A note on asymptotic behaviors of stochastic population model with Allee effect

Applied Mathematical Modelling 35 (2011) 4611–4619

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A note on asymptotic behaviors of stochastic population model with Allee effect Qingshan Yang ⇑, Daqing Jiang School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, PR China

a r t i c l e

i n f o

Article history: Received 31 March 2010 Received in revised form 6 March 2011 Accepted 15 March 2011 Available online 23 March 2011

a b s t r a c t This paper analyzes the asymptotic behaviors of the stochastic population model with the Allee effect. We apply the Feller’s test to obtain the criteria of the asymptotic behaviors for this model and compare our results with those obtained by Lyapunov functions. At last, the numerical simulations conform our results. Ó 2011 Elsevier Inc. All rights reserved.

Keywords: Allee effect Asymptotic stability Feller’s test Itô’s formula

1. Introduction The increasing development in transportation accelerates the invasions of alien species, which may cause considerable threats to the world’s ecosystems [1–4]. As the alien species cause catastrophic ecological impacts, artificial control of these species is needed. A treatment of such problems is eradication, which refers to total elimination of alien species from a geographical area, which usually requires very costly efforts. In contrast to eradication, considerable progress in conservation biology has been made to understand the population biology of low-density and their extinction (e.g. [5–7]), and it is revealed that the Allee effect is one of the most important factors affecting the low-density population, which is pertinent to extinction via eradication. The Allee effect is first proposed by Allee and his collaborators in 1931 [8]. It describes a scenario that populations at low numbers are affected by a relationship between population growth rate and density. Because of the decrease in reproduction or survival when conspecific individuals are not numerous enough, Allee effect could cause the growth rate negative; on the other hand, species’ ranges are the results of successful past invasion. Thus, much attention is paid to Allee dynamics in the research on the extinction of the ending species and the formulation of the species’ geographic range (e.g. [9,10]). Ackleh et al. considered the following population model with the Allee effect in [11]:

   dNt Nt Nt ¼ cNt 1  1 ; dt T K

ð1:1Þ

N0 ¼ n0 ; where Nt is the population size at time t, T is the minimal population size, K is the environmental carrying capacity and c is the intrinsic growth rate. However, given that population systems are often subject to environmental noise [12], it is ⇑ Corresponding author. E-mail address: [email protected] (Q. Yang). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.03.034

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important to discover whether the presence of a such noise affects these results. Suppose the parameter c is stochastically perturbed, with

c ! c þ aB_ t ; where B_ t is white noise and a represents the intensity of the noise. Then this environmentally perturbed system may be described by the Itô equation

   Nt Nt dNt ¼ Nt 1   1 ðcdt þ adBt Þ; T K

ð1:2Þ

N0 ¼ n0 ; where Bt is the 1-dimensional standard Brownian motion with B0 = 0, and n0 is positive constant. In paper [13], authors prove the existence and uniqueness of positive solutions of (1.2), and discusses the stability in L2 and L1 norm by Lyapunov functions. In this paper, we utilize the canonical probability methods [14,15] to illustrate the asymptotic behaviors in almost sure sense according to the drift and diffusion coefficients and compare our results with those in [13]. The paper is arranged as follows. In Section 2, we first introduce a useful result on Feller’s test and discuss the asymptotic behaviors of (1.2) under different cases. The comparison between our results and those in [13] is also made. In Section 3, we represent the numerical simulations to validate our results.

2. Asymptotic behaviors of positive solution In this section, we discuss the asymptotic behaviors of solutions in (1.2) in the criteria of drift and diffusion coefficients, and compare our results with those in [13]. 2.1. A result on Feller’s test Before analyzing the asymptotic behavior of (1.2), we first introduce a result on the Feller’s test [14] which is applied in Section 2.2. Let I = (l, r), 1 6 l < r 6 +1. We consider the following one-dimensional, time-homogeneous stochastic differential equation

dX t ¼ bðX t Þdt þ rðX t ÞdBt ;

ð2:1Þ

X0 ¼ x and assume that the coefficients

ð1Þ

r : I ! R, b : I ! R satisfy the following conditions:

