Dynamics of a stochastic predator-prey model with Beddington DeAngelis functional response

Transcription

1 SCIENTIA Series A: Mahemaical Sciences, Vol , Universidad Técnica Federico Sana María Valparaíso, Chile ISSN c Universidad Técnica Federico Sana María 22 Dynamics of a sochasic predaor-prey model wih Beddingon DeAngelis funcional response Hoang The Tuan a, Nguyen Hai Dang b and Van Khu Vu b Absrac. This paper is concerned wih a sochasic predaor-prey model wih Beddingon DeAngelis funcional response. The exisence of he global soluion and he ulimae boundedness of momens of he soluion are proved. Moreover, we also esimae he average in ime of he soluion.. Inroducion In recen years, here have been many aemps o invesigae sochasic ecomodels. For a sysemaic review, we refer o L. J. S. Allen []. In [3] and [9], auhors gave lower and upper growh rae of he soluion o he sochasic Loka-Volerra model. x = diag x,, x n b Ax d σxdb, where x = x,, x n T, b = bi n, A = a ij n n, σ = σ ij n n. They also esimaed he average in ime of he momen of he soluion. Especially, he asympoic behavior of he classical predaor-prey perurbed by whie noise was sudied more deail in [4], [5] and []. On anoher direcion, he deerminisic predaor-prey model wih Beddingon- DeAngelis funcional response dx = x Cy A Bx d.2 dy = y D α βx γy d Ex α βx γy have received many aenion. Here, x and y respecively represen he densiies of he prey and he predaor a ime, while A is he inrinsic growh rae of he prey, D is he moraliy rae of he predaor,c is he feeding parameer, E is he conversion efficiency parameer, B and F are he inraspecies inerference parameers, β is a 2 Mahemaics Subjec Classificaion. Primary 6H, 92D25. Key words and phrases. Predaor prey model; Beddingon DeAngelis funcional response; Brownian moion; sochasic differenial equaion. 75

2 76 HOANG THE TUAN, NGUYEN HAI DANG AND VAN KHU VU food weighing facor, ha correlaes inversely wih he prey densiy a which feeding sauraion occurs and α is a normalizaion coefficien ha relaes he densiies of he predaor and prey o he environmen in which hey inerac. All hese parameers are posiive. In [2], [6] and [7], a complee classificaion of he global dynamics of his model was done. However, so far here seems no sudy on he sochasic predaor-prey model wih Beddingon-DeAngelis funcional response. Suppose he funcional response is perurbed by whie noise, hen Equaions.2 becomes a sochasic differenial equaions.3 dx = x Cy σxy A Bx d dy = y D α βx γy Ex α βx γy α βx γy db ρxy d α βx γy db, where σ and ρ are real consans. The goal of his paper is o prove ha Equaions.3 have he following properies Wih probabiliy, he soluion wih he iniial value x, y R 2 will remain in R 2. 2 Every momen of he soluion is ulimaely bounded. 3 The average in ime of he oal populaion densiy is lower and upper bounded by posiive consans, ha is, here exis wo posiive consans l and L such ha, l lim inf xs ys ds lim sup xs ys ds L, a.s. These resuls reveals he imporan role ha environmenal noise plays in populaion dynamics. In [], he auhors show ha he predaor or boh species will be exincive in some cases. Bu, in his paper, he auhors proved ha a iny amoun of sochasic noise can make boh species survive. The paper is organized as follows. The firs secion, auhors inroduce he sochasic predaor-prey model wih Beddingon-DeAngelis funcional response. The secion 2, we sudy he non-explosion of he soluion o.3 and momen esimaion. Secion 3 give a esimaion for a growh rae of he soluion and illusrae his resul by numerical soluion. Moreover, we also esimae he average in ime of x y. In secion 4, we give some conclusions. 2. The non-explosion of he soluion and momen esimaion Throughou his paper, we le Ω, F, {F }, P wih he filraion {F } saisfying he usual condiions, i.e., i is increasing and righ coninuous while F conains all P null ses. Le B, is a scalar Brownian moion defined on his probabiliy space. Denoe by R 2 he se {x, y R 2, x >, y > }. Obviously, he coefficiens of Equaions.3 are locally Lipschiz coninuous bu do no saisfy he linear growh condiion. However, we have

