# 47 Habitat capacity

Villy Christensen; Marta Coll; Jeroen Steenbeek; Joe Buszowski; David Chagaris; and Carl J. Walters

## Why are species where they are?

To bridge the gap between ecosystem models and species distribution models, the spatial-temporal explicit module Ecospace includes a habitat capacity model^{[1]} that addresses the central question, “why are species where they are?” The overarching assumption in the habitat capacity model is that they are where they are because they prefer certain combinations of environmental and ecological conditions.

Prior to the inclusion of the habitat capacity model in Ecospace, species distribution models and ecosystem models offered limited capabilities to work jointly to produce needed integrated assessments: assessments that take both food web dynamics and spatial-temporal environmental variability into account. The habitat capacity model is fairly simple and its integration in EwE mainly implied replacing a binary habitat variable with a continuous habitat suitability factor, where the area that species can feed in each cell is determined by functional responses to multiple environmental factors. This modification builds on the fact that animal populations have lower local impacts as the size of their forage area increases. The habitat capacity model offers the ability to drive foraging capacity from multiple physical, oceanographic, and environmental factors such as depth, bottom type, temperature, salinity, oxygen concentrations, etc., which have cumulative impacts on the ability of functional groups to forage. Since cell capacity is calculated for every functional group at every time step, this modification makes the model fully temporal and spatially dynamic.

## Using relative habitat capacity to predict spatial abundance

A reasonably simple and tractable way around the binary parameterization of habitat definition is to define a continuous relative habitat capacity *C _{rcj}* for each group

*j*in each cell (with row and column)

*r,c*, where

*C*varies from 0.0 to 1.0, and is calculated for each cell as a function of a vector of habitat attributes

_{rcj}*H*=

_{rc}*(H1,H2,…Hv)*of that cell, i.e.

_{rc}*C*=

_{rcj}*f*. For example

_{i}(H_{rc})*H1*might be water depth,

*H2*might be proportion hard bottom,

*H3*might be summer water temperature, etc. Figure 1 provides a schematic overview of the basic calculations in the habitat capacity model.

**Figure 1. Schematic diagram of the habitat capacity model calculations with four (hypothetical) environmental preference functions (any number of functions is possible). During model run, cell-specific environmental parameter values can be read from data layers for each time step, and a cell-specific habitat capacity value is estimated as the product of the environmental preference values. No weighting is used, but weighting can be considered by altering the shapes of the environmental preference functions.**

The proportion of a cell that a species (or functional group) can use is thus a continuous value from 0 to 1, and allows inclusion of as many environmental factors as needed to define the foraging capacity of a cell for a species in an Ecospace model.

If the functions *f _{j}()* are chosen carefully,

*C*can be updated over time with relatively little computational cost, for example by loading time-varying values of

_{rcj}*H*generated by other models or remotely sensed data for physical or biophysical change, and implemented using the spatial-temporal data framework of Ecospace, (see User Guide chapter for how-to).

_{rc}^{[2]}

In order to use the *C _{rcj}* habitat assessments, the

*C*values have to be linked to trophic interaction dynamics to specify how

_{rcj}*C*impacts food consumption and predation rates. A simple and reasonable way to represent this linkage is available through the basic foraging arena equations used to predict trophic interaction (food-web biomass flow) rates Eq. 1.

_{rcj}For this, the consumption rates *Q _{ij}* are based on the foraging arena theory (see chapter), where the biomass of prey

*i*is split between a vulnerable (

*V*) and a non-vulnerable (

_{ij}*B*) component. The transfer rate, called vulnerability (

_{i}-V_{ij}*υ*) between the two fractions determines the vulnerable biomass at time interval

_{ij}*dt,*

[latex]\frac{dV_{ij}}{dt}=v_{ij} \ (B_i-V_{ij})-v_{ij} \ V_{ij}-\frac{a_{ij} \ V_{ij} \ B_j}{D_j}\tag{1}[/latex]

where *a _{ij}* is the effective search rate for the predator

*j*, and D

_{j}represents loss of time searching due to handling time for the predator. The vulnerability parameter

*υ*is a function of the maximum increase in predation mortality under the given predator/prey conditions (see vulnerability multiplier chapter). High values of

_{ij}*υ*imply large proportions of biomass (

_{ij }*B*) vulnerable to predator

_{i}*j*(

*V*), and thus imply

_{ij}*V*=

_{ij }*B*, and that the predator

_{i}*j*is far from its carrying capacity with regards to prey

*i*.

