A regional organism exchange model for simulating fish migration

EtOLOfl]ItUL mODELLInG ELSEVIER

Ecological Modelling 74 (1994) 255-276

A regional organism exchange model for simulating fish migration E. Reyes a,,, F.H. Sklar b, J.W. Day c a Dept. of Oceanography and Coastal Sciences, Louisiana State University, Baton Rouge, LA 70803, USA b Belle W. Baruch Institute for Marine Biology and Coastal Research, University of South Carolina, Columbia, SC 29442, USA c Coastal Ecology Institute, Louisiana State University, Baton Rouge, LA 70803, USA

(Received 13 October 1992; accepted 31 August 1993)

Abstract

A grid-based, spatially-explicit Regional Organism Exchange (ROE) model is presented as a framework for integrating aquatic ecosystems and fish population processes at the landscape level. Active fish movements across a grid cell boundary were predicted, based on environmental tolerance ranges. The model was designed to be easily modified for any aquatic system, migratory life-stage, or trophic community. R O E was specifically developed to understand how large-scale physical patterns (i.e., tidal and freshwater intrusions) and landscape biological processes (i.e., primary production and foraging behavior) control migration of stenohaline fishes in the estuarine lagoon of Laguna de Terminos, Mexico. A migration response matrix for temperature, salinity, food availability, birth, and mortality was used to control cell-to-cell population movements. Internal cell processes included logistic population growth, trophic interactions, and ecosystem feedback parameters. Output data maps from the R O E model showed how population spatial distributions were linked to spatial and temporal patterns of water quality. However, the most significant parameter affecting long-term population stability was birth rate; an internal cell variable. It was concluded that the simulation of large, density-dependent, spatial processes such as migration can be understood with a grid-based mechanistic R O E model because its rule-based design for movement allowed organisms to respond to ecological processes and adjust to changing environmental conditions. Key words: Fish migration

* Corresponding author. Present address: Chesapeake Biological Laboratory, Center for Environmental and Estuarine Studies, P.O. Box 38, Solomons, MD 20688, USA. 0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0068-E

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I. Introduction

A spatially-explicit modeling framework for the simulation of active (as opposed to passive) migration of motile organisms as a function of population and ecosystem parameters has the enticing potential to give environmental regulators, wildlife managers, and landscape ecologists the power to realistically evaluate and predict the impacts of natural and human-induced disturbances on large geographic regions over long time scales. As a result, there is a great amount of interest in the measurement of spatial patterns (Turner and Gardner, 1991) using new concepts such as fractals (Milne, 1992) and predictability (Costanza and Maxwell, 1994). The classical European study of landscape ecology (Neef, 1967; Troll, 1968) now includes the development of spatially-explicit landscape models that incorporate management and regional planning (Risser et al., 1984; Sklar et al., 1994a). The heightened awareness of the lack of spatial models in ecology (Costanza and Sklar, 1985) and the recognition that ecological processes can affect population dynamics (Carpenter, 1988; Brandt et al., 1992) has fostered a number of spatial population models for invertebrates (Show, 1979; Kareiva and Shigesada, 1983), fish (DeAngelis and Yeh, 1984; Pola, 1985; Brandt et al., 1992), birds (Pulliam et al., 1992), and ungulates (Graham, 1986; Saarenmaa et al., 1988). However, many of these models are either 100% rule-based and thus lack ecological feedbacks or they only simulate processes within a single cell and thus lack cell-to-cell exchanges. Although models based entirely on within-cell interactions (i.e., spatially-implicit models) are computationally less complex and less prone to spatial error propagation than spatially-explicit models (Sklar et al., 1994b), they tend to be incomplete since they simulate either population or ecosystem processes but rarely both. Scientists need to develop spatially-explicit models, including both within-cell and across-cell feedbacks to integrate population and ecosystem processes, increase the management potential of spatial population models, and test theories at spatial and temporal scales (O'Neill, 1989; Weins, 1989). Previous migration models, based on mechanical facsimiles and cybernetic models, focused on force and counterforce movements in relation to muscle and body structure (DeAngelis and Yeh, 1984). These models were expensive to run and did not integrate population and ecosystem processes or predict the movement of an entire population within a heterogeneous environment. We wanted to develop a more holistic and less expensive approach, similar to what was done in the field of dynamic spatial landscape models (SEar and Costanza, 1990). This paper explores changes in spatial distribution of organisms by defining population density as a function of spatially-explicit ecosystem variables and temporally-explicit biological variables. We will show how animal motility can be described in terms of dynamic numerical vectors, as suggested by Nisbet and Gurney (1982), how non-linear "rules" for behavior can influence these vectors, and how these types of problems can be solved with differential or finite-difference equations (Rohlf and Davenport, 1969; Okubo, 1980). The Regional Organism Exchange (ROE) model was created in order to

