Directory UMM :Data Elmu:jurnal:M:Mathematical Biosciences:Vol166.Issue1.Jul2000:

Mathematical Biosciences 165 (2000) 97±114
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Dynamics of prey moving through a predator ®eld: a model of
migrating juvenile salmon
James H. Petersen a,*, Donald L. DeAngelis b
a

US Geological Survey, Biological Resources Division, Western Fishery Research Center, Columbia River Research
Laboratory, Cook, WA 98605, USA
b
US Geological Survey, Biological Resources Division, Florida Caribbean Science Center, Department of Biology,
University of Miami, Coral Gables, FL 33124, USA
Received 28 June 1999; received in revised form 10 March 2000; accepted 31 March 2000

Abstract
The migration of a patch of prey through a ®eld of relatively stationary predators is a situation that
occurs frequently in nature. Making quantitative predictions concerning such phenomena may be dicult,
however, because factors such as the number of the prey in the patch, the spatial length and velocity of the
patch, and the feeding rate and satiation of the predators all interact in a complex way. However, such
problems are of great practical importance in many management situations; e.g., calculating the mortality

of juvenile salmon (smolts) swimming down a river or reservoir containing many predators. Salmon smolts
often move downstream in patches short compared with the length of the reservoir. To take into account
the spatial dependence of the interaction, we used a spatially-explicit, individual-based modeling approach.
We found that the mortality of prey depends strongly on the number of prey in the patch, the downstream
velocity of prey in the patch, and the dispersion or spread of the patch in size through time. Some
counterintuitive phenomena are predicted, such as predators downstream capturing more prey per predator
than those upstream, even though the number of prey may be greatly depleted by the time the prey patch
reaches the downstream predators. Individual-based models may be necessary for complex spatial situations, such as salmonid migration, where processes such as schooling occur at ®ne scales and a€ect system
predictions. We compare some results to predictions from other salmonid models. Ó 2000 Elsevier Science
Inc. All rights reserved.
Keywords: Prey patchiness; Juvenile salmon; Management models; Migration; Columbia River; Northern pikeminnow

*

Corresponding author. Tel.: +1-509 538 2299; fax: +1-509 538 2843.
E-mail address: jim_petersen@usgs.gov (J.H. Petersen).

0025-5564/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 0 0 ) 0 0 0 1 7 - 1

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1. Introduction
Interactions between prey and their predators do not always take place in the ideal closed
homogeneous system implicitly assumed by simple models. Frequently a population of prey or
predators is moving in a de®nite direction through a ®eld of a relatively stationary population of
the other. A common situation is for patches of migrating prey to pass through a gauntlet of
predators. This type of predator±prey interaction presents diculties for modeling, as the interactions are transient and inherently inhomogeneous in space. Thus, it is dicult to apply standard
predator±prey models such as simple Lotka±Volterra equations.
However, the dynamics of a prey population migrating through an array of predators is important to understand for practical reasons. For example, the case of special interest in this paper
is migrating salmon. Stocks of Paci®c salmon (Oncorhynchus spp.) have disappeared from about
40% of their historical breeding range in Washington, Oregon, Idaho, and California, and 74% of
extant stocks face a ÔhighÕ or ÔmoderateÕ risk of extinction [1,2]. The decrease in salmon numbers
has been attributed to many factors ± a decline in the spawning habitat for adult salmon because
of mining, logging, and other development, increased pressure from commercial and sport ®sheries, construction of hydroelectric projects on rivers such as the Columbia and Lower Snake, and
ocean conditions [2]. However, one of the main sources of mortality to salmon is predation by
relatively stationary piscivorous ®sh, such as the northern pikeminnow (Ptychocheilus oregonensis;
previously called northern squaw®sh) during the downstream migration of juvenile salmon. The

