n Species impulsive migration model with Markovian switching

Journal of Theoretical Biology 307 (2012) 62–69

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n Species impulsive migration model with Markovian switching Zijian Liu a,n, Shouming Zhong a,b, Zhidong Teng c a b c

School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu Sichuan 610054, PR China Key Laboratory for NeuroInformation of Ministry of Education, University of Electronic Science and Technology of China, Chengdu Sichuan 610054, PR China College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, PR China

H I G H L I G H T S c c c

We investigate an n species stochastic impulsive migration model. The mode of dispersal in the present model is very novel and is very popular in nature. Species’ global positivity, ultimate boundedness in mean and extinction in mean are derived.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 18 September 2011 Received in revised form 1 May 2012 Accepted 2 May 2012 Available online 15 May 2012

An n species stochastic impulsive migration Lotka–Volterra model with Markovian switching in N different patches is presented and studied in this paper. By constructing appropriate Lyapunov functions, some sufficient conditions on the global positivity, the ultimate boundedness in mean and the extinction in mean are established. Real world examples are provided to illustrate the validity of our results. A discussion is given in the end. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Stochastic differential equation Generalized Itoˆ formula Ultimate boundedness in mean Extinction in mean

1. Introduction Owing to natural enemies, competition, seasonal alternatives or deterioration of patches of the environment, species dispersal (or migration) to and from two or more patches is common. In general, dispersal is mainly represented by one or other of the following three categories: (i) Dispersal occurs at any time, simultaneously between any two patches, i.e. continuous bidirectional dispersal, such as activities of some animals on their habitat which is separated by road or stream; activities of some birds between two neighborhood hills and so on. (ii) Dispersal occurs at some fixed time, simultaneously between any two patches, i.e. impulsive bidirectional dispersal. For example, in mating season, some male will search for mate among patches at some transitory time slots; Graziers feed their cattle or sheep group by group in one meadow, etc. All of these cases can be referred as the type of impulsive bidirectional dispersal. (iii) Dispersal is a migration, i.e. impulsive unilateral diffusion. Examples include the migration of some kinds of organisms from cold patches to warm patches with seasonal alternative, migration of some kinds of fish from ocean to their birthplace and so on.

n

Corresponding author. Tel.: þ86 15882059791; fax: þ86 991 8583556. E-mail address: [email protected] (Z. Liu).

0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.05.001

Many empirical studies and monographs on population dispersal systems of type (i) have been published (see Beretta et al., 1987; Beretta and Takeuchi, 1987, 1998; Freedman and Takeuchi, 1989; Kuang and Takeuchi, 1994; Mahbuba and Chen, 1994; Wang and Chen, 1997; Zhang and Wang, 2002; Beretta and Takeuchi, 1998; Lu and Takeuchi, 1992; Teng and Lu, 2001; Cui, 2006; Zhang and Teng, 2008; Liu and Zhong, 2010 and references cited therein). For example, in Teng and Lu (2001), Teng and Lu have investigated the following single-species nonautonomous dispersal model with delays: ! Z 0 dxi ðtÞ ¼ xi ðtÞ ai ðtÞbi ðtÞxi ðtÞci ðtÞxi ðttðtÞÞ ki ðt,sÞxi ðt þ sÞ ds dt s þ

n X

Dij ðtÞðxj ðtÞxi ðtÞÞ,

j¼1

where ai denotes the intrinsic growth rate of patch i, bi and ci are the density-dependent coefficients of patch i, ki ðt,sÞ is a integrable kernel, i.e. bounded on ðt,sÞ A ½0,1Þ  ½s,0, continuous with respect to t A ½0,1Þ and integrable with respect to s A ½s,0 with nonnegative constant s. Dij(t) represents the dispersal rate from patch j to patch i at time t, the dispersal established in this model is continuous and bidirectional, that is, the dispersal occurs all of the time and happens simultaneously between any two patches i and j. In recent years,

Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

some population dynamical models with impulsive bidirectional dispersal have been proposed and studied (see Wang and Liu, 2007; Jiao et al., 2011, 2009; Shao, 2010 and references cited therein). For instance in Wang and Liu (2007), the authors studied the following autonomous impulsive diffusion single species model: 8 dY 1 ðtÞ k1 > > > < dt ¼ r 1 Y 1 ðtÞ lnY ðtÞ 1 t ant, dY 2 ðtÞ k2 > > > ¼ r 2 Y 2 ðtÞ ln : dt Y 2 ðtÞ

