Representation of Multistanza Life Histories in Ecospace Models for Spatial Organization of Ecosystem Trophic Interaction Patterns

e Ecospace model for spatial organization of trophic interactions has seen limited use for evaluation of policies such as marine protected areas, partly because of concern about representing key indicator populations only by spatial biomass distributions. !e software has been improved to include spatial representation of age structure for such species, by means of the Ecosim “multistanza” population submodel, which assumes similar diet compositions, predation risk, and vulnerability to fishing over blocks or stanzas of fish ages. A computationally efficient version of Ecospace now preserves the multistanza age structure over spatial habitat and ecosystem biomass maps, evaluating body growth and mortality rates as spatial averages weighted by relative biomass use of each model spatial cell. A more computationally intense version divides each multistanza population into spatial packets (an individual-based model approach) for more precise analysis of how movement patterns and movement histories over mosaics of trophic opportunities and risks affect population performance and variability. !e two approaches give surprisingly similar predictions of abundance patterns over both time and space, agreeing well in case-study applications to the Gulf of Mexico and California coast with each other and with nonspatial Ecosim predictions. !e Ecopath with Ecosim (EwE) software is widely used for synthesis of information on trophic interactions and for screening of ecosystem-management options that may alter trophic interactions (Christensen and Walters, 2004). !e Ecopath component of the software is used to manage basic input data (abundances, diet compositions, productivity) and to provide static mass-balance analyses. !e Ecosim component provides dynamic predictions and capabilities similar to those of singlespecies assessment models for parameter estimation by statistically based fitting to historical time-series data. !e realism of Ecosim models has recently been improved dramatically through inclusion of “multistanza” population-dynamics accounting. Multistanza accounting replaces the simple biomass-dynamics relationships of Ecopath and early Ecosim versions with detailed, ageand size-structured populationdynamics relationships for key species within the ecosystem. !e different age ranges (stanzas) can be represented as having distinct diet preferences and vulnerabilities to predation and fishing mortality (Walters and Martell, 2004; Walters et al., 2008). !e EwE software package also includes a spatial modeling scheme called Ecospace, which replicates the Ecopath-Ecosim biomass variables over a grid of spatial cells and represents mixing of biomass (diffusion, advection, and seasonal migration) among the cells (Walters et al., 1999; Walters, 2000). Ecospace was originally intended to provide only crude equilibrium predictions of how spatially oriented management policies and spatially explicit distributions of fishing effort might alter trophic-interaction patterns (e.g., through trophic-cascade effects within protected areas). Ecospace can also make dynamic predictions (see Walters and Martell, 2004: BULLETIN OF MARINE SCIENCE, VOL. 86, NO. 2, 2010 440 279, box 11.2) and has been used to explore spatial management options for a wide variety of ecosystems (see, e.g., Pitcher et al., 2002; Gribble, 2003; Martell et al., 2005; Le Quesne et al., 2008), but when the multistanza population dynamics capability was added to Ecosim, only a crude, equilibrium approximation for age structure was included in the Ecospace software. At the time, we assumed that representing full ageand size-structure dynamics on each of the many grid cells (1000 or more) typically used in Ecospace representations would require massive computational and memory capacity and prevent rapid simulations needed in workshop situations. !e crude-approximation approach that was implemented allowed very fast solutions but was recognized to limit severely the ability of Ecospace to represent the spatial habitat shifts that typically accompany trophic ontogeny (different stanzas very often use different spatial cells). !ese ontogenetic shifts can be critical in evaluating impacts of policies, such as marine protected areas, that often only affect certain stanzas in the life cycles of the key species. !e inclusion of multistanza dynamics will permit tracking of the changes in abundance and growth of key species over both space and time, while representing other ecosystem functional groups by means of simpler biomass-dynamics equations with spatial mixing processes. Here, we describe two approaches to integration of multistanza population-dynamics predictions into the Ecospace framework. Full representation and solution of multistanza dynamics will allow for more realistic, and yet still computationally efficient, Ecospace models. !e first approach is based on making multistanza ageand size-structured predictions for an overall spatial region, then predicting distribution of stanza biomasses over spatial cells by means of predicted proportional cell use from continuous spatial mixing models. !e second, much more detailed, approach is to divide each multistanza age cohort into a large number of subcohorts or “packets,” essentially an individual-based modeling approach (Van Winkle et al., 1993), then to predict movement patterns of these packets over the spatial map. !e first approach is computationally efficient and therefore facilitates multiple model runs in policy-screening and gaming situations, whereas the second is intended mainly to check scenarios developed with the first approach for impacts of complex spatial effects such as differential body-growth rates of fish in different cells or in subareas of the spatial grid. !ese two new multistanza approaches greatly improve the capability of Ecospace to deal with policy questions not only about marine protected areas but also about changes in essential fish habitat available to each stanza for species with complex spatial and trophic ontogenies. A R  M L H  E R  G E M S  B D.