conversion of rivers, such as the Columbia, into a series of reservoirs may have increased the
habitat and population size of such predators, and thus increased the predation-related mortality
of salmon.
Fishery managers in the Columbia River Basin use various ÔpassageÕ models to simulate predation on juvenile salmon in rivers. Current passage models assume that large reservoirs can be
treated as one or a few large homogeneous areas or partitions (modeling approaches are reviewed
elsewhere [3,4]). The number of juvenile salmon is represented by a single variable in each partition. Thus salmonids are assumed to be evenly distributed throughout the model partitions of
the reservoirs and average predation rates are applied throughout these large areas.
We believe that such models lack the spatial resolution to realistically represent predator±prey
interactions during the migration of salmon smolts. In real situations, predators will be exposed to
a transient patch of prey. If a patch of prey is spatially narrow, a given stationary predator may be
exposed to high densities of prey for only a few hours, even though the patch of prey may be in the
reservoir for days. As the prey population changes in total size and spatial distribution through
time, each predator will be exposed to a di€erent temporal pattern of prey density. Thus, the
assumptions of spatial ÔmixingÕ of predators and prey implicit in most models may not even be
approximately met. This, in combination with the fact that predators will be satiated by suciently high densities of prey (swamping), results in very complex dynamics that are poorly
represented by models that treat a reservoir as one or a few well-mixed ÔpoolsÕ. Models used by
®shery managers may need to include the density and narrowness of smolt patches, for example, if
such features a€ect total predation.
We propose that an e€ective way to model such a system is through the spatially-explicit, individual-based (SEIB) modeling approach. In this approach, each predator and prey is individually modeled. This type of model allows a great deal of ¯exibility in building realism into the

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99

models, particularly in capturing the degree of spatial resolution needed to represent interactions
in space. Further, even though computer simulations are necessary to derive results from SEIB
models, we demonstrate that important theoretical generalizations can be derived from such
models, in addition to their use in applied population ecology. For example, the model that we
used produces a non-linear `swamping' e€ect on predators at high prey density through ®rst
principles, whereas this e€ect is generally described with type II or type III functional response
models [5,6].
In this paper, we formulate a model for a patch of prey moving through a predator ®eld. We
ask how the various characteristics of the prey patch, such as number of prey in the patch, its
speed of movement, and its spatial size and con®guration, a€ect the total mortality on the prey
and the amount of prey caught by individual predators. Field data on Columbia and Snake River
salmon smolts and their main predator, the northern pikeminnow, were used to parameterize the
model. However, the model presented here is relatively simple. We were looking only for general
properties of the system at this stage.

2. Model development

We developed a simulation model for the movement of a population of prey through an area
occupied by a spatial distribution of predators and the predation interaction. The particular
application in mind is a river reach containing predators distributed at relatively ®xed locations
along the reach and feeding on juvenile salmon migrating downstream through the reach. Most
juvenile salmon enter reservoirs during brief periods during the day and we call such a group a
`patch'. Our main purpose is to calculate the expected feeding rates of the predators when exposed
to a moving patch of these prey and to calculate the e€ect of predation on the survivorship of the
prey. The theoretical river reach was con®gured as a one-dimensional series of contiguous ÔcellsÕ
with prey entering the river reach at the upriver end (Cell #1) and migrating downriver through
successive cells. This is intended to resemble a reservoir such as the John Day Reservoir on the
Columbia River, depicted in Fig. 1. The prey and predator populations were modeled on an
individual-by-individual, or individual-based approach. Predators were not allowed to move
between cells and there was no predator emigration or mortality, the time scale of the simulation
being assumed to be on the order of days.
The purpose of the model here was to predict the spatial pattern of predation along the reach.
The simulations described below were conducted with one predator per cell, but can be generalized to higher densities. For example, predictions of mortality of prey in a patch moving
through the reach should scale with the number of predators as long as the number of prey is kept
in the same proportion to the number of predators. We do not consider possible predator density
e€ects in this model, but will address this question in future work.
The simulation model was approximately con®gured to match northern pikeminnow predation

during summer on juvenile chinook salmon migrating through John Day Reservoir on the lower
Columbia River. John Day Reservoir has been the site of numerous predation studies [7±9] and it
has been used as a reference for basin-wide comparisons of pikeminnow predation [10]. These
studies were used to parameterize the model.

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Fig. 1. (a) John Day Reservoir on the Columbia River; and (b) model structure.

Northern pikeminnow and other predators occur in shallow, nearshore areas where they encounter juvenile salmonids [11]. Our model reach was 100 cells in length (Cell #1 to Cell #100),
the cells being identical and each 1 km long, 20 m wide, and 5 m deep. No internal structure was
assumed within the individual cells; that is, prey and predators were assumed to be Ôwell-mixedÕ
within each cell. The prey had no spatial refuge in the model and all prey mortality was through
predation. We assumed that water velocity, Vw (m/s) was the same in all cells along the reach and
was calculated from data on ¯ow F (m3 /s) into the cell and cell size: Vw ˆ F /(Cell Width ´ Cell
Depth).
2.1. Migration of prey
We simulated single patches of prey moving downstream through the predator ®eld at hourly