DY 1 ¼ d1 ðY 2 ðtÞY 1 ðtÞÞ, DY 2 ¼ d2 ðY 1 ðtÞY 2 ðtÞÞ, t ¼ nt, where ri and ki (i¼1, 2) are the intrinsic growth rate and the carrying capacity of the ith patch, respectively. di (i¼1, 2) is the dispersal rate in the ith patch. The pulse diffusion occurs at every t period (t is a positive constant). Obviously, in this model, species Y inhabits two patches and at each pulse disperses from each of the two patches to the other, i.e. impulsive bidirectional dispersal. However, in all of these investigated dispersal models considered so far, there are few papers to consider the total impulsive migration system, i.e. impulsive unilateral diffusion (type (iii)) system. Practically, in the real ecological system, with seasonal alternative, some kinds of organisms will migrate from cold patches (or food resource poor patches) to warm patches (or food resource rich patches) in search of a better habitat to inhabit or breed, fish will go back from ocean to their birthplace to spawn and so on. Therefore, it is a very basic problem to research this kind of impulsive migration systems. In our previous work (Liu et al., 2010, 2011; Zhang et al., 2011), a two-patch impulsive diffusion periodic single-species logistic model (see Liu et al., 2010) and two-patch prey impulsive unilateral diffusion periodic predator–prey models (see Liu et al., 2011; Zhang et al., 2011) have been proposed and studied and some interesting results have been established. If patches are used for brief stays to supplement energy for the migrating species or the patch that the species moved back to is not where it had just come from, multi-patch dynamical behaviors of the migrating species should be studied. Hence, in this paper, we generalize our previous two-patch impulsive diffusion model to N patches. Since the patch choice of the species is stochastic, it can be regarded as a taking finite values with finite states Markov chain system. Therefore, the purpose of this paper is to research the stochastic impulsive migration model with Markovian switching. We mainly study the effects of the intrinsic growth rate and the inner and outer competitive coefficients to the population’s survival and extinction. This paper is organized as follows. In Section 2, we detail our model and present some preliminaries. In Section 3, we state and prove the global positivity, the ultimate boundedness in mean and the extinction in mean of system (2.2). Real world examples are provided to illustrate the validity of our results in Section 4. Finally, a discussion is given. 2. The model Before exhibiting our model, we first of all interpret some nomenclature which will be used throughout the paper. Nomenclature a finite positive integer space with S ¼ f1; 2, . . . ,Ng positive integer space real number space positive real number space a time interval with L ¼ ½t0þ ,1Þ n-dimensional real number space Rn x n-dimensional column vector, its transpose is denoted by xT

S Zþ R Rþ L

63

Rnþ

a positive cone with Rnþ ¼ fx ¼ ðx1 ,x2 , . . . ,xn Þ A Rn : xm 4 0, for all 1 rm r ng A A ¼ ðajl Þnn is an n  n real matrix, its transpose is denoted by AT þ the largest eigenvalue for an n  n real symmetric lmax ðWÞ þ matrix W with lmax ðWÞ ¼ supx A Rnþ ,9x9 ¼ 1 xT Wx 99 I |

the Euclidean norm unit matrix empty set

Let frðtÞ, t Z 0g be a right-continuous Markov chain on the probability space ðO,F, UÞ taking values in the finite space S ¼ f1; 2, . . . ,Ng with generator P ¼ ðpij ÞNN given by ( pij D þ oðDÞ if ia j, P½rðt þ DÞ ¼ j9rðtÞ ¼ i ¼ 1 þ pii D þoðDÞ if i ¼ j, where D 4 0, limD-0 oðDÞ=D ¼ 0. Here pij Z0 ð8i,j A S, i ajÞ is the transition rate from mode i to mode j, while

pii ¼ 

N X

pij :

ð2:1Þ

j ¼ 1,j a i

Consider a system of n species consisting of N different habitats(patches). Suppose that the N different patches have similar sizes (which aims to guarantee the population densities do not change instantaneously within the different patches) and the migration of the species to the N patches is stochastic and instantaneous, our model then takes the form 8 _ ¼ diagðx1 ðtÞ,x2 ðtÞ, . . . ,xn ðtÞÞ½bðrðtÞÞ þ AðrðtÞÞxðtÞ, t A ðtk1 , tk , xðtÞ > < xðtkþ Þ ¼ Gðrðtk Þ,rðtkþ ÞÞxðtk Þ, > : xðt þ Þ ¼ x 40, rðt þ Þ ¼ r A S, 0

0

0

0

ð2:2Þ where  xðtÞ ¼ ðx1 ðtÞ,x2 ðtÞ, . . . ,xn ðtÞÞT A Rn , each component xm ðtÞðm ¼ 1; 2, . . . ,nÞ of xðtÞ represents the population density of the mth species. xm ðt0þ Þ ¼ xm0 40 is the initial population density of the species m, where the initial time is t0 ¼ 0.  For a fixed iA S, each element bm(i) of bðiÞ ¼ ðb1 ðiÞ,b2 ðiÞ, . . . , bn ðiÞÞT represents the intrinsic growth rate of the mth species in patch i, AðiÞ ¼ ðajl ðiÞÞnn are the coefficients which describe the relationships between species j and species i in the ith patch, where j ¼l denotes the inner relationship of the jth patch(or lth patch).  Gði,jÞ ¼ diagðg 1 ði,jÞ,g 2 ði,jÞ, . . . ,g n ði,jÞÞ, each component g m ði,jÞ of Gði,jÞ denotes the remaining rate of species m after the migration from patch i to patch j.  ftk , k ¼ 1; 2, . . .g is a given positive number sequence satisfying 0 ¼ t0 o t1 o t2 o    o tk o tk þ 1 o    with limk-1 tk ¼ þ 1 and denotes the kth switched time, i.e. the time of transition of the mode from rðtk Þ ¼ i to rðtkþ Þ ¼ j a i, with tkþ ¼ limD-0 ðtk þ DÞ. Remark 2.1. Actually, different values of the components of the matrix AðiÞ ¼ ðajl ðiÞÞnn imply different relationships between species j and species l (1) ajl o0 and alj o 0, i.e. one species’ activity has a disadvantageous effect on the other. This implies a competition type relationship between species j and species l. (2) ajl o0 but alj 40 (or alj o0 but ajl 40), i.e. one species’s activity has a disadvantageous effect on the other, but an

