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TRANSIENT PROBABILITIES FOR A SIMPLE BIRTH-DEATH-IMMIGRATION PROCESS
UNDER THE INFLUENCE OF TOTAL CATASTROPHES
RANDALL J. SWIFT (Received 27 August 2000)
Abstract.The transient probabilities for a simple birth-death-immigration process are considered. Catastrophes occur at a constant rate, and when they occur, reduce the pop- ulation to size zero.
2000 Mathematics Subject Classification. Primary 60J80.
1. Introduction. In this note, a simple birth-death-immigration process is consid- ered, which is influenced by total catastrophes which, when they occur, reduce the population size to zero. Population processes under the influence of various types of catastrophes have been studied by Bartoszynski et al. [2], Brockwell et al. [3], and Kyriakidis [4,5].
The process is formulated by lettingN(t)represent the size of the population at timetand
Pn(t)=P
N(t)=n|N(0)=0
. (1.1)
As in the simple birth-death process, births and deaths occur proportional to the population size with a birth rateλ >0 and a death rateµ >0. Immigration will occur independent of population size with rateα >0. Further, the occurrence of a catastro- phe is also independent of population size and will occur at a rateγ >0. Thus, the process can be described by the following transition rates:
Transition Rate
i→i+1 λi+α (i≥0)
i→i−1 µi, (i≥1)
i →0 γ (i≥1)
The special case ofγ=1 was recently considered by Kyriakidis [4], whoobtained the stationary probabilities for this process. The transient probabilities of a simple immigration-catastrophe process, whereλ=0 andµ=0, was obtained by Swift [6]. In the next section, the transient probabilities for the general case are derived.
690 RANDALL J. SWIFT
2. The transient probabilities. The standard argument using the forward Kolmogorov equations shows thatPn(t)satisfies
Pn(t)=(n+1)µPn+1(t)+
(n−1)λ+α
Pn−1(t)−
n(µ+λ)+α+γ
Pn(t), (2.1) P0(t)=γ∞
i=1
Pi(t)+µP1(t)−αP0(t). (2.2)
Now ∞
i=1
Pi(t)=1−P0(t) (2.3)
sothat
P0(t)=γ+µP1(t)−(α+γ)P0(t). (2.4) Letting
ψ(s,t)= ∞ k=0
Pk(t)sk (2.5)
be the probability generating function (PGF) for the system, it follows from the stan- dard generating function method, thatψ(s,t)satisfies the partial differential equation
∂ψ(s,t)
∂t =(λs−µ)(s−1)∂ψ(s,t)
∂s +α(s−1)ψ(s,t)+γ
1−ψ(s,t)
. (2.6) Since we are considering a process with immigrations, we can assume, for the sake of simplicity, the initial conditionP0(0)=1. This givesψ(s,0)=1.
As in most birth-death processes, the solution of the partial differential equation (2.6) fo rψ(·,·)depends upon the values of the parametersλandµ. The PGFψ(·,·) is given below for three important cases. The method of solution of (2.6) is standard and indeed, it is interesting to note that these solutions can also be obtained using a computer algebra system such as Mathematica.
Case2.1(λ=µandλ=0). The PDE (2.6) has solution ψ(s,t)=
1 (α+γ)λ−αµ
×
γλ
(1−s)λ λ−µ
γ/(λ−µ)
2F1
αλ+γλ−αµ
λ2−λµ ,λ−µ+γ
λ−µ ,(λ+α)(λ−µ)+γλ λ(λ−µ) ,λs−µ
λ−µ
+(s−1)γ/(λ−µ)(λs−µ)−α/λ+γ/(µ−λ) (µ−λ)(λs−µ) µ−λs+λ(s−1)e(λ−µ)t
α/λ+γ/(λ−µ)
×
(µ−λ)(s−1) λ(s−1)+(µ−λs)e−(λ−µ)t
−γ/(λ−µ)
×
(α+γ)λ−αµ
−γλ
λ
λ(s−1)+(µ−λs)e−(λ−µ)t γ/(λ−µ)
×2F1
αλ+γλ−αµ
λ2−λµ ,λ−µ+γ
λ−µ ,(λ+α)(λ−µ)+γλ
λ(λ−µ) ,(µ−λs)e−(λ−µ)t
, (2.7)
where2F1(a,b;c,z)is the hypergeometric function defined by
2F1(a,b;c,z)= ∞ k=0
(a)k(b)k
(c)k
zk
k! (2.8)
(cf. Agarwal [1]).
