Evaluating dispersion strategies in growth models subject to geometric catastrophes
Valdivino Vargas Junior, Fábio Prates Machado, Alejandro Roldan-Correa
EEVALUATING DISPERSION STRATEGIES IN GROWTH MODELSSUBJECT TO GEOMETRIC CATASTROPHES
VALDIVINO VARGAS JUNIOR, F ´ABIO PRATES MACHADO,AND ALEJANDRO ROLD ´AN-CORREA
Abstract.
We consider stochastic growth models to represent population dynamicssubject to geometric catastrophes. We analyze different dispersion schemes after catas-trophes, to study how these schemes impact the population viability and comparing themwith the scheme where there is no dispersion. In the schemes with dispersion, we considerthat each colony, after the catastrophe event, has d new positions to place its survivors.We find out that when d = 2 no type of dispersion considered improves the chance ofsurvival, at best it matches the scheme where there is no dispersion. When d = 3, basedon the survival probability, we conclude that dispersion may be an advantage or not,depending on its type, the rate of colony growth and the probability that an individualwill survive when exposed to a catastrophe. Introduction
Catastrophes and spatial restrictions are among biological and environmental forcesthat drive the size dynamics of a population. These forces can reduce the population sizeor even eliminate it. Dispersion of the survivors is a possible strategy that could help toincrease the population viability.Models for population growth (a single colony) subject to catastrophes are consideredin Brockwell et al. [2] and later in Artalejo et al. [1]. In these models, the size of acolony increases according to a birth and death process subject to catastrophes. When acatastrophe strikes, the colony size is reduced according to some probability law and thesurvivors remain together in the same colony without dispersion. These authors study theprobability distribution of the first extinction time, the number of individuals removed, thesurvival time of a tagged individual, and the maximum population size reached betweentwo consecutive extinctions. For a comprehensive literature overview and motivation seefor example Kapodistria et al. [7].
Date : December 23, 2020.2010
Mathematics Subject Classification.
Key words and phrases.
Branching processes, catastrophes, population dynamics.F´abio Machado was supported by CNPq (303699/2018-3) and Fapesp (17/10555-0) and AlejandroRoldan by Universidad de Antioquia. a r X i v : . [ m a t h . P R ] D ec VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 2
Schinazi [11] and Machado et al [10] study stochastic models for population growthwhere individuals gather in independent colonies subject to catastrophes. When a catas-trophe strikes a colony the survivors disperse, trying to build new colonies that maysucceed settling down depending on the environment they encounter. Schinazi [11] andMachado et al [10] conclude that for these models dispersion is the best strategy. LatterJunior et al [6] focused on models that combine two types of catastrophes (binomial andgeometric) and reached the conclusion that dispersion may not be the best strategy. Theyobserve that the best strategy depends on the type of catastrophe, the spatial restrictionsthat the colony must deal with and the individual survival probability when it is exposedto a catastrophe. Machado et al [9] considered a general set up for growth rates and typesof catastrophes but with severe spatial restrictions in the sense that every colony, afterthe catastrophe event, has only up to d spots to move to.The common point between these papers [6, 9, 10] is the way the survivors dispersewhen their colony is stricken by a catastrophe. They consider that each survivor chooseindependently and with equal probability among the options they have to disperse. In thispaper we study a variety of dispersion possibilities and show that the survival probabilityof the whole population may be influenced by the scheme the individuals use when thedispersion occurs.In Section 2 we define and characterize four models for the growth of populations subjectto catastrophes considering different types of dispersion. In Section 3 we compare the fourmodels introduced in Section 2. Finally, in Section 4 we prove the results presented inSections 2 and 3. 2. Growth models
Artalejo et al. [1] present a model for a population which sticks together in one colony,without dispersion. That colony gives birth to new individuals at rate λ >
0, whilecatastrophes happen at rate µ . If at a catastrophe time the size of the population is i , itis reduced to j with probability µ ij = (cid:26) q i , j = 0 pq i − j , ≤ j ≤ i, where 0 < p < q = 1 − p . The form of µ ij represents what is called Geometriccatastrophe . Disasters reach the individuals sequentially and the effects of a disaster stop
VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 3 as soon as the first individual survives, if there is any survivor. The probability of nextindividual to survive, given that everyone fails up to that point, is p .The population size (number of individuals in the colony) at time t is a continuoustime Markov process { X ( t ) : t ≥ } that we denote by C ( λ, p ). We assume µ = 1 and X (0) = 1.Artalejo et al. [1] use the word extinction to describe the event that X ( t ) = 0, for some t >
0, for a process where state 0 is not an absorbing state. In fact the extinction timehere is the first hitting time to the state 0. Throughout this paper we say that a processsurvives if the extinction probability is strictly smaller than one.
