

constant over time (the term Population Balance Equation

refers to the fact that the overall population mass may be

distributed diﬀerently but, in total, stays in balance over

time).

Emergence. The evolution of the ﬁsh density distributions

is described from the perspective of the individual ﬁsh and

no ﬁxed food web structure is prescribed. This implies

that most of the parameters of SPRAT can be observed

in individual ﬁsh and that the trophic structure of the

ecosystem emerges dynamically from the model.

Interaction. In SPRAT, ﬁsh interact with each other mainly

through feeding on each other. In particular, the ﬁsh seek

out locations, where the concentration of potential prey is

high.

Sensing. We model ﬁsh to have perfect information about

their environment within a certain radius. In particular,

we assume that they are able to assess, in which direction

the maximum prey concentration lies within their radius

of perfect information.

Prediction. If prey abundance levels fall below a certain

level within the radius of perfect information, predatory

ﬁsh will apply simple strategies (swim in a ﬁxed direction)

to ﬁnd more suitable feeding grounds.

Stochasticity. SPRAT is completely deterministic.

Details. In the SPRAT model, ﬁsh of species κ= 1, . . . , n

are represented by the average carbon mass distribution

m[κ]: [0, tmax]×Ω→R,(t, x, y, r)7→ m[κ](t, x, y, r)(1)

with the combined space-size domain

Ω=ΩS×[rmin, rmax]. (2)

Here, tmax ∈R>0describes an arbitrary time limit for the

model, ΩSis a two-dimensional polygon domain represent-

ing the vertically integrated ocean, and rmin and rmax are

the minimal and maximum carbon content masses of indi-

vidual ﬁsh in the model, respectively. With carbon mass,

we refer to the absolute mass of the carbon content of the

dry mass of ﬁsh (both the mass distribution m[κ]and the

body size dimension rare carbon masses). By speaking of

average carbon mass, we mean that for any volume V⊂Ω,

the carbon mass of species κcontained in that volume is

given by RVm[κ](t, x, y, r)d(x, y, r). The unit of m[κ]is

kg C m−2(kg C)−1= m−2. Note that the model uses a

continuous size dimension instead of discrete size classes

as often found in ﬁsh models used for end-to-end modeling

purposes (e.g., ???). Tables B.5 and B.6 in Appendix B

provide an overview of all parameters of SPRAT.

Figure 2 provides a conceptual visualization of the dis-

tribution functions described by Equation 1. For each

ﬁsh species κ, at every point (x, y)of the spatial domain

Figure 2: Conceptual visualization of the ﬁsh mass distributions

in SPRAT. At each spatial point (x, y), there is a one-dimensional

biomass distribution along the size dimension r.

(i.e., the vertically-integrated ocean), there is a one-dimen-

sional distribution of ﬁsh biomass along the body size di-

mension. This one-dimensional distribution describes how

much mass m[κ](t, x, y, r)of individuals of a certain mass

rthere is in the system, at the current point in time t.

Our model assumes constant ratios for carbon mass to

dry mass (C[κ]

C/d) and dry mass to wet mass (C[κ]

d/w) for each

species. For convenience, we deﬁne M[κ]to denote the

wet mass distribution of species κand u[κ]to denote the

average individual count density of the respective species.

The evolution of the distributions m[κ]is governed by

the following system of mass-conserving population bal-

ance equations:

∂

∂t m[κ]+∂

∂x q[κ]

xm[κ]+∂

∂y q[κ]

ym[κ]+∂

∂r g[κ]m[κ]=H[κ](3)

The vector q[κ]= (q[κ]

x, q[κ]

y)is the spatial advection veloc-

ity of species κwith unit m s−1,g[κ]is a growth rate with

unit kg C s−1, and H[κ]is a source term with unit kg C

m−2(kg C)−1s−1= m−2s−1. The spatial derivatives (∂x

and ∂y) describe the transport of ﬁsh in space with ve-

locity q[κ](due to both passive advection by currents and

due to active locomotion). Correspondingly, the deriva-

tive in the size dimension (∂r) describes growth with the

dynamic growth rate g[κ]. The source term H[κ]expresses

how ﬁsh biomass is introduced to and removed from the

system. In particular, the source term handles ﬁshing as

well as the mass re-distribution occurring during feeding

and reproduction.

In the following sections, we specify the functional forms

of the terms in Equation 3 to include the concepts shown

in the overview of the SPRAT model in Figure 1.

2.1.1. Length-Weight Relationship

In some contexts, we need the lengths of individuals

instead of their weight. To convert wet mass M(in kg) to

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SPRAT:ASpatially-ExplicitMarineEcosystemModelBasedonPopulationBalanceEquationsArneN.Johansona,b,,AndreasOschliesa,WilhelmHasselbringb,BorisWormcaGEOMARHelmholtzCentreforOceanResearch,DüsternbrookerWeg20,24105Kiel,GermanybKielUniversity,DepartmentofComputerScience,24098Kiel,GermanycDalhousieUniversi...

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