timesteps. This agrees with the observation that juvenile salmon in the Columbia River pass dams
and enter the up-river end of reservoirs as patches, primarily during evening and night hours
(Fig. 2, [12±14]). Simulated prey entered the upstream end of the model reach in a diel pulse (or
moving patch) and migrated downriver through each successive cell. The percent of smolts PH
entering each hour H (0±23) of the day into the reach (based on [12]) was modeled through a given
day in the form shown in Fig. 2(a).
Longitudinal individual prey position Posi each hour i of the simulation was
Posi ˆ Posiÿ1 ‡ Vprey …Vsd …0:5 ÿ URN††;

…1†

where Posiÿ1 is the smolt position last hour, Vprey (km/h) is a predicted rate of migration, Vsd was
the standard deviation of Vprey , and URN is a uniform pseudorandom number (0 < URN < 1). At
each time step the prey was in one of the hundred spatial cells. In each step prey could remain
within their current cell or move to a cell further downriver, but were not allowed to migrate to an
upriver cell. We used an analysis of individual fall chinook salmon in the Snake River to specify

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101

Fig. 2. (a) Percentage of smolts that enter the cell farthest upstream as a function of the hour of day. (b) Probability of
predator attacking a smolt as a function of fullness of the predator's stomach.

prey velocity, Vprey , as a function of smolt length, Lprey , date, and river ¯ow, F ([15]; see
Appendix A; Table 1 for parameter values). At the start of a simulation, individual salmon smolts
were randomly assigned lengths between Lmin (90 mm) and Lmax (120 mm). The distribution of ®sh
lengths is an important assumption, because a ®sh's mean downstream velocity is a function of its
length, and thus a patch of smolts will tend to spread out spatially through time.
2.2. Feeding behavior of the predator
The representation of the predator was based on information for the northern pikeminnow,
which, similar to other predators [16,17], capture a few smolts during brief feeding bouts [18]. If
smolts were present in a cell, predation on these prey was modeled by estimating the number of

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Table 1
Constants and parameters for nominal simulations

Parameter or constant (units)
3

F (m /s)
Lmin (mm)
Lmax (mm)
Wa
Wb
b0
b1
b2
b3
Vsd (km/day)
Vpred (km/day)
c1 (mmÿ1 )
c2 (mm)
Rdist (m)
Wpred (g)
a1
a2

a3
a4

Description [source]

Value

River ¯ow [22]
Minimum prey length [9]
Maximum prey length [9]
Prey mass parameter [9]
Prey mass parameter [9]
Prey migration speed parameter [15]
Prey migration speed parameter [15]
Prey migration speed parameter [15]
Prey migration speed parameter [15]
Standard deviation of prey swimming speed [15]
Rate of predator swimming speed [27,47]
Capture success parameter [estimate]
Capture success parameter [estimate]

Reaction distance of predator [24,54]
Predator weight [7,27]
Predator evacuation rate parameter [53]
Predator evacuation rate parameter [53]
Predator evacuation rate parameter [53]
Predator evacuation rate parameter [53]

150
90
120
8.91 ´ 10ÿ6
3.031
)15.43
0.014
0.053
0.076
3.73
2.4
0.01
50
2.0
800
0.001
0.39
1.57
0.26

prey encountered by a predator, determining randomly which speci®c smolts were encountered,
deciding if an encountered smolt was attacked, and deciding if the attacks were successful. The
rate at which salmon were encountered by a predator, e (prey/h), was
e ˆ Svol Dprey ;

…2†

where Svol (m3 /h) is the volume of water searched by a predator and Dprey (prey/m3 ) is the average
density of prey in the cell at a given time period (hour). Details of the values of Svol used in the
model are discussed in the Appendix A.
A stochastic process determined the realized number of smolts encountered during an hour by a
particular predator. For a mean predicted encounter rate in a cell, e, the probability of encountering k smolts was computed as a Poisson process [17,18]. The realized smolt encounters
were randomly distributed among the available smolts currently in a cell so that each smolt had an
equal chance of being encountered by a predator.
The probability of attack following an encounter, Aprob , was assumed to be related to satiation
of the predator, which can be monitored by relative gut fullness [19]. We assumed that a northern
pikeminnow whose gut was empty or nearly empty would always attack an encountered salmon
(Aprob ˆ 1:0), whereas a predator whose gut was relatively full would be satiated and would not
attack (Aprob ˆ 0:0; [20,21]). Individuals with full guts may also be incapable of consuming a
relatively large prey item such as a juvenile salmonid. Aprob decreases linearly, as relative gut
fullness increases (see Fig. 2(b)). The probability of consumption of a smolt by the predator, Cprob ,
was estimated by multiplying the attack probability Aprob by a constant, CAPprob , describing the
probability of an attack being successful (see Appendix A). Consumption events were thus
probabilistic and in a simulation a prey individual was captured if a number chosen by a