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Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

advantageous effect on itself. This represents a predator–prey type relationship between species j and species l. (3) ajl 40 and alj 4 0, i.e. one species’ activity has an advantageous effect on the other. This implies a cooperative relationship between species j and species l. We have supposed that the system is composed of N different patches. When t A ðtk , tk þ 1 , all the species live in patch i, because of the change of the environment, the species will migrate to patch j, the remaining rate after the migration is G(i,j). Then the species will live in patch j during the period t A ðtk þ 1 , tk þ 2 . When the environment changes again, they will migrate to another patch, which admits a Markov switching property. The following two definitions are necessary for the study of our main results. Definition 2.1. System (2.2) is said to be ultimately bounded in mean if there is a positive constant H such that for any initial value x0 A Rnþ , solution xðtÞ of system (2.2) has the property that lim sup E9xðtÞ9 rH: t-1

Definition 2.2. System (2.2) is said to be extinct in mean if for any initial value x0 A Rnþ , solution xðtÞ of system (2.2) has the property that lim sup E9xðtÞ9 ¼ 0: t-1

Throughout this paper, the following assumption is of major importance. (H) The positive remaining rate Gðrðtk Þ,rðtkþ ÞÞ of the migration satisfies k

g m ðrðtk Þ,rðtkþ ÞÞ r bm

and

1 Y

Proof. Since the coefficients of the system are locally Lipschitz continuous in x, it is known (see e.g. Mao, 1999, Theorem A.2) that for any given initial value x0 A Rnþ there is a unique maximal local solution xðtÞ on t A ½t0þ , re Þ, where re is the explosion time. To show this solution is global, we need only to prove that re ¼ 1 a.s. From system (2.2), we know that the solution x(t) is positive for any given initial value x0 A Rnþ on t A ½t0þ , t1 , hence xðt1þ Þ is also positive. Therefore, we can obtain that the solution xðtÞ of system (2.2) is positive for all t A ½t0þ , re Þ by deduction. Now, we will prove that re ¼ 1 a.s. Let k0 4 0 be sufficiently large such that each component of x0 is no larger than k0. For each integer k Z k0 , define the stopping time

rk ¼ re 4infft A ½t0þ , re Þ : xm ðtÞ Z kg,

ð3:2Þ

where throughout this paper we set inf | ¼ 1. Obviously, rk is increasing as k-1. Set r1 ¼ limk-1 rk , hence r1 r re a.s. Define a function V : Rnþ  S-R þ by Vðx,iÞ ¼

n X

cm ðiÞxm :

ð3:3Þ

m¼1

For any k Z k0 , there exists ik such that rk A ðtik 1 , tik . By the generalized Itoˆ formula, we get for any t A ðtik 1 , tik  Z t4rk EVðxðt4rk Þ,rðt4rk ÞÞ ¼ EVðxðtiþk 1 Þ,rðtiþk 1 ÞÞþ E LV ðxðsÞ,rðsÞÞ ds: tik 1

ð3:4Þ Here LV is a mapping from Rnþ  S to R defined by LVðx,iÞ ¼ xT CðiÞbðiÞ þ xT ½CðiÞAðiÞ þAT ðiÞCðiÞx=2þ

bk r M o 1,

N X

pij Vðx,jÞ:

ð3:5Þ

j¼1

k¼1 k

solution xðtÞ of system (2.2) defined on t A L, and the solution will remain in Rnþ with probability one, namely xðtÞ A Rnþ for all t A L almost surely.

k

for any k A Z þ , m ¼ 1; 2, . . . ,n and b ¼ max1 r m r n fbm g. Especially, G(i,i)¼I is defined. Remark 2.2. Under normal circumstances, the remaining rate G(i,j) of the species is less than unity after migration from patch i to patch j because of the death during the migration caused by starvation, tireness, hunting or other disadvantageous factors. However, in this paper, we allow that the remaining rate gm(i,j) of some species m is larger than unity, which may be caused by human intervention. We Q k only need that the assumption 1 k ¼ 1 b rM o1 is satisfied.

P Setting p ¼ maxi A S f 9pij 9 : j A Sg. It follows from condition (3.1) that there is a constant K 1 4 0 such that LVðx,iÞ r K 1 9x9 þ p

N X

Vðx,jÞ:

j¼1

Let K 2 ¼ maxfcm ðiÞ=cm ðjÞ : 1 rm r n and i, j A Sg, and K 3 ¼ minfcm ðiÞ : 1 r mr n, iA Sg:

3. Main results

Then, by the definition of V, for any i, j A S, we have In this section, we study the global positivity, the ultimate boundedness in mean and the extinction in mean of system (2.2). Since the mth state xm(t) of system (2.2) is the population density of the mth species, it should be nonnegative. Moreover, the coefficients of system (2.2) do not satisfy the linear growth condition though they are locally Lipschitz continuous, so the solutions of system (2.2) may explode to infinity at a finite time (Mao, 1994, 1999). Therefore, it is necessary to establish some conditions under which the solutions of system (2.2) are not only positive but also not explode to infinity at any finite time. Theorem 3.1. Assume that assumption (H) holds. In addition, if there are positive numbers c1 ðiÞ,c2 ðiÞ, . . . ,cn ðiÞ, iA S such that þ lmax ðCðiÞAðiÞ þ AT ðiÞCðiÞÞr 0,