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understand how changing land use, hydrology, or climate interact with biogeochemical and population processes to alter "migrating" renewable natural resources. Specifically, the ROE was developed to understand the mechanics or "rules" of fish growth and migration in relation to water quality. Our objective is to describe the conceptual development of a ROE model in terms of: (1) annual fish migration patterns for an open-water estuarine lagoon; (2) interpolation of spatial data bases for model initialization; (3) construction of rules for fish movement in relation to growth, mortality, water quality, and food availability, and (4) the model's sensitivity to physical and biological parameters. In the process, we will also demonstrate how a ROE model integrates simulations of population movements with process-based ecological models to produce a better understanding of landscape processes.

2. Numerical techniques for simulating migration There are two major categories of mathematical models applicable to animal movements. Those that use differential calculus to solve diffusion equations and those that use probability functions to create diffusion. Both can be modified to incorporate ecosystem processes. The difference is that differential models add parameters to the equations, making them larger and more complex, while probability models modify the existing probabilities as a function of multiple interactions. We will briefly describe both because we use the former approach to structure equations, while using the latter to change parameter coefficients as a function of habitat suitability. General advection-diffusion (differential) models take an entire population approach. In these types of models, inert particles (e.g., salt or passive plankton movement) are spatially distributed, according to Fick's three-dimensional diffusion equation (Okubo, 1980):

2AA( , y,

02

+--

0y2

y, z )

)

,

-:-A(x, y

(1)

where A is the particle concentration at any particular point in 3-D space (x, y, z), at a time-step of dt, as a function of kx.y and z (the particle-specific diffusion coefficients). Despite their insensitivity to environmental conditions, diffusion models are used in fisheries management (Jones, 1959, 1976; cited in Mullen, 1989) to describe fish dispersion and local population dynamics (Okubo, 1980) because all migration has an inert component, the mathematics are well understood, and because techniques for ecosystem-based fisheries models are not fully developed

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E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

(Carpenter, 1988). The ROE model combines the diffusion approach with the ecosystem approach by using physiological tolerance ranges as surrogates for diffusion coefficients (discussed below). Spatial grids are often used to solve diffusion equations (Power and MacCleave, 1983; Power, 1984, 1986; Pola, 1985). This technique approximates differential diffusion equations by deriving finite difference equations for each grid-cell in the study area. This results in a large number of simultaneous equations which are solved at each time step. How does solving between this type of diffusion model differ from solving in the ROE model? The former is a continuous function that considers only mass balance while the latter is a discrete function that considers mass balance in response to habitat changes (i.e., active migration). In contrast to diffusion models, basic probability models, also known as random-walk models, simulate individual migration tracks and are often based on the assumption that the probability of discrete movements for a single animal, during each time step, is the same in every direction (DeAngelis and Yeh, 1984). For example, if an animal is initially situated at a point (i, j) in a two-dimensional grid of uniformly spaced points, the organism can potentially move with equal probability (p = 0.25) in any of four directions to the points (i, j - 1), (i, j + 1), ( i - 1, j) or (i + 1, j). It is possible to modify these models so that a particular direction is preferred (e.g., Pi,j-1 = 0.7) as long as the total p does not exceed 1.0. The probability distribution for a random-walk model is the probability for arrival at a given radial distance from the starting point as long as radial symmetry is assumed (Marsh and Jones, 1988). Thus, a population of fish starting from a common initial point and moving according to equal probability random walk can show no average spatial displacement in any direction. In this case, the net result is a population that behaves similarly to one modeled with simple advection-diffusion equations (Saila and Shappy, 1963; Hilborn, 1990). To add realism and a more directional spatial displacement to probability models, there has been an effort to incorporate ecological controlling functions in the form of spatially-explicit and temporally articulate probability distributions into their design (DeAngelis and Yeh, 1984; Marsh and Jones, 1988). Introduction of these "biased" rules to modify movement implies complex decision-making on the part of the organism. In fact, some of these more sophisticated probability models can produce fairly accurate simulations of an organism's response to heterogeneous environmental conditions (Saila and Shappy, 1963; Kareiva and Shigesada, 1983; Pulliam et al., 1992). However, simulation experiments using these modified random-walk models tend only to be "snap-shots" of specific spatial scenarios. Thus, generalizations and extrapolations from these models to other landscapes tend to be difficult (Rohlf and Davenport, 1969; Smith, 1974). In summary, the diffusion and probability approaches differ significantly because the former simulates population densities as a function of all spatial interactions, while the latter simulates individual organism movement as a function of its previous location in space, keeping a record of individual tracks. The advantage of the diffusion approach is that it calculates distribution densities across an entire landscape. The advantage of the probability approach is that it