Theorem 2.1 (Artalejo et al. [1]) . Let X ( t ) a process C ( λ, p ) , with λ > and < p < .Then, extinction event occurs with probability ψ A = min (cid:26) − pλp , (cid:27) . The geometric catastrophe would correspond to cases where the decline in the pop-ulation is halted as soon as any individual survives the catastrophic event. This maybe appropriate for some forms of catastrophic epidemics or when the catastrophe has asequential propagation effect like in the predator-prey models - the predator kills preyuntil it becomes satisfied. More examples can be found in Artalejo et al. [1], Cairns andPollett [3], Economou and Gomez-Corral [4], Thierry Huillet [5] and Kumar et al. [8].Based on the previous model we next define three models with dispersion on T + d .2.1. Growth model with dispersion on T + d . Let T + d be an infinite rooted tree whosevertices have degree d + 1, except the root that has degree d . Let us define a process withdispersion on T + d , starting from a single colony placed at the root of T + d , with just oneindividual. The number of individuals in a colony grows following a Poisson process ofrate λ >
0. To each colony we associate an exponential time of mean 1 that indicates whenthe geometric catastrophe strikes a colony. The individuals that survived the catastropheare dispersed between the d neighboring vertices furthest from the root to create newcolonies. Among the survivors that go to the same vertex to create a new colony at it,only one succeeds, the others die. So in this case when a catastrophe occurs in a colony,that colony is replaced by 0,1, ... or d colonies. We consider three types of dispersion: • Optimal dispersion:
Individuals are distributed, from left to right, in order tocreate the largest possible number of new colonies. If r individuals survive to a VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 4 catastrophe, then the number of colonies that are created equals min { r, d } . Letus denote the process with optimal dispersion by C od ( λ, p ). • Independent dispersion:
Each one of the individuals that survived the catas-trophe picks randomly a neighbor vertex and tries to create a new colony at it.When the amount of survivors is r , the probability of having y ≤ min { d, r } verticescolonized is T ( r, y ) d r (cid:18) dy (cid:19) , where T ( r, y ) denote the number of surjective functions f : A → B , with | A | = r and | B | = y . Let us denote the process with independent dispersion by C id ( λ, p ). • Uniform dispersion:
For every r , the amount of survivors, each set of numbers r , r , . . . , r d ∈ N ( occupancy set of numbers ), solution for r + r + · · · + r d = r has probability (cid:0) d + r − r (cid:1) − . So, the probability of having y ≤ min { d, r } verticescolonized when the amount of survivors is r is (cid:0) r − y − (cid:1)(cid:0) d + r − r (cid:1) (cid:18) dy (cid:19) . Let us denote the process with uniform dispersion by C ud ( λ, p ).The C od ( λ, p ), C id ( λ, p ) and C ud ( λ, p ) are continuous-time Markov processes with statespace N T d . For each of these processes we say that it survives if with positive probabilitythere are colonies for any time in that process. Otherwise, we say that the process diesout .We denote by ψ od ( ψ id and ψ ud ) the extinction probability for C od ( λ, p ) ( C id ( λ, p ) and C ud ( λ, p ), respectively) process. By coupling arguments one can see that the extinctionprobability, ψ od ( ψ id and ψ ud ), is a non-increasing function of d , λ and p . Remark 2.2.