ð3:1Þ

where CðiÞ ¼ diagðc1 ðiÞ,c2 ðiÞ, . . . ,cn ðiÞÞ. Then, for any system parameter bðÞ and any given initial value x0 A Rnþ , there is a unique

K 2 Vðx,iÞ ¼

n X

K 2 cm ðiÞxm Z

m¼1

n X

cm ðjÞxm ¼ Vðx,jÞ,

ð3:6Þ

m¼1

and 9x9 r

n X

xm ¼

m¼1

n X

cm ðiÞxm =cm ðiÞ rVðx,iÞ=K 3 :

m¼1

Hence, there is a constant K 4 4 0 such that LVðx,iÞ r K 1 Vðx,iÞ=K 3 þ pNK 2 Vðx,iÞ rK 4 Vðx,iÞ:

ð3:7Þ

Therefore, we have EVðxðt4rk Þ,rðt4rk ÞÞ rEVðxðtiþk 1 Þ,rðtiþk 1 ÞÞ Z t EVðxðs4rk Þ,rðs4rk ÞÞ ds: þ K4 tik 1

ð3:8Þ

Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

65

Using the well-known Gronwall inequality, we get

Therefore,

EVðxðt4rk Þ,rðt4rk ÞÞ r EVðxðt

LVðx,t,iÞ r et ð½9cðiÞ9 þ9CðiÞbðiÞ9 þ pi 9x9l9x9 =2Þ r K 5 et , P where pi ¼ j a i pij 9cðjÞ9 and

þ þ ik 1 Þ,rð ik 1 ÞÞ

rb

ik 1

t

2

expfK 4 ðttik 1 Þg

EVðxðtik 1 Þ,rðtik 1 ÞÞ expfK 4 ðttik 1 Þg:

ð3:9Þ

Setting qðtÞ ¼ EVðxðtÞ,rðtÞÞ and taking the Dini-right-upper derivative of q(t), we have 8 þ > D qðtÞ rK 4 qðtÞ for t A ðti1 , ti , i ¼ 1; 2, . . . , > < i1 þ qðti1 Þ r b qðti1 Þ, ð3:10Þ > > þ : qðt Þ ¼ EVðx0 ,r 0 Þ:

ð3:19Þ

K 5 ¼ max½9cðiÞ9 þ 9CðiÞbðiÞ9 þ pi 2 =ð2lÞ: iAS

Denote ( K~ ¼ max

k 1 Y

sup

)

bi , 1 ,

k Z 2, k A Z þ i ¼ 1

0

it follows from (3.18) and (3.19) that Eq. (3.10) implies that qðtÞ r b

i1

qðti1 Þ expfK 4 ðtti1 Þg,

t A ðti1 , ti , i ¼ 1; 2, . . . :

ð3:11Þ

þ þ þ EVðx,t,rðtÞÞ r EVðxðtk1 Þ, tk1 ,rðtk1 ÞÞ þ E

iY k 1

rb

bl EVðx0 ,r0 Þ expðK 4 tÞ rM

l¼1

 EVðx0 ,r 0 Þ expðK 4 tÞ,

ð3:12Þ

þE

for all t A L. Hence we have

rb

rb

It turned out from Theorem 3.1 that under assumption (H) and condition (3.1), the solution of system (2.2) will be positive with probability one. These properties of positivity and nonexplosion are essential for a population system. On the other hand, due to the limit of the resources, the property of the ultimate boundedness is more desired, which can be guaranteed by the following result. Theorem 3.2. Assume that assumption (H) holds. In addition, condition (3.1) is strengthened by þ

ð3:15Þ

iAS

Then there exists a positive number H 4 0 such that lim supE9xðtÞ9 r H,

ð3:16Þ

t-1

for any solution x(t) of system (2.2) with the initial value x0 A Rnþ . Proof. By Theorem 3.1, the unique solution x(t) of system (2.2) will remain in Rnþ for all t A L with probability one. Define ðx,t,iÞ A Rnþ  L  S,

ð3:17Þ

T

where cðiÞ ¼ ðc1 ðiÞ,c2 ðiÞ, . . . ,cn ðiÞÞ . For any t A L, there exists a positive integer k A Z þ such that t A ðtk1 , tk . By the generalized Itˆo formula, we have Z t þ þ þ Þ, tk1 ,rðtk1 ÞÞþ E LVðxðsÞ,s,rðsÞÞ ds, EVðxðtÞ,t,rðtÞÞ ¼ EVðxðtk1 tk1

