Larval transport and population dynamics of intertidal barnacles: a coupled benthic/oceanic model.
A major goal of ecological research is to understand what controls the abundance and distribution of populations, and in turn, to understand how populations and communities might respond to physical changes in the environment. To accomplish this goal with marine populations, it is critical to understand the effects of physical processes on biological processes. The life cycle of many marine organisms inhabiting the intertidal zone is complex, consisting of a planktonic larval stage followed by a sessile adult stage. Recent studies have shown that transport processes in the offshore water column affecting planktonic larvae can directly influence recruitment and the population dynamics of sessile adults (Gaines et al. 1985, Shanks 1986, Shanks and Wright 1987, Roughgarden et al. 1988, Jacobsen et al. 1990, Farrell et al. 1991, Pineda 1991, Roughgarden et al. 1991a, Miller 1992). The research presented in this paper involves theoretical models designed to provide an initial understanding of the role of offshore circulation patterns and physical oceanography in the population dynamics of coastal species as exemplified by intertidal barnacles. The models extend previous theory (Roughgarden and Iwasa 1986, Possingham and Roughgarden 1990, Roughgarden et al. 1994) to account for both cross-shelf and alongshore advection and the presence of offshore fronts along the central California coast.
This coast is similar to other major upwelling areas located on eastern ocean margins such as the western coasts of South America and Africa. Cold, nutrient-rich waters brought to the surface from depth support areas of high biological productivity. The role of coastal upwelling in the distribution of biological substances is undoubtedly significant (Ebert and Russell 1988, Mackas et al. 1991, Smith and Lane 1991). Moreover, the population dynamics of many marine organisms are affected by the offshore and onshore transport of water across the continental shelves (Peterson et al. 1979, Bailey 1981, Yoshioka 1982, Smith et al. 1986, Fraga et al. 1988, Peterson et al. 1988). Specifically, a longstanding problem facing marine ecologists is explaining the large fluctuations in the abundance of stock populations caused by the pattern of recruitment of their larvae in coastal areas with high productivity. It is hypothesized that the population dynamics of intertidal barnacle species along the central California coast reflect the dynamics of the California Current System and wind-induced upwelling (Farrell et al. 1991, Roughgarden et al. 1991a).
The present modeling effort extends those in Roughgarden et al. (1985) and Possingham and Roughgarden (1990) on the population dynamics of marine organisms with complex life cycles. The primary purpose of each of these models was to develop a predictive theory of barnacle population dynamics that included both pre-settlement and post-settlement processes. In keeping with this goal, we expand upon a preliminary model presented in Roughgarden et al. (1994), which incorporates an additional physical feature of offshore circulation into the modeling framework. As cold, upwelled water moves offshore perpendicular to the coast, it intersects the southward-flowing surface water of the California Current to produce a frontal boundary. It is hypothesized that this frontal boundary acts as a convergence zone at which organisms such as barnacle larvae accumulate (Roughgarden et al. 1991a). The location of the front moves depending on the strength of the winds driving coastal upwelling. When the winds are strong, the front is pushed far from shore and cross-shelf transport at the surface carries intertidal larvae away from the shore and their settling areas. When the winds are weak, cross-shelf transport is reduced and the front moves closer to shore. If relaxation occurs for an extended time, the front can actually collide with the coast and deposit its larvae, producing a recruitment event. This hypothesis may offer a solution to the recruitment problem in central California, potentially explaining both the timing of recruitment at the coast and why it occurs in discrete pulses.
Evidence for this recruitment hypothesis comes from a combination of plankton samples, recruitment data, and oceanographic data. The analysis of plankton samples confirms that a front separates pelagic barnacles from intertidal barnacles (Gaines et al. 1985, Roughgarden et al. 1988). In addition, recruitment data for the intertidal barnacles Balanus glandula and Chthamalus spp. during 1988 show that recruitment pulses near Monterey, California corresponded each time to an event when a warm, low-salinity water mass arrived at shore and replaced the cold, high-salinity upwelled water (Farrell et al. 1991, Roughgarden et al. 1994). The characteristics of these water masses indicated they were California Current offshore oceanic water. Direct evidence also suggests there is a spatial gradient in recruitment caused by variations in upwelling intensity and position of an offshore front. During the upwelling season, recruitment of larvae is negatively correlated with the distance from shore of an upwelling front (Miller 1992). When the front is close to shore, larvae recruit on a more continuous basis. As the front moves farther from shore, recruitment significantly decreases.
Additional evidence, data of sea-surface temperature off the California coast from Advanced Very High Resolution Radiometer (AVHRR) satellite imagery and nearshore advection of surface water from High Frequency (HF) radar provide a general picture of front movement and surface flow during the upwelling season. AVHRR satellite images of sea-surface temperature illustrate front location and movement during upwelling/relaxation events along the coast between the Monterey Peninsula and Point Sur. Current vectors off of Granite Canyon during the upwelling season show surface flow to the southwest during upwelling and surface flow back toward the coast during relaxation (Shkedy et al. 1995). Images of sea-surface temperature during upwelling and relaxation events coincide with changes in surface flow. This information illustrates the physical transport processes that potentially affect recruitment.
Interdisciplinary modeling studies of the transport of biogenic material between coastal environments and offshore waters have been receiving increased attention in recent years (see review by Wroblewski and Hofmann 1990). Coupled physical/biological models that include both oceanographic data and life history information have been developed and solved numerically for a number of marine systems (Wroblewski 1982, Davis 1984, Hofmann 1988, Capella et al. 1992). Additionally, analytical solutions of the two-dimensional advection - diffusion - mortality equation have been used to estimate larval dispersal and larval wastage between release and recruitment sites (Hill 1990, 1991, McGurk 1989). However, the influence of larval dispersal on recruitment in a time-dependent, two-dimensional model with a moving offshore boundary has not yet been studied.
Initially, we present and further develop a one-dimensional prototype model (Roughgarden et al. 1994) that incorporates a larval phase influenced by advection and diffusion in the water column together with an offshore reflecting boundary representing an upwelling front. Exploration of the model demonstrates the effect of cross-shelf flow and upwelling/relaxation events on the population dynamics of a coastal barnacle species, offering initial support of the hypothesis that recruitment of larvae to coastal adult populations is determined by oceanographic events (Roughgarden et al. 1991a). The simple one-dimensional analysis allows us to inspect the main features of the model and to show the types of results that can be achieved as well as some model limitations. We subsequently extend the model to a more realistic two-dimensional coupled benthic/oceanic model that includes spatial dimensions for both the larval and adult populations, providing a mechanism to study the interaction between offshore flows and alongshore flows in a simple physical representation of upwelling/relaxation [ILLUSTRATION FOR FIGURE 1 OMITTED]. This modeling framework presents an idealized view of the California Current system and its influence on population dynamics in the rocky intertidal zone along the coast. The simple representation of Ekman transport is an appropriate starting point for the model given the main assumptions that the larvae remain at one depth level in the Ekman layer, and cannot control their movement in the water column. As a proviso, we offer this study as a numeric analysis preliminary to a more realistic time-dependent treatment in the future.