As the optimal dispersion maximizes the number of new colonies wheneverthere are individuals that survived from the latest catastrophe, that type of dispersion isthe one which maximizes the survival probability. Moreover, for d = 2 and d = 3, thesurvival probability for the model with independent dispersion is larger or equal than thesurvival probability for the model with uniform dispersion. The reason for that is becausethe cumulative distribution function of the number of new colonies created right after acatastrophe for the model with independent dispersion is smaller than the analogous forthe model with uniform dispersion. In conclusion for d = 2 and d = 3, VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 5 ψ od ≤ ψ id ≤ ψ ud . (2.1)The next results present necessary and sufficient conditions for population survival ofthe process C o ( λ, p ). Theorem 2.3.
The process C o ( λ, p ) survives ( ψ o < ) if and only if λ > − pp . Moreover ψ o = min (cid:26) , − pλp (cid:27) . Remark 2.4.
From Theorems 2.1 and 2.3 one sees that extinction probabilities for themodels C ( λ, p ) and C o ( λ, p ), are the same ( ψ A = ψ o ). This is a big surprise and don’t seeany intuitive reason for this. As a consequence, when d = 2, for fixed λ and p , it is notpossible to increase the survival probability by any type of dispersion. Theorem 2.5.
The process C o ( λ, p ) survives ( ψ o < ) if and only if p > λ + 12 λ + 2 λ + 1 . Moreover ψ o = min (cid:40) , λ + 12 λ (cid:34) − (cid:115) λp + 4 − p ( λ + 1) p (cid:35)(cid:41) . Theorem 2.6.
The process C i ( λ, p ) survives ( ψ i < ) if and only if p > λ + 2 λ + 2 λ + 2 . Moreover ψ i = min (cid:26) , (1 − p )( λ + 2) λp ( λ + 1) (cid:27) . Theorem 2.7.
The process C i ( λ, p ) survives ( ψ i < ) if and only if p > λ + 32 λ + 3 λ + 3 . (2.2) Moreover ψ i = min (cid:40) , λ (cid:34) − ( λ + 3) + (cid:115) ( λ + 3)( pλ + 4 λ + 6 − p ) p ( λ + 1) (cid:35)(cid:41) . VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 6
Theorem 2.8.
The process C u ( λ, p ) survives ( ψ u < ) if and only if ln( λ + 1) < λ [ λ p + (4 p − λ + 2 p ]2( λ + 1) p . Moreover ψ u = min (cid:26) , λ (1 − p )( λ + 2)( λ + 1) λp − p ( λ + 1) ln( λ + 1) (cid:27) . Theorem 2.9.
The process C u ( λ, p ) survives ( ψ u < ) if and only if p ( λ + 1) λ ( λp + 1) (cid:20) λ + 2 − λ + 1) λ ln( λ + 1) (cid:21) > . (2.3) Moreover ψ u = min (cid:40) , (cid:34) √ ∆ − ( k + k ) + ( m + m ) ln( λ + 1) k − m ln( λ + 1) (cid:35)(cid:41) , where ∆ = ( m + m ) ln ( λ + 1) − k + k )( m + m ) + 2 βm ] ln( λ + 1) + [( k + k ) + 4 βk ] ,β = 1 − pλp + 1 , k = − p ( λ + 1)(5 λ + 6) λ ( λp + 1) , k = p ( λ + 1)( λ + 12 λ + 12) λ ( λp + 1) ,m = − p ( λ + 1) ( λ + 3) λ ( λp + 1) and m = 6 p ( λ + 1) ( λ + 2) λ ( λp + 1) . Dispersion as a survival strategy
Towards being able to evaluate dispersion as a survival strategy we define λ c ( p ) := inf { λ : C ( λ, p ) survives } ,λ oc ( d, p ) := inf { λ : C od ( λ, p ) survives } ,λ ic ( d, p ) := inf { λ : C id ( λ, p ) survives } ,λ uc ( d, p ) := inf { λ : C ud ( λ, p ) survives } . Remark 3.1.
