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Fish mortality in reservoirs depends on a multitude of interactive factors. An important component of this is the predation rate which in turn is dependent on the number and behavior of predators, size of prey, genetic disposition of prey, disease, stress from dam passage, and degree of smoltification. Theory presented here approximates mortality processes in reservoirs while future versions of CRiSP will deal with these interactions. At the current time mortality involves the predation rate and travel time. These factors in turn depend on flow, predator density, and river temperature. An interaction between predator density and reservoir volume is provided as a switchable function to represent the impact of confining predators in a smaller volume when a reservoir is lowered. (Fig. 34).
The theoretical framework for describing reservoir mortality in the current model uses the time fish spend in a river segment and the segment rate of mortality. The basic equation describing the rate of mortality as a function of time is
(57)
where
- S = measure of smolt density in the river segment and can be taken as the total number in the segment
= mortality rate from all causes.
In the present model two causes of mortality are identified: predation and gas bubble disease. CRiSP.1 assumes the rates of each are independent and this is expressed by the equation
(58)
where
- Mp = mortality rate from predation with units of time-1
- Mn = mortality rate from nitrogen supersaturation with units of time-1
- S = number of smolts leaving reservoir per day (smolts reservoir -1)
- = combined mortality rate as used in eq (62).
Fish enter and leave river segments every day and spend differing amounts of time in a segment as described by the migration equations. Thus, on a given day the group of fish leaving a segment may have entered on different days and thus have different residence time in the segment. To describe the number of fish that survive a river segment on a daily basis CRiSP.1 solves eq (57) for each group, identified by when they entered the segment and when they exited. The solution is
(59)
where
- S0 (tj | ti) = potential number of fish that enter the segment on day ti and survive to leave the segment on day tj
- S (tj | ti) = actual number of fish that enter the segment on day ti and leave on day tj.
Applying an elementary property of integrals the integral is expressed
(60)
In general, the numerical form of the integral is
(61)
where
t = reservoir computational time increment.
The resulting equation for the number of fish passing through each river segment as a function of when it entered the segment is expressed
(62)
The input term S0 (tj | ti) expressing the potential number that exit on day tj given then entered the segment on day ti can be expressed
(63)
where
- N (ti) = number of fish that enter the river segment on day ti
- P (tj | ti) = probability that a fish entering on day ti survives to exit on day tj.
- This probability is defined by eq (50).
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Columbia River Salmon Passage Model CRiSP.1.5 Theory, Calibration & Validation Manual
Copyright © 1996, Columbia Basin Research. All rights reserved.
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