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

259

offers spatial control of cell-to-cell migration probabilities. Mullen (1989) suggested combining both models with the use of a "variable diffusivity" model, in which the local environment affects local population dynamics by creating unique diffusion coefficients for each spatial coordinate.

3. A landscape approach to migration 3.1. Study area

Notable among tropical coastal lagoons with a long-term environmental and biological database is Laguna de Terminos at the base of the Yucatan Peninsula (Fig. 1; summarized in Yfifiez-Arancibia and Day, 1988). Laguna de Terminos is Mexico's largest coastal lagoon. It is a shallow (mean depth 3.5 m) 2500-km 2 coastal system that includes open water, mangroves, and freshwater marshes. Laguna de Terminos supports Mexico's largest and most economically important shrimp fishery (Yfifiez-Arancibia and Aguirre-Leon, 1988) and many marine organisms use the lagoon as nursery grounds (Yfifiez-Arancibia and Day, 1982, 1988). Laguna de Terminos has a strong net east to west water flow, caused by prevailing south-eastern trade winds (Gierloff-Emden, 1977). This circulation pattern creates semi-permanent gradients in salinity, turbidity, nutrient levels and sediment types, and promotes spatial/temporal assemblages of foraminifera, benthic macrofauna, fish and shrimp (Day et al., 1982). Fish life history strategies in Laguna de Terminos in relation to ecological and environmental parameters such as salinity, turbidity, and temperature have been statistically developed to identify common ecological tolerance ranges (YfifiezArancibia et al., 1980, 1985a,b, 1986, 1988a). Using these data as ordination criteria for the classification of dominant fish species, Y~ifiez-Arancibia et al. (1986) identified four fish assemblages as "functional groups". Each functional group had a distinctive migration pattern; these were categorized by Y~fiezArancibia et al. (1986) into four categories: marine stenohaline and euryhaline seasonal visitors, permanent residents and occasional visitors. We use the migration pattern of the marine stenohaline pigfish, Orthopristis chrysoptera, whose habitat utilization patterns are discussed by Diaz-Ruiz et al. (1982), as a test of our ROE modeling approach. O. chrysoptera lives on the seagrass beds around Carmen Island and Puerto Real Inlet (Fig. 1) as a fingerling, then migrates through the lagoon to the freshwater areas (salinity < 10 ppt) near the Candelaria River as a juvenile. During peak river discharge, the adult pigfish moves to saltier parts of the lagoon at Puerto Real Inlet or to shallow waters offshore (salinity > 33 ppt) to reproduce. 3.2. Numerical techniques

A digital version of the Laguna de Terminos landscape was created by overlaying a grid of 14 rectangular (100 km 2) cells over the lagoon georegions (Fig. 1B).

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

260 g 20

IB' SO

~J"

40"

50'

91a 20' io +

, --_--'~~Carmen

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~

-

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t)~..

•

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, ,"/

so'

4o'

"-

.....

i- ......

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Fig. 1. (A) Localization map for Laguna de Terminos, Mexico. Arrow indicates migration pattern for O. chrysoptera (Diaz-Ruiz et al., 1982). (B) Spatial grid of the ROE model where each cell is a 100-kin2 (14"7.143 km) area of shallow (3.5 m), open water habitat. Exchanges across inlet and riverine boundaries occur in cells 1, 4, and 11, all other exchanges occur within the lagoon. Dots indicate field stations used to calculate spatially-explicit environmental forcing functions. Letters indicate sampling stations that were used for validation. Map was rotated to accommodate for a vertical presentation of the ROE model cells.

