48 Habitat capacity

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_rcj varies from 0.0 to 1.0, and is calculated for each cell as a function of a vector of habitat attributes H_rc = (H1,H2,…Hv)_rc of that cell, i.e. C_rcj = f_i(H_rc). For example 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_rcj can be updated over time with relatively little computational cost, for example by loading time-varying values of H_rc 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).^[2]

In order to use the C_rcj habitat assessments, the C_rcj values have to be linked to trophic interaction dynamics to specify how C_rcj 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.

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_ij) and a non-vulnerable (B_i-V_ij) component. The transfer rate, called vulnerability (υ_ij) between the two fractions determines the vulnerable biomass at time interval 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_ij is a function of the maximum increase in predation mortality under the given predator/prey conditions (see vulnerability multiplier chapter). High values of υ_ij imply large proportions of biomass (B_i) vulnerable to predator j (V_ij), and thus imply V_ij = B_i, and that the predator 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_j is growth efficiency (corresponding to the production/consumption ratio, which can vary as predators grow in size), v_ij is prey vulnerability exchange rate, and a_ij is the rate of effective search by the predator. Note that in this model, vulnerable prey density V_ij is represented by the foraging arena equation Eq. 1), which simplified can be expressed when there is only one prey type i as

[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_rcj, i.e.,

[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_j increases in locales with less foraging arena area.

Importantly, including C_rcj as a modifier in the a_ij·B_j / C_rcj predation rate term results in the equilibrium predator biomass (B_j for which dB_j/dt=0) being proportional to Crcp, 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_rcj, other factors (prey biomasses B_i and mortality rates Z_j) being equal over space. We could, of course, also had assumed that variation in habitat capacity also affects the vulnerability exchange rates v_ij, search rates a_ij, and predation rates Z_j (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 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_rck variation are assigned at the start of each Ecospace simulation by assuming that these biomasses are proportional to the C_rcj. If there are nw water cells, such that overall biomass density for group k across the grid is given by nwB_i* where B_i* is the Ecopath base biomass for group 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_ju* is the spatially invariant initial predator abundance obtained by noting that applying the C_rcj correction in Eq. 4 results in

[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_rcj, Eq. 8 can be written as:

[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_j. 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

[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_u to make simulations come as close as possible to predicting the base Q_u. The rate of effective search a_u is further adjusted upward by the multiplicative factor Q_m Q_oj’/(Q_m Q_oj’-1) to account for handling time effects in order to create type II functional response effects by setting a low ratio of maximum (Q_m) to base feeding rate (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_1jB_rcj to the right and m_2jB_rcj to the left. Absent orientation implies m_1j=m_2j=m_j, where m_j is an (input) expected dispersal rate.^[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_1j and m_2j have to be varied so as to meet the balance condition

[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_rcj, then adjusting the exit rate for the cell with higher C_rcj to m_j times the capacity ratio. Thus for example if C_rc+1j  >C_{rcj, m1j} to the right is set to m_j and m_2j to the left is set to m_jCrcj/C_rc+1j so that m_2j will be very small if C_rcj << C_rc+1j , i.e. movement into the low capacity cell will be severely restricted.

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.