Asymptotic behaviour of the stochastic Lotka Volterra model

Transcription

1 J. Math. Anal. Appl Asymptotic behaviour of the stochastic Lotka Volterra model Xuerong Mao, Sotirios Sabanis, and Eric Renshaw Department of Statistics and Modelling Science, University of Strathclyde, Glasgow G XH, UK Received March 3 Submitted by M. Iannelli Abstract This paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process, namely the Lotka Volterra model. The stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart. Indeed, the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations. 3 Elsevier Inc. All rights reserved. Keywords: Lotka Volterra model; Brownian motion; Stochastic differential equation; Moment boundedness; Asymptotic behaviour. Introduction Deterministic subclasses of the Lotka Volterra model are well-known and have been extensively investigated in the literature concerning ecological population modelling. One particularly interesting subclass describes the facultative mutualism of two species, where each one enhances the growth of the other, represented through the deterministic equations ẋ t = x t b a x t a x t, ẋ t = x t b a x t a x t * Corresponding author. address: xuerong@stams.strath.ac.uk X. Mao. -7X/$ see front matter 3 Elsevier Inc. All rights reserved. doi:.6/s-7x3539-

2 X. Mao et al. / J. Math. Anal. Appl for a and a positive constants. The associated dynamics have been developed by, for example, Boucher, He and Gopalsamy and Wolin and Lawlor 9. Now in order to avoid having a solution that explodes at a finite time, a a is required to be smaller than a a. To illustrate what happens when the latter condition does not hold, suppose that a = a = α and a = a = β i.e., we have a symmetric system and α <β. Moreover, let us assume that b = b = b and that both species have the same initial value x = x = x >. Then the resulting symmetry reduces system to the single deterministic differential equation ẋt = xt b α βxt whose solution is given by b xt = α β b αβx x e. bt Now the assumption that α <β causes xt to explode at the finite time t = b {lnb α βx ln α βx }. Nevertheless, this can be avoided, even when the condition a a <a a does not hold, by introducing stochastic environmental noise. Let Ω, F, {F t } t,p be a complete probability space with filtration {F t } t satisfying the usual conditions, i.e., it is increasing and right continuous while F contains all P -null sets see Mao 6. Moreover, let wt be a one-dimensional Brownian motion defined on the filtered space and R n ={x Rn : x i > forall i n}. Finally, denote the trace norm of a matrix A by A = tracea T A where A T denotes the transpose of a vector or matrix A and its operator norm by A =sup{ Ax : x =}. Now consider a Lotka Volterra model for a system with n interacting components, which corresponds to the case of facultative mutualism, namely ẋ i t = x i t b i a ij x j, i n. This equation can be rewritten in the matrix form ẋt = diag x t,..., x n t b Axt, t, where xt = x t,..., x n t T, b = b i n and A = a ij n n. Stochastically perturbing each parameter a ij a ij σ ij ẇt results in the new stochastic form ẋt = diag x t,..., x n t b Axt dt σxtdwt, t. 3 Here σ = σ ij n n, and we impose the condition H { σii >, if i n, σ ij, if i j. For a stochastic differential equation to have a unique global solution i.e., no explosion in a finite time for any given initial value, the coefficients of Eq. are generally required

3 X. Mao et al. / J. Math. Anal. Appl to satisfy both the linear growth condition and the local Lipschitz condition cf However, the coefficients of Eq. 3 do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of Eq. 3 may explode at a finite time. Under the simple hypothesis H, the following theorem shows that this solution is positive and global. Theorem Mao et al. 8. Let us assume that hypothesis H holds. Then, for any system parameters b R n, A R n n and any given initial value x R n, there is a unique solution xt to Eq. 3 on t. Moreover, this solution remains in R n with probability, namely xt R n for all t almost surely. The above result reveals the important role that environmental noise plays in population dynamics. The idea that even a tiny amount of stochastic noise can suppress an imminent deterministic explosion in a number of co-habiting species brings a whole new dimension into the study of population modelling.. Asymptotic moment estimation Since Eq. 3 does not have an explicit solution, the study of asymptotic moment behaviour is essential if we are to gain a deeper understanding of the underlying process. This paper is essentially a continuation of the moment results derived by Mao et al. 8. Theorem. Let the system parameters b R n and A R n n be given, and assume that hypothesis H holds. Then, for any θ,, there exists a positive constant K θ such that, for any initial value x R n, the solution of Eq. 3 has the property t lim sup t t E xi θ s ds K θ. Proof. Define a C -function V : R n R by Vx= xi θ. According to Itô s formula, dv xt = θx i b i θxi θ a ij x j σ ij x j dwt. θθ xi θ x j σ ij dt