T h e f i s h d e n s i t y i n e a c h cell w a s t r e a t e d as a s t a t e v a r i a b l e (i.e., a p o p u l a t i o n i n a fixed region of space changing only due to births, deaths, immigration and e m i g r a t i o n ) s u c h t h a t g r o w t h a n d m o v e m e n t p a r a m e t e r s i n a cell f l u c t u a t e d as a

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

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f

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,

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~'~

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044 l

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Fig. 2. An example of a single cell and its possible interactions with adjacent cells in the R O E model. Cumulative suitability within a cell was influenced by population and ecological processes where F = food (Chl. a in mg m - 3 ) , S = salinity (S), T = temperature (°C). The three lower graphs show how cumulative suitability index (CSI) relates to tolerances and optimum for each of the suitability indices for marine stenohaline fish groups (Lara-Dominguez and Y~fiez-Arancibia, Programa E P O M E X , unpubl, data).

function of local environmental conditions (Fig. 2). A finite-difference equation was used to quantify fish population density in each cell (Eq. 2). The environmental conditions (see Forcing Functions, Section 3.3) were incorporated into the fish density equation as migratory response coefficients. The model was initialized with a population confined to a single two-dimensional cell. The number of fish crossing from one cell to another depended on population size (P) and a migration

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

262

parameter (K), such that unidirectional flow across a single boundary (KP) could be summed for all four sides of each cell and expressed as:

dPi,~ dt = ( Ki- ,,j,ij)(Pi-1,j) + ( Ki + 1,j,io)( Pi+ 1,j) + ( Kij-1,ij)( Pi,j-1) + ( K i,j + 1.i,j )( Pi.j +1 ) - ( K i.j,i- 1,j ) ( Pi,j ) + ( K i,j.i +1 ,j ) ( Pi,j )

+ ( K i,j,i,j-, )( Pi,j ) + ( K i,j,i,j +1 ) ( Pi ,j )'

(2)

where K,,i+t,i0 is the fish flow parameter from cell i,j + 1 to cell i,j and Pi,j is the population density for cell i,j at time t (SEar et al., 1985). The fish migration parameter (K) varied between 0 and 1 as a function of environmental conditions, cell-specific physiological tolerance ranges for the fish population (Lara-Dominguez and Yfifiez-Arancibia, Programa EPOMEX, unpubl. data), and potential pathways between adjacent grid ceils. We created an index similar to the equations of Nisbet and Gurney (1982) and Ollason (1987) for spatial interactions in patchy environments, to normalize physiological tolerance ranges and evaluate environmental suitability (U.S. Fish and Wildlife Service, 1981). This index was designed to allow for variable diffusivity (Mullen, 1989). It allowed us to create simple migration rules such that the "decision" of a population or subgroup to migrate depended on the interaction of fish density with its environmental (i.e., salinity) and physiological (e.g., birth rate) tolerances, requirements, and optimums. Each environmental and physiological factor was indexed between 0 and 1, weighted according to specific stenohaline ecological tolerances, and expressed as a combined suitability index (CSI). This CSI was then used to determine the stenohaline fish migration parameter (K) of the source cell such that: K = 1 - CSI.

(3)

The CSI was based on the food supply (F, chlorophyll a, mg m-3), salinity (S) and temperature (T, °C) according to the equation: CSI = (S*T*F2) 1/4

(4)

(U.S. Fish and Wildlife Service, 1981). The exponent of 1/4 normalized the equation between 0 and 1 and F was squared as a weighting factor that stressed the relative importance of food. The CSI, as a spatially-explicit forcing function, conceptualized classical diffusion coefficients as a population's "need" to leave a cell as a function of that population's optimum physiological requirements and tolerance ranges. Although not implemented in this version of the ROE model, one could add temporal variability to CSI, thereby increasing its effectiveness in capturing age-specific tolerance ranges. A unique characteristic of the ROE model is the combination of the migration equation (Eq. 2) with more "classical" population parameters (Gause, 1934; Hardin, 1960). Daily birth (/3) and death (p,) rates were based on a simplified

E. Reyes et al. / Ecological Modelling 74 (1994) 2 5 5 - 2 7 6

263

Table 1 Fish migration equations and coefficients P(t) = P ( t - dt) + ( ( K i _ ld, i,j)(Pi _ l,j) + (Ki + l,i, id)(Pi + l,j) + (Kid - 1,i,j)(ei,j - 1) + ( g i , j , i , j + 1 ) (Pi,j + l ) - ( g i , j , i - 1,/)(Pid) - (Ki,y.i + ld)(Pi,/) - (Kid - 1,i./)(Pi,j) - (Ki.y, id + 1)(Pi,j) + b(P) m(P))*dt -