Case2.2(λ=µandλ=0). In this case, the PDE (2.6) has solution ψ(s,t)=γeγ/λ(1−s)
λ(1−s) ∞
1 z−α/λe−γz/λ(1−s)dz− e−γt λ(1−s)
1 1+(1−s)tλ
α/λ
×
λ(s−1)+γ
1+γλt(1−s)∞
1 z−α/λe−γ(t+1/λ(1−s))zdz
.
(2.9)
Case2.3(λ=0 andµ=0). In this case, the PGF is given as
ψ(s,t)= 1 µ
e−αs/µ
γeα/µ ∞
1 z−(1+γ/µ)e−α(1−s)z/µdz +e−γt
µeα(1−(1−s)eµt)/µ−γeα/µ ∞
1 z−(1+γ/µ)e−α(1−s)eµt/µdz
.
(2.10) Using the PGF, the probability of extinction,P0(t), can be obtained in each of the above cases.
Case2.1(λ=µandλ=0).
P0(t)=ψ(0,t)
= 1
(α+γ)λ−αµ
λ−µ λe(λ−µ)t−µ
α/λ+γ/(λ−µ) µ−λ λ−µe(−λ+µ)t
γ/(µ−λ)
×
αµ−(α+γ)λ+γλ
λ λ−µe(−λ+µ)t
γ/(λ−µ)
×2F1
α(λ−µ)+λγ
λ(λ−µ) ,λ−µ+γ
λ−µ ,(λ+α)(λ−µ)+γλ
λ(λ−µ) , µ
µ−λe(λ−µ)t
+γλ λ
λ−µ γ/(λ−µ)
×2F1
α(λ−µ)+λγ
λ(λ−µ) ,λ−µ+γ
λ−µ ,(λ+α)(λ−µ)+γλ λ(λ−µ) , µ
µ−λ
.
(2.11) Case2.2(λ=µandλ=0).
P0(t)=eγ/λ
1 1+λt
α/λ
e−γ(1+λt)/λ−1 λ
(γ+γλt) ∞
1 z−α/λe−γ(t+1/λ)zdz
+γ λ
∞
1 z−α/λe−(γ/λ)zdz
.
(2.12)
692 RANDALL J. SWIFT Case2.3(λ=0 andµ=0).
P0(t)=e−γt µ
µe−α(etµ−1)/µ−γeα/µ ∞
1 z−(γ+µ)/µe−(αetµ/µ)zdz
+γeα/µ µ
∞
1 z−(γ+µ)/µe−(α/µ)zdz.
(2.13)
We note here that these expressions forP0(t), ast→ ∞, withγ=1, reduce tothe stationary probabilities obtained by Kyriakidis.
The probabilitiesPn(t), fo rn≥1 can be obtained by expandingψ(s,t)as a power series ins. However, the nature of the representations (2.7), (2.9), and (2.10) fo rψ(s,t) makes this a formidable task. Alternatively,P0(t)can be used in (2.4) toobtainP1(t) then (2.1) can be used recursively, toobtainPn(t)forn≥1.
References
[1] R. P. Agarwal,Generalized Hypergeometric Series, Uttar Pradesh Scientific Research Com- mittee, Allahabad, India, Asia Publishing House, Bombay, 1963.MR 31#3636.
[2] R. Bartoszynski, W. J. Buehler, W. Chan, and D. K. Pearl,Population processes under the influence of disasters occurring independently of population size, J. Math. Biol.27 (1989), no. 2, 167–178.MR 90h:92017. Zbl 715.92022.
[3] P. J. Brockwell, J. Gani, and S. I. Resnick,Birth, immigration and catastrophe processes, Adv. in Appl. Probab.14(1982), no. 4, 709–731.MR 84d:60107. Zbl 496.92007.
[4] E. G. Kyriakidis,Stationary probabilities for a simple immigration-birth-death process under the influence of total catastrophes, Statist. Probab. Lett.20(1994), no. 3, 239–240.
CMP 1 294 110. Zbl 801.60073.
[5] E. G. Kyriakidis and A. Abakuks,Optimal pest control through catastrophes, J. Appl. Probab.
26(1989), no. 4, 873–879.MR 90m:92066. Zbl 688.60069.
[6] R. J. Swift,A simple immigration-catastrophe process, Math. Sci.25(2000), no. 1, 32–36.
CMP 1 771 175.
Randall J. Swift: Department of Mathematics, Western Kentucky University, Bowling Green, KY42101, U SA
E-mail address:[email protected]
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