When 0 < λ c ( p ) < ∞ for 0 < p < , the graph of λ c ( p ) splits the parametricspace λ × p into two regions. For those values of ( λ, p ) above the curve λ c ( p ) there issurvival in C ( λ, p ) with positive probability, and for those values of ( λ, p ) below the curve λ c ( p ) extinction occurs in C ( λ, p ) with probability 1. The analogous happens also for λ oc ( d, p ), λ ic ( d, p ) and λ uc ( d, p ). For d = 2 and d = 3, from inequality (2.1) it follows that λ oc ( d, p ) ≤ λ ic ( d, p ) ≤ λ uc ( d, p ) . VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 7
For an illustration, see Figures 1 and 2. λ —— λ uc (2 , p ) —— λ ic (2 , p ) —— λ oc (2 , p ) —— λ c ( p ) p Figure 1.
Graphics of λ c ( p ), λ oc (2 , p ), λ ic (2 , p ) and λ uc (2 , p ). λ —— λ uc (3 , p ) —— λ ic (3 , p ) —— λ oc (3 , p ) —— λ c ( p ) p Figure 2.
Graphics of λ c ( p ), λ oc (3 , p ), λ ic (3 , p ) and λ uc (3 , p ).From Remark 2.4 and the fact that the extinction probabilty, ψ od ( ψ id and ψ ud ), is anon-increasing function of d , λ and p , one sees that ψ o ≤ ψ o = ψ A . When analysing thecritical parameters one sees (Figure 2) that λ oc (3 , p ) < λ c ( p ). This shows that when d = 3,for all 0 < p <
1, the optimal dispersion is a superior strategy when compared to thenon-dispersion scheme studied in Artalejo et al. [1].
VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 8
However, dispersion is not always a better scenary for population survival, as one cansee in Figure 2. Observe that: λ ic (3 , p ) ≤ λ c ( p ) ⇐⇒ p ≤ p i = 12 (3.1)and λ uc (3 , p ) ≤ λ c ( p ) ⇐⇒ p ≤ p u ≈ . , (3.2)where the values p i and p u are obtained by plugging λ = λ c ( p ) = − pp in equations (2.2)and (2.3), respectively, taken as equality.In the region bounded by the curves λ ic (3 , p ) and λ c ( p ), for 0 < p < p i , we obtain from(3.1) that ψ i < ψ A = 1. In this case, independent dispersion is a better strategy than non-dispersion . On the other hand, in the region bounded by the curves λ ic (3 , p ) and λ c ( p )when p i < p <
1, we obtain that ψ A < ψ i = 1 and that non-dispersion is a better than independent dispersion . These two latter observations were also presented in Junior etal [6].Analogously, we can conclude from (3.2) that uniform dispersion is a better strategythan non-dispersion in the region bounded by λ uc (3 , p ) and λ c ( p ) when 0 < p < p u . Theopposite ( non-dispersion is a better strategy than uniform dispersion ) holds in the regionbounded by λ uc (3 , p ) and λ c ( p ) when p u < p < C ( λ, p ) and C i ( λ, p )(or C u ( λ, p )) survives. This question is answered by the following propositions. Proposition 3.2.
Assume that ψ i < and ψ A < . Then, ψ i < ψ A if and only if p < λ + 1)3 λ + 5 . (3.3)Proposition 3.2 is a direct consequence of Theorems 2.1 and 2.7.From (3.1) and Propo-sition 3.2 we can conclude that independent dispersion is a better strategy compared to non-dispersion , when the parameters ( λ, p ) fall in the gray region of Figure 3. The oppo-site ( non-dispersion is a better strategy than independent dispersion ) holds in the yellowregion. Remark 3.3.