4 X. Mao et al. / J. Math. Anal. Appl Moreover, it is easy to show that θx i b i a ij x j θx i b i a ij x i x j and θ θxi θ x j σ ij θ θx θ i σ ii. As a result, we obtain dv xt θ b i x i θ xi θ a ij x i x j θ θ σ ii xθ i dt σ ij x j dwt. 5 Furthermore, by taking into consideration that fact that the polynomial θ θ θ b i x i a ij x i x j σii xθ i has an upper positive bound, say K θ, inequality 5 yields where V xt Mt = θ θ θ xi θ σii xθ i ds V x σ ij x j dws K θ ds Mt, 6 is a real-valued continuous local martingale vanishing at t =. Taking expectations on both sides of 6 then results in where E t x θ i ds V x Kθ t, ˆσθ θ ˆσ = min { σii, i n}. The required assertion follows immediately.

5 X. Mao et al. / J. Math. Anal. Appl Pathwise estimation Theorem 3. Let us assume that hypothesis H holds. Moreover, let the system parameters b R n, A R n n and the initial value x R n be given. Then, there exists a K>,which is independent of x but not necessarily of the system parameters, such that n lim sup ln x i t t t λ minσ T σ xs ds K a.s., 7 where λ min σ T σ is the smallest eigenvalue of the matrix σ T σ. Proof. For each i n, applying Itô s formula to lnx i t results in d ln x i t = b i a ij x j σ ij x j dt σ ij x j dwt, which implies that ln x i t = ln x i where M i t = σ ij x j dws b i a ij x j σ ij x j ds M i t, 8 is a real-valued continuous local martingale vanishing at t = with quadratic form Mi t, M i t = σ ij x j ds. Fix, arbitrarily. For every integer k, using the exponential martingale inequality cf. Mao 6, Theorem.7. we have { P sup M i t Mi t, M i t > } t k ln k k. An application of the well-known Borel Cantelli lemma yields that, with probability one, M i t Mi t, M i t ln k sup t k holds for all but finitely many k. In other words, there exists an Ω i Ω with PΩ i = such that for any ω Ω i an integer k i = k i ω can be found such that M i t Mi t, M i t ln k, t k,

6 6 X. Mao et al. / J. Math. Anal. Appl for any k k i ω. Thus Eq. 8 results in ln x i t ln x i b i a ij x j σ ij x j ds ln k 9 for t k i ω and k k i ω whenever ω Ω i.nowletω = n Ω i. Clearly PΩ =. Moreover, for any ω Ω,letk ω = max{k i ω: i n}. Then, for any ω Ω, it follows from 9 that ln x i t ln x i b i a ij x j σ ij x j ds n ln k for all t k and k k ω. Note that n n σ ij x j = σx. Thus n ln x i t t σx ds Since n ln x i b i b i a ij x j σx K, a ij x j σx ds n ln k. for some positive constant K, it follows that for ω Ω we have n ln x i t t n σx ds ln x i Kt n ln k, for t k and k k i ω. Consequently, for any ω Ω,ifk t k and k kω, t n ln x i t which implies that n lim sup ln x i t t t ln n x i lim sup k k σx ds ln n x i k σx ds n k ln k K = K K n ln k, k

7 X. Mao et al. / J. Math. Anal. Appl almost surely. On noting that σx = x T σ T σx λ min σ T σ x, it then follows that lim sup t n ln x i t λ min σ T σ t Letting tend to zero yields the required assertion. xs ds K a.s. Theorem. Let us assume that hypothesis H holds. Then, for any system parameters b R n, A R n n and any initial value x R n, ln n lim sup x i t n a.s. t lnt Proof. For each i n, applying Itô s formula to e γt lnx i t for γ>results in e γt ln x i t = ln x i e b γs i a ij x j σ ij x j ds where M i t = e γs γ e γs ln x i s ds M i t, σ ij x j dws is a real-valued continuous local martingale vanishing at t = with quadratic form Mi t, M i t = e γs σ ij x j ds. Fix any, and θ>. For every integer k, on using the exponential martingale inequality we have { P M i t e γk M i t, M i t } > θeγk ln k k θ. sup t k By the Borel Cantelli lemma we observe that there exists an Ω i Ω with PΩ i = such that for any ω Ω i an integer k i = k i ω can be found such that M i t e γk M i t, M i t θeγk ln k for all t k and k k i ω. Thus Eq. leads to

