05-Catch1

# 05-Catch1 - THE CATCH PROCESS Usually we cannot harvest all...

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THE CATCH PROCESS Usually we cannot harvest all the fish from a population all at the same time. Instead, we catch fish over some period of time and gradually diminish the size of the population. Now we will explore a model for this catch process. Suppose we have a cohort of fish in isolation, and in the absence of fishing 80% survive each time period. N t 1 + 0.80 N t = Now suppose we start harvesting fish with fishing gear that removes 50% of the fish each time period. The percentage of fish that survive both forms of mortality is given by the product of the two survival fractions, 80% x 50% = 40% . This follows from a basic rule of probability. If A and B are two independent events , then the probability that both events occur is equal to the product of the individual probabilities, P( A and B ) = P( A ) x P( B ) . We cannot apply this rule to the mortality probabilities (as opposed to the survival probabilities) because an individual can only die once. Death by natural causes and death by fishing are not independent events. Below is a hypothetical example showing how the abundance of a cohort is affected by different mortality sources operating individually and then operating together. In the first case the survival rate is from natural mortality only, S M 0.80 = . In the second case the survival rate is from fishing mortality only, S F 0.50 = . In the third case the survival rate is from both natural mortality and fishing mortality. The initial abundance for the cohort is N 0 1000 = . It is impossible for there to be no natural mortality, but this is a hypothetical example. Deaths, both sources M only F only Both sources N_SM t 1000 800 640 512 = N_SF t 1000 500 250 125 = N_SMF t 1000 400 160 64 = N 0 N_SMF t - 0 600 840 936 = t 0 1 2 3 = Consider the numbers of fish dying in the different scenarios. During 1st period During 2nd period The two mortality processes "compete" for fish. (Deaths if M only) + (Deaths if F only) (Deaths if M and F). Deaths if M only: Deaths if F only: Deaths if M and F: 200 500 600 360 750 840 FW431/531 Copyright 2008 by David B. Sampson Catch1 - Page 33

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We know how many fish die. It is simply N_Deaths N 0 N_Survivors - = . But how do we determine how many of these deaths represent catches, and how many represent natural deaths? One way to answer this is by working with the instantaneous mortality rates. Natural Survival
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05-Catch1 - THE CATCH PROCESS Usually we cannot harvest all...

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