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Resumenes 98
The dynamic of logistic stochastic system it is modeled by mean of a
deterministic differential equation that describes the probability distribution
of the stochastic variable Nt (population size) for each t ≥ 0 .
Let ( N t ) a stochastic process of births and deaths [1] with
t≥0
rates λn and β n respectively, the change of the probability distribution
function (p.d.f) correspondent to population size n, when time t changes, it
is modeled by the followings differential equations:
dP
dt
( n,
t)
( n − 1,
t)
λ + P(
n + 1,
t)
β − P(
n,
t)(
λ + β )
= P
n−1
n+
1
dP
dt
( 1,
t)
dP
dt
= P
( 0,
t)
= P
( 1,
t)
β1
( 2,
t)
β − P(
1,
t)(
λ + β )
2
where P(n, t) is the probability that in the instant t ≥ 0 the population size
let n.
This p. d. f., it is studied when the population is in equilibrium state,
that is, in the instant when the births and deaths rates are equal. Therefore,
it has that:
with
dP
dt
( n,
t)
= 0
( n 1,
t)
λ + P(
n + 1,
t)
β = P(
n,
t)(
λ + )
P β
− n−
1
n+
1
The method presented by Daniels [4] for the p.d.f., called
Saddlepoint approximation is presented; after some transformations it is
obtained the truncated saddlepoint approximation f m ( x)
, [5. 6], resulting
the normal distribution for m = 2 and the Saddlepoint approximation for
determine the population distribution with m = 3. In [5] it is recommend its
use and they employ a approximation by truncation for the Saddlepoint
approximation of p.d.f.
Concretes examples applied for snail gardens, badgers and foxes
populations are given, using the approximation by truncation p.d.f.,
showing the goodness adjustment for different approximations effectuated
according the parameter involves and using approximate and exact
cumulants.
Referencias
[1] Bailey, N. T. 1964. “The Elements of Stochastic Processes”, Wiley, New York.
[2] Daniels, H. E. 1954. Saddlepoint approximations in statistics, Ann. Math.
Statist., 25: 631-650.
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