8 8 X. Mao et al. / J. Math. Anal. Appl e γt ln x i t ln x i γ e γk e γs b i e γs ln x i s ds a ij x j σ ij x j ds e γs σ ij x j ds θeγk for t k and k k i ω whenever ω Ω i, which can be rewritten as e γt ln x i t ln x i γ θeγk e γs b i ln k ln k a ij x j e γk s σ ij x j ds e γs ln x i s ds. Now let Ω = n Ω i. Clearly PΩ =. Moreover, for any ω Ω,letk ω = max{k i ω: i n}. Then, for any ω Ω, it follows from that n e γt ln x i t e γs nθeγk b i ln k γ ln x i e γs a ij x j e γk s σ ij x j ds ln x i s ds for all t k and k k ω. Since for positive constant K b i a ij x j σ ij x j γ lnx i K, x R n, we have n n e γt ln x i t ln x i K γ eγt K γ nθeγk ln k

9 X. Mao et al. / J. Math. Anal. Appl Fig.. A sample path of ln x i t/ lnt produced by generating 6 points with time step = 5 and initial condition x = x = 5. for all t k and k k ω. Consequently, for any ω Ω,ifk t k and k kω, it follows that ln n n x i t e γk ln x i K lnt lnk γ nθeγ ln k, which implies that ln n lim sup x i t nθeγ a.s. t lnt By letting, θ andγ, we then obtain ln n lim sup x i t n a.s. t lnt as required. Figure illustrates the above theoretical results by highlighting the bounded nature of the process here b i =, a ij = andσ ij = for every i, j =,. Note the controlling influence of the downward surges. The conclusion of Theorem is very powerful since it is universal in the sense that it is independent both of the system parameters b R n and A R n n, and of the initial value x R n. It is also independent of the noise intensity matrix σ as long as the noise exists in the sense of hypothesis H. However, since estimation is based on the multiplication n x i t, it would be better to use instead the norm xt. To do so we need additional conditions on the noise intensity matrix.. Pathwise estimation with additional conditions imposed on σ Improved results concerning the pathwise behaviour of the solution can be achieved by introducing somewhat more restrictive assumptions. A numerical example given at the end

10 5 X. Mao et al. / J. Math. Anal. Appl of this section shows that applying our theoretical results to specific practical situations is a relatively simple procedure. Consider a new hypothesis in which we assume the existence of two constants λ and ρ, with ρ >λ, such that { diagx,...,x n σ x λ x, x R n H, x T diagx,...,x n σ x ρ x 6, x R n. Then we can use this to develop a new suite of theorems. Theorem 5. Let us assume that hypothesis H holds, and that there exist two positive constants λ and ρ, with ρ >λ, such that hypothesis H also holds. Moreover, let the system parameters b R n, A R n n and the initial value x R n be given. Then, with probability, lim sup t t δ ρ λ ln xt where µ = max{ a ij, b i : i, j n} and δ = nµ nµnµ. nµ xs nµ δ ds δ ρ λ µδ ρ λ, Proof. Define a C -function V : R n R by Vx= ln x. Then applying Itô s formula yields dv xt = x xt diagx,...,x n b Ax dt { x trace diagx,...,x n σ x x T x diagx,...,x n σ x } dt x xt diagx,...,x n σ x dwt. Now Eq. can be rewritten in the form dv xt { = x b i xi x xi xi a ij x j σ ij x j x xi } σ ij x j dt 3

11 X. Mao et al. / J. Math. Anal. Appl x xi Moreover, taking into account that < x i σ ij x j dwt. x i x i n x results in x xt diagx,...,x n b Ax µ n x, x R n, where µ = max{ a ij, b i : i, j n}. Consequently, Eq. becomes where V xt V x M t = x µ n x λ x ds x x T diagx,...,x n σ x ds M t, 5 xi σ ij x j dws = x xt diagx,...,x n σ x dws is a real-valued continuous local martingale vanishing at t = with quadratic form M t, M t = x x T diagx,...,x n σ x ds. Fix any >. By the exponential martingale inequality we have that for every integer k { P M t M t, M t > ln k } k. sup t k Since k= k converges, the application of the Borel Cantelli lemma proves that for almost all ω Ω there exists a random integer k ω such that for all k k ω sup M t t k M t, M t lnk, which implies M t M t, M t lnk on t k.