ð3:18Þ where LV is a mapping from

Rnþ

K 5 es ds

k1 k2

EVðxðtk2 Þ, tk2 ,rðtk2 ÞÞ Z t Z tk1 K 5 es dsþ K~ E K 5 es ds

b

k1 k2

b

ð3:14Þ

 L  S to R defined by

LVðx,t,iÞ ¼ Vðx,t,iÞ þet ðxT CðiÞbðiÞ þ xT CðiÞAðiÞxÞ þ

N X j¼1

By condition (3.15), there is 2

xT CðiÞAðiÞx ¼ xT ½CðiÞAðiÞ þ AT ðiÞCðiÞx=2 r l9x9 =2:

tk1

EVðxðtk2 Þ, tk2 ,rðtk2 ÞÞ þ K~ E

r M  EVðx0 , t0þ ,r 0 Þ þ K~ E

That is, r1 ¼ 1 a.s. It therefore implies that re ¼ 1 a.s. This completes the proof. &

Vðx,t,iÞ ¼ et cT ðiÞx,

t

tk2

Let k-1, then t-1, one can obtain that

l9max lmax ðCðiÞAðiÞ þ AT ðiÞCðiÞÞ o 0:

K 5 es ds

tk2

Z

þ K~ E

ð3:13Þ

Pðr1 o 1Þ ¼ 0:

tk1

k1

tk1

M  EVðx0 ,r 0 Þ expðK 4 tÞ ZEVðxðt4rk Þ,rðt4rk ÞÞ Z Pðrk r tÞcm ðiÞxm ZPðrk r tÞK 3 k:

t

Z t EVðxðtk1 Þ, tk1 ,rðtk1 ÞÞ þ E K 5 es ds tk1   Z tk1 k1 þ þ þ EVðxðtk2 Þ, tk2 ,rðtk2 ÞÞ þ E K 5 es ds rb

Eqs. (3.9), (3.11) and assumption (H) yield EVðxðt4rk Þ,rðt4rk ÞÞ r

Z

Z

t

Z

t

tk2

K 5 es ds

K 5 es ds

0 t

r M  EVðx0 , t0þ ,r 0 Þ þ K~ K 5 e :

ð3:20Þ

Noting that 9xðtÞ9 rVðx,t,rðtÞÞ=ðK 3 et Þ, therefore we obtain lim sup E9xðtÞ9 rlim supðM  EVðx0 , t0þ ,r 0 Þ þ K~ K 5 et Þ=ðK 3 et Þ t-1

t-1

r K~ K 5 =K 3 , which means (3.16) holds. The proof of Theorem 3.2 is completed. & Theorem 3.2 shows that the property of the ultimate boundedness of the species is because of the limit of the resources. But species may also become extinct under some special circumstances such as resource shortages or major environmental changes. Hence, for system (2.2), we have the following result that all the species will be extinct in mean. Theorem 3.3. Assume that assumption (H) and condition (3.1) hold. In addition, condition   g9max max fbm ðiÞg þ pii ð1K 2 Þ o 0 ð3:21Þ iAS

1rmrn

is satisfied. Then any solution x(t) of system (2.2) will be extinct in mean with the initial value x0 A Rnþ . Proof. It follows from Theorem 3.1 that the unique solution x(t) of system (2.2) will remain in Rnþ for all t A L with probability one. Define Vðx,iÞ ¼

n X

cm ðiÞxm ,

ðx,iÞ A Rnþ  S:

ð3:22Þ

m¼1

pij Vðx,t,jÞ:

For any t A L, there exists k A Z þ such that t A ðtk1 , tk . By the generalized Itˆo formula, we have Z t þ þ EVðxðtÞ,rðtÞÞ ¼ EVðxðtk1 Þ,rðtk1 ÞÞ þ E LVðxðsÞ,rðsÞÞ ds: ð3:23Þ tk1

66

Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

Here LV is a mapping from Rnþ  S to R defined by (3.5). By conditions (2.1), (3.1) and (3.21), for any t A ðtk1 , tk , we have the following estimation: LVðx,iÞ ¼ xT CðiÞbðiÞ þ xT ½CðiÞAðiÞ þ AT ðiÞCðiÞx=2 þ

N X

pij Vðx,jÞ

j¼1 n X

r

bm ðiÞcm ðiÞxm þ

N X

pij Vðx,jÞ þ pii Vðx,iÞ

j ¼ 1,j a i

m¼1

r maxfbm ðiÞgVðx,iÞ þ

N X

pij K 2 Vðx,iÞ þ pii Vðx,iÞ

j ¼ 1,j a i

r maxfbm ðiÞgVðx,iÞpii K 2 Vðx,iÞ þ pii Vðx,iÞ r gVðx,iÞ:

Therefore, we have þ þ Þ,rðtk1 ÞÞ þðgÞ EVðxðtÞ,rðtÞÞ r EVðxðtk1

Z

t

EVðxðsÞ,rðsÞÞ ds,

tk1

ð3:25Þ by inequality (3.24). Using the well-known Gronwall inequality, we obtain þ þ Þ,rðtk1 ÞÞ expfgðttk1 Þg EVðxðtÞ,rðtÞÞ r EVðxðtk1

rb

k1

EVðxðtk1 Þ,rðtk1 ÞÞ expfgðttk1 Þg:

ð3:26Þ

Setting qðtÞ ¼ EVðxðtÞ,rðtÞÞ and taking the Dini-right-upper derivative of q(t), we have 8 þ > D qðtÞ rgqðtÞ for t A ðti1 , ti , i ¼ 1; 2, . . . , > < i1 þ Þ r b qðti1 Þ, qðti1 ð3:27Þ > > : qðt þ Þ ¼ EVðx0 ,r 0 Þ: 0

