BRANCHING PROCESS WITH

Transcription

1 Chapter 2 A MODIFIED MARKOV BRANCHING PROCESS WITH IMMIGRATION 2. Introduction Several physical and biological populations exhibit a branching character in their evolutionary behaviour in the sense that each individual (also called a particle ) of one type lives for some amount of time and then splits into several individuals of the same type or of different types. These processes are called cascade processes or reproductive processes or branching processes. Two important factors which characterize these processes are the random life-time of an individual and the random number of its off-springs. By considering various assumptions on the life-time and the number of off-springs, several models have The content of the first model of the chapter is published in the Proceedings of the International Conference on Mathematical Computer Engineering (ICMCE23), pp. -8. ISBN and the content of the second model of the chapter has been communicated for publication in the international journal Far East Journal of Mathematical Sciences. 4

2 2.. Introduction 5 been proposed and analysed very extensively in the past (see, for example, Harris [34], Srinivasan [83], Mode [57], Athreya and Ney [] and Assmussen and Hering []). One class of models is the class of Markov branching processes. A Markov branching process is a continuous-time branching process in which all particles behave independently and identically; and all particles have a common exponential life-time and a common off-spring probability generating function. There is basically one important aspect which is inherent in biological growth, namely the immigration of particles of the same type from outside the population. Adke [2] has considered a birth, death and migration processes. Foster [3] and Pakes [64] have separately studied a discrete branching process with immigration occurring only when the population size is. Aksland [3] has studied a simple birth-and-death process with state-independent immigration. Yamazato [2] has analysed a continuous time branching process with state-dependent immigration in which once the population size reaches the state it sojourns in that state for an exponentially distributed time and at the end of it the population instantaneously resurrects to some positive state according to some probability distribution. Bartoszynski et al. [4] have considered Markov branching processes under the influence of disasters that arrive independently of the present population size and derived an integral equation for the probability of extinction. Chen and Renshaw [23] have studied Markov branching processes with instantaneous immigration. Pakes [66] has studied the problem of constructing a Markov branching process with instantaneous resurrection. Swift [89] has obtained transient probabilities for a simple birth-death-immigration process subject to Catastrophes that occur at a constant rate. Li and Chen [54] have studied a Markov branching process with state-dependent immigration and resurrection. Di Crescenzo et al. [27] have studied birth-death processes subject to catastrophes and analysed the probability distribution of the first effective catastrophe occurrence time. Recently, Li and Liu [56] have analysed Markov branching processes with immigration-migration and resurrection. Li et al. [55] have studied the asymptotic properties of the Markov branching process with immigration. In the above models all particles have no age restriction in branching. However branching requires a specified amount of time. Parthasarathy [67] considered a Markov branching process in which each particle surviving at least up to a definite time T splits into several particles at a later time t (t>t) while all particles whose life-time ends before the time T leave no descendents.

3 2.. Introduction 6 He calls T as the maturity age of the particles. This type of maturity dependency in Markov branching processes has not been studied so far with immigration and catastrophes. To fill this gap, we investigate in the present chapter two models of Markov branching processes which incorporates immigration and age-dependent branching. In the first model the immigration is assumed to be state-independent and the second model it is assumed to be state-dependent. The organization of the chapter is as follows: In section 2.2 we consider the model I of the present chapter. The subsection 2.2. describes the first model of a modified Markov branching process with state-independent immigration. Inter-connected integral equations for conditional probability generating functions are formulated in the subsection We obtain explicit transient expressions for some of the characteristics of the population in the subsection We consider the second model of the present chapter in section 2.3. The model is described in the subsection Interconnected integral equations are derived in the subsection Explicit expression for the mean number of particles at any time t is found in the subsection In section 2.3.4, we provide the analysis of the point process of resurrection. For analysing the behaviour of the two models, we present the model of Yamazato [2] in section 2.4 and obtain an explicit expression for the mean number of particles at any time t. We also obtain an explicit expression for the transient probability P (t) of the population size is at time t given that the population size at time t= is. A numerical illustration is provided in section 2.5 to compare the two models of the present chapter with the model of Yamazato [2] and highlight the impact of the maturity-dependent branching process with immigration and disasters.