Initial value for P ( t ) = 500000 individuals in Puerto Real cell (Cell D). P ( t ) = Fish population in cell at time t P ( t - d t ) = population at time t - d t ( K i l,j,i,jXPi - l,j)= Fish population from adjacent cell ( K i 1,j,i,j)= 1 - C S I i _ I , / ( K i , L i _ 1,jXPi,j)= Fish population to adjacent cell (Ki,j,i_l, j) = 1 - CSIi,j b(P) = b*P b = (O.O071*(FiZ, j)) m(P) = m*P

m = 0.0051 CSIi,j = (Sid * Ti5 * Fi2j) 1/4 Combined suitability index for cell i,j CSIi,/= (Si,i * Ti,j * 'Fi2j) 1/4 Si, j = salinity index for cell i,j. Salinity value is geometrically determined. T/d = temperature index in cell i,j. Temperature value is geometrically determined. 'F/,/= food index in cell i,j. Food value in cell i,j is: Fi, j = p C h l - aPi, j + YPi,j

p = 40 oz=l y = 0.001 Chl is chlorophyll value determined geometrically Geometric algorithm Z = - ((a*i + b * j - d ) / c )

Z = parameter value i = coordinate value on x-axis j = coordinate value on y-axis From 3 pre-defined stations: a b c d

= = = =

jI*(Z2 - Z3) + j2*(Z3 - Z1) + j3*(Z1 - Z2) Z l * ( i 2 - i3) + Z 2 " ( i 3 - il) + Z 3 * ( i l - i2) if* (j2 - j3) + i2" (j3 - j l ) + i3" (jl - j2) il* ( j 2 * Z 3 - j 3 * Z 2 ) + i2" (j3*Z1 - j l * Z 3 ) + i3" ( j l * Z 2 - j 2 * Z 1 )

Nisbet and Gurney such that:

(1982) equation

aPi, dt

for population

flux i n a p a t c h y e n v i r o n m e n t

[dM, dM ], = (/3-/x)P-

T

+--~-J" ]

w h e r e t h e n e t p o p u l a t i o n flux, d P / d t , f o r e a c h cell i , j s p a t i a l l y a r t i c u l a t e m i g r a t i o n f o r e a c h axis ( M i a n d M j ) .

(5) is a l s o a f u n c t i o n

of

264

E. Reyes et al. /Ecological Modelling 74 (1994) 255-276

In this version of the ROE model, mortality rates were kept constant, and birth rates were partitioned into two components (for sensitivity analysis): an intrinsic birth rate and a birth rate affected by the food supply (i.e., fecundity:food ratio). The birth rate was modified by the fecundity:food ratio to normalize the fish population response to available food and allow reproduction capacity to increase as a function of the environment in a cell (Table 1). The birth rate was also controlled by the recruitment period (i.e., season). In this version of the model, reproduction was allowed to occur between June and December. Simple ecological processes (Odum, 1983) and feedbacks (Margalef, 1968, 1982) were added to the ROE model by allowing the fish population in each cell to affect food availability by eating, and by excreting nutrients to the water column, according to the equation:

Fi,j = pCh,,j - aP~.~ + 3'P~,j,

(6)

where F,,j is the food stock in cell i, j, Chi, j is the chlorophyll a, and Pi,j is the fish population. The coefficient p was used to convert chlorophyll to carbon, a converted population density into food stock deletion due to feeding, and 3' converted fish excretions to food stock (Table 1). Nutrients were not modeled per se. Rather, the nutrient-to-chlorophyll conversion constant (g) was estimated (Table 1), multiplied by the stock of fish, and added back to the chlorophyll state variable with each iteration of the model. By incorporating this dynamic as a pseudo-feedback loop we were able to decrease the plankton in cells with high fish densities while spatially increasing nutrients for primary production.

3.3. Forcing functions and system features As discussed, environmental conditions and ecological processes in Laguna de Terminos were used to affect fish growth and migration within and across cells. Due to the shallowness of the lagoon, we assumed a homogeneous water column. Salinity and temperature were chosen as fish movement regulators because of their physiological significance (Lagler et al., 1977), their ease of measurement, and their conservative behavior. Only one non-conservative forcing function was used because our goal was to simply evaluate the ROE conceptual framework. Field data for salinity, temperature, and chlorophyll concentrations from three locations (see Fig. 1; Y~fiez-Arancibia et al., 1988a) were used to extrapolate across the lagoon. As forcing functions, salinity and temperature distributions provided the physical rules for fish migration while chlorophyll, as the ecological driving force, emphasized the degree to which organisms were tied to their food supply. Given a subset of spatial coordinates for any empirical parameter, all other values (z) were solved by a linear kriging such that:

Ai + Bj + Cz + D = O ,

(7)

where any three noncolinear points with Cartesian coordinates (i~, Jl, Zl), (i2, J2, z2) and (i3, J3, -73)were expanded by minors for every row and solved by its

E. Reyes et aL / Ecological Modelling 74 (1994) 255-276

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E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

determinants to estimate conditions in neighboring ceils. The general form of such an equation:

[jl(Z2--Z3) +j2(z3--Zl) +j3(Zl-Z2)]i +[z,(i2-i3)

+ z2(i3-i,)

+ Za(il-i2)]J

+ [i1( J2 - J 3 ) + i2(J3 - J l ) + i3(Jl - J 2 ) ] z + [il(J2z 3 --J3z2) + i2(J3z 1 --JlZ3) +i3(JlZ 2 --J2Zl)] = 0

(8)

pro.duced seasonally-averaged "contour" maps for each environmental forcing function (Fig. 3). The simulation language " S T E L L A ®'' (Richmond et al., 1987), a user-friendly object-oriented application for the solution of complex differential equations (Costanza, 1986), was used to develop and test the R O E modeling concepts on an Macintosh Ilci computer. Simulations were run for five years using Eulers' integration method, a dt of one day, and forcing functions repeated annually. A density of 500000 fish was set as the total initial population (approx. 0.08 g / m 2, YfifiezArancibia, pers. commun., 1991) and all organisms started their growth and migration in cell 11 (Fig. 1). The geometric algorithm for kriging the forcing functions (Eq. 8) was validated with field data from Yfifiez-Arancibia and Day (1982) and calculated with field data from Ygtfiez-Arancibia et al.. (1988b).

4. Results

In general, the empirical data distributions of Yfifiez-Arancibia and Day (1982) showed significant correspondence with the geometrically calculated spatial distributions for the environmental forcing functions used in the R O E model (Table 2). Monthly forcing functions, shown as landscape maps (Fig. 3), indicated that spatial gradients for salinity and chlorophyll varied more often than temperature. The annual temperature ranged only from 23 ° to 32°C, while salinity ranged from 0 to 40, and chlorophyll ranged from 0.9 to 5.8 mg m -3. These spatial and temporal patterns indicated that salinity decreased during the later part of the year in the southern sections due to a combination of high river discharge and northern

Table 2 Correlation coefficientsfor environmental parameters at different sampling stations (predicted values vs. field data a) Cell A Cell B Cell C Cell D Salinity 0.956 (11) b 0.894 (12) 0.853 (12) 0.809 (12) Temperature 0.943 (9) 0.947 (9) 0.963 (10) 0.937 (12) Food (Chl. a) 0.749 (16) 0.703 (19) 0.738 (22) 0.787 (18) Location of calibration cells are indicated in Fig. 1. a Y~fiez-Arancibiaand Day (1982). b Parentheses indicate number of samples.

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

267

BOO000 550000' "0

500000"

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45000 0 '

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~

350000

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Fig. 4. The total stenohaline fish population densities in Laguna de Terminos as predicted by the ROE model had a stable seasonal pattern that agreed with data collected by Diaz-Ruiz et al. (1982).

frontal passages (Y~fiez-Arancibia et al., 1988a), while chlorophyll a decreased during the dry season (i.e., late spring-early summer) in the northern sections. Generally, spatial patterns appeared to be controlled by river discharge and tidal exchange. High-salinity, low-chlorophyll waters were prevalent for the northern region while low-salinity, high-chlorophyll waters were found mostly in the south. Total population densities stabilized at a yearly average of 413 740 and ranged from a monthly minimum of 229 386 to a maximum of 589 748 by the end of the spawning season. The spatial distribution of the fish population throughout the landscape varied from a minimum of 28 individuals per cell (0.005% of total population at that time), obtained in cell 1, to a maximum (345 141, 58% of total population) in cell 7 near the area of river discharge. Two approaches were used to evaluate the R O E model stability and sensitivity. First, we plotted the total population size during the 5-year simulation (Fig. 4) and found that the population behaved in a stable sinusoidal manner, declining rapidly during the first part of the year, when there are no births, and increasing steadily in the latter part of the year as new organisms were recruited. These results indicated long-term replicability for the whole population and realistic temporal cycles. Second, we evaluated population response characteristics in relation to changing birth rates and fecundity : food ratios. This showed that the most sensitive factor was birth rate. A stability zone was found where the birth rate and fecundity : food ratio interact to maintain the population around the initial number (Fig. 5). The slope of this zone showed that a 10% increase in fecundity : food ratio was needed to compensate for a 7.9% decrease in birth rate. The most convenient way to view the results of the R O E model was by creating monthly population distribution maps. These maps not only showed the population's spatial distribution among cells, they indicated the spatial seasonalities and annual replicability of the population's migration patterns. They were also used to validate the principles of the R O E model (simulated fish densities and environmental conditions must match spatially-explicit empirical data). There were some differences during the first two months of the first year in comparison with the other years (Fig. 6) which we attribute to the initial lumping