— For functional groups not represented by multistanza population dynamics accounting, Ecospace represents biomass (B) dynamics over a set of spatial cells (k) with the spatially discretized rate formulation dB dt e Q Z B m B m B ik i ik ik ik ikk k ik ik k ik k = + ^ h (1) where Bik is the biomass of functional group i in spatial cell k; ei is conversion efficiency of food intake by group i into net production; Qik is total food consumption WALTERS ET AL.: MULTISTANZA LIFE HISTORIES IN ECOSPACE MODELS 441 rate by group i in spatial cell k; Zik is total mortality rate of group i biomass due to predation, fishing, etc.; mikk ́ is instantaneous movement rate of group i biomass from cell k to cell k ;́ and mik ́k is movement rate of group i biomass from cell k ́ to cell k. All of the terms on the right hand side of Eq. 1, except ei, are treated as dynamically variable over time so as to reflect changes in food availability (Qik), fishing effort and predation risk (Zik), and seasonal changes in movement patterns (mikk ́). Food consumption rates Qik are calculated as sums over prey types j (i.e., Qik = ΣjQjik). Likewise, total mortality rates are calculated as sums over predator types and fishing fleets f: Zik = Moi + Σf Fifk + ΣjQijk/Bik , where Moi is unexplained mortality rate, the fishing rate components Fifk by fleets f are predicted from spatial distributions of fishing effort for each “fleet” f over the grid cells k, and the Qijk/Bik ratios represent predation rate components of M (i.e., Mijk = Qijk/Bik) calculated from predator j consumption rates Qijk. !e Ecospace grid cells are arranged as a rectangular grid with rows r and columns c, so that each cell k exchanges biomass directly only with those cells k ́ that are in adjacent rows and columns. If cell k represents row r, column c, then k ́ is restricted to cells (r – 1,c), (r + 1,c), (c – 1,r), and (c + 1,r). Exchanges at the map perimeter are set to zero, except for groups that are assumed to be advected across the map, in which case biomasses at the map boundary are set to constant (Ecopath base estimate) values. A critical feature of Ecospace is that trophic interactions are not treated as occurring randomly over space within each grid cell. !e Ecosim “foraging arena” formulation (Walters et al., 1997; Christensen and Walters, 2004) is used for predicting the Qijk, and the formulation is based on the assumption that animals can exhibit highly organized patterns of space use at much finer spatial scales than the size of typical model cells used for Ecospace simulations. !e idea is that behavior (e.g., seeking of safe microhabitats) leads to the exchange of individuals between “safe” and “vulnerable” behavioral states, either continuously over time or in temporally restricted feeding bouts (Walters and Christensen, 2007). !e Ecosim equations for predicting Qijk allow for classic predator-prey functional response types—type I if no handling time or foraging time adjustment is needed, type II if handling times and or foraging time adjustments limit per-capita food-consumption rates, and type III if predator rates of search decrease when prey densities are low or prey spend less time foraging when their densities are low—but the foraging-arena equations depart from classical functional response predictions in calculating prey densities not as average total biomass in each cell but rather as effective biomass densities in the restricted arenas where foraging typically occurs and where such local densities can be strongly affected by densities of competing predators (ratio-dependence effect). Local depression of available prey biomass can occur whether or not predation affects overall grid-wide prey densities. !e default assumption for spatial mixing in Ecospace is for the mikk ́ rates to be equal over all active (nonland) cell faces, representing simple groupor species-specific diffusion processes. Ecospace allows users to improve upon this assumption in four important ways. First, passive advection by currents can be represented by modification of the m’s by means of vertically averaged velocity fields provided by physical models. Second, each spatial cell can be assigned a distinct user-defined habitat type, and each group can be designated to use one or more such types. Given these designations, dispersal rates are modified so as largely to prevent movement into cells with “bad” habitat types, and movement rates for animals currently in bad cells are increased in the direction of more suitable cells. !is convention is particuBULLETIN OF MARINE SCIENCE, VOL. 86, NO. 2, 2010 442 larly important for simulation of behaviors that move larval and juvenile fishes from offshore spawning areas into coastal nursery areas, which can often involve using behavioral tactics such as vertical migration and movement only on incoming tides in conjunction with oriented swimming. !ird, dispersal rates can be modified to represent movements oriented toward seasonally varying preferred spatial positions; movement can be more strongly oriented (m’s reduced more in directions away from preferred positions) to simulate seasonal migration patterns. Under this option, scaling parameters for north-south and east-west orientation can be specified by model users to concentrate biomasses more or less tightly around the preferred locations. Fourth, movement rates (and optionally directions) can be linked to indices of fitness (food consumption rate Q/B, mortality rate Z) to concentrate animals in more favorable cells (Martell et al., 2005). Fitness-driven dispersal typically causes dispersal rates to be density dependent, leading to large-scale patterns such as range contraction when overall abundance declines. Typical Ecospace models developed to date have represented 20–60 functional groups (denoted i) on spatial grids with 20–50 rows and columns (k cells) and simulated time horizons on the order of 50 yrs, commonly run with monthly time steps. !e Eq. 1 system therefore has on the order of 10,000–150,000 i,k elements and must be solved by some very efficient implicit numerical procedure so that new results can be generated quickly for management-policy comparisons and gaming. We have chosen to use a fully implicit, second-order backward differentiation algorithm. Such implicit algorithms have the valuable property of being numerically stable even at very large time steps and even when some of the system variables can change very rapidly (when the system is numerically “stiff”). !e “fast” variables therefore do not force the whole solution method to use a very short time step (e.g., minutes to hours for models with high spatial mixing rates and fast variables like small phytoplankton); instead, the implicit integration method “discards” fast variation, essentially treating fast variables as remaining near a moving equilibrium with respect to changes in the slower variables that affect them (e.g., phytoplankton is treated as remaining near equilibrium with respect to changes in zooplankton biomass). !at is, the moving equilibria for fast variables are assumed to be good estimates of average variable values over whatever complex, cyclic patterns such variables may exhibit over time if shorter solution time steps were used. R  E M P D A.—Selected biomass groups can be designated life-history stanzas within single-species populations. In such cases, the Ecosim differential equation representation for biomass change (Eq. 1) is replaced by a monthly-difference equation system, with full age-structured accounting for population age and size structure at monthly age increments. !e basic accounting relationships are exp N N Z 12 , , , a t a t s t 1 1 = + + ^ h (2) W q W , , , a t a a t a t 1 1 a t = + + + (3)