REVIEW OF PREVIOUS MODELS
An early attempt at a coupled benthic/oceanic model of population dynamics took the form of a metapopulation model with the adults contained in local sites that communicate with a common larval pool (Roughgarden and Iwasa 1986). This model includes an initial representation of spatial structure for the larval phase. The larvae from each adult population combine to produce one common larval pool. This model may be appropriate to an estuary or enclosed body of water where the larval life-span exceeds the time for horizontal mixing of the water mass. However, for the coastal ocean a more realistic treatment includes spatial dimensions for both the adult and larval phases.
The proposed extensions are partially based on a two-dimensional model that has been analyzed numerically by Possingham and Roughgarden (1990). In this version of the model, larvae are produced by the adults, transported in the offshore water column by advection and diffusion, and settle at the local populations, where they metamorphose into adults. The population dynamics of this system are represented by simultaneous equations for the adult and larval phases. Previously Possingham and Roughgarden (1990) analyzed only alongshore flow, with no cross-shelf transport. The larvae in the offshore water column are advected from north to south along the coast in a flow field suggestive of the southward-flowing component of the California Current. The results showed how the strength of the alongshore flow field, the amount of available suitable habitat, and the initial conditions interact to influence species persistence.
Roughgarden et al. (1994) extended these previous models to introduce the hypothesis that recruitment pulses result from the approach and eventual collision of upwelling fronts with the intertidal zone. They incorporated a new frontal feature that depicts the line of demarcation between upwelled water and oceanic water. In the one-dimensional analytical analysis, the front is represented mathematically as a reflecting boundary located at a position offshore. The distance offshore of the reflecting frontal boundary is determined by the strength of the prevailing northwest winds driving coastal upwelling. Recruitment pulses are produced when relaxation occurs for an extended time and the front collides with the shoreline, depositing its larvae. Unlike typical diffusion equations, this system of equations shows that the location of the offshore boundary need not be a fixed constant but may be time dependent. In the following section we introduce an improved grouping of parameters from the preliminary Roughgarden et al. (1994) example and explore the model, both analytically and numerically, in greater detail.
Advection and diffusion
Roughgarden et al. (1994) consider a one-dimensional model involving onshore - offshore larval diffusion and offshore advection. The adult population is situated at a fixed point along the coast. Larvae are released from the adult population on the coast and are diffused laterally, by turbulence in the ocean. The mortality rate of the larvae is significant. The adult dynamics are explained first, followed by the larval dynamics. The rate of change of the number of adult barnacles at the coast at time t, B(t), is
dB(t)/dt = cF(t)L(0, t) - [Mu]B(t), (1)
where F(t) is the amount of free space available at time t, L(0, t) is the larval concentration at the coast (x = 0) at time t, c is the proportionality constant for the rate of larval settlement, and [Mu] is the death rate of the adult barnacles. The space constraint is represented by the following equation
A [equivalent to] F(t) + aB(t). (2)
The total amount of space, A, is identically equal to the amount of free space, F(t), and the amount of occupied space, aB(t), at time t, where a is the average basal area of an adult individual.
The rate of change of the larval population, L(x, t), with respect to time, is described by the partial differential equation
[Delta]L(x, t)/[Delta]t = -U[Delta]L(x, t)/[Delta]x + K[[Delta].sup.2]L(x, t)/[Delta][x.sup.2] - [Lambda]L(x, t). (3)
The equation includes a longitudinal dispersion coefficient, K, a constant offshore advection rate, U, and a larval mortality rate, [Lambda]. K is not a function of x. The second-order partial differential equation requires two boundary conditions. The coastal boundary condition, representing larval flux at the coast, is
UL(0, t) - K[Delta]L(0, t)/[Delta]x [[where].sub.x = 0] = mB(t) - cF(t)L(0, t), (4)
where m equals the rate of production of larvae by adult barnacles. The offshore, reflecting boundary condition represents the upwelling front at a point [x.sub.f], along the x axis,
UL([x.sub.f], t) - K[Delta]L([x.sub.f], t)/[Delta]x [[where].sub.x = [x.sub.f]] = 0. (5)
For the simplest case of the model, offshore advection, U, is set equal to zero, and larvae are transported by diffusion only.
Because these equations cannot be solved analytically in their present form, two numerical approaches have been taken. The first is an approximation based on a scaling argument (Roughgarden et al. 1994), and the second is a finite difference method. We will address the approximation method in this section, and the numerical method in the following section.
The one-dimensional model can be solved analytically to some extent by using an approximation based on different time scaling for larvae and adults. The adult population, B(t), is provisionally treated as a constant, independent of time, in the larval equations. This assumption is justified through the use of age-averaged parameters in the adult population and scaling arguments. Barnacle adults grow much more slowly than the water column processes operate. The entire larval life cycle is [approximately]4 wk long, compared to an average adult life-span of [greater than] 1 yr. The adults are therefore treated as stationary during each 4-wk period of the model, and then changed according to the outcome of the larval dynamics. With this assumption, the equation for L(x, t) is solved by letting L(x, t) come to equilibrium for a fixed B, i.e., solve for L(x, t) = L(x, B(t)), and then stepping the equation for dB(t)/dt. Eventually the system will reach equilibrium for both the larvae and the adults.
The number of larvae at the coast, L(0), will eventually build up to a steady state, the point where larval settlement balances larval production. The equilibrium solution to the diffusion equation (Eq. 3) with boundary conditions (Eqs. 4 and 5) describing the larval distribution for a fixed adult population size, B, and upwelling front position [x.sub.f], is
[Mathematical Expression Omitted],
[Mathematical Expression Omitted],
where the water column parameters group as
[q.sub.1] = U + [square root of 4K[Lambda] + [U.sup.2]] (8)
[q.sub.2] = U - [square root of 4K[Lambda] + [U.sup.2]] (9)
[q.sub.3] = [e.sup.[q.sub.2][x.sub.f]/2K] [q.sub.1] (10)
[q.sub.4] = [e.sup.[q.sub.1][x.sub.f]/2K] [q.sub.2]. (11)
The adult population can then be stepped according to the equation
dB/dt = B(-[Mu] + 2(A - aB)cm([q.sub.3] - [q.sub.4])/(2(A - aB)c + [q.sub.2])[q.sub.3] - (2(A - aB)c + [q.sub.1])[q.sub.4]). (12)
The positivity requirement, p, for the equilibrium adult population, [Mathematical Expression Omitted], is
p = (-[Mu] + 2Ac(m - [Mu])q) [greater than] 0 (13)
where q is composed of water column properties
q = [q.sub.3] - [q.sub.4]/[q.sub.2][q.sub.3] - [q.sub.1][q.sub.4]. (14)
This approximate solution is useful in a number of ways. It provides a control for the numerical analysis, is computationally efficient, and allows some analytical results to assist with phenomenological interpretation, especially concerning the conditions for population extinction as a function of the front position and strength of offshore advection.