3 DYNAMICS OF A STOCHASTIC PREDATOR-PREY MODEL 77 Theorem 2.. For any given iniial value x, y R 2, here is a unique soluion x o Equaions.3 on and his soluion will remain in R 2 almos surely. Proof. Since he coefficiens of Equaions.3 are locally Lipschiz coninuous, for any given iniial value x, y R 2, here is a unique local soluion x on [, τ e, where τ e is he explosion ime. The soluion is global if τ e = a.s. For each k N, we define he sopping ime τ k = inf{, x y < k or x y > k} wih convenion inf Ø =. τ k is increasing, so we pu τ = lim τ k. Denoe k by V x, y he funcion x y ln x ln y. Obviously V x, y for all x >, y >. By Io formula, [ x Cy dv x, y = A Bx α βx γy y Ex D α βx γy 2 2 σy ρx 2 α βx γy ] 2 d σx y ρy x db. α βx γy I is easy o show ha, here exis wo consans K, K 2 such ha x A 2 2 Cy Ex σy ρx y D K x, and α βx γy α βx γy 2 α βx γy 2 K 2 for all x >, y >. Thus, here exiss K > such ha Cy Ex x A Bx y D α βx γy α βx γy 2 2 σy ρx 2 α βx γy 2 K x K 2 bx 2 < K x, y R 2. Consequenly, for any >, we have 2. EV x τ k, y τ k =V x, y E τk [ xs Cys A Bxs α βxs γys ys 2 2 Exs σys ρxs D α βxs γys 2 α βxs γys ] 2 ds τk V x, y E Kds = V x, y KE τ k. Suppose τ < wih a posiive probabiliy. I implies he exisence of wo posiive consans ɛ and T > such ha P{τ < T } > 2ɛ. Hence, here is k N such ha P{τ k < T } > ɛ for any k > k. Pu m k = k ln k k ln k, hen m k as k and V τ k m k. By

4 78 HOANG THE TUAN, NGUYEN HAI DANG AND VAN KHU VU 2., m k ɛ m k P{τ k < T } EV T τ k KET τ k V x, y KT V x, y k > k. Le k we ge a conradicion KT V x, y. Hence, τ = wih probabiliy. The proof is complee. Theorem 2.2. For any θ >, here exiss K θ > such ha for any given iniial value x >, y >, lim sup E x θ s y θ s K θ. Proof. By Lyapunov s inequaliy E X r r E X p p < r < p, i is sufficien o prove he heorem for θ > 2. Pu V x, y = Ex Cy θ. In view of Io formula, 2.2 de θd V x, y = e θds LV x, y θdv x, y d where θe θd Ex Cy θ Eσ Cρxy α βxs γys db, LV x, y = θexcy θ EAx EBx 2 θ CDy 2Ex Dy Eσ Cρ 2 x 2 y 2 α βx γy 2. θ Eσ Cρ 2 x 2 y 2 Noe ha 2Ex Cy α βx γy 2 θ Eσ Cρ 2 2E γ 2 x x >, y >. Therefore, M > can be found such ha for all x >, y >, 2.3 θdv x, y LV x, y =θex Cy θ EA Dx EBx 2 D θ Eσ Cρ 2 x 2 y 2 2Ex Cy α βx γy 2 M. Define he sopping imes k = inf{ : x y > k}, k N. By virue of 2.3, aking expecaions on boh sides of 2.2 yields Ee θd k V x k, y k V x, y M θd E e θd k Le k, we ge or which implies e θd EV x, y V x, y M θd eθd, EV x, y V x, y e θd M θd e θd, lim sup EV x, y M θd. By combining his wih he inequaliy x θ y θ E θ C θ V x, y, he proof is complee. The following resul is a direc corollary of his heorem.