If we consider how *Ecosim* represents biomass dynamics (exclusive of spatial mixing effects), trophic interaction and fishery effects are modelled by equations of the basic form (looking at only one prey type to simplify the equation)

[latex]\frac{dB_j}{dt}=\frac{g_j \ a_{ij} \ v_{ij} \ B_j \ B_i}{2 \ v_{ij}+a_{ij} \ B_j}-Z_j \ B_j\tag{}[/latex]

where *Z _{j}* is total instantaneous mortality rate of

*j*,

*g*is growth efficiency (corresponding to the production/consumption ratio, which can vary as predators grow in size),

_{j}*v*is prey vulnerability exchange rate, and

_{ij}*a*is the rate of effective search by the predator. Note that in this model, vulnerable prey density

_{ij}*V*is represented by the foraging arena equation Eq. 1), which simplified can be expressed when there is only one prey type i as

_{ij}[latex]V_{ij}=\frac{v_{ij} \ B_i}{2 \ v_{ij}+a_{ij} \ B_j}\tag{3}[/latex]

where predation pressure in a cell depends on the foraging arena area in that cell. If we assume that variation in relative habitat capacity for the predator means variation in the foraging arena area over which a species can forage successfully, we can include variation in relative habitat capacity in the model by dividing the denominator *a _{ij}·B_{j}* term by relative habitat size or capacity

*C*, i.e.,

_{rcj}[latex]V_{ij}=\frac{v_{ij}\ B_i}{2\ v_{ij} + a_{ij} \ B_j / C_{rcj}}\tag{4}[/latex]

In effect, this assumption concentrates predation activity into smaller relative areas when *C* (foraging arena size) is small, so as to drive down vulnerable prey densities *V _{ij}* more rapidly as

*B*increases in locales with less foraging arena area.

_{j}Importantly, including *C _{rcj}* as a modifier in the

*a*predation rate term results in the equilibrium predator biomass (

_{ij}·B_{j }/ C_{rcj}*B*for which

_{j}*dB*=0) being proportional to

_{j}/dt*C*rcp, i.e.,

[latex]B_j=(g_j \ v_{ij} \ B_i / Z_j-2 \ v_{ij}/a_{ij}) \ C_{rcj}\tag{5}[/latex]

That is, using the *C _{rcj}* as modifiers of the foraging arena consumption rate equation results in spatial patterns of biomass of consumers being proportional to

*C*, other factors (prey biomasses

_{rcj}*B*and mortality rates

_{i}*Z*) being equal over space. We could, of course, also had assumed that variation in habitat capacity also affects the vulnerability exchange rates

_{j}*v*, search rates

_{ij}*a*, and predation rates

_{ij}*Z*(and if so, added minor changes to the code to implement these assumptions), but the default assumption is that the dominant cause of “poor” or relatively small habitat capacity is lack of usable foraging arena area. As such, the basic change made to the rate equations is a simple division of the denominator terms for predator search term, by-arena vulnerable prey density equations, by the capacity values

_{j}*C*.

_{rcj}The new model is made compatible with earlier Ecospace* *models by providing the option to derive capacity directly from presence/absence of habitats. In this case, habitat maps and habitat preferences are directly converted to a capacity map for each functional group. Cells that contain a preferred habitat will receive a full capacity of 1, other cells will receive a capacity of (almost) 0. The implementation in Ecospace further ensures that it is optional for every group in a model to use habitat maps and/or habitat capacity to drive distributions.