An indication of the stability of this positive equilibrium is obtained by linearizing the model around the extinction solution, B = 0, with respect to a perturbation along the equilibrium trajectory for L as a function of B. From the linearized model one can determine whether the adult population will increase when rare for various combinations of advection strengths and front positions. The linearized differential equation for the adult population (Eq. 12) leads to sigmoid growth, with an initial rate of increase, [r.sub.0],
dB/dt [[where].sub.B = 0] = [r.sub.0]B, (15)
[r.sub.0] = p/1 + 2Acq. (16)
The adult population, B, comes to equilibrium at
[Mathematical Expression Omitted].
The condition for increase when rare (dB/dt [[where].sub.B = 0] [greater than] 0) coincides with the condition for the existence of a positive interior solution. Therefore, extinction is an unstable solution when an interior positive solution exists, and numerical analysis has suggested that the system does converge to the interior solution.
Because Eq. 12 exhibits sigmoid growth, it is instructive to compare it with the logistic equation
dN(t)/dt = r(1 - N(t)/[K.sub.1])N(t), (18)
where we identify
[Mathematical Expression Omitted].
The adult barnacle population, exhibiting a growth pattern similar to that of the logistic equation, will eventually reach equilibrium at
[K.sub.1] = p/2ac(m - [Mu])q, (20)
with r, the eigenvalue of the adult equation, defined as
[Mathematical Expression Omitted].
The solution to the logistic equation is an S-shaped curve determined by r, the intrinsic rate of increase, and [K.sub.1], the carrying capacity (the subscript 1 is used to distinguish carrying capacity, [K.sub.1], from K, the eddy-diffusion coefficient). When the population size, N, is small, the population growth rate is close to r, however, as N approaches [K.sub.1], the rate slows significantly and tends to zero. Fig. 2, which illustrates both dB/dt and dN/dt, shows that the logistic approximation to dB/dt is in fact extremely close.
If the water column processes of cross-shelf advection, U, and front location, [x.sub.f], change slowly relative to the speed at which the larval population comes to equilibrium, one could take U = u(t) and [x.sub.f] = [x.sub.f](t) as time-dependent parameters (c.f. Roughgarden et al. 1994). However, the fluctuation over time in these parameters can be very fast, changing daily, and it remains to be seen if the scaling argument here still has validity in such conditions. Therefore, a numerical analysis of the model using an implicit finite difference algorithm is developed in the next section that does not use any scaling arguments.
Full time-dependent numerical analysis
To solve this system of equations numerically the equations are approximated using an implicit centered-space, centered-time finite difference scheme ??(see Appendix for details). In the scenario that has been explored, a pulse of larvae is released into the water column. The pattern of recruitment is then monitored as a simple scenario of upwelling and relaxation events are simulated. From these simulations, it can be seen directly how the continuous movement of a front and corresponding change in offshore advection rate affect larval recruitment and subsequently the adult dynamics.
In the time-dependent solution, we introduce a moving upwelling front into the model. The position of this front varies between 0.1 km and 20.1 km offshore. This assumption is necessary for the numerical analysis in order to preserve one cell of the larval solution space as the front approaches the coast. The simple scenario developed has the front moving in to the coast and then back offshore 20 km every 25 d. The front moves 100 m every 1.5 h. This pattern is repeated for 6 mo. This hypothetical scenario is loosely based on observations from 2 yr of AVHRR satellite images showing sea-surface temperature, from which we monitor front movement, combined with daily surface temperature recordings taken from shore. While the front moves offshore, we incorporate a fixed offshore advection; while the front moves onshore, offshore advection is set to zero. The diffusion coefficient remains constant throughout the entire period. This scenario offers a phenomenological model of a sequence of upwelling/relaxation episodes.
TABLE 1. Physical and biological parameters used in model. Physical parameters were obtained from data on local conditions off the coast of central California. Biological parameters were obtained from field data on the intertidal barnacle Balanus glandula.
Physical and biological parameters
B(y, t) = adult barnacles per 100 m length of coastline;
L(x, y, t) = larvae per 100 [m.sup.2];
F(y, t) = free space available for adults per 100 m length of coastline;
A(y) = total available space for adults per 100 m length of coastline;
a = space occupied by one barnacle = 0.0001 [m.sup.2] (maximum 1,000,000 adults/100 [m.sup.2]);
[Lambda] = mortality rate of larvae = 5%/d = 0.00000056 [s.sup.-1];
[Mu] = mortality rate of adults = 2.5%/d = 0.00000028 [s.sup.-1];
m = larval production rate of adults = 0.0000067 [s.sup.-1];
[l.sub.0] = larval concentration outside of north boundary = 500 larvae/100 [m.sup.2];
K = eddy-diffusion coefficient = 10 [m.sup.2] [s.sup.-1];
U = offshore advection rate varies between 0 and 0.20 m [s.sup.-1];
V = alongshore advection rate varies between 0 and 0.20 m [s.sup.-1];
c = larval settlement coefficient = 0.0000056 [s.sup.-1];
[x.sub.f] = position of upwelling front (metres from coast)
The parameter values used in this model are typical of the barnacle, Balanus glandula, that is found throughout the high rocky intertidal zones of the west coast of North America. These values and their dimensions, taken directly from field data, are listed in Table 1 (see also Possingham and Roughgarden 1990). In the numerical cases, the parameters are nondimensionalized using a characteristic length of 100 m, a characteristic area of substrate at the coast of 100 [m.sup.2] (e.g., a strip of coastline 100 m long and 1 m high), and a characteristic time of 3 h. The larval production rate, m, is an age-integrated average determined from Hines (1978). The larval settlement coefficient, c, was found by Gaines et al. (1985) by measuring the larval settlement rate for a known concentration of larvae at the coast. The adult mortality rate is age independent. The value used here is representative of Balanus glandula mortality rates under crowded conditions (Gaines and Roughgarden 1985). The larval mortality rate is also an age-integrated average (Pyefinch 1948). The eddy-diffusion coefficient, K, is an approximation of the average dispersion rate of passive particles in the ocean. It is not an absolutely correct description because dispersion in the ocean depends upon the scale of the physical processes involved (Okubo 1978). However, dye experiments by Okubo (1971) show that the approximate K of 10 [m.sup.2]/s is a good approximation for the spatial and time scales over which barnacle larvae disperse off the central California coast. The offshore advection rate, U, ranges from 0 to 20 cm/s in our simulations. At 0 cm/s, there is no offshore advection occurring, such as in a relaxation event. At 20 cm/s, the strength of offshore advection is typical of steady upwelling conditions along the open coast.