4 2.2. Model I Model I 2.2. Description of the model We consider a modified Markov branching process with state-independent immigration. To be specific, a biological population is considered in which each individual lives for a random length L of time and at the end of its life-time splits into a random number of identical off-springs if L > T, where T is a positive constant. The individual leaves no descendants if L < T. The descendants behave independently and identically to the ancestor. We assume that the life-time of each individual is governed by the probability density function f (t) given by f (t)=λe λt,λ>, t>. (2.2.) Let h(s) be the off-spring probability generating function and h(s)= p + p j s j. (2.2.2) j=2 We note that h ()= jp j (2.2.3) j=2 and h () represents the mean number of off-springs of an individual. The branching process is called sub-critical, critical or super-critical according as h ()<, h ()= or h ()>. The growth of the population is supplemented by immigration of particles in batches independently from outside the population which occur according to a Poisson process with rate γ>. The number of particles in each batch is a random variable ξ having state space{, 2, 3, } and which has a common probability generating function defined by q(s)= q k s k. (2.2.4) k=

5 2.2. Model I 8 Let X(t) be the number of individuals of the population present at time t. Then the stochastic process{x(t), t } is the Markov branching process with state-independent immigration and age-dependent branching The Probability Generating Functions To study the process X(t), we define the following conditional probability generating functions: G (s, t)=e{s X(t) X()=}, (2.2.5) G (s, t)=e{s X(t) X()=}. (2.2.6) Case (i) t T : If X()=, then either an immigration occurs or no immigration takes place before t. Then we have G (s, t)=e γt +γ e γu q(g (s, t u))du. (2.2.7) If X()=, then the three mutually exclusive and exhaustive events are possible: e : No event takes place up to time t. e 2 : The first event that occurs after time t= and before time t is that the life of the particle which existed at time t= ends. e 3 : The first event that occurs after time t = and before time t is that a batch immigration occurs. Then we get G (s, t)= se (λ+γ)t +λ e (λ+γ)u G (s, t u)du

6 2.2. Model I 9 +γ e (λ+γ)u k= q k {G (s, t u)} k+ du. (2.2.8) Case (ii) t>t : If X()=, then either an immigration occurs or no immigration takes place before t. Then we have G (s, t)=e γt +γ e γu q(g (s, t u))du, (2.2.9) If X()=, then three mutually exclusive and exhaustive events are possible: e : No event takes place up to time t. e 2 : The first event that occurs after time t= and before time t is that the life of the particle which existed at time t= ends. The time of occurrence of this event may be either before the maturity time T or after the maturity time T. If it is before the time T, then no off-spring is produced. If it is after the time T, then a random number of off-springs are produced. e 3 : The first event that occurs after time t = and before time t is that a batch immigration occurs. Then we get T G (s, t)= se (λ+γ)t +λ e (λ+γ)u G (s, t u)du +λ e (λ+γ)u p G (s, t u)+ T +γ e (λ+γ)u k= p j {G (s, t u)} j du j=2 q k {G (s, t u)} k+ du. (2.2.)

7 2.2. Model I The mean number of individuals We define the following conditional means: m (t)=e{x(t) X()=}, (2.2.) m (t)=e{x(t) X()=}. (2.2.2) If t T, then differentiating (2.2.7) and (2.2.8) with respect to s and putting s=, we get respectively m (t)=γq () e γu m (t u)du, (2.2.3) m (t)=e (λ+γ)t +λ e (λ+γ)u m (t u)du +γ[q ()+] e (λ+γ)u m (t u)du. (2.2.4) If t>t, then differentiating (2.2.9) and (2.2.) with respect to s and putting s=, we get respectively m (t)=γq () e γu m (t u)du, (2.2.5) T m (t)=e (λ+γ)t +λ e (λ+γ)u m (t u)du+λp e (λ+γ)u m (t u)du T +λh () e (λ+γ)u m (t u)du+γ[q ()+] e (λ+γ)u m (t u)du. (2.2.6) T Using the Heaviside function, (2.2.4) and (2.2.6) yield m (t)=e (λ+γ)t +λ [ H(u T)]e (λ+γ)u m (t u)du +λp H(u T)e (λ+γ)u m (t u)du +a e (λ+γ)u H(u T)m (t u)du

8 2.2. Model I 2 +(b+γ) e (λ+γ)u m (t u)du, for all t>, (2.2.7) where a=λh () and b=γq (). From (2.2.3) and (2.2.5), we get m (t)=b e γu m (t u)du, for all t>. (2.2.8) Using Laplace transform, (2.2.7) and (2.2.8) give m (θ)= θ+λ+γ +λm (θ) [ ] e (θ+λ+γ)t + λp m (θ)e (θ+λ+γ)t θ+λ+γ θ+λ+γ Solving (2.2.9) and (2.2.2), we get m (θ)= m (θ)= From (2.2.2), we get + ae (θ+λ+γ)t (b+γ) θ+λ+γ m (θ)+ θ+λ+γ m (θ); (2.2.9) m (θ)= b θ+γ m (θ). (2.2.2) b (θ+λ+γ)(θ+γ) bλ (b+γ)(θ+γ) [a(θ+γ) bλ( p )]e (θ+λ+γ)t, (2.2.2) (θ+γ) (θ+λ+γ)(θ+γ) bλ (b+γ)(θ+γ) [a(θ+γ) bλ( p )]e (θ+λ+γ)t. (2.2.22) m (θ)= (θ+λ+γ)(θ+γ) [ bλ+(b+γ)(θ+γ)+[a(θ+γ) bλ( p )]e (θ+λ+γ)t (θ+λ+γ)(θ+γ) b ] = b n= n j= ( ) n [a(θ+γ) bλ( p )] j e j(θ+λ+γ)t {bλ+(b+γ)(θ+γ)} n j j (θ+λ+γ) n+ (θ+γ) n+ = b n= n n j( n j j= k= )( n j k )