2

r ðxÞ > 0; 8x 2 I;

ð2Þ 8x 2 I;

Z

9e > 0;

xþe

xe

1 þ jbðyÞj dy < 1; r2 ðyÞ

ð2:2Þ

and fixing some c 2 I, the scale function is defined:

pðxÞ ¼

Z c

x

 Z exp 2

n c

 bðfÞdf dn; r2 ðfÞ

x 2 R:

Now, we present the following useful Proposition. Proposition 2.1 [15]. Assume that conditions (1) and (2) hold, and let X be a nonexplosive solution of (2.1) in I, with X0 = x 2 I. We distinguish four cases: (a) p(l+) = 1, p(r) = +1. Then

    P sup X t ¼ r ¼ P inf X t ¼ l ¼ 1 tP0

tP0

and for any y 2 I, we have

Pf9t 2 ð0; 1Þ; X t ¼ yg ¼ 1: (b) p(l+) > 1, p(r) = +1. Then

n o P lim X t ¼ l ¼ Pfsup X t < rg ¼ 1: t!1

tP0

(c) p(l+) = 1, p(r) < +1. Then

Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

4613

  n o P lim X t ¼ r ¼ P inf X t > l ¼ 1: t!1

tP0

(d) p(l+) > 1, p(r) < +1. Then

n o n o pðrÞ  pðxÞ : P lim X t ¼ l ¼ 1  P lim X t ¼ r ¼ t!1 t!1 pðrÞ  pðlþÞ 2.2. Asymptotic behaviors of positive solution In [13], the existence and uniqueness theorem is proved and it is known that if the initial value 0 < n0 < T, then 0 < Nt < T for any t P 0 and if T < n0 < K, then T < Nt < K for any t P 0. It reveals that the initial value determines the boundary of the solution of Eq. (1.2). If n0 = 0, T or K, then Nt  0, T or K, respectively by the uniqueness of solutions. Therefore, we just consider the asymptotic stability under the nontrivial cases. Now, we apply Proposition 2.1 to show the asymptotic behaviors of positive solutions of Eq. (1.2). Theorem 2.1 (1) When 0 < n0 < T, we have that if a2c2 6 1, then

n o n o P lim Nt ¼ T ¼ P inf Nt > 0 ¼ 1; t!1

t

if 1 6 a2c2 < KT , then K

n o n o pðþ1Þ  pðxÞ ; P lim Nt ¼ 0 ¼ 1  P lim Nt ¼ T ¼ t!1 t!1 pðþ1Þ  pð1Þ if

2c a2

P KT , then K

  n o P lim Nt ¼ 0 ¼ P sup Nt < T ¼ 1: t!1

t

(2) When T < n0 < K, we have that if a2c2 6  KT , then T

  n o P lim Nt ¼ T ¼ P sup Nt < K ¼ 1; t!1

if

 KT T

t

2c

6 a2 <

KT , K

n

then

P lim Nt ¼ K t!1

if

2c a2

o

n o pðþ1Þ  pðxÞ ; ¼ 1  P lim Nt ¼ T ¼ t!1 pðþ1Þ  pð1Þ

P KT , then K

n o n o P lim Nt ¼ K ¼ P inf N t > T ¼ 1: t!1

t

Nt Proof. (1) We first consider the case when 0 < n0 < T. Let X t ¼ log TN and applying Itô formula, we see that t

dX t ¼ UðX t Þ½WðX t Þdt  adBt ; where UðxÞ ¼

ðKTÞex þK , Kð1þex Þ

WðxÞ ¼

a2 2

ð2:3Þ

 UðxÞ 

ex 1 ex þ1

 c.

n o R x ðyÞ Rx Let qðxÞ ¼ exp 2 0 aW 2 UðyÞ dy , pðxÞ ¼ 0 qðyÞdy, and by computation, we have that

2

Z

0

x









WðyÞ 2c 2cT K 2cT 2K  T  2 lnð1 þ ex Þ  2 þ 2 ln 2 dy ¼ 1 þ 2 x þ 2  ln ex þ ln a K T a2 UðyÞ a ðK  TÞ a ðK  TÞ K  T ¼ ln