Consider the processes C ( λ, p ) and C i ( λ, p ). From Theorems 2.1 and 2.7and Proposition 3.2 it follows that, • For λ > VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 9 λ —— λ ic (3 , p ) —— λ c ( p ) —— Equality in (3.3) p Figure 3.
Independent dispersion vs non-dispersion . In the gray region, ψ i < ψ A . In the yellow region, ψ i > ψ A . – If p ∈ (0 , λ +32 λ +3 λ +3 ], then ψ A = ψ i = 1. – If p ∈ ( λ +32 λ +3 λ +3 , λ +1 ], then ψ i < ψ A = 1. – If p ∈ ( λ +1 , λ +1)3 λ +5 ), then ψ i < ψ A < – If p = λ +1)3 λ +5 , then ψ i = ψ A = λ +32 λ ( λ +1) < – If p ∈ ( λ +1)3 λ +5 , ψ A < ψ i < • For λ = 1 – If p ∈ (0 , ], then ψ A = ψ i = 1. – If p ∈ ( , ψ A < ψ i < • For λ < – If p ∈ (0 , λ +1 ], then ψ A = ψ i = 1. – If p ∈ ( λ +1 , λ +32 λ +3 λ +3 ], then ψ A < ψ i = 1. – If p ∈ ( λ +32 λ +3 λ +3 , ψ i < ψ A < Proposition 3.4.
Assume that ψ u < and ψ A < . Then, ψ u < ψ A if and only if (cid:20)(cid:18) p ( λ − λp (cid:19) m + m (cid:21) ln( λ + 1) < (cid:18) p ( λ − λp (cid:19) k + k − λpλp + 1 . (3.4) VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 10
Proposition 3.4 is a consequence of Theorems 2.1 and 2.9. From (3.2) and Proposi-tion 3.4 we can conclude that uniform dispersion is a better strategy compared to non-dispersion , when the parameters ( λ, p ) fall in the gray region of Figure 4. The opposite( non-dispersion is a better strategy than uniform dispersion ) holds in the yellow region.Analogous to what is presented in Remark 3.3 we have λ ≈ .
18 splitting the cases. λ —— λ uc (3 , p ) —— λ c ( p ) —— Equality in (3.4) p Figure 4.
Uniform dispersion vs non-dispersion . In the gray region, ψ u <ψ A . In the yellow region, ψ u > ψ A . Example 3.5.
Consider the processes C (4 , p ) and C u (4 , p ). From Theorems 2.1 and 2.9and Proposition 3.4 follows that • If p ∈ (cid:16) , −
375 ln(5) (cid:105) , then ψ A = ψ u = 1. • If p ∈ (cid:16) −
375 ln(5) , (cid:105) , then ψ u < ψ A = 1. • If p ∈ (cid:16) , −
225 ln(5)676 −
375 ln(5) (cid:17) , then ψ u < ψ A < • If p = −
225 ln(5)676 −
375 ln(5) , then ψ i = ψ A < • If p ∈ (cid:16) −
225 ln(5)676 −
375 ln(5) , (cid:17) , then ψ A < ψ u < Remark 3.6.
Observe that the model with only one colony has a catastrophe rate of1 while the multiple colonies model has a catastrophe rate of n if there are n colonies.Moreover, a catastrophe is more likely to wipe out a smaller colony than a larger one. VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 11
On the other hand multiple colonies give multiple chances for survival and this may be acritical advantage of the multiple colonies model over the single colony model. Thereforeour analysis shows that for d = 3, dispersion is an advantage or not for population survivaldepending on the dispersion type, λ and p .4. Proofs
Lemma 4.1.