9 2.2. Model I 22 [a(θ+γ) bλ( p )] j e j(θ+λ+γ)t b n j k λ n j k (b+γ) k (θ+γ) k = n= (θ+λ+γ) n+ (θ+γ) n+ n n j j= k= j ()( n n j ( ) l j k l= )( j l ) b l+n+ j k λ l+n j k (b+γ) k a j l ( p ) l e j(θ+λ+γ)t = (θ+λ+γ) n+ (θ+γ) n++l j k n= n n j j= k= j ( )( n n j ( ) l j k b n+ j k+l λ l+n j k (b+γ) k a j l ( p ) l e j(θ+λ+γ)t (θ+λ+γ) n+ δ n++l j k,+ ( ) ( ) n+l j k+r λ r δ n++l j k, r (θ+λ+γ) On inversion, (2.2.23) gives r= l= )( j l ) n++l j k+r. (2.2.23) m (t)=e (λ+γ)t n= n n j j= k= j ( )( n n j ( ) l j k l= )( j l ) δ n++l j k, n! b n+ j k+l λ l+n j k (b+γ) k a j l ( p ) l H(t jt)(t jt) n + ( ) ( ) n+l j k+r λ r n+l j k+r+ (t jt) δ n++l j k, r (2n+l j k+r+)!. (2.2.24) Similarly, using (2.2.22), we get r= m (θ)= (θ+λ+γ) [ ] bλ+(b+γ)(θ+γ)+[a(θ+γ) bλ( p )]e (θ+λ+γ)t (θ+λ+γ)(θ+γ) = = n= [ bλ+(b+γ)(θ+γ)+{a(θ+γ) bλ( p )}e (θ+λ+γ)t] n n= n j= (θ+λ+γ) n+ (θ+γ) n ( ) n{bλ+(b+γ)(θ+γ)} n j {a(θ+γ) bλ( p )} j e j(θ+λ+γ)t j (θ+λ+γ) n+ (θ+γ) n

10 2.2. Model I 23 = n= n n j j= k= j l= ( n j )( n j k )() j l ( ) j l (bλ)n j k (b+γ) k a l b j l λ j l ( p ) j l e j(θ+λ+γ)t (θ+λ+γ) n+ (θ+γ) n k l = n= n n j j= k= j l= ( n j )( n j k )() j l ( ) j l (bλ)n j k (b+γ) k a l b j l λ j l ( p ) j l e j(θ+λ+γ)t (θ+λ+γ) n+ ( ) n+k+l λ δ n k l, + ( δ n k l, ) (θ+λ+γ) n k l+r θ+λ+γ n n j j ( )( )() n n j j = j k l n= j= ( ) j l (bλ)n k l (b+γ) k a l ( p ) j l e j(θ+λ+γ)t (θ+λ+γ) n+ ( ) n k l+r δ λ r n k l,+ ( δ n k l, ) r On inversion, (2.2.25) yields r= k= l= (θ+λ+γ) n k l+r. (2.2.25) m (t)=e (λ+γ)t n= n n j j= k= j l= ( n j )( n j k )( j l ) ( ) j l (bλ) n k l (b+γ) k a l ( p ) j l H(t jt)(t jt) n δ ( ) n k l, n k l+r λ r (t jt) n k l+r + ( δ n k l, ) n! r (2n k l+r)!. (2.2.26) r= By settingγ= in (2.2.26), we get back the result of Parthasarathy [67]: m (t)=e λt H(t nt){λh ()(t nt)} n. (2.2.27) n! n=

11 2.3. Model II 24 Further, by setting T=, we get back the classical result: m (t)=e λ(h () )t. (2.2.28) 2.3 Model II 2.3. Description of the model We consider a maturity-dependent Markov branching process with state-dependent immigration. As in model I, each individual of a population lives for a random length L of time and at the end of its life-time splits into a random number of identical off-springs if L>T, where T is a positive constant. The individual leaves no descendants if L<T. The descendants behave independently and identically to the ancestor. We assume that the life-time of each individual is governed by the probability density function f (t) given by f (t)=λe λt,λ>, t>. (2.3.) Let h(s) be the off-spring probability generating function which is given by h(s)= p + p j s j. j=2 Suppose that m is the mean number of off-springs of an individual. Then we obtain m=h ()= jp j. (2.3.2) j=2 The branching process is called sub-critical, critical or super-critical according as m <, m= or m>. We assume that catastrophes arrive in the population according to a Poisson process with rate η and each catastrophe wipes out the population instantaneously.