Eq. (3.27) implies that i1

qðtÞ r b

qðti1 Þ expfgðtti1 Þg,

t A ðti1 , ti , k ¼ 1; 2, . . . : ð3:28Þ

It follows from (3.26), (3.28) and assumption (H) that EVðxðtÞ,rðtÞÞ r

k 1 Y

common pattern involves flying north in the spring to breed in the temperate or Arctic summer and returning in the fall to wintering grounds in warmer regions to the south. Lots of kinds of birds have such behaviors, for example barn swallow, cuckoo, mallard, swan, goose and so on. Many types of fish also have migratory behaviors in nature. Fish usually migrate because of diet or reproductive needs or some other unknown reasons. Examples include forage fish, which often make great migrations between their spawning, feeding and nursery grounds; six species of pacific salmon: chinook, coho, sockeye, chum, pink, and Cherry, which are the best-known anadromous fish; a number of large marine fishes, such as the tuna, migrate north and south annually, following temperature variations in the ocean and so on. Mathematically, all of these species’s dynamical behaviors with migration mentioned above can be modeled simply by a class of differential equations with impulsive diffusion, for example the model (2.2) investigated in the present paper. If the migratory species have three or more patches that can be taken (for a brief stay to supplement energy or inhabitation), dynamical behaviors of the species should be considered in N ðN Z 3Þ patches, and the choice of patches will be stochastic. As far as the number of the species is concerned, two or more species are also involved in our model. Single species impulsive migration phenomenon (such as barn swallow) and two species competition impulsive migration phenomenon (such as chub and Bighead Carp Aristichthys nobilis, which have similar foods, habitats and migration time) and such species’ dynamics can be modeled as follows. To illustrate the validity of the results from our mathematical model, here, we consider a three species predator–prey system with Markovian switching in two different patches; in which, two prey species and one predator species are denoted by x1 ðtÞ, x2 ðtÞ, and x3 ðtÞ, respectively. Species x3 prey on species x1 and x2, however, species x1 and x2 are reciprocal. Let xðtÞ ¼ ðx1 ðtÞ, x2 ðtÞ,x3 ðtÞÞT . Then the impulsive migration system with Markovian switching can be written as 8 _ ¼ diagðx1 ðtÞ,x2 ðtÞ,x3 ðtÞÞ½bðrðtÞÞ þ AðrðtÞÞxðtÞ, xðtÞ > < xðtkþ Þ ¼ Gðrðtk Þ,rðtkþ ÞÞxðtk Þ, > : xðt þ Þ ¼ x 40, rðt þ Þ ¼ r A S, 0

0

bl EVðx0 ,r0 Þ expðgtÞ r M

t A ðtk1 , tk ,

0

0

ð4:1Þ

l¼1

 EVðx0 ,r 0 Þ expðgtÞ,

ð3:29Þ

for all t A L. Noting that 9xðtÞ9 rVðx,iÞ=K 3 , therefore we have lim sup E9xðtÞ9 rlim supM  EVðx0 ,r 0 Þ expðgtÞ=K 3 ¼ 0: t-1

t-1

This completes the proof.

&

Remark 3.1. It can be seen from system (2.2) that the coefficients bðrðtÞÞ and A(r(t)) are only associated with the changing state rðtÞ ¼ i A S, but not related to the changing time t. However, for a fixed patch i, the intrinsic growth rate b(i) and the competitive coefficients A(i) may not always be invariable with the changing time t. Hence, considering the coefficients like b(t,r(t)) and A(t,r(t)) will be more close to the real world, which will be studied in our future work. 4. Examples Bird migrations are very common and are made in response to changes in food availability, habitat or weather. The most

where rðtÞ A S ¼  f1; 2g  is a right-continuous Markov chain gener6 ated by  P ¼ 6, 6, 6 . Hence we have the transfer matrix 0:6 P ¼ 0:4, Choose tk tk1 ¼ 0:05  10 years ¼ 0:5 year (10 0:6, 0:4 . years is the time unit we choose in this example and there are two migrations by the birds every year) and 0 1 0 1 0:5 0:4 0:4 0:6 0:3 0:5 B 0:3 0:4 0:3 C B 0:2 0:5 0:4 C Að1Þ ¼ @ A, Að2Þ ¼ @ A, 0:3

0

1:3 B Gð1; 2Þ ¼ @ 0 0

0:2 0

0:8 0

0:2

0:2 0

1

0 C A, 0:7

0

0:7 B Gð2; 1Þ ¼ @ 0 0

0:1

0 0:7 0

0:3 0

1

0 C A: 0:6

ð4:2Þ

From the selection of the coefficients A(1) and A(2) we note that the relationship between x1 species and x2 species is reciprocal coexistence because the effects of these two species are positive (a12 ð1Þ ¼ 0:4 4 0 and a21 ð1Þ ¼ 0:34 0; a12 ð2Þ ¼ 0:34 0 and a21 ð2Þ ¼ 0:2 4 0). Also note that g 1 ð1; 2Þ ¼ 1:3 4 1, which shows that the number of x1 species is increased after the migration from patch 1 to patch 2. Gð1; 2Þ 4Gð2; 1Þ tells us that the danger of the migration from patch 1 to patch 2 is less than one from patch 2 to patch 1.

Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

If we further choose Cð1Þ ¼ Cð2Þ ¼ diagð5, 10, 10Þ, then we have þ lmax ðCð1ÞAð1Þ þ AT ð1ÞCð1ÞÞ ¼ 1:2633,

and þ lmax ðCð2ÞAð2Þ þ AT ð2ÞCð2ÞÞ ¼ 2:6236:

Hence, l ¼ maxi A f1;2g lmax ðCðiÞAðiÞ þ AT ðiÞCðiÞÞ ¼ 1:2633 o0, which satisfies conditions (3.1) and (3.15). Suppose the species live in patch 1 as the initial state and have a migration in every impulsive time tk , because the transfer probability is 0.6, then for the remaining rate of migration Gði,jÞ, i,j A f1; 2g, we have Q1 Q1 k k=2 o 1, assumption (H) is satisfied. k¼1b r k ¼ 1 ð1:3  0:7Þ Hence, all the conditions of Theorems 3.1 and 3.2 are satisfied, then for any system parameter bðÞ and any given initial value x0 A Rnþ , system (4.1) has a unique solution x(t) on t A L and x(t) will be ultimately bounded in mean. See Fig. 1(a) and (b). Note that the system parameter bðÞ can be chosen as an arbitrary number in Theorem 3.2. It is easy to understand that system (4.1) is ultimately bounded if the intrinsic growth rate bðÞ o0. However, if bðÞ 40, because of the existence of the predator, prey species x1 and x2 are bounded; the loss during the migration and the inner competition guarantees the boundedness of species x3. Consequently, the selection of the intrinsic growth rate bðÞ does not affect the boundedness of the three species, but it affects the size of every species. þ

67

On the extinction in mean of system (4.1), we choose A(1), A(2), G(1,2), G(2,1), C(1), and C(2) as above and 0 1 0 1 0:05 0:03 B 0:02 C B 0:02 C bð1Þ ¼ @ A, bð2Þ ¼ @ A: 0:01 0:01 From the definition of K2 we get K2 ¼1. Therefore, we have   max fbm ðiÞg þ pii ð1K 2 Þ ¼ 0:01 o0, g ¼ max i A f1;2g

1rmr3

then all the conditions of Theorem 3.3 are satisfied. Hence, system (4.1) will be extinct in mean. See Fig. 2(a) and (b). Fig. 2 verifies the validity of Theorem 3.3. Meanwhile, it follows from Fig. 2 that even if there is an increase of the number of the species after the migration (for example, the number of x1 species is increased after the migration from patch 1 to patch 2), it also admits to be extinct as time goes if all the intrinsic growth rates of the species are negative. But, if we change the negative intrinsic growth rates b(1) and b(2) chosen above to positive, i.e. 0 1 0 1 0:5 0:3 B 0:2 C B 0:2 C bð1Þ ¼ @ A and bð2Þ ¼ @ A: 0:1 0:1 Then we have   max fbm ðiÞg þ pii ð1K 2 Þ ¼ 0:5 40, g ¼ max i A f1;2g

1rmr3

3

3.5

2.5

x1

3

x2 x3

2.5

x1

2

x2 x3

1.5

x

x

2

1

1.5

0.5

1

0

0.5 0

–0.5

0

1

2

3

4 5 6 t (10 years)

7

8

9

10

0

1

2

3

4 5 6 t (10 years)

7

8

9

10

1.2 1

1.4

0.8

1.2

x3

x3

1 0.8

0.6 0.4 0.2

0.6

4

0.4 0.2

2 x2

0 3

2.5

2 x1

1.5

1

0

Fig. 1. (a) The ultimate boundedness in mean of the time series of system (4.1) for any bðÞ A Rn and A(1), A(2) taken from (4.2). (b) The ultimate boundedness in mean of the phase of system (4.1) for any bðÞA Rn and A(1), A(2) taken from (4.2).

–5

0 –0.2 2

1.5

1

0.5

0

5

0 x2

x1 Fig. 2. (a) The extinction in mean of the time series of system (4.1) for bð1Þ ¼ ð0:05, 0:02, 0:01ÞT , bð2Þ ¼ ð0:03, 0:02, 0:01ÞT and A(1), A(2) taken from (4.2). (b) The extinction in mean of the phase of system (4.1) for bð1Þ ¼ ð0:05, 0:02, 0:01ÞT , bð2Þ ¼ ð0:03, 0:02, 0:01ÞT and A(1), A(2) taken from (4.2).

Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

which does not satisfy condition (3.21) of Theorem 3.3. We also illustrate the trends of the solutions of system (4.1), see Fig. 3. While, from Fig. 3(a) and (b), we observe that the densities of the three species will tend to zero as t-1, i.e. all the three species will be extinct in mean as time goes despite the positive intrinsic growth rates of the species. However, if we keep the negative growth rates b(1) and b(2) unchanged and transform the competitive coefficients A(1) and A(2) to 0 1 0 1 0:5 1:3 0:5 0:5 1:4 0:4 B 1:1 0:4 0:4 C B 1:2 0:2 0:3 C Að1Þ ¼ @ A: A and Að2Þ ¼ @ 0:2 0:1 0:1 0:3 0:2 0:1 ð4:3Þ

3.5 x1

3

x2 x3

2.5 2 x

68

1.5 1 0.5 0

Other coefficients G(1,2), G(2,1), C(1) and C(2) are not changed, then we have

0

1

2

3

4

þ lmax ðCð1ÞAð1Þ þ AT ð1ÞCð1ÞÞ ¼ 14:5066,

5 6 t (10 years)

7

8

9

1.4

and

1.2

þ lmax ðCð2ÞAð2Þ þ AT ð2ÞCð2ÞÞ ¼ 11:3751:

Thereby, l ¼ maxi A f1;2g lmax ðCðiÞAðiÞ þ AT ðiÞCðiÞÞ ¼ 14:5066 40, which does not satisfy condition (3.1) of Theorem 3.1. But the illustration in Fig. 4(a) and (b) shows that the solutions of system (4.1) are also extinct in mean. Comparing the choice of A(1) and A(2) here to the previous example, we note that the reciprocal relationship between x1 species and x2 species is strengthened and the inner competition þ

1 x3

0.8 0.6 0.4 0.2

3

0 2.5

2.5

x2

1 0.5

0

1

2

3

1.5

1

0.5

0

4

0 x2

Fig. 4. (a) The extinction in mean of the time series of system (4.1) for bð1Þ ¼ ð0:5, 0:2, 0:1ÞT , bð2Þ ¼ ð0:3, 0:2, 0:1ÞT and A(1), A(2) taken from (4.3). (b) The extinction in mean of the phase of system (4.1) for bð1Þ ¼ ð0:5, 0:2, 0:1ÞT , bð2Þ ¼ ð0:3, 0:2, 0:1ÞT and Að1Þ, Að2Þ taken from (4.3).

x3

1.5

0

2

2

x1

x1

2 x

10

4

5

6

7

8

9

10

of all the three species is weakened. But Fig. 4 tells us all the three species will also be extinct in mean as time goes despite these. Figs. 3 and 4 show that even if the conditions of Theorem 3.3 are not satisfied, three species will also be extinct, which bring forward a further question for us to search for some better and exact conditions that guarantee the validity of the Theorem.

t (10 years) 1.4

5. Discussion

1.2

In this paper we have established an n species impulsive migration model with Markovian switching in N different patches. By constructing appropriate Lyapunov functions, we have analyzed the global positivity, the ultimate boundedness and the extinction of the species. This paper tells us that under what environment conditions the species are not only positive but also not explode to infinity at any time and under what environment conditions the species will be extinct as time goes. It is worth noting that the model (stochastic dispersal) studied in this paper is new and firstly presented. In most dispersal models that have been investigated so far, the choice of the species to the patches is fixed, and unalterable. However, many facts show that due to the deterioration of patch of the environment or human intervention, the migratory species move from one patch to another, will not go back to the previous one when natural enemy or winter comes. Or the migratory species find another more suitable habitat than the previous one during the

x3

1 0.8 0.6 0.4 0.2 0 2.5

0 2 2

1.5

1

0.5

0

4

x2

x1 Fig. 3. (a) The extinction in mean of the time series of system (4.1) for bð1Þ ¼ ð0:5, 0:2, 0:1ÞT , bð2Þ ¼ ð0:3, 0:2, 0:1ÞT and A(1), A(2) taken from (4.2). (b) The extinction in mean of the phase of system (4.1) for bð1Þ ¼ ð0:5, 0:2, 0:1ÞT , bð2Þ ¼ ð0:3, 0:2, 0:1ÞT and A(1), A(2) taken from (4.2).

Z. Liu et al. / Journal of Theoretical Biology 307 (2012) 62–69

migration, they will also not go back to the previous one. All of these indicate that, in a long time, the choice of the species to the patches during the species diffusion (or migration) is alterable. The model presented in this paper just makes up this point. Besides, Remark 2.2 tells us another message that we allow the remaining rate gm(i,j) of some species m is larger than unity, but Q k only need the assumption 1 k ¼ 1 b r M o 1 is satisfied, which has not been involved in the previous literature. These two points show that our model presented in this paper is valuable and worthy of studied further. But unfortunately, due to shortage of the analysis techniques on the stochastic model, the permanence of the species has not been studied at the present paper. This is still an open problem. Meanwhile, we have mainly considered the case that the coefficients A and b are constants in a given patch, which do not change with time t as well as the population densities x. This assumption is a little rough. We have noted that the coefficients A and b may also be affected by the changing time (Remark 3.1) and population densities, which can be regarded as a function with time t and population densities x. Then the first equation of our model will take the form _ ¼ diagðx1 ðtÞ,x2 ðtÞ, . . . ,xn ðtÞÞ½bðt,x,rðtÞÞ þ Aðt,x,rðtÞÞxðtÞ: xðtÞ This is also still an open problem and we will study this model in our future work.

Acknowledgment I am very grateful to the two referees for their very helpful suggestions and careful reading of the manuscript. This research was supported by National Basic Research Program of China (2010CB732501) and the National Natural Science Foundation of PR China (10961022 and 10901130). The National Natural Science Foundation of China (Grant No. 31170338 and Grant no. 30970305). References Beretta, E., Takeuchi, Y., 1987. Global stability of single-species diffusion Volterra models with continuous time delays. Bull. Math. Biol. 49, 431–448.

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