Let { Z n } n ≥ be a branching process with Z = 1 , whose offspring distribu-tion has probabilty generating function given by g ( s ) = p + p s + p s + p s and withextinction probability denoted by ψ . Then i ) ψ < if and only if p + 2 p + 3 p > .ii ) Assume p (cid:54) = 0 . If p + 2 p + 3 p > then ψ = 12 − − p p + (cid:115)(cid:18) p p (cid:19) + 4 p p .iii ) Assume p = 0 . If p + 2 p > then ψ = p p .iv ) Assume p = β and p y = k y + m y ln ν, y = 1 , , . Then ψ < if and only if ( m + 2 m + 3 m ) ln ν < − k − k − k . If ( m + 2 m + 3 m ) ln ν < − k − k − k and p (cid:54) = 0 then ψ = 12 (cid:34) − √ ∆ − k − m − ln νk + m ln ν (cid:35) where ∆ = ( m + m ) ln ν + 2[( k + k )( m + m ) + 2 βm ] ln ν + [( k + k ) + 4 βk ] .v ) Take k = m = 0 in iv ) . Then ψ < if and only if ( m + 2 m ) ln ν < − k − k . Besides, if ( m + 2 m ) ln ν < − k − k and p (cid:54) = 0 then ψ = βk + m ln ν . Proof of Lemma 4.1.
VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 12 i ) ψ < µ = g (cid:48) (1) > g (cid:48) (1) = p + 2 p + 3 p .ii ) ψ is the smallest non negative solution of g ( s ) = s . That is, one must solve theequation ( s − p s + ( p + p ) s − p ) = 0 .iii ) ψ is the smallest non negative solution of g ( s ) = s . Then when p = 0, one has tosolve the equation ( s − p s − p ) = 0 .iv ) The first part follows immediately from i ). For the second part, observe that in ii ) ψ = 12 (cid:34) − (cid:112) ( p + p ) + 4 p p − p p (cid:35) .v ) The first part follows from i ) and the second part follows from iii ). (cid:3) Remark 4.2.
The probability distribution of the number of survivals right after thecatastrophe (but before the dispersion) is given by P ( N = 0) = β, P ( N = n ) = αc n , n = 1 , , . . . , where β = 1 − pλp + 1 , α = ( λ + 1) pλ ( λp + 1) and c = λλ + 1 . Observe that β = 1 − (1 + α ) c (1 − c ) . (4.1)For details see Machado et al [9, section 2.2]. Proof of Theorem 2.3.
Observe that for d = 2 we have p = P ( N = 0) = β, p = P ( N = 1) = αc and p = 1 − β − αc. Next, apply Lemma 4.1, part iii ). (cid:3) Proof of Theorem 2.5.
Here d = 3, then we have p = P ( N = 0) = β, p = P ( N = 1) = αc, p = P ( N = 2) = αc and p = 1 − β − αc − αc . Next, applying Lemma 4.1, part ii ) VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 13 ψ = 12 − (cid:18) αc − β − αc − αc (cid:19) + (cid:115)(cid:18) αc − β − αc − αc (cid:19) + 4 β − β − αc − αc = 12(1 − β − αc − αc ) (cid:104) − (1 − β − αc ) + (cid:112) (1 − β − αc ) + 4 β (1 − β − αc − αc ) (cid:105) . The result follows when one considers β = 1 − pλp + 1 , α = ( λ + 1) pλ ( λp + 1) and c = λλ + 1 . (cid:3) Proof of Theorem 2.6.