12 2.3. Model II 25 The growth of the population is supplemented by immigration of particles from outside the population which occur whenever the population size is zero. We assume that the sojourn time of the population in the state zero is exponentially distributed with mean. We assume that at the expiry of this sojourn time, a batch of k immigrants join the γ population with probability q k and we define the probability generating function of the batch size by q(s)= q k s k. (2.3.3) k= Let X(t) be the number of individuals of the population present at time t. Then the stochastic process{x(t), t } is the maturity-dependent Markov branching process with state-dependent immigration subject to disasters The Probability Generating Function of X(t) We define the following conditional probability generating functions: G (s, t)=e{s X(t) X()=}, (2.3.4) Using the total probability theorem, we obtain G (s, t)=e{s X(t) X()=}. (2.3.5) G (s, t)=e γt +γ e γu q(g (s, t u))du. (2.3.6) To find the integral equation for G (s, t), we have two cases: Case: t T In this case, we note that (i) no event occurs up to time t or (ii) the first event that occurs in (, t) is that the life-time of the particle which existed at time t= ends before

13 2.3. Model II 26 the time t with out producing any off-spring or (iii) the first event that occurs in (, t) is that a disaster occurs at some time in (, t). Then, by total probability law, we get Case: t>t G (s, t)=e (λ+η)t s+(λ+η) e (λ+η)u G (s, t u)du. (2.3.7) In this case, we note that (i) no event occurs up to time t so that there is exactly one individual existing at time t or (ii) the first event that occurs in (, t) is that the life-time of the particle which existed at time t= ends before the time T with out producing any off-spring or (iii) the first event that occurs in (, t) is that a disaster occurs at some time in (, t) wiping out the population or (iv) the first event that occurs in (, t) is that the lifetime of the particle which existed at time t= ends in (T, t) producing a random number of individuals governed by the probability generating function h(s). Then, by applying the total probability law, we obtain T G (s, t)=e (λ+η)t s+λ e (λ+η)u G (s, t u)du+η e (λ+η)u G (s, t u)du +λ e (λ+η)u p G (s, t u)+ T j=2 p j (G (s, t u)) j du. (2.3.8) The mean number of individuals We define the following conditional means: m (t)=e{x(t) X()=}, (2.3.9) m (t)=e{x(t) X()=}. (2.3.)

14 2.3. Model II 27 Differentiating (2.3.6) with respect to s and putting s=, we get m (t)=b e γu m (t u)du, for t>. (2.3.) Differentiating (2.3.7) with respect to s and putting s=, we get m (t)=e (λ+η)t + (λ+η) e (λ+η)u m (t u)du, t T. (2.3.2) Differentiating (2.3.8) with respect to s and putting s=, we get T m (t)=e (λ+η)t +λ e (λ+η)u m (t u)du+η e (λ+η)u m (t u)du +λ e (λ+η)u [p m (t u)+h ()m (t u)]du, t>t. (2.3.3) T Using Heaviside function, we write (2.3.2) and (2.3.3) together as m (t)=e (λ+η)t +λ [ H(u T)]e (λ+η)u m (t u)du +η e (λ+η)u m (t u)du +λ H(u T)e (λ+η)u [p m (t u)+h ()m (t u)]du. (2.3.4) Using Laplace transform, (2.3.) gives Similarly, (2.3.4) yields m (θ)= bm (θ) (θ+γ). (2.3.5) m (θ)= ( ) e (θ+λ+η)t θ+λ+η +λ m (θ) θ+λ+η