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103

pseudorandom number generator was less than Cprob . Captured smolts were removed from the
migrating patch.
A captured smolt had a mass Wprey , calculated from its assigned length (see Appendix A). At
the end of each hour i, the mass of food in a predator's gut, STcont;i (g), was updated:
STi ˆ STiÿ1 ‡ I ÿ E;

…3†

where STcont ,iÿ1 (g) is the mass remaining from the previous hour, I (g) is the prey mass ingested,
and E (g) is the mass evacuated during one hour. Calculation of gut evacuation per hour is described in the Appendix A.
We assumed that predators preferred juvenile salmon exclusively when they were present in a
cell, [18, unpublished analyses], but alternative prey were eaten when salmonids were not present.
When salmonids were present in a cell, the prey mass ingested during an hour, I, was the sum of
individual smolts eaten. When salmon were absent from a particular cell, predators consumed
alternative prey only 80% of the time, since northern pikeminnow often have empty guts [9]. If a
predator consumed alternative prey during an hour, the mass consumed was a random fraction of
the proportion of the gut that was empty.
2.3. Simulations
We conducted two simulation experiments. The ®rst series of simulations was conducted to
demonstrate the response of individual predators to a `patch' of migrating prey. For all simulations, `patch' refers to a single group of juvenile salmon that migrate through a river reach. In the
impounded portion of the Columbia River, the largest proportion of diel passage of juvenile
salmon at a dam generally occurs over a few hours during the night [12±14,18]. The second set of
simulations examined how patch dynamics in¯uenced mortality rates of a patch of smolts.
Simulations were conducted with parameters that were assumed to be representative of the
pikeminnow±smolt system (Table 1).

3. Results
3.1. Individual predator feeding behavior
We simulated a pikeminnow's response to a brief pulse (patch) of smolts to examine how
capture rate varied with time and prey density for a bout-feeding predator. First, we followed the
response of the predator in Cell #1, the cell farthest upstream, to rapidly changing smolt density.
No smolts entered Cell #1 for 83 h, 130 smolts were present in the cell at hour 84, and 870 smolts
were present at hour 85. During hour 84, smolt density was relatively low (1.3 smolts per 2000 m3 ),
but pikeminnow predation rate was highest (1.8 smolts captured per h; Fig. 3) compared to other
hours. During hour 85, smolt density increased to 9.6 smolts per 2000 m3 , but the average
predation rate declined since the predator in the cell was satiated, having captured multiple prey
(maximum ˆ 3; 5 replicate runs) when ®rst exposed to smolts during hour 84. Despite the fact that
more prey were encountered when density was highest in hour 85, few attacks were initiated
because of predator satiation. Prey density declined gradually over hours 86±96 as prey migrated

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J.H. Petersen, D.L. DeAngelis / Mathematical Biosciences 165 (2000) 97±114

Fig. 3. Capture rate and prey density during the passage of a patch of juvenile salmon through Cell #1 (average of ®ve
replicate simulations). The patch was created with 130 prey at hour 84 and 870 prey at hour 85.

out of the cell, and the predation rate also declined (Fig. 3). Using smolt density alone to predict
predation could give an erroneous estimate of the average capture rate of smolts per hour, without
some information about the recent feeding history of the predator or its gut fullness. For example,
predation rate during hour 90 in the simulation (Fig. 3) was one-ninth the rate during hour 84 (0.2
vs. 1.8 smolts per h, respectively), but the prey density was not signi®cantly di€erent between these
two hours (approximately 1.3 prey per 2000 m3 ).
Next, an individual predator (800 g; maximum gut capacity approximately 45 g) located in Cell
#50 (midpoint of the reservoir) was followed before, during, and after a smolt patch passed
through the cell (Fig. 4). At this stage of the smolts' passage through the reservoir, the prey

Fig. 4. Gut fullness over time for an individual predator in cell #50 in a 100-cell simulation, including passage of a
patch of juvenile salmon through the cell (histogram). A patch of 2000 smolts entered the reach (Cell #1) during the ®rst
24 h of the simulation. Smolts were captured by the simulated predator during hours 116, 129, 137, 144, 150, and 153
(downward pointing arrows).