2c  x  2cT K a2 ðKTÞ ð1þa2 Þx e þ KT e

ðex

þ 1Þ

2



By the first equality in (2.4), we note that qðxÞ  exp

2cT 2K  T þ 2 ln 2: ln a2 ðK  TÞ K  T n

2c a2

ð2:4Þ

o x when x ? 1. Therefore, p(+1) < +1 if and only if a2c2 < KT  KT . K K

Similarly, the second equality in (2.4) implies that p(1) > 1 if and only if By Proposition 2.1, we distinguish three cases: if a2c2 6 1, then

2c a2

> 1.

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Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

n o n o P lim X t ¼ þ1 ¼ P inf X t > 1 ¼ 1; t!1

2c

if 1 < a2 <

KT , K

t

then

n

o n o pðþ1Þ  pðxÞ ; P lim X t ¼ 1 ¼ 1  P lim X t ¼ þ1 ¼ t!1 t!1 pðþ1Þ  pð1Þ if

2c a2

P KT , then K

  n o P lim X t ¼ 1 ¼ P sup X t < þ1 ¼ 1: t!1

t

By definition of Xt and discussion above, we have that if

2c a2

6 1, then

n o P lim Nt ¼ T ¼ Pfinf Nt > 0g ¼ 1; t!1

2c

if 1 6 a2 <

KT , K

t

then

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2

0

0.5

1

1.5

2

2.5

2

2.5

Fig. 1. y = 0.5, c = 1, a = 1, n = 500, L = 5.

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0

0.5

1

1.5

Fig. 2. y = 0.5, c = 0.5, a = 1, n = 500, L = 5.

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Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

n o n o pðþ1Þ  pðxÞ ; P lim Nt ¼ 0 ¼ 1  P lim Nt ¼ T ¼ t!1 t!1 pðþ1Þ  pð1Þ if

2c a2

P KT , then K

  n o P lim Nt ¼ 0 ¼ P sup Nt < T ¼ 1: t!1

t

t (2) When T < n0 < K, let X t ¼ log KN and applying Itô formula again, we have that N t T

b ðX t Þdt  adBt ; b ðX t Þ½ W dX t ¼ U b ðxÞ ¼ b ðxÞ ¼ where U  W By computation, we have that KT TK

2

Z 0

x

KþTex , 1þex

ð2:5Þ

2 ca

2

b ðxÞ  U



ex 1 . ex þ1











WðyÞ 2cT 2c K 2c K  2 lnð1 þ ex Þ  2 ln 1 þ þ 2 ln 2 dy ¼ 1 þ 2 x þ 2  ln ex þ a a T T a2 UðyÞ a ðK  TÞ 2c

¼ ln

2c T Þx a2 ðKTÞ

ð1þ

ðex þ KT Þa2 e



ðex þ 1Þ2

  K þ 2 ln 2 ln 1 þ a2 T

2c

ð2:6Þ

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

2

2.5

Fig. 3. y = 0.5, c = 0, a = 1, n = 500, L = 5.

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

Fig. 4. y = 0.5, c = 0.5, a = 1, n = 500, L = 5.

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Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

By the same way as in the proof of the first case, it is seen that p(+1) < +1 if and only if a2c2 < KT and p(1) > 1 if and only K if a2c2 >  KT . T By Proposition 2.1 again, we have that if a2c2 6  KT T , then

n o n o P lim X t ¼ þ1 ¼ P inf X t > 1 ¼ 1; t!1

t

if  KT < a2c2 < KT , then T K

n o n o pðþ1Þ  pðxÞ ; P lim X t ¼ 1 ¼ 1  P lim X t ¼ þ1 ¼ t!1 t!1 pðþ1Þ  pð1Þ if

2c a2

P KT , then K

  n o P lim X t ¼ 1 ¼ P sup X t < þ1 ¼ 1: t!1

t

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0 0

0.5

1

1.5

2

2.5

Fig. 5. y = 0.5, c = 0.25, a = 1, n = 500, L = 5.