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

268

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function of population (intrinsic birth rate [individuals recruited per individual per day]) and ecosystem processes ( f e c u n d i t y : f o o d ratio). Arrows show p a r a m e t e r zones where fish population densities were stable for m o r e than 5 years. T h e diagonally-oriented d a s h e d line is the slope the p a r a m e t e r c o n t o u r s would have if both p a r a m e t e r s affected population densities equally.

of all individuals into the Puerto Real celt (cell 11). We tested different startup cells as initial conditions for intra-annual variability. The effect of the startup point on the population disappears rather quickly (Table 3). For example, in the second column of Table 3, uppermost section: the same cell A can be compared on two runs with different initial conditions, in this case, all the initial population starts in cells D and B. Using daily data for the first three years, the population density in cell A/D and A / B produced a correlation coefficient of 0.998, showing the recurring presence patterns that result from different initial conditions. Otherwise, the spatial patterns produced by the R O E model had a high degree of annual replicability (Table 4). Comparmg the same cell (e.g., cell B) for year 3 with years 1 and 2 yielded correlation coefficients of 0.8 and 0.93 respectively. These indicated that the population density in the same cell as a yearly signal is consistent. The high correlations between the R O E model output of environmental data and empirical observations (Table 2) indicated that environmental gradients and their persistence over the seasonal cycle (Table 4) were realistically simulated. These correlations also indicated the importance of the marine and freshwater inputs as overall indicators of ecosystem change and habitat type. The results of the R O E model showed a clear migration pattern (Fig. 6). Quantifying the population's spatial aggregation, by identifying the maximum

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

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E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

270

Table 3 Correlation coefficients as a function of population size for a 3-year simulation in relation to initial conditions (i.e., the cell used to store the population at dt = 0) Comparison cell/ Startup cell

A/A

A/B

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A/D

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1 0.996 1

1 0.998

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1 0.989

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p o p u l a t i o n at a given day, i n d i c a t e d that t h e s e s t e n o h a l i n e o r g a n i s m s have spatial p r e f e r e n c e s as a function o f t i m e (Fig. 7). S i m u l a t e d distribution m a p s i n d i c a t e d that a c o m m u n i t y o f s t e n o h a l i n e fish in L a g u n a de T e r m i n o s w e r e likely to b e evenly d i s p er s ed d u r i n g t h e first part of th e year, a g g r e g a t e d in the c e n t r a l p a r t o f the l a g o o n for a c o u p l e of m o n t h s , a n d t h e n c o n c e n t r a t e d o n t h e e a s t e r n side o f t he l a g o o n during t h e latter half of t h e year. E x c e p t for t h e first few m o n t h s o f the first year, this m i g r a t o r y p a t t e r n (Fig. 7B) m a t c h e d t h e o n e d e s c r i b e d by D i a z - R u i z et al. (1982) for the s t e n o h a l i n e pigfish, Orthopristis chrysoptera (Fig. 7A).

Table 4 Correlation coefficients as a function of population size indicating the ROE model stability Cell A Year 2 Year 3

Year 1 0.995 0.991

Cell B Year 2 0.997

Year 1 0.78 0.80

Cell C Year 2 0.934

Year 1 0.618 0.637

Cell D Year 2 0.999

Year 1 0.901 0.776

Year 2 0.924

Location of cells are indicated in Fig. 1. Sample size n = 1095. All correlations were highly significant (p > 0.001) in one-way anovas.

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

271

5. Discussion The ability to model fish movements in relation to water quality is an important new management tool, particularly in coastal areas where land-water interactions are changing rapidly (Copeland et al., 1983). As the standing stock of humans increases, the economic pressures to harvest coastal forests, impound wetlands, and pollute estuaries also increase (Odum et al., 1988). We invoked a landscape approach for modeling fish migration because it can be used to quantify some "balance" between ecology and economy (Costanza et al., 1991), identify cumulative impacts (Turner and Gardner, 1990), understand spatial and temporal controls (Hyman et al., 1990), and predict ecological change (Costanza et al., 1990). We showed that numerical simulation techniques applied across a network of interactive cells are more realistic than regionally averaged analytical solutions, due to their increased effectiveness (i.e., explanatory power) resulting from high spatial articulation (Costanza and Sklar, 1985).