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105

numbers have decreased and dispersion has increased the spatial spread of the patch. Before
smolts arrived at the cell, the predator fed on alternate prey and the mass of food in the gut varied
from near zero to about 26 g. Gut fullness averaged 23% (range 0±58%) before the smolt patch
arrived in Cell #50 (Fig. 4). Smolts began to arrive in Cell #50 during hour 96 of the simulation
and smolt density increased to a maximum of 128 smolts per cell during hour 133 (Fig. 4). Density
declined after hour 135 as smolts migrated farther downriver and out of Cell #50. The predator in
this cell ®rst captured a prey individual during hour 116 and captured a total of six smolts during
passage of the prey patch. As smolts were captured, the predator's average gut fullness increased,
reaching a maximum of 42 g (92% of maximum capacity). While smolts were present in the cell,
the predator's average gut fullness was 54% (range 14±92%). After passage of the patch, the
predator's gut fullness declined, averaging 16% during the last 200 h of the simulation.
To better understand the e€ects of patch size, we investigated how pikeminnow in di€erent
parts of the reach responded to smolt patches of variable initial size upstream. We simulated
single prey patches starting at the upriver end with di€erent numbers of smolts and we computed
the cumulative number of prey consumed by a predator in the ®rst (Cell #1) and the penultimate
cell (Cell #99); patterns in other cells were intermediate between these extremes. In Cell #1, cumulative predation increased rapidly as the initial upstream patch size increased, reaching a
maximum of about seven smolts consumed by the predator in this cell (Fig. 5(a)). The pattern in
Cell #99 was very di€erent ± cumulative predation stayed low until patch size reached about 600
smolts, increased rapidly as patch size went from 700 to about 1700 smolts, and peaked at about
15 prey consumed when the initial upstream patch size was 4000 (Fig. 5(a)).
Consumption di€erences between individual predators along the reservoir were due to the
spread or dispersion of smolt patches as they moved through the reach. Dispersion of patches
caused the temporal sequence of smolt densities to vary for predators in di€erent sections of the
reach. If the patch entering the reservoir was small ( 2000 smolts), per
capita predation rate was high in both the upriver and downriver cells but more smolts were
consumed by individual predators at the downriver location. There are two reasons for this. First,
predator satiation occurs, so that predation per unit time is limited for a given predator, even at
very high prey densities. Thus a predator at an upstream cell can take only limited advantage of
the extremely high prey densities in its cell. Second, the spread of the patch during passage meant
that downstream cells were exposed to prey over longer time periods than upstream cells. Hence,
total number of prey consumed was higher in Cell #99 than in Cell #1.

3.2. Patch dynamics and reservoir-wide smolt mortality
Per-patch mortality during passage through the reservoir was the cumulative result of consumption by all predators, each of which had a complex feeding response (see Fig. 5(a)). The total
number of smolts consumed throughout the simulated reach increased as patch size increased
(Fig. 5(b)). Percent mortality was highest for small patches (>97% mortality for patches of less

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J.H. Petersen, D.L. DeAngelis / Mathematical Biosciences 165 (2000) 97±114

Fig. 5. Number of juvenile salmon eaten by predators as a function of patch size (average of ®ve replicate simulations).
(a) shows the cumulative number of smolts eaten by an individual predator in Cell #1 (squares) and a predator in Cell
#99 (triangles). (b) shows the total number of smolts consumed by 100 predators during passage of the patch through
the reach (dots) and the percent mortality for the patch (solid line).

than 600 smolts). Per-prey probability of mortality declined rapidly as patch size increased,
especially for patches with more than 2000 smolts (Fig. 5(b)). As patch size increased some
pikeminnow became satiated and a higher proportion of smolts could thus pass through cells
while predators were satiated. As the size of the smolt patches became very large (about 3000
smolts), the percent mortality appeared to decline as 1/(patch size) (Fig. 5(b)).
Smolt patches consist of many individuals that may migrate at di€erent rates, thus causing the
patch to spread during its downriver movement. The standard deviation of smolt migration rate,
Vsd , was varied to cause smolt patches to spread at di€erent rates. Increasing Vsd produced a
greater variation in migration rates among individual smolts, thus causing patches to spread more
quickly; smaller values for Vsd made patches more cohesive during passage through the reach.
For small patches of 200 smolts, changes in the rate of patch spread had little e€ect on the
overall mortality through the reservoir (Fig. 6; mortality > 97%). However, mortality varied
greatly with the rate of patch spread when patches had 1000 or 2000 smolts (Fig. 6). Cohesive
patches (e.g. Vsd < 3 km/day) experienced a lower mortality rate than patches that were spatially