## Setting initial adjusted biomasses

In going from Ecopath to Ecospace, it is assumed that the Ecopath base biomasses represent the average over all modeled cells of the cell-specific biomasses. This means that Ecospace biomass densities can be much higher in favourable spatial cells if there are relatively few such cells. Initial biomass densities *B _{rcj}(0)* reflecting the

*C*variation are assigned at the start of each

_{rck}*Ecospace*simulation by assuming that these biomasses are proportional to the

*C*. If there are

_{rcj}*nw*water cells, such that overall biomass density for group

*k*across the grid is given by

*nw*B

_{i}* where

*B*is the Ecopath base biomass for group

_{i}**j*, the initial spatial biomass densities are assigned as

[latex]B_{rcj}(0)=(C_{rcj}/ T \ C_j) \ nw \ B_j^*\tag{6}[/latex]

where *TC _{j}* is a total capacity index over the grid for group

*j*, i.e.,

[latex]TC_j=\sum\limits_{r,c}C_{rcj}\tag{7}[/latex]

and the sum over *r* and *c* is over all *nw* water cells in the spatial grid. Note that *TC _{j }*<<

*nw*implies severe concentration of group

*j*biomass on few cells.

## Correction of search rate and vulnerability parameters for spatial overlap patterns

Spatial concentration of biomass for any group implies a requirement to adjust the rates of effective search *au* and vulnerability exchange rates *vu* for all foraging arenas *u *that are used by group *j* and its predators *j’*, since without such adjustments predicted predation rates (using foraging arena equations from Ecosim) at the higher local densities would be artificially increased from the rates implied by Ecopath base consumptions. In order to make this adjustment, the rates are set so that the total consumption for each arena link is the same in Ecospace as in Ecosim, scaled up to the total number of water cells. This implies the condition

[latex]n_w \ Q_u= (a_u \ v_u \sum\limits_{r,c} B_{rcj} \ B_{rcj}^{'})\ / \ (2 \ v_u+a_u \ B_j^*)\tag{8}[/latex]

Here, *Q _{u}* is the Ecosim base biomass flow rate for arena link

*u*,

*B*‘

_{rcj}is the initial predator abundance (biomass for non-stanza groups or sum of numbers at age times length squared for multi-stanza groups) for cell (

*r,c*), and

*B*is the spatially invariant initial predator abundance obtained by noting that applying the

_{ju}**C*correction in Eq. 4 results in

_{rcj}[latex]B_j^* = (n_j \ B_{Ecosim,j}^{'}) / TC_j^{'}\tag{9}[/latex]

Here *B’ _{Ecosim,j}* is the Ecosim initial predator abundance. Using the assumed relationships above, between initial

*B*,

_{rcj}*B’*

_{rcj }and

*C*, Eq. 8 can be written as:

_{rcj}[latex]Q_u = (a_u \ v_u \ B_{ju}^{**} \ B_j^*) \ / \ (2 \ v_u+a_u \ B_j^*)\tag{10}[/latex]

where B_{ju}** is the prey-predator “incidence weighted” mean prey biomass divided by *B*_{j}* for link *u* given by

[latex]B_{ju}^{**} = (B_j^* \ \sum\limits_{r,c}C_{rcj} \ C_{rcj}^{'}) / TC_j\tag{11}[/latex]

Note that this reduces to just *B*_{ju}* if all predator *C’*_{rcj}* *are near 1.0 for the same (*r,c*) cells where prey *C _{rcj }*are near 1.0, i.e. where there is strong spatial overlap of the prey and predator distributions, but can be much lower than for cases where predators occupy restricted spatial areas compared to the prey. Assuming the same vulnerability exchange rate

*vu*as in Ecosim (from total base consumption rate over all predators using arena

*u*) where

*k*is the user-supplied vulnerability multiplier (aka

*Vulmult*, see vulnerability chapter), Eq. 10 can be solved for

*a*

_{u}[latex]a_u = (2 \ v_u) \ / \ [B_{j'}^{*} \ v_u \ B_{ju}^{**} \ / \ (Q_u-1)]\tag{12}[/latex]

Unfortunately, this calculation fails if *v _{u }B*_{ju} / Q_{u}* < 1, which can happen with relatively low

*vu*settings and weak overlap between prey and predator such that

*B**

_{ju}is much less than

*B*. In that case, the assumed spatial distribution overlap pattern simply cannot support the total predation rate estimated for the link in Ecopath and Ecosim, and instead we simply set

_{j}[latex]v_u=1.001 \ Q_u / B_{ju}^{**}\tag{13}[/latex]

before solving for *a _{u}* in Eq. 12 so as to provide at least some large estimate of

*a*to make simulations come as close as possible to predicting the base

_{u }*Q*. The rate of effective search

_{u}*a*is further adjusted upward by the multiplicative factor

_{u}*Q*to account for handling time effects in order to create type II functional response effects by setting a low ratio of maximum (

_{m }Q_{oj’}/(Q_{m }Q_{oj’}-1)*Q*) to base feeding rate (

_{m}*Q*).