The first part can be seen as an application of Proposition 4.3 fromMachado et al [9]. As d = 2, ψ i < E (cid:34)(cid:18) (cid:19) N (cid:35) < . Observe that E (cid:0) s N (cid:1) = β + ∞ (cid:88) n =1 s n αc n = sc ( α − β ) + β − sc . (4.2)Then, considering Proposition 4.3 from Machado et al [9] we see that ψ i < c ( α − β ) + β − c < , then, substituting α, β and c accordingly we see that ψ i < p > λ + 2 λ + 2 λ + 2 . In order to obtain the extinction probability we consider Proposition 4.4 from Machado etal [9]. There we have that when d = 2, ψ i is the smallest non-negative solution of (cid:88) y =0 (cid:34) s y (cid:18) y (cid:19) ∞ (cid:88) n = y T ( n, y )2 n P ( N = n ) (cid:35) = s where T ( n, y ) = y (cid:88) i =0 (cid:20) ( − y (cid:18) yi (cid:19) ( y − i ) n (cid:21) . Then, to obtain ψ i we have to solve the equation β + 2 sα ∞ (cid:88) n =1 (cid:18) (cid:19) n αc n + s ∞ (cid:88) n =2 (cid:18) n − n (cid:19) αc n = s VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 14 or equivalently β + (cid:18) αc + c − − c (cid:19) s + αc (1 − c )(2 − c ) s = 0 . Then, substituting α, β and c accordingly, we obtain λp ( λ + 1) s + (2 p − λ p − λ − s + (1 − p )( λ + 2) = 0or equivalently ( s − λp ( λ + 1) s + ( λ + 2)( p − . Finally, we have that ψ i = min (cid:26) , (1 − p )( λ + 2) λp ( λ + 1) (cid:27) . (cid:3) Proof of Theorem 2.7.
Proposition 4.3 from Machado et al [9] ( d = 3) is to be consideredin order to prove the first part. First of all observe that ψ i < E (cid:34)(cid:18) (cid:19) N (cid:35) < . Using equation (4.2) and applying Proposition 4.3 from Machado et al [9] we see that ψ i < c α − β ) + β < (cid:16) − c (cid:17) . Then, substituting α, β and c accordingly, we see that ψ i < p > λ + 32 λ + 3 λ + 3 . In order to obtain the extinction probability we consider Proposition 4.4 from Machado etal [9]. There we have that when d = 3, ψ i is the smallest non-negative solution of β + 3 αs (cid:34) ∞ (cid:88) n =1 (cid:16) c (cid:17) n (cid:35) + 3 αs (cid:34) ∞ (cid:88) n =2 (2 n − n c n (cid:35) + αs (cid:34) ∞ (cid:88) n =3 (3 n − · n + 3)3 n c n (cid:35) = s. So, we have to solve2 αc s + 6 αc (1 − c ) s + (3 αc + c − − c )(3 − c ) s + β (1 − c )(3 − c )(3 − c ) = 0 , or equivalently (see equation (4.1)), solve( s − αc ) s + (6 αc − αc ) s + 2 αc − αc + 9 αc + 2 c − c + 18 c −
9] = 0 . VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 15
Substituting α and c , equivalently we have to solve( s − (cid:20) s + ( λ + 3) λ s + ( p − λ + 9 λ + 9)2( λ + 1) pλ (cid:21) = 0obtaining ψ i = min (cid:40) , λ (cid:34) − ( λ + 3) + (cid:115) ( λ + 3)( pλ + 4 λ + 6 − p ) p ( λ + 1) (cid:35)(cid:41) . (cid:3) Proof of Theorem 2.8.
Observe that the process C u ( λ, p ) behaves a branching process.Next we show that the distribution of Y , the number of new colonies right after a catas-trophe, satisfies item v ) of Lemma 4.1. First we see that P ( Y = 0) = P ( N = 0) = 1 − pλp + 1 . For n (cid:54) = 0, P ( Y = y | N = n ) = (cid:18) y (cid:19)(cid:18) n − y − (cid:19) n + 1 , y = 1 , y ≤ n. Then, P ( Y = 1) = ∞ (cid:88) n =1 P ( N = n ) P ( Y = 1 | N = n ) = − α − αc ln(1 − c )and P ( Y = 2) = ∞ (cid:88) n =2 P ( N = n ) P ( Y = 2 | N = n ) = α (2 − c )1 − c αc ln(1 − c ) . Substituting α and c we see that P ( Y = 1) = − p ( λ + 1) λ ( λp + 1) − p ( λ + 1) λ ( λp + 1) ln (cid:18) λ + 1 (cid:19) and P ( Y = 2) = p ( λ + 1)( λ + 2) λ ( λp + 1) + 2( λ + 1) pλ ( λp + 1) ln (cid:18) λ + 1 (cid:19) . The result follows after identifying the parameters k y , m y and ν in item v ) of Lemma 4.1. (cid:3) Proof of Theorem 2.9.