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107

Fig. 6. Mortality (% dead; average of ®ve replicate simulations) for patches of 200, 1000, and 2000 smolts migrating
through a 100-cell reach as a function of the rate of patch spread. The rate of patch spread is the standard deviation of
individual smolt migration rate (see text); higher rates mean that the patch spreads more rapidly or has less cohesion.
The nominal rate of patch spread is indicated by the arrow.

more dispersed (e.g. Vsd > 5 km/h), since predators throughout the reach were satiated by the
leading edge of the cohesive patch and more smolts escaped predation during passage (see Fig. 5).
As the rate of patch dispersion increased, mortality increased rapidly and approached an asymptote that appeared to be dependent on the initial patch size (Fig. 6). The nominal rate of patch
spread used was 3.73 km/day ([18]; 1993 data), although both larger and smaller values were
observed [18] during other years (5.95 km/day, 1992 data; 2.61 km/day, 1991 data). For intermediate (1000 smolts) and large (2000 smolts) patches, the rate of patch mortality through the
reach was quite sensitive to small changes from the nominal value (Fig. 6).
The e€ects of ¯ow and patch size on smolt mortality were simulated by varying ¯ow, F, from
850 to 17,000 m3 /s and patch size from 200 to 4000 smolts per patch. Typical discharge through
John Day Reservoir during July ranges from about 1500 to 5000 m3 /s, but ¯ows >34 000 m3 /s
have been recorded at the Dalles Dam [22]. We assumed that ¯ow in¯uenced only the migration
rate of juvenile salmon in the model, and not other aspects of the predator±prey interaction such
as capture success.
The highest percent mortality for smolt patches occurred with the lowest ¯ows and the smallest
smolt patches (Fig. 7). When smolt patches entering the reservoir contained 200 smolts, mortality
decreased from >99±60% as ¯ow increased from 850 to 17 000 m3 /s (Fig. 7). As patch size increased, however, the in¯uence of ¯ow diminished. For larger patches, the highest mortality
occurred at intermediate ¯ows. When a patch contained 4000 smolts, mortality increased from
28% at 850 m3 /s to 45% at 11 000 m3 /s, but then decreased to 39% at the highest ¯ow of 17 000 m3 /s.
These results have practical value, suggesting that arti®cial augmentation of river ¯ow rate, and
thus smolt migration velocity, may provide the highest bene®t when smolt patches are relatively

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J.H. Petersen, D.L. DeAngelis / Mathematical Biosciences 165 (2000) 97±114

Fig. 7. Mortality (% dead) as a function of river ¯ow (´ 1000 m3 /s) and patch size (number of smolts) during passage
through the simulated reach.

small; when smolt patches are large, increasing the volume of ¯ow produced relatively small
changes in the overall mortality.

4. Discussion
Our model results are interesting in light of empirical data on salmon smolts. Field studies and
laboratory observations have shown that northern pikeminnow respond rapidly to transient
patches of juvenile salmon by often capturing several smolts during a brief feeding bout [18,23±
25]. Predation rates in our modeling studies were sensitive to the dynamics of smolt patches and
interactions between patch characteristics and river conditions such as ¯ow. The size of patches
entering the reservoir caused the mortality per patch to range from about 30% to 100%. Small
patches stimulated feeding by individual predators and smolt mortality was very high. Mortality
per unit smolt declined rapidly as patch size increased since large patches satiated predators
throughout much of the simulated reservoir. High rates of patch dispersion resulted in predators
being exposed to densities of prey below satiation levels over long periods of time. Thus prey
patches with high prey numbers and high dispersion tended to su€er greater overall mortality than
patches with high prey numbers but smaller rates of dispersion (Fig. 6).
Our model was developed to explore the general importance of prey patchiness, bout feeding,
and predator satiation for a migrating prey in an idealized river reach. On a complexity scale we
consider our model to lie somewhere between a highly simpli®ed analytic model and a detailed
model designed to simulate a speci®c system. We left out many known details of a real reservoir
like John Day Reservoir, such as spatial variation in the density of predators. On the other hand,
we included some realistic detail of predator satiation, as this has important qualitative e€ects on
the interaction. Although we were not attempting to simulate an actual system, our model produced gut fullness patterns, capture sequences of smolts, and rates of predation that corresponded
quite closely to ®eld measurements [9,18,26,27].
Northern pikeminnow consumption rate of smolts in the simulations was a complex relation
between prey density, bout feeding behavior by predators on smolts, and the recent foraging

