01-Mortality

01-Mortality - MORTALITY: a mathematical model for the...

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MORTALITY: a mathematical model for the death (survival) process Let N(t) denote the number of animals in some closed population. In this population there is no immigration, no emigration, and no reproduction. The only thing that can happen to change the number of animals is death. Suppose some fraction μ die during one time interval; the fraction surviving is S 1 μ - = N t 1 + ( ) S N t ( ) = 1 μ - ( 29 N t ( ) = Let μ 0.3 = ( i.e. S = 0.7 ) and N(0) = 1000. 0 1 2 3 4 5 200 400 600 800 1000 t N(t) t 0 1 2 3 4 5 = N t ( ) 1000.0 700.0 490.0 343.0 240.1 168.1 = N t 1 + ( ) N t ( ) 0.7 0.7 0.7 0.7 0.7 = Note that in reality there cannot be fractions of animals. This model is a continuous approximation to a process that is fundamentally discrete. Algebraically, we have the following: N 1 ( ) S N 0 ( ) = N 2 ( ) S N 1 ( ) = S S N 0 ( ) ( ) = N 3 ( ) S N 2 ( ) = S S S N 0 ( ) ( ) = Here t can be any positive number, not just an integer. and in general N t ( ) S t N 0 ( ) = In fisheries we usually write the equation for N(t) in an equivalent exponential form: exp(X) = e X e=2.71828. .. N t ( ) N 0 ( ) exp M - t ( ) = To see the equivalence, let's convert S t to exponential form. Observe that ln S t ( 29 t ln S ( ) = ln(X) = log e (X) FW 431/531 Copyright 2008 by David B. Sampson Mortality - Page 4
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This means that S t exp t ln S ( ) ( ) = e ln(X) = X The exponentiation function is the inverse of the logarithm function, and vice versa. Exponentiation undoes the logarithm; the logarithm undoes the exponentiation. Now substitute the right hand side into the equation N t ( ) S t N 0 ( ) = ==> S t N 0 ( ) N t ( ) = exp t ln S ( ) ( ) N 0 ( ) = N t ( ) N 0 ( ) exp ln 1 S - t = ln(1/X) = - ln(X) Let M = ln(1/S) ==> N t ( ) N 0 ( ) exp M - t ( ) = We're done! This coefficient M is usually described as the instantaneous rate of mortality. We have the following relationships: M ln 1 S = ln S ( ) - = ln 1 μ - ( 29 - = In the numerical example μ = 0.3 and M = 0.3567. .. In general, if μ is small (say, less than 0.2), then M 2245 μ . Here are some examples: ln 1 0.20 - ( ) - 0.2231 = ln 1 0.10 - ( ) - 0.1054 = ln 1 0.05 - ( ) - 0.0513 = One reason for working with instantaneous rates is that it is easy to change the time-step, by simple scaling. Implicit in the examples above is that each value of μ , M, and S has an associated time-step, such as "per year" or "per month". To convert from an annual instantaneous rate to a monthly instantaneous rate, we just divide by 12. So, for example, an annual instantaneous mortality rate of 1.2 per year is equivalent to a monthly instantaneous rate of 0.1 per month. Note that we cannot convert between the annual M and an approximate value for the annual μ ,using the rule M 2245 μ, but we can convert the monthly M to the approximate monthly μ , which is about 10% mortality per month. Review of Some Mathematics
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01-Mortality - MORTALITY: a mathematical model for the...

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