15 2.3. Model II 28 + ηm (θ) θ+λ+η + λe (θ+λ+η)t( p m (θ)+h ()m (θ) ). (2.3.6) θ+λ+η Substituting (2.3.5) in (2.3.6) and simplifying, we obtain m (θ)= θ+γ (θ+γ)(θ+λ+η)+{λb( p ) a(θ+γ)} e (θ+λ+η)t b(λ+η), (2.3.7) where a=λh () and b=γq (). Then, by substituting (2.3.7) in (2.3.5), we get m (θ)= b (θ+γ)(θ+λ+η)+{λb( p ) a(θ+γ)} e (θ+λ+η)t b(λ+η). (2.3.8) Assuming that the binomial expansion is valid in a domain, (2.3.8) yields m (θ)= (θ+γ)(θ+λ+η) [ b(λ+η)+{a(θ+γ) bλ( p )}e (θ+λ+η)t (θ+λ+η)(θ+γ) b ] [ b(λ+η)+{a(θ+γ) bλ( p )} e (θ+λ+η)t] n = b (θ+λ+η) n+ (θ+γ) n+ n= = b n= n j= ( ) n j ( j ( ) )a j k k (θ+γ) k b j k λ j k ( p ) j k e (θ+λ+η) jt b n j (λ+η) n j j k (θ+λ+η) n+ (θ+γ) n+ k= = = n= n= n j= n j= j ()( n j ( ) )a j k k λ j k ( p ) j k e (θ+λ+η) jt b n k+ (λ+η) n j j k (θ+λ+η) n+ (θ+γ) n+ k k= j ()() n j ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k+ (λ+η) n j j k k= (θ+λ+η) n+ [ δ n+ k, + ( δ ] n+ k,) (θ+γ) n+ k

16 2.3. Model II 29 n j ( )() n j = ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k+ (λ+η) n j j k n= j= k= (θ+λ+η) n+ δ n+ k,+ ( δ n+ k,) (+ (γ λ η) (θ+λ+η) )n+ k n j ( )() n j = ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k+ (λ+η) n j j k n= j= k= (θ+λ+η) δ n+ k,+ ( δ n+ k,) ( ) n +k (γ λ η) r n+ (θ+λ+η) n+ k r (θ+λ+η) r n j ()() n j = ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k+ (λ+η) n j j k n= j= k= δ n+ k, ( δ n+ k,) ( ) n +k (γ λ η) r (θ+λ+η) n++ (θ+λ+η) 2n+2 k r (θ+λ+η) r n j ()() n j = ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k+ (λ+η) n j j k n= j= k= δ n+ k, ( ) n+ k+r (θ+λ+η) n++ ( δ (λ+η γ) r n+ k,) r (θ+λ+η). (2.3.9) 2n+2 k+r Taking inverse transform on both sides of (2.3.9), we get r= r= r= m (t)=e (λ+η)t n= n j= k= j ( )() n j ( ) j k j k δ n+ k, n! a k λ j k ( p ) j k b n k+ (λ+η) n j H(t jt)(t jt) n ( ) n k+r (λ+η γ) r (t jt) n+ k+r + ( δ n+ k, ) r (2n+ k+r)!. (2.3.2) r=

17 2.3. Model II 3 Considering (2.3.7) and by binomial series expansion in negative powers of (θ+λ+η), we obtain [ b(λ+η)+{a(θ+γ) bλ( p )} e (θ+λ+η)t] n m (θ)= (θ+λ+η) n+ (θ+γ) n n= = n= n j= = = = ( ) n j ( j ( ) )a j k k (θ+γ) k b j k λ j k ( p ) j k e (θ+λ+η) jt b n j (λ+η) n j j k (θ+λ+η) n+ (θ+γ) n k= n= n= n= n j= n j= n j= j ()( n j ( ) )a j k k λ j k ( p ) j k e (θ+λ+η) jt b n k (λ+η) n j j k (θ+λ+η) n+ (θ+γ) n k k= j ()() n j ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k (λ+η) n j j k k= (θ+λ+η) n+ [ δ n k, + ( δ ] n k,) (θ+γ) n k j ( )() n j ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k (λ+η) n j j k k= (θ+λ+η) n+ δ n k,+ ( δ n k, ) (θ+λ+η) n k (+ (γ λ η) (θ+λ+η) )n k n j ( )() n j = ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k (λ+η) n j j k n= j= k= (θ+λ+η) δ n k,+ ( δ n k,) ( ) n+k (γ λ η) r n+ (θ+λ+η) n k r (θ+λ+η) r n j ( )() n j = ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k (λ+η) n j j k n= j= k= δ n k, ( δ n k, ) ( ) n+k (γ λ η) r (θ+λ+η) n++ (θ+λ+η) 2n+ k r (θ+λ+η) r r= r=

