On a periodic like behavior of delayed density- dependent branching

On a periodic like behavior of delayed density- dependent branching processes. Tetsuo Fujimagari1 . Periodically changing time series data for some animal ...
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AICME II abstracts

Ecological applications of theoretical and simulation ...

Ecological applications of theoretical and simulation ...

AICME II abstracts

where

On a periodic like behavior of delayed densitydependent branching processes 1

Tetsuo Fujimagari . Periodically changing time series data for some animal populations have been attractive and suggestive to many theoretical ecologists. Especially, the population dynamics of the vole in Hokkaido, Japan ([2]), are no doubt very important among them in that the survey was done in extensive areas of Hokkaido for nearly 30 years. Stimulated by those time series data, we have considered some probabilistic models which are intuitively simple and can describe at least qualitatively in a probabilistic way the population dynamics such as the vole. (n) {ξiKj

Assume that ; n = 0, 1, 2, . . . , K > 0 and i, j = 1, 2, . . .} is a sequence of independent nonnegative integer-valued random variables, and assume that the distribution of them : (n)

pk (i, K) = P (ξiKj = k)

(k = 0, 1, 2, . . .)

d−1 Z

Kn = K0 β Zn−1 +αZn−2 +···+α

n−d

(n = 0, 1, 2, . . .),

z, i, K0 and d are positive integers, 0 < β ≤ 1, α > 0, and Zn+1 = 0 whenever Zn = 0. We call the process {Zn ; n = 0, 1, 2, . . .} as a delayed (n) density-dependent branching process (DDBP). Then, ξiKj is the number of offsprings of the j- th mother of the n-th generation in the environment with a carrying capacity K when the total size of the n-th generation is i. {pk (i, K); k ≥ 0} is the offspring probability distribution. (n) The mean offspring number m(i, K) = E(ξiKj ) is assumed here to satisfy the condition: m(i, K) ≤ 1 for i > K , which relates to a top-down effect over a carrying capacity K. These processes DDBP have been studied numerically, which has shown some interesting aspects of their behavior. In carrying out their simulations, we assumed for the density-dependence specifically one of three ways. A periodic like behavior with a random period and a random amplitude is shown to emerge more likely about the DDBP with a longer delay and/or a deeper effect of a population regulation.

depends on i and K, satisfying the conditions: p0 (i, K) + p1 (i, K) < 1 and pk (i, K) 6= 1

(k ≥ 2).

Define the random process {Zn ; n = 0, 1, 2, . . .} recursively as Z0 = z, Zn = 0 (n = −1, . . . , −d), and Zn+1 =

i X

(n)

ξiKn j for Zn = i (i > 0),

j=1

References [1] Fujimagari, T., 1999, Controlled branching processes as a stochastic model of population growth, A. Vijayakumar & M. Sreenivasan, eds., Stochastic Processes and their Applications, Narosa, New Delhi, 31-38. [2] Saitoh, T., N. C. Stenseth & O. N. Bjørnstad, 1998, The population dynamics of the vole Clethrionomys rufocanus in Hokkaido, Japan, Res. Popul. Ecol., 40, 61-76.

1

Department of Mathematics, Kanazawa University, Kanazawa 920-1192, Japan (email: [email protected]).

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