5 DYNAMICS OF A STOCHASTIC PREDATOR-PREY MODEL 79 Corollary 2.. Equaions.3 is sochasically ulimaely bounded in he sense ha for any ɛ >, here is a posiive consan H = He such ha for any iniial value x, y R 2, he soluion has he propery ha lim sup P{x y > H} < ɛ. 3. Pahwise esimaion In his secion, we always denoe by x, y he soluion o Equaions.3 wih he iniial value x, y R 2. The following heorem give a esimaion for he growh rae of he soluion. Theorem 3.. lim sup Ex Cy ln Eσ Cρ2 EBγ 2. Proof. Le < θ < D and pu V x, y = Ex Cy. In view of Io formula, 3. e θ V x, y =V x, y e θs Exs A θ Bxs CD θys ds θs Eσ Cρxsys e α βxs γys dbs. Pu M = Eσ Cρxsys eθs α βxs γys dbs. I is also known ha M is a realvalued coninuous local marigale vanishing a = wih quadraic form M, M = e 2θs Eσ Cρxsys 2 α βxs γys 2 ds. For each λ >, i follows from he exponenial marigale inequaliy ha P { sup M λe θk M, M > eθk k λ ln k} k 2. By he Borel-Canelli lemma, here exiss an Ω Ω wih PΩ = such ha for any ω Ω, here exiss a k = k ω N saisfying M λe θk M, M < eθk λ ln k k, k k. On he oher hand, for any k, e θk e θs Eσ Cρxsys 2 e θs Eσ Cρ 2 M, M 2 ds α βxs γys γ 2 x 2 sds, which resuls in 3.2 M e θs λeσ Cρ 2 γ 2 x 2 sds eθk λ ln k k, k k.

6 8 HOANG THE TUAN, NGUYEN HAI DANG AND VAN KHU VU I is easy o show ha for λ < EBγ 2 Eσ Cρ 2, Eσ Cρ2 3.3 K λ = sup ExA θ Bx D θy λ x,y R 2 γ 2 x 2 <. Thus, from 3., 3.2 and 3.3 we have e θ V x, y V x, y K λ Consequenly, for all k k and k, e θs ds eθk λ ln k k, k k. V x, y V x, y e θ K λ θ e θ eθk ln k. λ Obviously, if k k and k k, he following inequaliy holds V x, y ln e θ lnk Le k, we ge lim sup leing θ, λ EBγ2 Eσ Cρ 2. V x, y K λ K λ θ θ lnk eθ λ V x, y ln ln k lnk. eθ. The required asserion follows from λ Example 3.2. We illusrae he above resuls by he following example. Consider.3 wih A=, B=2,C=, D=,E=,F=2, α = β = γ =, σ = ρ = 2. This numerical soluion is displayed in figure. Figure. The graph of funcion Ex Cy ln In he sequel, we esimae he average in ime of x y. Theorem 3.3. There exiss L > such ha lim sup xs ys ds L a.s.

7 DYNAMICS OF A STOCHASTIC PREDATOR-PREY MODEL 8 Proof. Pu V x, y = Ex Cy. By Io formula, 3.4 [ ] V x, y V x, y = Exs A Bxs CDys ds M = D V xs, ysds M ɛ M, M, where M = Eσ Cρxsys α βxs γys vanishing a = wih quadraic form M, M = Exs A D Bxs 2 Eσ Cρxsys ɛ 2 ds α βxs γys db is a real-valued coninuous local marigale I is easy o chose ɛ sufficienly small such ha Eσ Cρxsys 2 α βxs γys 2 ds. EBx 2 Eσ Cρxy2 ɛ 2 ɛx 2 x, y R 2, α βx γy which implies 2 Eσ Cρxy 3.5 M 2 = sup ExA D Bx ɛ x,y R 2 α βx γy 2 <. On he oher hand, V x, y > x, y R 2. Thus, i follows from 3.4 and 3.5 ha 3.6 D V xs, ysds V x, y M 2 M ɛ M, M. Applying he exponenial maringale inequaliy [8, Theorem 7.4, p.44] yields, { P sup {M ɛ M, M } > ln k } k ɛ k 2 By Borel- Canelli lemma, for almos all ω, here exiss a number k = k ω such ha for all k > k and k, M ɛ M, M < ln k ɛ, which implies ha for k k As a resul, 3.7 lim sup ln k M ɛ M, M k ɛ M ɛ M, M lim k ln k k ɛ =.