18 2.3. Model II 3 n = n= j= k= δ n k, (θ+λ+η) n++ ( δ n k,) Inverting (2.3.2), we obtain j ()() n j ( ) j k a k λ j k ( p ) j k e (θ+λ+η) jt b n k (λ+η) n j j k ( ) n k+r (λ+η γ) r r r= (θ+λ+η) 2n+ k+r. (2.3.2) m (t)=e (λ+η)t n= n j= k= j ( )() n j ( ) j k j k a k λ j k ( p ) j k b n k (λ+η) n j H(t jt)(t jt) n δ ( ) n k, n k+r (λ+η γ) r (t jt) n k+r + ( δ n k, ) n! r (2n k+r)!. (2.3.22) r= Settingγ= andη= in (2.3.22), we get back the result of Parthasarathy [67]: m (t)=e λt H(t nt){λh ()(t nt)} n. (2.3.23) n! n= Setting further T=, we get back the classical result of the Markov branching process: m (t)=e λ(h () )t. (2.3.24) The point process of resurrection Consider the stochastic process X(t). The maturity-dependent Markov branching process X(t) is subject to disasters and the growth of the population is supplemented by immigration of particles from outside the population which occur whenever the population size is zero. Yamazato [2] has termed this type of immigration as state-dependent immigration since it has the property that once the population size reaches the state it sojourns in that state for an exponentially distributed time and at the end of it the

19 2.3. Model II 32 population instantaneously resurrects to some positive state according to some probability distribution. Pakes [66] has termed the above type of immigration as resurrection since the process regenerates from state with a random number of immigrants. Li and Liu [56] have considered a modified Markov branching process incorporating with both statedependent immigration-migration and resurrection and obtained explicit expressions for the extinction probabilities and mean extinction times. Chen et al. [22] have obtained extinction probabilities for two interacting branching collision processes where an ordinary Markov branching process strongly interacts with a collision branching process. We now consider the stochastic point process Z(t) generated by the sequence of random time points at which resurrection takes place. We note that Z(t) is in fact a marked stochastic point process. We proceed to obtain the stationary first-order product density of the point process Z(t). We define ξ = lim t P{X(t)= X()=}; ξ = lim t P{X(t)= X()=}. Then, from (2.3.6), we get ξ = q(ξ ). (2.3.25) Similarly, from (2.3.8), we get ξ = λξ ( ) e (λ+η)t + ηξ λ+η λ+η +λh(ξ ) λ+η e (λ+η)t. (2.3.26) Suppose that Then we obtain where q(s)= s 2 s, h(s)= 2 (+ s2 ). ξ ξ = ( c) + c(+ξ2 ), (2.3.27) 2 ξ 2 c= λ λ+η e (λ+η)t.

20 2.4. Yamazato Model 33 Simplifying (2.3.27), we get cξ 3 2(c+)ξ2 + (3c+2)ξ 2c=. (2.3.28) Factorizing (2.3.28), we get (ξ )[cξ 2 (c+2)ξ + 2c]=. (2.3.29) Therefore the roots are The smallest root lying in [, ) is ξ =, c+2± (c+2) 2 8c 2. 2c ξ = c+2 (c+2) 2 8c 2. 2c Hence the stationary first-order product density of the point process Z(t) is given by lim t h(t)=ξ γ= γ 2c [c+2 (c+2) 2 8c 2 ]. (2.3.3) 2.4 Yamazato Model Yamazato [2] has considered a continuous time branching process which allows random immigration whenever the population size is zero. This model is a particular case of our II model when we remove age-dependency and disaster in the II model. To be specific, we put T= andη= in (2.3.8) and get G (s, t)=e λt s+λ e λu p G (s, t u)+ j=2 p j (G (s, t u)) j du, (2.4.)

21 2.4. Yamazato Model 34 where G (s, t) is given by (2.3.6). Differentiating (2.4.) with respect to s and putting s=, we get m (t)=e λt +λ e λu [p m (t u)+h ()m (t u)]du. (2.4.2) We can solve (2.4.2) together with (2.3.8). Taking Laplace transform on both sides of (2.4.2), we get Solving (2.3.2) and (2.4.3), we get m (θ)= θ+λ +λ[p m (θ)+h ()m (θ)]. (2.4.3) θ+λ m (θ)= m (θ)= b (θ+λ a)(θ+γ) λp b, (2.4.4) θ+γ (θ+λ a)(θ+γ) λp b, (2.4.5) where a=λh () and b=γq (). These equations can also be obtained from (2.3.4) and (2.3.5) by putting T = and η =. Using binomial series, (2.4.4) gives Inverting (2.4.6), we obtain m (θ)= n= ( ) n+r λ n p n bn+ (λ a γ) r. (2.4.6) r (θ+λ a) 2n+r+2 r= m (t)=e (λ a)t n= r= ( ) n+r λ n p n bn+ (λ a γ) r t 2n+r+. (2.4.7) r (2n+r+)! Similarly (2.4.5) gives m (θ)= θ+λ a + n= ( ) n+r λ n p n bn (λ a γ) r. (2.4.8) r (θ+λ a) 2n+r+ r=