1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0

0.5

1

1.5

Fig. 6. y = 1.5, c = 1, a = 1, n = 500, L = 5.

2

2.5

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Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

Therefore, we have that by definition of Xt if a2c2 6  KT T , then

  n o P lim Nt ¼ T ¼ P sup Nt < K ¼ 1; t!1

if

 KT T

2c

< a2 <

t KT , K

n

then

P lim Nt ¼ K t!1

if

2c a2

o

n o pðþ1Þ  pðxÞ ¼ 1  P lim Nt ¼ T ¼ ; t!1 pðþ1Þ  pð1Þ

P KT , then K

n o n o P lim Nt ¼ K ¼ P inf N t > T ¼ 1: t!1

t

The proof of Theorem 2.1 is completed. h Since for any t P 0, Nt is uniformly bounded, convergence in Lp norm is obtained by Theorem 2.1 and the dominated convergence theorem for any p > 0. 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0

0.5

1

1.5

2

2.5

2

2.5

Fig. 7. y = 1.5, c = 0.5, a = 1, n = 500, L = 5.

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 0

0.5

1

1.5

Fig. 8. y = 1.5, c = 0, a = 1, n = 500, L = 5.

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Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

Corollary 2.1 (1) When 0 < n0 < T, we have that if a2c2 6 1, then for any p > 0,

lim EjNt  Tjp ¼ 0;

t!1

if

2c a2

P KT , then for any p > 0, K

lim EjNt jp ¼ 0:

t!1

(2) When T < n0 < K, we have that if a2c2 6  KT , then for any p > 0, T

lim EjNt  Tjp ¼ 0;

t!1

if

2c a2

P KT , then for any p > 0, K

lim EjNt  Kjp ¼ 0:

t!1

Remark 2.1. In [13], authors discuss the stability of (1.2) by Lyapunov functions, and conclude that 2

(1) in the case of 0 < n0 < T, if c < a2, then limt?1E(Nt  T)2 = 0 and if c > a2 , then limt!1 EN 2t ¼ 0. a2 a2 (2) in the case of T < n0 < K, if c <  ðKTÞ , then limt?1E(Nt  T)2 = 0 and if c > ð3KTÞðKTÞ , then limt?1E(Nt  K)2 = 0. T 2TK Comparing with the results above, we could obtain the accurate results by the canonical probability methods. However, since Feller’s test is just valid for one-dimensional, time-homogeneous stochastic differential equation, the Lyapunov functions seems more efficient in high dimensional cases. 3. Numerical simulations At last, we give the simulation results to substantiate the analytical findings. The numerical simulation is given by the following stochastic Runge–Kutta numerical difference scheme [16]:

dY t ¼ bðY t Þdt þ rðY t ÞdBt ; Y 0 ¼ x; ðnÞ

ð3:1Þ

ðnÞ ðnÞ þ bðY ðnÞ Y mþ1 ¼ Y m m ÞMt n þ rðY m Þ  nm þ   pffiffiffiffiffiffiffiffi  i 1 1 h  ðnÞ ðnÞ pffiffiffiffiffiffiffiffi r Y ðnÞ þ r Y r Y  Mt ðn2m  Mtn Þ;  n m m m 2 Mtn

   where bðyÞ ¼ cy 1  Ty Ky  1 ,



rðyÞ ¼ ay 1  Ty

 y K

 ðnÞ    1 , Y 0 ¼ y, Mt n ¼ nL, {nm, m 6 n} is i.i.d, nm  N 0; nL .

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3

0

0.5

1

1.5

Fig. 9. y = 1.5, c = 0.5, a = 1, n = 500, L = 5.

2

2.5

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Q. Yang, D. Jiang / Applied Mathematical Modelling 35 (2011) 4611–4619

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3

0

0.5

1

1.5

2

2.5

Fig. 10. y = 1.5, c = 0.25, a = 1, n = 500, L = 5.

