
---
title: "Overview of model concepts and design"
author: |
  | Michael R. Heath (m.heath@strath.ac.uk)
  | University of Strathclyde, Department of Mathematics and Statistics,
  | Glasgow, Scotland, UK
date: "`r format(Sys.Date(), '%B %d, %Y')`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Overview of model concepts and design}
  %\VignetteEngine{knitr::rmarkdown}
---

\


# Summary

* **StrathE2E2** is a model of the 'big-picture', whole ecosystem effects of hydrodynamics, temperature, nutrient additions, and fishing 
on continental shelf marine food webs.

* **StrathE2E2** has two linked parts - a fishing fleet model and an ecology model. The fishing model integrates harvesting, discarding and seabed 
disturbance rates across a range of gears and passes the results into the ecology model.

* The ecology model is a network of coupled ordinary differential equations representing the rates of change in organic detritus, dissolved inorganic 
nutrient, and coarse guilds of living biomass spanning microbes to megafauna. The equations include representations of feeding, metabolism, reproduction, active migrations, advection and mixing. Environmental driving data include temperature, irradiance, hydrodynamics, and nutrient inputs from rivers, atmosphere and ocean boundaries.

\


## See also the following supporting documents:
[Origin of the model (previous versions)](StrathE2E2_Model_origin.html)   
[Ecology model description](StrathE2E2_Ecology_model.pdf)  
[Fishing fleet model description](StrathE2E2_Fishing_fleet_model.pdf)  
[Parameter optimization, sensitivity and Monte Carlo analysis](StrathE2E2_Optimization_sensitivity_Monte_Carlo_methodology.pdf)  
[StrathE2E2 implementation for the North Sea](StrathE2E2_North_Sea_model.pdf)

\


## 1 Introduction

The effects of anthropogenic or natural pressures applied to any part of an ecosystem are eventually felt everywhere to some extent through the phenomenon known as 
a 'trophic cascade' (Pace et al., 1999). Cascading effects are attenuated or amplified as they propagate through the food web, depending on the nature of the 
pressure and details of the ecology (Heath et al., 2014). Diagnosing the type and magnitude of pressures that an ecosystem can sustain before being fundamentally 
altered requires simulation with mathematical models that aim to represent the key ecological components and processes which govern cascades.

\

StrathE2E2 models both bottom-up and top-down trophic cascades in shelf-sea ecosystems, spanning inorganic and organic nutrients through to birds and mammals. 
The model takes a highly macroscopic, view of ecology, aggregating over the many microscopic details of taxonomy, demography and spatial structure. The aim is 
to represent the gross dynamics with a tolerable parameter count so as to enable 'big-picture' strategic scenario analyses. The model is supported by functiins enabling 
computational parameter optimization, sensitivity analysis and estimation of credible intervals of model outputs. The scheme 
comprises a fishing fleet model and an ecology model with coupling between the two.


### 2 Ecology model general description

The ecology model is a network of mass conserving coupled ordinary differential equations (ODEs) describing spatially averaged rates of change in state variables 
representing organic detritus, dissolved inorganic nutrient, and living biomass. To simplify the description we can think of the variables as being divided between 
two coupled sub-networks: a predator-prey network - the food web - and a nutrient recycling network. Between the two, all marine life-forms are explicitly or 
implicitly accounted for, but aggregated into coarse groups or 'guilds' defined mainly by feeding characteristics and diet preferences (**Figure 1**). All state 
variables, except macrophytes, are expressed solely in terms of nitrogen mass, since this element is the most commonly limiting in temperate shelf seas. Macrophytes 
are expressed in terms of both nitrogen and carbon mass with dynamic stoichiometry since these organisms have an exceptional capacity to seasonally absorb and 
store nitrogen.

![](images/FOOD_WEB_map_r.png)

**Figure 1**. Schematic of the food web compartments of the StrathE2E2 model. Green arrows represent advection, mixing and migration; orange arrows represent 
fishery-related fluxes; black arrows represent biological fluxes. Red labelled components are active migrators whilst blue are subject to passive advection 
and mixing and black are anchored. Pale blue boxes represent quantities that are exported from the model whilst yellow are imported. The model also includes 
fluxes from living components to ammonia, detritus and corpses due to excretion, defecation and death but these are not shown for clarity. Also for clarity, 
birds, pinnipeds and cetaceans are combined as a single box but in the model are separate entities. The abbreviation "Macrop." is shorthand for macrophytes.