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272

E. Reyes et al. / Ecological Modelling 74 (1994) 255-276

We succeeded in implementing a regional organism exchange (ROE) model across a network of interacting cells, and in producing a landscape description of population movement in relation to biological and ecological processes. The simulated migratory patterns (Fig. 7B) resembled those described by Diaz-Ruiz et al. (1982). We showed how simulated population migration data can be validated against empirical data (e.g., Fig. 7 and Table 3), and we demonstrated how ecological forcing functions at a few points can be spatially interpolated for input into a grid-like landscape model. As a result, we conclude that the variable-diffusivity concept (Mullen, 1989) is an effective way to combine probability and mechanistic landscape models of fish migration. Although the ROE design was rather simplistic, since it assumed that in relation to fish migration, only three spatially-explicit landscape parameters (salinity, temperature, and food availability) were representative of the water quality and ecological conditions in Laguna de Terminos, it demonstrated that fishes have deterministic physiological triggers and environmental cues that drive migratory habits (McCleave et al., 1984). Other studies have shown that age and size influence feeding, migration, and reproductive capacity (Lagler et al., 1977). The results from the sensitivity analysis on birth rate and fecundity:food ratio corroborate these ideas (Fig. 5). The success of a population to persist in the model through time depended on the number of individuals recruited into the population and the environmental carrying capacity (implied by the fecundity:food ratio). This modeling approach, simplified by only focusing on adult fish and keeping mortality constant, was realistic despite the significance that variations in predation mortality can exert on recruitment success (Houde, 1989a,b). The value of increasing the realism by varying mortality and partitioning its impact into starvation and predation components through ontogenetic and seasonal time scales will have to be left for future research. Given that the ROE model results were realistic and sensitive to environmental conditions, our assumptions and conceptual approach appear appropriate. Three ecological parameters proved sufficient for Laguna de Terminos, however, for greater spatial resolution or for more complex landscapes it may be necessary to include: (1) additional environmental parameters (e.g., sediments type) for better control of organism flow from cell-to-cell; (2) cell-specific biological parameters (e.g., unique nursery cells) for better control of organism processes within a cell; (3) life stage changes (growth and sexual maturity); and (4) increased trophic interactions for more realistic ecological structure. The power of the ROE model is that it can be used to test landscape scale hypotheses and ask questions which explore biological, physiological, and ecological processes in relation to population movements. Desktop computers are now powerful enough to easily implement a ROE model. This will enhance our ability to find solutions for important sustainability problems, such as, biodiversity and ecological-economic development. As a management tool for evaluating the impacts of ecological and land-use changes on migrating organisms, the ROE model is particularly well suited. Boundary cells capture both impacts and the ROE approach distributes impacts

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273

physically and biologically. T h e R O E model's potential for large scale m a n a g e m e n t has us working on a n u m b e r of structural (e.g., cell size) and mechanistic (e.g., primary production) enhancements. T h e s e include: (1) the design for a "suitability index" that can switch f r o m static (one tolerance range t h r o u g h o u t the year) to dynamic (temporal variation of the tolerance range), (2) m o r e realistic mortality parameters, (3) m o r e detailed nutrient cycling, and (4) e x p a n d e d trophic interactions (i.e., ichtyoplankton and juvenile fish).

Acknowledgments W e are especially grateful for the c o m m e n t s and suggestions of Drs. Alejandro Yftfiez-Arancibia, Daniel L. Childers and Joel V a n A r m a n . W e would also like to thank A n a L a u r a L a r a - D o m i n g u e z for all the fish physiological information f r o m her dissertation material and Steve H u t c h i n s o n for his help with the p r o g r a m m i n g code. This study was f u n d e d by U.S. M a n and Biosphere P r o g r a m ( G r a n t No. 1753-900559) and N S F grants No. BSR-8906269 and No. BSR-8514326. Dr. Reyes was s u p p o r t e d with an scholarship by Direcci6n G e n e r a l de A s u n t o s del Personal Acad6mico, U N A M , Mexico. This article is co-listed as contribution n u m b e r 978 of the Belle W. Baruch Institute for Marine Biology and Coastal R e s e a r c h at the University of South Carolina and as a publication of the South Florida W a t e r M a n a g e m e n t District.

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