Observe that the process C u ( λ, p ) behaves a branching process.Next we show that the distribution of Y , the number of new colonies right after a catas-trophe, satisfies item iv ) of Lemma 4.1. First we see that P ( Y = 0) = P ( N = 0) = 1 − pλp + 1 . VALUATING DISPERSION STRATEGIES IN GROWTH MODELS 16
For n (cid:54) = 0: P ( Y = y | N = n ) = (cid:18) y (cid:19)(cid:18) n − y − (cid:19)(cid:18) n + 22 (cid:19) , y = 1 , , , y ≤ n. Then, P ( Y = 1) = ∞ (cid:88) n =1 P ( N = n ) P ( Y = 1 | N = n ) = 3 α (2 − c ) c + 6 α (1 − c ) c ln(1 − c ) , P ( Y = 2) = ∞ (cid:88) n =2 P ( N = n ) P ( Y = 2 | N = n ) = 3 α ( c − c + 12 α (2 c − c ln(1 − c )and P ( Y = 3) = ∞ (cid:88) n =3 P ( N = n ) P ( Y = 3 | N = n ) = α ( c − c + 12) c (1 − c ) + 6 α (2 − c ) c ln(1 − c ) . Substituting α and c we see that P ( Y = 1) = 3 p ( λ + 1)( λ + 2) λ ( λp + 1) + 6 p ( λ + 1) λ ( λp + 1) ln (cid:18) λ + 1 (cid:19) , P ( Y = 2) = − p ( λ + 1)(5 λ + 6) λ ( λp + 1) − p ( λ + 1) ( λ + 3) λ ( λp + 1) ln (cid:18) λ + 1 (cid:19) and P ( Y = 3) = p ( λ + 1)( λ + 12 λ + 12) λ ( λp + 1) + 6 p ( λ + 1) ( λ + 2) λ ( λp + 1) ln (cid:18) λ + 1 (cid:19) . The result follows after identifying the parameters k y , m y and ν in item iv ) of Lemma 4.1. (cid:3) References [1] J.R.Artalejo, A.Economou and M.J.Lopez-Herrero. Evaluating growth measures in an immigrationprocess subject to binomial and geometric catastrophes.
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Probability in the Engineering and Informational Sciences (1), 79-111 (2016).[8] Nitin Kumar, Farida P. Barbhuiya, Umesh C. Gupta. Analysis of a geometric catastrophe modelwith discrete-time batch renewal arrival process. RAIRO-Oper. Res. (5) 1249-1268 (2020).[9] F.P.Machado, A. Roldan-Correa and V.V. Junior. Colonization and Collapse on HomogeneousTrees. Journal of Statistical Physics , 1386–1407 (2018).[10] F.P.Machado, A. Roldan-Correa and R.Schinazi. Colonization and Collapse.
ALEA-Latin AmericanJournal of Probability and Mathematical Statistics , 719-731 (2017).[11] R.Schinazi. Does random dispersion help survival? Journal of Statistical Physics , , (1), 101-107(2015).(Valdivino Vargas Junior) Institute of Mathematics and Statistics, Federal University ofGoias, Campus Samambaia, CEP 74001-970, Goiˆania, GO, Brazil
Email address : [email protected] (F´abio P. Machado) Statistics Department, Institute of Mathematics and Statistics, Uni-versity of S˜ao Paulo, CEP 05508-090, S˜ao Paulo, SP, Brazil.
Email address : [email protected] (Alejandro Rold´an) Instituto de Matem´aticas, Universidad de Antioquia, Calle 67, no53-108, Medellin, Colombia
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