\

Each ODE comprises a set of rate-of-change terms representing a variety of biological and physical processes (**Box 1**, **Box 2** for macrophytes). Biological 
terms describe the balance between gains due to assimilation of food, and losses due to mortality and metabolism. Some components of the food web (planktivorous 
and demersal fish; suspension/deposit feeding and carnivore/scavenge feeding benthos) are resolved into life-stages and for these the equations also include the 
balance between gains due to recruitment and losses due to developmental progression or spawning. In addition, all components of the model are, in principle, 
replicated across homogeneous spatial compartments. To facilitate this each ODE also includes terms representing sinking, advection, mixing and migration flows 
through the system. 

![](images/Equation_box_1.png)

\

![](images/Equation_box_2.png)

\

\

The spatial structure is highly stylised, consistent with the coarse guild-definitions of the living and chemical components of the system. Two horizontally distinct 
but interconnected bathymetric/hydrographic zones are distinguished - a shallow, vertically mixed zone mostly influenced by tides and freshwater inputs, and a deeper, 
potentially seasonally stratified zone mostly influenced by exchange with an external ocean (**Figure 2**). For convenience we refer to these as the inshore and 
offshore zones respectively, though there is no necessity for the inshore zone to be adjacent to the coast - in principle it could represent a shallow offshore bank. 
The water column in the offshore zone is divided vertically into two compartments or layers, whilst the inshore zone is represented by a single compartment. 
Seabed habitats are represented by exposed rock and up to three compartments of different sediment properties in each zone, each defined by median grain size and 
natural disturbance rates. State variables are resolved hierarchically to spatial compartments with the largest (in terms of body size) and/or most mobile guilds 
being represented at the coarsest spatial resolution (**Table 1**). The nominal sea surface area of the model domain is 1m^2^ sea surface area. Hence, the units of 
all the state variables (mM nitrogen) are also scaled to a domain of 1m^2^.

![](images/model_physical_structure_r.png)

**Figure 2**. Schematic showing the horizontal and vertical spatial structure of the model. The compartments S0-S3 and D0-D3 refer to inshore/shallow and 
offshore/deep seabed sediment habitats respectively. S0 and D0 are rock habitats which reflect, rather than absorb, settling material back into the water 
column.

\

**Table 1**. Ecology model state variables and spatial hierarchy.

![](images/TABLE_1.PNG)

\


#### 2.1 Predator-prey connections, demography and mortality 

Ingestion of prey by a predator is governed by a preference matrix and a standard type II response in which per-unit-biomass predator consumption 
rates increase asymptotically towards a Q10 temperature-dependent maximum with increasing prey concentration (**Box 1**). A proportion of ingested 
food becomes new body mass in the predator. The remainder is divided equally between fluxes to organic detritus and ammonia, to represent defecation 
of undigested material and food-dependent metabolism. Background (non-feeding) metabolism increases with temperature but with a higher Q10 than 
maximum uptake rates, so the net result is that productivity, i.e. production rate per unit biomass, will exhibit a dome-shaped response to 
temperature.

\


#### 2.2 Nutrient recycling network

Six forms of organic detritus are represented in the recycling network: suspended material, labile and refractory sediment material, 'macrophyte 
debris', 'corpses', and 'discards'. Both the suspended and sediment fractions implicitly include dissolved and particulate organic matter and 
associated bacterial flora, and are formed in the living food web by defecation and density-dependent mortality fluxes from plankton and the 
larval stages of fish and benthos. Corpses are produced by density-dependent mortality of fish, benthos and top predators, and the decay of 
discards. The latter are a short-lived, special form of detritus generated as a by-product of fishery harvesting. Macrophyte debris is created 
by wave and density-dependent destruction of living macrophyte forest biomass. All forms of detritus are regarded as a potential food source 
for detritivorous and scavenge feeding guilds of living organisms.

\

The dynamics of each detritus and dissolved nutrient category are governed by an ODE in which the rate-of-change terms correspond to the 
production and consumption rates elsewhere in the food web, plus physical flows between spatial compartments. Q10 temperature-dependent 
coefficients govern transformations between different forms of detritus, conversions of detritus into ammonia (mineralisation); ammonia to 
nitrate (nitrification), and nitrate to nitrogen gas (denitrification). To complete the biogeochemical cycle, nitrate and ammonia are 
re-absorbed into the food web by phytoplankton and macrophytes, governed by light and temperature-dependent uptake responses.

\


#### 2.3 Interior and boundary fluxes: sinking, advection, mixing and migration terms

Passive transport (sinking, advection, and mixing) and, where appropriate, active migration terms in each ODE form the links between the 
vertical and horizontal spatial compartments of the model, and connections to the world outside the model domain. 