ONE-DIMENSIONAL MODEL ILLUSTRATIONS
Diffusion only, analytical case
In Fig. 3 we show the effect of the distance of the front from shore on the steady-state abundance of barnacle adults, expressed as percent cover of the benthic surface. This case assumes only diffusion and no advection, and the front is a fixed distance from shore. The larval and adult populations grow until an equilibrium state is reached in which the number of larvae produced equals the number of larvae settling on shore. We found that as the front approaches closer than 15 km from shore, the recruitment rate to the shore increases resulting in a higher equilibrium abundance. Beyond 15 km from shore, the effect of the front on the larval and adult populations is negligible. When no front is present, the values for the adult population and larval distribution agree with Possingham and Roughgarden (1990). In addition, the presence of an upwelling front allows the adult population to remain viable when there are higher diffusion coefficients and higher death rates.
Advection and diffusion, analytical case
Next we consider results from the analytical solution to an approximation of the model based on scaling arguments. As in the previous case, the front remains at a fixed distance from shore, but advection is now included. Fig. 4 shows a typical equilibrium distribution of larvae between the coast and the front during weak upwelling conditions. The larvae are distributed between the coast and the front in a U shape due to the production of larvae at the coast and the advection-driven accumulation of larvae at the front. If offshore advection were increased, the accumulation of larvae at the front would be more pronounced, but the total number of larvae at equilibrium would decrease.
The time to steady state is controlled by the mortality rate of the larvae. Once at steady state, the point at which advection is balanced by diffusion is 100 m toward shore from the front. In other words, the larval zone of accumulation is [approximately] 100 m wide when K = 10 [m.sup.2]/s and U = 0.1 m/s. This value would increase, i.e., the zone of accumulation would widen, if local advection at the front was weaker than at the other portions leading up to the front. The distribution of larvae changes, increasing the number of larvae at the coast, as the upwelling front approaches the coast. In Fig. 5 we consider the effect of the interaction of strength of upwelling and front position on the adult population. For five different equispaced advection levels (U = 0.0-2.0 cm/s) we determine the percent cover of adults over a range of front positions. In each case, the position of the front ranges from at the coast to 50 km offshore. The combination of advection levels and front positions show a threshold level at which a front is necessary to keep the population viable. With a stationary front, a viable equilibrium adult population can only be maintained for weak advection levels. Specifically, the front must be very close to shore when the advection rate reaches 2.0 cm/s or greater for the population to remain viable over time.
For the one-dimensional model, Fig. 6 shows curves of adult population size, B, as a function of time for a stationary front located 100 km offshore and varying advection strengths. The population approaches equilibrium, or carrying capacity, in a manner similar to that predicted from the logistic equation. Likewise, Fig. 7 suggests density dependence by the change in adults per time step as the population approaches equilibrium. As determined earlier, the model is highly sensitive to changes in advection strength, and an increase in advection strength results in a decrease in adult population equilibrium size.
Criterion for population viability
The conditions for the adult population to increase or decrease when rare were determined for various advection strengths and front positions using a linear stability analysis with respect to a special perturbation from the extinction solution. In Fig. 8 we show that the extinction solution is unstable because dB/dt is positive, allowing the population to increase when rare, for the range of advection strengths and front positions below the curve. This curve is plotted directly from Eq. 15. When advection strength is 0 cm/s, the sign of dB/dt is independent of the front position. When the strength of advection is [greater than] 0 cm/s, the sign of dB/dt depends upon the front position. For advection rates [less than] 1.7 cm/s, the sign of dB/dt is always positive. For advection rates [greater than] 1.7 cm/s, the sign of dB/dt is positive as the front position approaches the coast and negative as the front position moves offshore. In all of these cases, diffusion is constant at 10 [m.sup.2]/s.
Advection and diffusion, numerical case
Next we look at the results from the full time-dependent numerical analysis of the advection and diffusion model. In this case the front position is no longer fixed. A numerical experiment is considered in which 50 000 larvae are introduced into the water column adjacent to the coast and the pattern of recruitment pulses is monitored, as a scenario of upwelling and relaxation events is simulated. The front moves from the coast out to 20 km offshore and back to the coast every 25 d. When the front moves offshore, advection is set to 2.5 cm/s, and when it moves onshore advection relaxes to 0 cm/s. The initial condition for the adult population varies between 0 and 500 000 adults/100 [m.sup.2]. Diffusion is held constant at 10 [m.sup.2]/s. Fig. 9 shows the approach to a steady-state cycle in percent cover by adult barnacles resulting from a regular sequence of upwelling and relaxation events. Fig. 10 shows the recruitment pulses that would be observed along the shore as a result of this scenario. Recruitment begins as the front approaches the shore, and peaks after the front has collided (to within 0.1 km) with the shore. Recruitment patterns approach an identical cycle for all initial conditions. There is a very sharp decline in recruitment after the front has reached the coast and begins to move offshore. This decline is caused by the change in offshore advection from 0 to 2.5 cm/s. This result differs slightly in shape from the illustration of Roughgarden et al. (1994), where a sinusoidal pattern through time was used to vary both front movement and advection strength.
We further expand the model to incorporate larval transport in the alongshore direction and a spatial dimension for the adult population. The two-dimensional model includes an adult population situated along a stretch of hypothetical coastline influenced by larval recruitment and benthic mortality coupled to an offshore larval pool influenced by diffusion, advection, larval mortality, and a moving upwelling front. Larvae enter and exit the water column through larval production by adults, recruitment, and flow through the north and south boundaries. Recruitment of larvae into the adult population is a function of the abundance of larvae at the coast, which in turn is determined by water column processes.
In the two-dimensional model, solved using numerical techniques, both larval and adult distributions are presented in various flow regimes. In order to better understand the influence of the transport mechanism in the model, the runs progress in complexity from a very simple representation to a somewhat more realistic representation of transport processes in the upper Ekman layer of the ocean. Initially the population dynamics of adults are considered when there is a stationary offshore front. These control cases can be checked directly against the one-dimensional numerical solution, which in turn can be checked against the one-dimensional analytical solution. We show how constant offshore and alongshore mean flows ultimately cause the population to go extinct when a stationary front is located well offshore. A phenomenological account of upwelling and relaxation is then superimposed on this system. Distribution and recruitment patterns are shown for a series of combinations of offshore and alongshore advection.