8 82 HOANG THE TUAN, NGUYEN HAI DANG AND VAN KHU VU Combining 3.6 and 3.7 ges lim sup lim sup V xs, ysds M 2, a.s. Thus D xs ys ds L a.s, where L = M 2 D min{e, C. The proof is complee. } Theorem 3.4. There exiss l > such ha lim inf Proof. By Io formula xs ys ds l a.s ln x ln x = σ 2 y 2 s α βxs γys 2 ds Cys A Bxs ds α βxs γys σys α βxs γys dbs M = wih quadraic form σys α βxs γys dbs is a coninuous marigale vanishing a = M, M = σ 2 y 2 s α βxs γys 2 ds σ 2 γ 2. By he srong law of large numbers [8, Theorem.3.4, p.2], M 3.9 lim a.s. On he oher hand, i follows from Theorem 3. ha 3. lim sup Combining 3.8, 3.9 and 3. yields lim inf ln x ln x a.s. Cys Bxs α βxs γys σ 2 y 2 s 2 2 ds A. α βxs γys Noe ha an H > can be found such ha Cy α βx γy σ 2 y Hy x, y R 2. α βx γy Thus, lim inf Bxs Hys ds A wih probabiliy. This inequaliy resuls in he required asserion.

9 DYNAMICS OF A STOCHASTIC PREDATOR-PREY MODEL Conclusion Compare wih Equaions. which is sudied in [3] and [9], Equaions.3 has more desired properies. I is due o he linear growh rae of he funcion xy. Namely, while Mao e al [9] show ha for θ < 3, here exiss α βx γy M θ such ha lim sup E [ n x θ i ] < M θ x R, n i= he soluion o Equaions.3 saisfies lim sup E[x θ y θ ] < K θ θ >, where K θ is only dependen on θ. In addiion, we show ha lim sup Eσ Cρ 2 ln n i=, insead of he esimaion lim sup x i EBγ 2 Sam [3]. The esimaion < l lim inf ys ds L < a.s. is also an ineresing propery. ln xs ys ds lim sup Ex Cy ln given by Du and xs References [] L. J. S. Allen, An Inroducion To Sochasic Processes Wih Applicaions o Biology, Pearson Educaion Inc., Upper Saddle River, New Jersey 23. [2] R.S. Canrell, C. Cosner, On he dynamics of predaorprey models wih he BeddingonDeAngelis funcional response, J. Mah. Anal. Appl [3] Du, N. H., Sam, V. H., Dynamics of a sochasic Loka-Volerra model perurbed by whie noise, J. Mah. Anal. Appl., 32426, [4] R. Z. Khasminskii and F. C. Klebaner, Long erm behavior of soluions of he Loka-Volerra sysem under small random perurbaions, Ann. Appl. Probab. 3, [5] R. Z. Khasminskii and F. C. Klebaner and R. Lipser, Some resuls on he LokaVolerra Model and is small random perurbaions, Aca Appl. Mah , 2 26 [6] T.W. Hwang, Global analysis of he predaorprey sysem wih BeddingonDeAngelis funcional response, J. Mah. Anal. Appl , [7] T.W. Hwang, Uniqueness of limi cycles of he predaorprey sysem wih BeddingonDeAngelis funcional response, J. Mah. Anal. Appl , [8] Mao, X., Sochasic Differenial Equaions and Applicaions, Horwood Publishing Chicheser 997. [9] Mao, X., Sabanis, S., Renshaw, E., Asympoic behaviour of he sochasic Loka-Volerra model, J. Mah. Anal. Appl, 28723, [] Rudnicki, R., Long-ime behaviour of a sochasic prey-predaor model, J. Mah. Anal. Appl, 823, a Hanoi Insiue of Mahemaics, 8 Hoang Quoc Vie, Cau Giay, Hanoi, Vienam. address: huan@mah.ac.vn Received , revised

10 84 HOANG THE TUAN, NGUYEN HAI DANG AND VAN KHU VU b Faculy of Mahemaics, Mechanics and Informaics, College of Science, Vienam Naional Universiy, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vienam. address: dangnh.mahs@gmail.com address: khu.vu@hus.edu.vn