Overview of model concepts and design

Michael R. Heath ()
University of Strathclyde, Department of Mathematics and Statistics,
Glasgow, Scotland, UK

May 19, 2020


Summary


See also the following supporting documents:

Origin of the model (previous versions)
Ecology model description
Fishing fleet model description
Parameter optimization, sensitivity and Monte Carlo analysis
StrathE2E2 implementation for the North Sea


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.

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.




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 1m2 sea surface area. Hence, the units of all the state variables (mM nitrogen) are also scaled to a domain of 1m2.

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.


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.


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.


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.