Advection and diffusion
In the two-dimensional, depth-averaged model both the larval and adult populations are functions of space and time. The larvae are advected and diffuse in the xy plane and the adult population is located along the y axis. The rate of change of the number of adult barnacles at the coast at time t and position y is
[Delta]B(y, t)/[Delta]t = c(y)F(y, t)L(0, y, t) - [Mu](y)B(y, t). (22)
The larvae settle into the adult population at a rate proportional to the amount of free space available, F(y, t), and the abundance of larvae at the coast, L(0, y, t). The coefficient, c(y), is a proportionality constant. The adults die at rate, [Mu](y). The area equation is
A(y) [equivalent to] F(y, t) + aB(y, t). (23)
The amount of free space at each point along the coast, F(y, t) is determined from the total space, A(y), less the occupied space, where a is the area occupied by an individual barnacle.
A transport equation that includes both offshore and alongshore advection and diffusion describes the larval population dynamics,
[Delta]L(x, y, t)/[Delta]t + U(x, y, t) [Delta]L(x, y, t)/[Delta]x + V(x, y, t) [Delta]L(x, y, t)/[Delta]y
= K([[Delta].sup.2]L(x, y, t)/[Delta][x.sup.2] + [[Delta].sup.2]L(x, y, t)/[Delta][y.sup.2]) - [Lambda](x, y, t)L(x, y, t). (24)
U(x, y, t) and V(x, y, t) are the current velocities in the x and y directions, respectively, that advect larvae. The larvae diffuse in both the x and y directions at a rate determined by K. The larval death rate is [Lambda].
A boundary condition is specified at the coast, representing the flux of larvae in an outwardly normal direction at the coast at a point y,
U(0, y, t)L(0, y, t) - K[Delta]L(0, y, t)/[Delta]x [[where].sub.x = 0]
= m(y)B(y, t) - c(y)F(y, t)L(0, y, t). (25)
The rate of larvae production by adults, m(y)B(y, t), less the rate at which larvae settle onto the coast, c(y)F(y, t)L(0, y, t), determines this flux, where m(y) is the larval production rate. The boundary condition at the offshore upwelling front is a reflecting boundary.
U([x.sub.f], y, t)L([x.sub.f], y, t) - K[Delta]L([x.sub.f], y, t)/[Delta]x [[where].sub.x = [x.sub.f]] = 0. (26)
There is no transport of larvae across this boundary; instead larvae are reflected back into the larval pool without loss.
Two additional boundaries that must be set for the larval transport equation are fluxes expressing the net movement of larvae in from the north and out from the south due to alongshore advection and diffusion. They are located at y = 0 and y = [y.sub.1]. The north boundary, y = 0, is
V(x, 0, t)L(x, 0, t) - K[Delta]L(x, 0, t)/[Delta]y [[where].sub.y = 0] = L(x, 0, t)V(x, 0, t), (27)
and the south boundary, y = [y.sub.1], is
V(x, [y.sub.1], t)L(x, [y.sub.1], t) - K[Delta]L(x, [y.sub.1], t)/[Delta]y [[where].sub.y = [y.sub.1]] = L(x, [y.sub.1], t)V(x, [y.sub.1], t). (28)
The number of larvae that enter through the north boundary depends upon the larval concentration at the boundary, L(x, 0, t), and the alongshore advection. Likewise, the number of larvae exiting through the south boundary depends upon the number of larvae at the border, L(x, [y.sub.1], t), and the alongshore advection. Alternatively, for a fixed larval source term at the north boundary, the north boundary condition becomes
V(x, 0, t)L(x, 0, t) - K[Delta]L(x, 0, t)/[Delta]y [[where].sub.y = 0] = [l.sub.0]V(x, 0, t), (29)
where [l.sub.0] represents the constant source of larvae. The equation for the south boundary remains the same,
K[Delta]L(x, [y.sub.1], t)/[Delta]y [[where].sub.y = [y.sub.1]] = 0. (30)
These equations are solved numerically with finite difference techniques.
To solve this two-dimensional system of equations numerically, an alternating direction semi-implicit method is used for the larval equations and a forward-time explicit method is used for the adult equations (Noye 1984; see Appendix for details). A solution space is defined where the adults are indexed along the y axis between 0 [less than] y [less than] [y.sub.1] and the larvae are indexed in the x-y plane defined by 0 [less than] x [less than] [x.sub.f] and 0 [less than] y [less than] [y.sub.1]. A grid is defined in this solution space by the intersection of lines separated by distance [Delta]x on the x axis and [Delta]y on the y axis. This solution space is 30 km from the coast to the reflecting front and 30 km along the coast between the north and south boundaries, and generally represents the area defined by the coastline between the Monterey Peninsula and Point Sur and a front located 30 km offshore. The solution of Eqs. 24-28 (given as larval concentrations) is determined at each point on this grid using known concentrations from the previous time step and concentrations from preceding points in the current time step. The adult population (Eqs. 22 and 23) is then approximated using larval concentrations at the coast at the end of that time step. Alternating directions through the solution space at each time step minimizes numerical error. Stability conditions require [Delta]x and [Delta]y to be 200 m and At to be 18 min when advection levels are [less than] 10 cm/s. For advection levels between 10 cm/s and 20 cm/s, the technique is stable when [Delta]x and [Delta]y are 100 m and [Delta]t is 10.8 s. Therefore, due to lack of computing power, very few runs are possible with advection levels [greater than] 10 cm/s.
To investigate the influence of offshore advection, alongshore advection, and an offshore front, a series of runs are performed. The runs progress in complexity from a stationary front and direct offshore advection to a more realistic combination of a moving from with both offshore and alongshore advection. In the first set of illustrations, the advection of larvae in from the north boundary equals the advection of larvae out from the south boundary. These results are then contrasted with illustrations using a set larval source term at the north boundary. The initial condition for each run is an adult population of 100 000 every 100 m along the coastline, i.e., an initial adult population occupying 10.0% of the available space. No larvae are initially present in the water column. This condition represents the time just before upwelling starts, when adults are present and reproductive activity is low. The peak reproductive period for B. glandula is in the spring and summer, coinciding roughly with the upwelling period in central California (Standing 1981). In the stationary offshore front simulations the front is positioned 30 km offshore. In the moving offshore front simulations, a simple scenario of onshore and offshore movement is employed, with the front oscillating between 0.2 km offshore and 30.2 km offshore. The front moves in to the coast and then back offshore 30 km with a period of 30 d. This pattern is repeated for 6 mo. Both offshore and alongshore advection are positive when the front moves out, and zero when the front relaxes back into shore. The diffusion coefficient remains constant throughout the entire run.