22 2.4. Yamazato Model 35 Inverting (2.4.8), we obtain m (t)=e (λ a)t + n= r= We can also write (2.4.7) and (2.4.9) in the following form: ( ) n+r λ n p n bn (λ a γ) r t 2n+r r (2n+r)!. (2.4.9) m (t)= 2be (λ a+γ)t 2 (λ a γ)2 + 4λp b sinh t (λ a γ) 2 + 4λp b 2 (2.4.) m (t)= e (λ a+γ)t 2 (λ a γ)2 + 4λp b cosh t (λ a γ) 2 + 4λp b (λ a γ)2 + 4λp b 2 (λ a γ) sinh t (λ a γ) 2 + 4λp b 2. (2.4.) Resolving m (θ) and m (θ) into partial fractions, we can also obtain m (t) and m (t) in the following form m (t)= be (λ a)t α β ( e αt e βt), (2.4.2) m (t)= e (λ a)t ( αe βt βe αt), (2.4.3) 2(α β) where α= 2 [ λ a γ+ (λ a γ)2 + 4λp b ], (2.4.4) β= 2 [ λ a γ (λ a γ)2 + 4λp b ]. (2.4.5) Suppose that X()=i and P i j (t)=p[x(t)= j X()=i]. Then we obtain P i (t+ )=P i (t)[ γ ]+ P i (t)λp, (2.4.6) j P i j (t+ )=P i j (t)[ jλ ]+ P i, j r+ (t)( j r+)λp r r= +P i, (t)γq j, j. (2.4.7)

23 2.4. Yamazato Model 36 Writing (2.4.6) and (2.4.7) in the form of differential equations, we get P i (t)= γp i(t)+ P i (t)λp, (2.4.8) j P i j(t)= jλp i j (t)+ P i, j r+ (t)( j r+)λp r r= +P i, (t)γq j, j. (2.4.9) Let us define the generating function G i (s, t)= P i j (t)s j. (2.4.2) j= Then, by using (2.4.8) and (2.4.9), we obtain G i (s, t) t = P i j(t)s j j= = γp i (t)+ P i (t)λp + jλp i j(t)+ j= j P i, j r+ (t)( j r+)λp r + P i, (t)γq j s j r= = γp i (t) λs s P i j (t)s j j= +λ j= j (l+)p i,l+ (t)p j l s j +γ l= P i, (t)q j s j j= = γp i (t) λs s P i j (t)s j j= +λ l= (l+)p i,l+ (t)p j l s j +γ j=l P i, (t)q j s j j= =λ[h(s) s] G i(s, t) s +γp i, (t)[q(s) ]. (2.4.2)

24 2.4. Yamazato Model 37 Defining the mean m i (t)=e [X(t) X()=i], (2.4.2) yields m i(t)=(a λ)m i (t)+bp i, (t). (2.4.22) Equation (2.4.22) agrees with Yamazato [2]. Equation (2.4.22) is subject to the initial condition m i ()=i. Taking Laplace transform on both sides of (2.4.22), we get [θ (a λ)]m i (θ)=i+bp i, (θ). Therefore, we get Putting i= in (2.4.23) and equating with (2.4.4), we get P, (θ)= m i (θ)= i+bp i, (θ) θ (a λ). (2.4.23) θ+λ a (θ+λ a)(θ+γ) λp b = = n= n= λ n p n bn (θ+λ a) n (θ+γ) n+ ( ) n+r λ n p n bn (λ a γ) r. (2.4.24) r (θ+λ a) 2n+r+ r= Inverting (2.4.24), we get explicitly P (t)=e (λ a)t n= r= ( n+r )λ n p n r bn (λ a γ) r t 2n+r (2n+r)!. (2.4.25) Resolving P, (θ) into partial fractions, we can also obtain P (t)= e (λ a)t α β whereαandβare given by (2.4.4) and (2.4.5). ( αe αt βe βt), (2.4.26)

25 2.5. A Numerical Illustration A Numerical Illustration In this section, we present a numerical illustration to highlight the impact of maturitydependent branching and state-dependent immigration under the influence of catastrophes on the growth of the population. We consider the models of Yamazato [2] and Parthasarathy [67] and compare the performance of the mean values of the present models with their mean values. To accomplish this, we first take the model of Yamazato [2] and our models and compute numerically the mean value m (t) as t varies from. to 2. and make a comparative picture. For this, we have chosen the parameter values as λ=.3,γ=.,η=., T=.3, p =.4, h ()=.2, q ()=2.3. In Table 2., we have presented the mean number of particles for the Yamazato model and that of the two models of the present chapter. The mean value for model II decreases and that of Yamazato model increases as t increases. This is as expected, since the model II has the occurrence of disaster added to Yamazato model. The rate of change of mean value for Yamazato model is very much higher in comparison with that of the II model as it is expected so, since the model II is maturity dependent. The rate of change of mean value for model I is higher than that of Yamazato since model I has independent immigration. Next, to study the impact of catastrophes on the growth of the population, we fix the time as t=2. and vary the catastrophe parameterηfrom. to.4 and choosing the same values for all the other parameters as for table 2., we obtain Table 2.2. In this table, we highlight the performance of model II in comparison with that of Yamazato [2]. We find in Table 2.2 that, as η increases from. to.4, equivalently, as the mean of the inter-occurrence interval of catastrophes decreases, the mean value at the time point t = 2 decreases, as expected. We compute m Y (2) = We find in Table 2.2 that, as η increases from. to.4, equivalently, as the mean of the inter-occurrence interval of catastrophes decreases, the mean value at the time point t=2 decreases, as expected.