\

Passive fluxes between water column compartments within the model domain are the product of the dynamic differences in concentrations between 
vertical or horizontal spatial compartments, scaled by hydrodynamic mixing coefficients supplied as time-varying parameters. Active vertical 
and horizontal migration fluxes of zooplankton, fish and top-predators between spatial compartments are modelled as if motivated by 
food - migration is directed up gradients in the ratio of preference-weighted prey:predator concentrations.

\

Differences in nutrient concentrations between sediment pore waters and the overlying water column generate a diffusion flux with the rate 
coefficient defined by sediment permeability. Disturbance also generates a nutrient flux, and is modelled as an instantaneous equilibration 
of pore-water and water column concentrations in the disturbed volume-fraction of each sediment layer. Three types of sediment disturbance 
are represented - bioturbation by deposit feeding benthos, natural erosion by bed shear stress, and fishing-related abrasion. Disturbance 
also generates a resuspension flux of labile sediment detritus to suspended detritus. Conversely, deposition of suspended detritus to become 
labile sediment material is regarded as a first-order rate process.

\

Influxes of material to the model domain from the world outside are defined by driving data-sets. These comprise hydrodynamic flows of 
dissolved nutrient, suspended detritus and phytoplankton from adjacent sea-regions, dry and wet deposition of atmospheric nutrient to the 
sea surface, nutrient inputs from river discharges and other unspecified sources (e.g. aquaculture), and the active immigration flux of 
migratory species. Boundary export fluxes, which are dynamic and computed by the model, comprise hydrodynamic losses of nutrient, suspended 
detritus and phytoplankton to adjacent sea-regions assuming conservation of fluid volume within the model domain, loss of gaseous nitrogen 
generated by denitrification, burial of refractory organic nitrogen in the sediments, emigration of migratory species, beach-cast of 
macrophyte debris, and extraction of biomass by fishing. 

\

The inputs to, and outputs from, the ecology model are summarised in **Table 2**.

\

**Table 2**. Ecology model parameters, input and outputs.

![](images/TABLE_2.PNG)

\


#### 2.4 Representation of fishing in the ecology model

Living biomass guilds considered vulnerable to targeted capture or incidental by-catch by fishing gears are the top-predators (birds, 
pinnipeds and cetaceans); planktivorous, demersal and migratory fish; carnivorous/scavenge and suspension/deposit feeding benthos, carnivorous 
zooplankton, and macrophytes.  In each case, the fishing process is represented in the ODE for each guild by a 'harvest ratio' (proportion of 
instantaneous biomass captured per unit time). Then, a proportion of the catch is directed to the discards class, comprising whole-animal 
rejects and viscera arising from at-sea processing of the remaining catch. Only the residual fraction of the catch weight is exported from 
the model to represent landings. 

\

In addition to the direct capture process, three collateral effects of fishing activity are represented in the ecology model - release of 
pore-water nutrients, resuspension of sediment detritus, and damage mortality of benthos, due to sea-bed abrasion by bottom-contact mobile 
fishing gears.  These processes are driven by the area-proportion of each seabed sediment habitat abraded per unit time by fishing gears.

\


### 3 Fishing fleet model description

The fishing fleet model is a static, matrix-based scheme which integrates across up to 12 different types of fishing gears to assemble the 
data on harvest ratios and discard rates for each guild and abrasion rates for each seabed habitat that are required as inputs to the ecology 
model. Key inputs are, for each gear type, the spatial distribution of activity density, catching power, selectivity, discard and at-sea 
processing rates for each ecology model guild, and contact rate with the seabed (**Table 3**). Activity density is defined as the deployment 
duration of a given gear per unit sea surface area in a given time interval, integrated across all vessels (units: m-2). The power of a gear 
is a measure of its efficiency at catching biomass of a given resource guild. The product of activity density and power is a quantity that 
we refer to as fishing effort. For a given resource guild, effort is proportional to the harvest ratio and so can be summed across gears.

\

**Table 3**. Fishing fleet model inputs and outputs.

![](images/TABLE_3.PNG)

\


## References

Beddington, J.R. (1975). Mutual interference between parasites or predators and its effect on searching efficiency. *Journal of Animal Ecology*, **51**, 331-340.

DeAngelis, D.L., Goldstein, R.A.. & O'Neill, R.V. (1975). A model for trophic interaction. *Ecology*, **56**, 881-892.

Heath, M.R., Speirs, D.C. & Steele, J.H. (2014). Understanding patterns and processes in models of trophic cascades. *Ecology Letters*, **17**, 101-114.

Pace, M.L., Cole, J.J., Carpenter, S.R. & Kitchell, J.F. (1999) Trophic cascades revealed in diverse ecosystems. *Trends in Ecology and Evolution*, **14**, 483-488.