Using the numerical simulation method given out above and Matlab software, we get the simulations of Eq. (1.2) under suitable parameters. The simulation results are shown in the following figures. In all simulations, we set T = 1, K = 2 and give the simulations in Figs. 1–5 when 0 < y < T, and also the simulations in Figs. 6–10 when T < y < K. In Fig. 1, we choose parameters such that a2c2 < 1, and all five curves approach to 1. In Fig. 2, the parameter a2c2 ¼ 1, and there are four curves approaching to 1, one seems uncertain. In Fig. 3, the parameter 1 < a2c2 < 0:5, and there are two curves approach 1, and three approach to 0. In Figs. 4 and 5, the parameter 1 < a2c2 > 0:5 and 1 < a2c2 ¼ 0:5 respectively, and all ten curves approach to 0. In Fig. 6, the parameter a2c2 < 1, and there are three curves approach to 1, two is uncertain. In Fig. 7, the parameter a2c2 ¼ 1, and there are four curves approach to 1, one is uncertain. In Fig. 8, the parameter 1 < a2c2 < 0:5, and there are three curves approach to 2, two approach to 1. In Fig. 9, the parameter 1 < a2c2 > 0:5, and all curves approach to 2. In Fig. 10, the parameter a2c2 ¼ 0:5, and there are three curves approach to 2, two are uncertain. Acknowledgements This work was supported by NSFC of China (No. 10971021), the Ministry of Education of China (No. 109051), the Ph.D. Programs Foundation of Ministry of China (No. 200918) and Young Teachers of Northeast Normal University (No. 20090104). References [1] A.M. Liebhold, W.L.D. MacDonald, D. Bergdahl, V. Mastro, Invasion by exotic forest pests: a threat to forest ecosystems, For. Sci. Mon. 30 (1995) 1–49. [2] P.M. Vitousek, C.M. D’Antonio, L.L. Loope, R. Westbrooks, Biological invasions as global environmental change, Am. Scientist 84 (1996) 468–478. [3] I.M. Parker, D.S. Simberloff, W.M. Lonsdale, M. Goodell, P.M. Wonham, P.M. Kareiva, M.H. Williamson, B. VonHolle, P.B. Moyle, J.E. Byers, L. Goldwasser, Impact: toward a framework for understanding the ecological effects of invaders, Biol. Invasions 1 (1999) 3–19. [4] D.S. Simberloff, R.N. Mack, W.M. Lonsdale, H.F. Evans, M. Clout, F.A. Bazzaz, Biotic Invasions: Causes, Epidemiology, Global Consequences and Control, Issues in Ecology, vol. 5, Ecological Society of America, Washington, DC, 2000. [5] R. Lande, Risks of population extinction from demographic and environmental stochasticity and random catastrophes, Am. Nat. 142 (1993) 911–927. [6] S. Nee, How populations persist?, Nature 367 (1994) 123–124 [7] P.A. Stephens, W.J. Sutherland, Consequences of the Allee effect for behavior ecology and conservation, Trends. Ecol. Evol. 14 (1999) 401–405. [8] W.C. Allee, Animal Aggregations, A Study in General Sociology, University of Chicago Press, Chicago, 1931. [9] T.H. Keitt, M.A. Lewis, R.D. Holt, Allee Effects, Invasion pinning, and species borders, Am. Nat. 157 (2) (2001) 203–216. [10] R.D. Holt, T.H. Keitt, M.A. Lewis, B.A. Maurer, M.L. Taper, Theoretical models of species’ borders: single species approaches, OIKOS 108 (2005) 18–27. [11] A. Ackleh, L. Allen, J. Carter, establishing a beachhead: a stochastic population model with an Allee effect applied to species invation, Theor. Popul. Biol. 71 (2007) 290–300. [12] X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in population dynamics, Stochastic Process. Appl. 97 (2002) 95– 110. [13] M. Krstic´, M. Jovanovic´, On stochastic population model with the Allee effect, Math. Comput. Model. 52 (2010) 370–379. [14] S. Xiping, W. Yongji, Stability analysis of a stochastic logistic model with nonlinear diffusion term, Appl. Math. Model. 32 (2008) 2067–2075. [15] I. Karatzas, S. Shreve, Brownian Motion and Stochastic Calculus, Springer Verlag, New York, 1988. [16] E.P. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations, Springer Verlag, Berlin, 1992.