26 2.5. A Numerical Illustration 39 Next, we fix the catastrophe rateη=.6 and vary the maturity T from. to.4 and compute the mean value m (2) for the model II and compare with that of Yamazato [2] model at the time point t=2 in table 2.3. We find in Table 2.3 that as the maturity level increases, the mean value decreases as expected and that the occurrence of catastrophes has decreased the mean value of Yamazato model. This result is as expected. Table 2.: Comparison of m (t) t m Y (t) m(i) (t) m(ii) (t) t m Y (t) m(i) (t) m(ii) (t) Table 2.2: Comparison of m (2) of model II with that of Yamazato [2] λ=.3, T=.3,γ=., p =.4, h ()=.2, q ()=2.3 η m II (2) η m II (2) η m II (2)

27 2.5. A Numerical Illustration 4 Table 2.3: Comparison of m (2) of model II with that of Yamazato [2] with the occurrence of catastrophes λ=.3,γ=.,η=.6, p =.4, h ()=.2, q ()=2.3 T m II (2) T m II (2) T m II (2) Table 2.4: Comparison of m (t) of model II with that of Yamazato [2] with out the occurrence of catastrophes λ=.3,γ=.,η=., p =.4, h ()=.2, q ()=2.3 T m II (2) T m II (2) T m II (2) We also study the effect of maturity alone on the population growth by takingη= in m (t) of model II and obtain the mean value at t=2 for maturity level ranging from to.4. We present the mean values in table 2.4. We find in Table 2.4 that as the maturity level increases, the mean value decreases as expected. We have computed m Y (2)= To study the impact of immigration and catastrophes on the population growth, we next consider a numerical comparison of the performance of our models with that of Parthasarathy [67]. In Table 2.5, we tabulate the mean values of the present models I and II with that of Parthasarathy [67] for t increasing from to 6. We observe that the rate of change in the mean value of model II is very much lower than that of Parthasarathy [67]. This is due to the effect of the occurrence of catastrophes. The rate of change of the mean value for model I is higher than that of Parthasarathy [67]. This is as expected since model

28 2.5. A Numerical Illustration 4 I includes immigration in it. Next we proceed to compare further the effect of maturity T and immigration in the population mean of our models. We take the same parameter values as in Parthasarathy [67] and observe that the population mean value is higher in our model as compared with that of Parthasarathy [67] (see Table 2.6, Table 2.7 and Table 2.8). This is because of the fact that the immigration takes care of the population growth and we also observe that the maturity slows down the growth in both the models. Further the growth is more in the super-critical case. Table 2.5: Comparison of m (t) with that of Parthasarathy [67] λ=.3, T=.3, h ()=.2 t m P (t) m(i) (t) m(ii) (t) t m P (t) m(i) (t) m(ii) (t) Table 2.6: Effect of maturity time T in the mean of m (t) in the subcritical case λ=2., T=.,.25,.5,.75 h () T m (P) (2) m(i) (2) m(ii) (2)

29 2.5. A Numerical Illustration 42 Table 2.7: Effect of maturity time T in the mean of m (t) in the critical case λ=2., T=.,.25,.5,.75 h () T m (P) (2) m(i) (2) m(ii) (2) Table 2.8: Effect of maturity time T in the mean of m (t) in the supercritical case λ=2., T=.,.25,.5,.75 h () T m (P) (2) m(i) (2) m(ii) (2)

30 2.5. A Numerical Illustration 43 Finally, we depict the performance of our models as compared with the model of Yamazato [2] in Figure 2.. In Figure 2.2, we highlight the performance of our models as compared with the model of Parthasarathy [67]. Figure 2.: Performance of the models compared to the model of Yamazato [2] Figure 2.2: Performance of the models compared to the model of Parthasarathy [67] We observe that the mean values m (I) (t) and m(ii) (t) of our I and II models are both much lower than that of Parthasarathy [67].