_{o}## Modification of spatial mixing rates to reflect movement toward preferred cells

For species with body sizes and mobility large enough to exhibit oriented dispersal and/or migration, it is reasonable to assume that dispersal rates between adjacent spatial cells are distorted so as to maintain abundance differences reflective of differences in habitat capacities between the cells.

Without such distortions or oriented movement, random dispersal between cells would greatly reduce abundance gradients created by the *C _{rcj}* capacity effects, and for species with restricted habitat use would result in too much biomass dispersing into unsuitable spatial cells so as to cause biomass to decrease substantially from Ecopath base biomasses, even without any changes in fishing pressure or predator abundances. For each border between cells, e.g. between cell (

*r,c*) and cell (

*r,c+1*) to its right,

*Ecospace*assumes instantaneous mixing rates

*m*to the right and

_{1j}B_{rcj}*m*to the left. Absent orientation implies

_{2j}B_{rcj}*m*where

_{1j}=m_{2j}=m_{j},*m*is an (input) expected dispersal rate.

_{j}^{[3]}In order to avoid smearing of the distribution, the dispersal rates are set so that

[latex]m_{1j} \ B_{rcj} = m_{2j} \ B_{rc+1j}\tag{14}[/latex]

Assuming biomasses are then to remain near or proportional to *C _{rcj}*, this balanced movement condition implies that the

*m*and

_{1j}*m*have to be varied so as to meet the balance condition

_{2j}[latex]m_{1j} \ / \ m_{2j} = C_{rc+1j} \ / \ C_{rcj}\tag{15}[/latex]

Ecospace meets this condition by setting the exit rate to *m _{j}* for whichever cell has lower capacity

*C*, then adjusting the exit rate for the cell with higher

_{rcj}*C*to

_{rcj}*m*times the capacity ratio. Thus for example if

_{j}*C*

_{rc+1j}>C_{rcj}_{, m1j}to the right is set to

*m*and

_{j}*m*to the left is set to

_{2j}*m*so that

_{jCrcj}/C_{rc+1j}*m*will be very small if

_{2j}*C*, i.e. movement into the low capacity cell will be severely restricted.

_{rcj}<< C_{rc+1j}Rounding off this chapter, the source publication^{[4]}study used simulation modeling to evaluate the sampling characteristics of the habitat capacity model, based on an artificial data set and a spatial food web model of a marine ecosystem. This was used to derive “true” distribution based on environmental preference for the functional groups in the model, and then evaluate the degree to which it is possible to recreate the “true” distributions from sampling. As part of this, the impact of sample size and uncertainty in key parameters was evaluated. We refer to the source publication for details, and note that the habitat suitability model can be used to address a suite of new ecological questions, such as the impact of habitat degradation due to coastal development, eutrophication and climate change. In most cases, it should be considered to use the habitat capacity facility instead of the pre-defined habitat approach.

**Attribution **This chapter is based on Christensen, V, M Coll, J Steenbeek, J Buszowski, D Chagaris, and CJ Walters. 2014. Representing variable habitat quality in a spatial food web model. Ecosystems 17(8): 1397-1412. https://doi.org/10.1007/s10021-014-9803-3. Reused with License Number 5757230625588 from Springer Nature. Rather than citing this chapter, please cite the source.

### Media Attributions

- From Figure 1 in Christensen et al. 2004

- Christensen, V, M Coll, J Steenbeek, J Buszowski, D Chagaris, and CJ Walters. 2014. Representing variable habitat quality in a spatial food web model. Ecosystems 17(8): 1397-1412. https://doi.org/10.1007/s10021-014-9803-3 ↵
- Steenbeek, J., Coll, M., Gurney, L., et al., 2013. Bridging the gap between ecosystem modeling tools and geographic information systems: Driving a food web model with external spatial–temporal data. Ecological Modelling 263, 139–151. https://doi.org/10.1016/j.ecolmodel.2013.04.027. ↵
- There is an IBM model dispersal rate estimator at https://ecopath.app ↵
- Christensen et al. 2014.
*op. cit*. ↵