TWO-DIMENSIONAL MODEL ILLUSTRATIONS
Diffusion only, stationary front
With no offshore or alongshore advection, the adult population increases quickly to equilibrium. As demonstrated in the one-dimensional model, a stationary front within 15 km of the coast significantly increases the equilibrium adult population size. In this case, the offshore front is located 30 km from the coast and has negligible influence on the adult dynamics. The adult population reaches an equilibrium cover (803 909 adults/100 m length of coastline) in agreement with the equilibrium obtained in the one-dimensional analytical solution. The approach to equilibrium is illustrated in Fig. 11. The larval distribution at equilibrium shows buildup of larvae at the coast, with larvae decreasing exponentially away from the coast [ILLUSTRATION FOR FIGURE 2 OMITTED]. With no advection, larvae remain within [approximately] 15 km of the coast. This scenario represents a baseline case with which to compare the effects of advection on adult and larval distributions.
Advection and diffusion, stationary front
When offshore advection is added to this scenario (stationary front located 30 km offshore), the adult population dynamics quickly change. For constant offshore advection strengths of [less than] 1.7 cm/s, the population can sustain itself. For example, when advection strength is 1.0 cm/s, the adult population approaches a positive equilibrium of 129 938 adults/100 m length of coastline [ILLUSTRATION FOR FIGURE 13 OMITTED]. The larval distribution is U-shaped, with a slight increase of larvae at the coast due to production from adults and a strong buildup of larvae at the offshore front [ILLUSTRATION FOR FIGURE 14 OMITTED]. This distribution is shown at day 360 to illustrate the slow approach to equilibrium. Both the adult and larval equilibrium distributions are consistent with values obtained from the one-dimensional analytical solution. However, as determined in the one-dimensional model, when constant advection levels are [greater than] 1.7 cm/s, the population cannot sustain itself and goes extinct. Larvae are pushed offshore and accumulate at the offshore front, but are unable to return to the coastal adult population. The addition of alongshore advection does not influence the equilibrium larval or adult distributions when there is a constant flow of larvae into the solution space equalling that out of the solution space.
Advection and diffusion, moving front
In the full time-dependent numerical analysis including offshore advection, alongshore advection, diffusion, and a moving offshore front, a series of scenarios are considered in which the front moves from 30 km offshore to the coast and back again every 30 d. In the first scenario, offshore advection is 7.5 cm/s when the front is moving offshore, and 0 cm/s when the front is moving onshore. Alongshore advection is 0 cm/s during both onshore and offshore movement. Adult population density oscillates as the front moves offshore and onshore, peaking each time the front reaches the coast [ILLUSTRATION FOR FIGURE 15 OMITTED]. As expected, the larvae are distributed symmetrically in the north/south direction and accumulate at the offshore front [ILLUSTRATION FOR FIGURE 16 OMITTED]. As before, an increase in strength of alongshore advection does not change the adult and larval distributions, since the supply of larvae entering at the north boundary equals the supply of larvae leaving at the south boundary. However, when offshore advection is increased, the adult population oscillates at lower densities.
In the final scenario the north boundary condition is changed to include a fixed larval source term. Now, the number of larvae entering at the north boundary is not equal to the number of larvae leaving through the south boundary, and the strength of alongshore advection affects the adult and larval distributions. Offshore and alongshore advection strengths are used that typify steady summer upwelling and relaxation conditions along the coast between Monterey Bay and Point Sur. In this case larvae enter through the north boundary from a base concentration of 500 larvae/100 m length of boundary, and exit through the south boundary at a rate dependent upon the concentration of larvae at that boundary and the strength of alongshore advection. Both offshore and alongshore advection are positive, 7.5 and 5.0 cm/s, respectively, when the front is moving offshore, and zero when the front is moving onshore. The adult population increases just inside the north boundary due to the influx of larvae [ILLUSTRATION FOR FIGURE 17 OMITTED]. The effect of alongshore advection is also apparent in the larval distribution. A snapshot at day 180 shows the diagonal transport of larvae to the southwest. Larvae are present across the entire plane; however, there are more larvae in the southwest half due to the transport of larvae in that direction [ILLUSTRATION FOR FIGURE 18 OMITTED]. In these cases, when the period of one upwelling/relaxation cycle is increased, for example from 30 d to 45 d, the adult population oscillates at a lower overall population density due to fewer larvae returning to the coast. Conversely, when the period is decreased, for example to 15 d, larvae are brought back to the coast at a faster rate, and the adult population will oscillate at a higher overall density.
The extension of previous population dynamic models to include cross-shelf transport and the existence and movement of upwelling fronts was motivated by field studies of the recruitment of intertidal barnacles in central California (Farrell et al. 1991). In this expanded model for the dynamics of coastal barnacle populations we have included the hypothesis that recruitment pulses result from the collision of upwelling fronts with the intertidal zone (Roughgarden et al. 1991a). The effects of the presence of an upwelling front and simulated upwelling and relaxation events on barnacle populations were studied with models using analytical and numerical methods. All of the simulations used biological parameters taken directly from field measurements of the intertidal barnacle, Balanus glandula, and physical parameters typical of conditions off the central California coast during the spring and summer upwelling months as identified by AVHRR satellite imagery of sea-surface temperature and HF-radar-derived current vectors of surface flow. Data show surface waters flowing toward the southwest during upwelling periods and back toward the coast during relaxation events. While the actual physics of transport in the Ekman layer is quite complicated, we developed a simple phenomenological representation of transport based on the assumption that intertidal larvae cannot control their movement in the water column and generally remain at one level in the Ekman layer. This representation allows us to explore potential larval transport mechanisms without using a three-dimensional circulation model.
We have presented the model scenarios in order of increasing complexity. We discuss results common to both the one-dimensional and two-dimensional models together, followed by results unique to the two-dimensional model. In the initial case in which larvae diffuse between the coast and a stationary front, both the one-dimensional [ILLUSTRATION FOR FIGURE 3 OMITTED] and two-dimensional [ILLUSTRATION FOR FIGURE 11 OMITTED] analyses illustrate the baseline effect of the interaction of eddy-diffusion, larval mortality, and front position on the adult dynamics. This situation may mimic periods in which larvae are present in the water column with no coastal upwelling occurring, such as in the early fall or late winter along the California coast. Tidal or other types of fronts may be present at this time. As a next step we incorporate advection into the model, simulating potential conditions during the spring and summer upwelling period. For advection and diffusion values representing steady upwelling conditions, the model is advection dominated and larvae accumulate at offshore fronts as seen in the one-dimensional [ILLUSTRATION FOR FIGURE 4 OMITTED] and two-dimensional [ILLUSTRATION FOR FIGURE 12 OMITTED] illustrations. These results are in agreement with studies showing the accumulation of biomass at oceanic frontal zones through physical processes (Dengler 1985, Smith et al. 1986, Traganza et al. 1987). Small increases in advection levels significantly affect the equilibrium adult population [ILLUSTRATION FOR FIGURE 5 OMITTED]. In these cases larvae accumulate at the offshore front and are unable to return to the coast to settle before dying. Most importantly, not only will increased advection levels cause a population to go extinct, but in the presence of advection [greater than] 1.7 cm/s, a front is necessary to keep the population viable. Constant along-shore flow in the two-dimensional model [ILLUSTRATION FOR FIGURE 13 OMITTED], however, does not change the adult or larval dynamics when the flux of larvae into the solution space equals that out of the solution space.