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109

success of the predator. Predators in di€erent parts of the simulated reach had di€erent average
rates of predation on salmon and consumed di€erent total numbers of salmon prey from the same
patch. Predators in the upriver section of the reach encountered a compact patch of smolts,
whereas predators further downriver encountered prey in the same patch that were more dispersed. The patch was also somewhat smaller when it reached downriver predators because of the
removal of smolts by successful upriver predators. Upriver versus downriver predators encountered quite di€erent smolt density patterns during passage of the same smolt patch.
Appropriate mechanisms and scales are critical in simulation models used to predict the
growth, mortality, or distribution of species in spatially-complex ecosystems. Model outputs are
often sensitive to prey patchiness, the mobility of predators and prey, and the scale of the simulation [28,29]. In large aquatic ecosystems, consideration of ®ne-scale mechanisms and processes
can signi®cantly change system-wide predictions about recruitment, predator growth, ecosystem
production, and prey mortality [30±32]. Including adequate mechanisms at the appropriate scales
for time and space is critically important in models that are used for making management decisions for rare or valuable species like salmon [33,34].
A review of management models for juvenile salmon in the Columbia River noted the critical
importance of data and mechanisms for estimating reservoir mortality [4]. Existing management
models include predation as a major source of smolt mortality [35,36], or mortality is modeled as a
constant rate per mile that varies with river ¯ow [4]. Mid-reservoir areas in current management
models are assumed to be homogeneous, smolt density generally changes only daily, and migrating smolts and predators are assumed to be mixed throughout mid-reservoir zones that are
often >50 km long. The local smolt patchiness that often exists cannot be accommodated within
such large partitions, which tend to smooth the temporal variability of prey density. Prey patch
dynamics may have to be combined with predator foraging behavior to model predation in
systems where some prey types form transient patches.
The results of the model simulations could have important implications for ®shery managers in
the Columbia River basin. For example, hatchery managers may consider how their release
procedures create batches of salmonids in the mainstream rivers when predation is known to be
an important source of mortality. Our results suggest that the total mortality for a release group
cannot be simply computed as a predation rate expanded across a population of predators.
Additional work on patch dynamics may enable us to better quantify the sources of smolt
mortality in dwindling salmon populations and recommend better management actions.
The model that we developed had an intermediate level of complexity to allow exploration of
smolt patchiness only. Additional details of smolt and predator behavior could be added to this
structure to explore questions about such factors as predator density and onshore-o€shore
movements of juvenile salmon. We used one predator per cell and assumed no predator-to-predator interactions. Northern pikeminnow have been shown to aggregate near hatchery releases of
juvenile salmonids [25,37,38] and there may be competition between predators as density increases
(J.H. Petersen, unpublished analyses). There is no evidence that predators track a patch of juvenile salmon very far down the river and this seems unlikely since most energy is obtained from
benthic-oriented organisms [7,11,26,27]. We did not allow a spatial refuge for juvenile salmon in
our model and the results would be most applicable to salmonid stocks that are rearing and
migrating nearshore. Several of the salmonids that migrate through the Columbia River gradually
move o€shore as they grow and approach smolti®cation [39], likely reducing the chance that they

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J.H. Petersen, D.L. DeAngelis / Mathematical Biosciences 165 (2000) 97±114