Comparisons to the logistic equation illustrate density dependence in the adult population due to a finite resource (space) in the benthic habitat and feedback on settlement by freespace availability [ILLUSTRATION FOR FIGURE 6 OMITTED]. Carrying capacity, however, is set by water column processes affecting larval supply, not solely by adult resource limitation. Ultimately, these water column processes determine whether or not the population will persist [ILLUSTRATION FOR FIGURE 7 OMITTED], suggesting that a combination of benthic and oceanic processes controls barnacle population dynamics in the rocky intertidal zone along the central California coast. Using the phenomenological representation of upwelling, we determine the criteria for the water column parameters that yield an unstable extinction solution (i.e., where the front is sufficiently close to shore and/or the advection is sufficiently weak). Adult barnacle populations can increase when rare for the range of advection strengths and front positions below the curve [ILLUSTRATION FOR FIGURE 8 OMITTED]. This system is nonlinear. The biology together with the fluid dynamics brings out two distinct solutions: a positive equilibrium solution and an extinction solution. What is new is that we have shown that there is a qualitative effect on the system that is dependent upon the front position and strength of advection. Typical combinations of front positions and advection strengths found along the coast would often lead to extinction in the model. These results suggest the existence of an additional component of the transport process influencing barnacle recruitment, leading to the next iteration of the model.
Relaxation events and a moving upwelling front are introduced into the modeling framework in a phenomenological context. Model scenarios of upwelling and relaxation events were simulated, causing recruitment pulses along the shore. As suggested in the hypothesis, recruitment pulses begin in both the one-dimensional [ILLUSTRATION FOR FIGURE 10 OMITTED] and two-dimensional [ILLUSTRATION FOR FIGURE 15 OMITTED] illustrations when the front is within a few kilometres of shore, and peak as the front is closest to shore. This pattern of recruitment agrees with the field observations by Farrell et al. (1991) and Roughgarden et al. (1994), in which increases in larval settlement coincided with the appearance of warm, fresh water at the coast (representing the arrival of the front between upwelled water and oceanic water). In a single-species analogy to the "storage effect" hypothesis (Warner and Chesson 1985), each pulse of larvae may allow the adult populations to persist until the next pulse arrives, despite unfavorable recruitment conditions due to high offshore advection.
The role of hydrodynamic features in the transport of benthic larvae may be critical in many marine systems. In a situation similar to the model scenario described above, the offshore dispersal of echinoderm larvae in the Ligurian Sea is limited by the presence of a density front. When the front is not present, larvae disperse much farther offshore, which can have an adverse affect on recruitment to adult populations (Pedrotti and Fenaux 1992). Ebert and Russell (1988) also implicate offshore transport due to intense upwelling in the loss of echinoderm larvae to local populations along the coasts of California and Oregon. Mechanisms of shoreward return can vary in each system. Along the central California coast, changes in surface water transport due to changes in the wind are implicated in the shoreward return of barnacle larvae.
The one-dimensional model represents a phenomenological account of the physics of these oceanographic events, and offers a prototype for predicting how changes in an offshore oceanographic regime can propagate to coastal ecosystems and potentially alter population dynamics. In the cases of the models discussed so far, the two-dimensional model analysis has provided a control to the one-dimensional results. Long-term effects of constant flow regimes on coastal populations have been considered. These simple scenarios help us to understand the qualitative effects of advection levels and front positions. However, during the upwelling season there are considerable temporal and spatial fluctuations in flow patterns, with current speeds often much greater than the sustainable levels indicated above. Therefore, short-term effects of these fluctuations are important to recruitment patterns and to overall population persistence. To explore these effects, the two-dimensional model allows us to proceed further and specify more realistic conditions.
In the next step of the model we include a continual supply of larvae from an external source that significantly affects local population dynamics. The impact of the larval supply depends on the advection into the local region and the concentration of larvae arriving from the external source. Field observations indicate higher larval settlement and adult population densities to the north of Monterey Bay near San Francisco than along the coast to the south of Monterey Bay. In this model scenario, the area just north of the solution space has a higher adult barnacle density than the area within the solution space. The transport of larvae through the north boundary increases when alongshore advection is high and decreases when it is low. Adults increase in the north, and larvae become unevenly distributed between the north and south boundaries, demonstrating transport in the southwest direction coinciding with HF radar current vectors and AVHRR satellite imagery of sea-surface temperature [ILLUSTRATION FOR FIGURES 17 AND 18 OMITTED]. In this model, advective forces typically dominate diffusive forces and lead to greater movement of larvae away from their original release site, and potentially to settle or colonize at distant habitats or die. When there are only diffusive forces present, larval retention is very high and there is potentially less mixing between coastal populations.
The models support the hypothesis that recruitment is negatively correlated along the coast with the distance from shore of the upwelling front, and that recruitment pulses occur when the front actually hits the coast (Farrell et al. 1991, Roughgarden et al. 1991a, 1994). Other physical mechanisms of shoreward transport to nearshore adult barnacle populations, including internal tidal bores and surface slicks produced by internal waves, have been proposed (Shanks 1983, 1986, Pineda 1991). However, these mechanisms are influenced by bottom topography, and it is not clear whether they are strong factors in the central California upwelling region. While simplistic in its physical representation, the two-dimensional model is intended as a next step toward understanding the role of physical transport processes in the variability in recruitment to coastal barnacle populations. It represents the continuing transition from modeling ecological processes on a local scale to modeling regional-scale phenomena. This approach of describing regional-scale processes with minimal complexity is also being utilized in terrestrial ecosystems models (Parton et al. 1988, Running and Coughlan 1988, Schimel et al. 1990, Running and Gower 1991). These studies and the present effort emphasize the importance of processes that span different spatial and temporal scales, and represent a growing trend in theoretical research in ecology as well as global climate research (Harvey and Schneider 1985a, b).
In the future, this two-dimensional model can be further used to explore the spatial pattern of recruitment and population abundance as a function of the spatial pattern of features in the offshore water column. A spatial gradient in recruitment of barnacles was found between San Francisco and Point Sur, California, and attributed to the distance from shore of the meander of the California Current (Miller 1992). During relaxation events, the front hit the coast more often in the north than in the south. This observation can be included in the model by incorporating a spatial gradient with the reflecting boundary close to shore in the north and farther from shore in the south. Also, alterations in physical and biological parameters, such as changes in offshore and alongshore advection, diffusion, reproductive output, and mortality due to changes in the global climate, can be explored using the two-dimensional model (Bakun 1990, Lubchenco et al. 1993). Although precise predictions are impossible, improved understanding of the effects of environmental anomalies through models that couple local and regional processes is possible.
Other physical transport processes such as eddies and meanders, which may provide a mechanism for larvae retention near shore, and jets and squirts, which may carry larvae away from shore, can be included in the model (Boicourt 1988). One way of accomplishing this task is to couple the population dynamic model to an actual physical - oceanographic circulation model. Laboratory primitive equation models based upon physical first principles (Zang 1993) and data assimilation simulation circulation models (Haidvogel et al. 1991) have been used to study the structure of upwelling flow and the evolution of upwelling fronts and filaments. This coupling could provide greater resolution of physical features and three-dimensional circulation. It may be sufficient, however, to minimize complexity, and continue to model depth-averaged surface flow. The main assumptions here are that there is relatively minimal vertical variance in the movement of the upper Ekman layer, where the intertidal barnacle larvae are located, and that the major forces in the system (i.e., surface transport due to upwelling and offshore fronts) are included in the model. Additionally, for predictive purposes, it is important to represent features such as fronts accurately, and this places great demands on circulation models in which specific fronts that develop along the California Current are horizontally unstable.
Alternatively, the two-dimensional model can be forced with real-time remote sensing data, and move from hypothetical situations to temporally and spatially specific occurrences. The application of remote sensing to regional-scale ecological studies is becoming increasingly popular (Roughgarden et al. 1991b). Images of sea-surface temperatures from the AVHRR on the NOAA satellites can be used to find actual front positions by the location of maximum gradient in sea-surface temperature. The movement of fronts onshore and offshore can be detected with daily images. Radar measurements of surface currents near shore are also available. Offshore and alongshore advection vectors can be obtained directly from HF radar located at Granite Canyon (Shkedy et al. 1995). A time series of these advection vectors and front locations can be assimilated into the model to yield predictions about the recruitment and population dynamics of barnacles that can be tested directly against benthic recruitment data along this stretch of coastline.
We sincerely thank E. Bjorkstedt, P. Ehrlich, B. Grantham, J. Koseff, S. Schneider, Y. Shkedy, and two anonymous reviewers for advice and valuable comments on earlier draft manuscripts. In addition, we thank R. Dial, G. Daily, J. Koseff, S. Monismith, and S. Schneider for discussions on various stages of model development. This work was supported by a grant from the National Science Foundation to J. Roughgarden (OCE-9115876).
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The one-dimensional larval equations are approximated using a centered-space, centered-time implicit finite difference scheme, and the adult equation is approximated using a simple forward-time finite difference scheme (see Noye 1984:231-250). For the reflecting boundary condition (front) and the mixed boundary condition (coast), values at those boundaries in the next-time step are approximated with a forward-time, centered-space differencing scheme, using the method of matching of truncation errors (see Noye 1984:176-182).
This scheme has no stability restrictions on the diffusion and advection coefficients or the space and time restrictions. However, a solvability condition is imposed, s [greater than or equal to] ((c/2) - 1), where s = k[Delta]t/[([Delta]x).sup.2] and c = u1[Delta]t/[Delta]x (see below for definition of k and u1). The initial larval distribution is entered by reading in the file "data.txt". The initial parameters are entered by an input file and output is written to the file "tex.file" and to the screen or a specified output file (i.e., program [less than] input [greater than] output).
The program was run on an IBM RT PC workstation and a Hewlett-Packard Apollo Series 400 computer. The spatio-temporal domain of this program changes with each time step to simulate a moving offshore front. The front is moved between 0.1 and 20 km offshore in a period of 25 d. For simulations using a stationary front, this program is modified so that the spatial domain remains the same throughout the entire run. The program can also be easily modified to reflect a shorter or longer time period and movement of the front farther offshore.
The two-dimensional larvae equations are approximated using semi-implicit finite difference scheme in which the density of larvae at each grid point is calculated from known values at the same time level and the previous time level. There are four different combinations of known values that can be used. The coefficients for the known grid points are presented in Table A1 for all four schemes. The order of these schemes alternates so that information flows in opposite directions and numerical error is reduced. For more details on the finite difference technique see Noye (1984:233-283). For the reflecting boundary condition (front), and the mixed boundary conditions (coast, north, and south), values at those boundaries were approximated with a forward-time, centered-space differencing scheme using the method of matching of truncation errors (Noye 1984:176-182). The adult equation was approximated using a forward time difference scheme.
The stability conditions for this scheme require that [s.sub.x] + [s.sub.y] [less than or equal to] 0.5, [c.sub.x] [less than or equal to] 2[s.sub.x], and [c.sub.y] [less than or equal to] 2[s.sub.y]., where [s.sub.x] = K[Delta]t/[([Delta]x).sup.2], [s.sub.y] = K[Delta]t/[([Delta]y).sup.2], [Lambda] = [Lambda][Delta]t/2, [c.sub.x] = u[Delta]t/[Delta]x, [c.sub.y] = v[Delta]t/[Delta]y. For the standard parameter set with a characteristic time of 3 h, characteristic length of 100 m, and a characteristic area of 1 [m.sup.2], these conditions force [Delta]t = 0.05 and [Delta]x = [Delta]y = 2 for advection levels [less than] 10.0 cm/s. For advection levels between 10 and 20 cm/s, these conditions force [Delta]t = 0.001 and [Delta]x = [Delta]y = 1.
The parameters are entered into the program by an input file called "data" and output is sent to the files "outadult" and "outlarvae." The program was run on a Hewlett-Packard Apollo Series 400 computer and an IBM Risc 6000 computer. For simulations with a moving front, the spatial domain of this program changes on a set time interval. The front is moved between 0.2 km from the coast and 30 km offshore with a period of 30 d. For simulations with a stationary front, this program is modified so that the spatial domain remains the same throughout the entire run. The program can also be easily modified to reflect a shorter or longer upwelling period and movement of the front farther offshore.
[TABULAR DATA FOR TABLE A1 OMITTED]
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|Author:||Alexander, Susan E.; Roughgarden, Jonathan|
|Date:||Aug 1, 1996|
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