will encounter shore-line oriented predators such as northern pikeminnow. A spatial refuge could
be modeled by including age-or size-related o€shore movements of juvenile salmon.
Our results were qualitatively di€erent from some recent model studies that included predation
e€ects on migrating juvenile salmon in rivers or streams. Jager et al. [40], for example, developed
an individual-based and spatially-explicit model (called ORCM) for fall chinook salmon and
compared simulation results with data from the Tuolumne River, California. Several aspects of
their model were similar to the model that we developed, including predation by ®sh and
movement of smolt. A major di€erence between these models was that smolt moved as `patches'
through predator-occupied cells in our model, whereas smolt movement in ORCM did not begin
as a structured patch. A general conclusion of Jager et al. was that temperature was more important than predation as a source of mortality on juvenile salmonids, although during some
years predation-related loss was high. Our model was much simpler than that of Jager et al. and
did not consider temperature as a source of mortality; however, the sensitivity of smolt mortality
to patch size and rate of spread in our model suggests that the dynamics of smolt movements may
be important considerations in these types of models.
The high rates of predation-related mortality that we observed for small patches of juvenile
salmonids could have some serious implications as salmon stocks decline. Several stocks of Paci®c
salmon are currently considered threatened or endangered, and numerous extinctions have occurred [1,2]. With too few adult salmon spawning within a reach, the numbers of juveniles produced at the spawning bed may be below some critical size (patch size) and few or no smolts will
survive to the ocean. This can be thought of as a depensatory type of mortality, because the
probability of predation per individual smolt increases as the number of smolt decreases. The
chance of extinction may remain high even with increasing numbers of adult salmon if no juvenile
salmonids survive the predator gauntlet. Fall chinook salmon within the Lower Snake River, for
example, have been listed as threatened and predation is likely the main source of mortality [11].
Additional work at a speci®c location would be needed on prey density, predator density, and the
rate of prey mortality to verify such a depensatory e€ect.
New technologies and methods are beginning to provide data at the appropriate scales of space
and time to study the movements of individual predators and prey and to characterize the dynamics of salmon patches. Hydroacoustics is being used to describe the distribution of juvenile
salmonids in the Columbia River Basin [39,41] and in other large aquatic systems [42±44]. Radiotagging of smolts (>100 mm) and pikeminnow has become feasible and movement patterns of
individual ®sh are being studied [45±47]. Analytical methods are being developed to apply these
new types of data on movement and patchiness in formulating and testing foraging models
[48±51].

Acknowledgements
We appreciate critical reviews and helpful comments from Michele Adams, Tony Ives, Tom
Poe, Rip Shively, and two anonymous reviewers. J.H.P. was supported by the Bonneville Power
Administration through contracts administered by Bill Maslen. D.DeA.Õs part in this work was
supported in signi®cant part by the Department of Interior's Critical Ecosystem Studies Initiative
and in part by the USGSÕs Florida Caribbean Science Center.

J.H. Petersen, D.L. DeAngelis / Mathematical Biosciences 165 (2000) 97±114

111

Appendix A
In this section, several of the functions used in the prey migration/predation model are described in more detail.
A.1. Speci®cation of smolt swimming velocity
Smolt downstream swimming velocity, Vprey (always >0), was represented as
Vprey ˆ …b0 ‡ b1 Lprey ‡ b2 F ‡ b3 DATE†=24;
where Lprey is the fork length of smolt (mm), F the water ¯ow (m3 /s), DATE the Julian day and b0 ,
b1 , b2 , and b3 are the constants parameters ([15], Table 1).
A.2. Predator search volume
Gerritsen and Strickler [52] derived equations for predators searching for mobile prey in a
three-dimensional space that we used in our simulations. Predators were assumed to have a threedimensional visual ®eld that extended 90° laterally from each eye and included prey above, below,
and forward. The radius of this half sphere was assumed to be the horizontal reaction distance
Rdist (m) of the predator. The volume searched Svol (m3 /h) by a predator was
2=3
2
2
†=Vprey †Š;
‡ 3Vprey
Svol ˆ 0:5‰p…Rdist
†……Vpred

where Vpred is the predator swimming speed (km/h) and Vprey is the average migration speed of
smolts (km/h).
A.3. Probability of a predator attack being successful
The probability of a predator attack on a smolt, CAPprob , given that it occurs, being successful,
is
CAPprob ˆ expfÿc1 …Lprey ÿ c2 †g;
where
Lprey ˆ prey length …mm†; and c1 and c2 are constants (see Table 1).
A.4. Prey mass as a function of standard length
Prey weight Wprey (g) is given as a function of prey length Lprey (mm) by the empirical expression,
b
;
Wprey ˆ Wa LWprey

where Wa and Wb are parameters [9].

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J.H. Petersen, D.L. DeAngelis / Mathematical Biosciences 165 (2000) 97±114

A.5. Predator gut evacuation rate
The predator's rate of evacuation of gut contents E (g/h) is given by the empirical formula for
the northern pikeminnow [53],
a4
E ˆ a1 fI a2 T a3 Wpred
g;

where I is the original mass ingested (g), T is the temperature (°C), Wpred is the predator weight (g)
and a1, a2, a3, a4 are the parameters [53].

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