12 5 X. Mao et al. / J. Math. Anal. Appl By taking into consideration inequality 5, assumption H, and the above results, we then obtain V xt V x µ n x ρ λ ρ x ds lnk. Now, since ρ >λ, we can choose small enough to ensure that ρ ρ>λ. Thus, for any δ>, we have the inequality µ n x δ ρ λ x nµ δ δ δ ρ λ µ. As a result, we obtain V xt t ρ λ ρ xs ds δ V x nµ δ δ ρ λ µ t lnk on t k. In particular, for almost all ω Ω, ifk t k and k k ω, it follows that V xt t ρ λ ρ xs ds t δ V x lnk k nµ δ k δ ρ λ µ k. Whence letting t so k, and then, results in lim sup t t δ ρ λ ln xt Since the right-hand side of this equation is minimised when δ = nµ nµnµ, nµ the required assertion follows. xs nµ δ ds δ ρ λ µδ ρ λ. Theorem 6. Let us assume that hypothesis H holds, and that there exist two positive constants λ and ρ, with ρ >λ, such that hypothesis H also holds. Moreover, let the system parameters b R n, A R n n and the initial value x R n be given. Then, with probability, ln xt lim sup ρ t lnt ρ λ. 6

13 X. Mao et al. / J. Math. Anal. Appl Proof. Let us define the following C -function V : R n R R such that Vx,t= e t ln x. Then applying Itô s formula yields dv xt,t = e t ln x e t x xt diagx,...,x n b Ax dt { e t x trace diagx,...,x n σ x x T x diagx,...,x n σ x } dt e t x xt diagx,...,x n σ x dwt. 7 Moreover, it is easy to show that x xt diagx,...,x n b Ax µ n x, x R n, where µ = max{ a ij, b i : i, j n}. As a result, Eq. 7 yields where V xt,t V x, M t = e s ln xs ds e s µ n xs λ xs ds e s xs x T diag x s,..., x n s σxs ds M t, 8 e s xs xt diag x s,..., x n s σxsdws is a real-valued continuous local martingale vanishing at t = with quadratic form M t, M t = e s xs x T s diag x s,..., x n s σxs ds. Given any >, θ>andα>, on exploiting the exponential martingale inequality once again, we can show that for almost all ω Ω there exists a random integer k ω such that, for all integer k k ω, M t e kα M t, M t θekα ln k, for all t kα.

14 5 X. Mao et al. / J. Math. Anal. Appl By taking into consideration inequality 8, hypothesis H, and the above results, we then obtain V xt,t V x, e s ln xs t ds e s µ n xs ρ λ e kα s ρ xs θe kα ln k ds 9 for all t kα. Now,sinceρ>λ, we can choose small enough to ensure that ρ > λ, namely, choose,ρ λ/ρ. Moreover, there exists a positive constant κ such that ln x µ n x ρ λ ρ x κ, x R n. Consequently, inequality 9 yields e t ln xt ln x κe t κ θekα ln k on t kα, which implies that ln xt e t ln x κ κ θekα t ln k on t kα. In particular, for almost all ω Ω, ifk α t kα and k k ω, it follows that ln xt lnt e k α ln x r κ κ lnk lnk θeα ln k on t kα. lnk Whence letting t so k too yields ln xt lim sup θ t lnt eα a.s. Finally, by letting θ, α and ρ λ/ρ, we obtain ln xt lim sup ρ a.s. t lnt ρ λ which is the required assertion. Let us now discuss a simple numerical example which not only demonstrates that the set of functions and parameters satisfying hypothesis H is not empty but also illustrates the estimation obtained by Theorem 6. Consider the case n = with b =, A = It is easy to see that diagx,x σ x 5 x, and σ =.

15 X. Mao et al. / J. Math. Anal. Appl since diagx,x σ x = x x x x and = x x3 x x x x x 3 x 5 x diagx,x σ x = x x3 x 8x x x x 3 x = x x x x x x. Similarly, the inequality x T diagx,x σ x 3 x 6 holds, since x T diagx,x σ x = x x x x x x and x T diagx,x σ x 3 x 6 = x 6 x5 x 5x x x3 x3 5x x x x 5 x6 = x 6 x5 x x x x3 x3 x x x x 5 x6 = x 3 x3 x x x x x x x x 3 x x. We have therefore proved that there exists a pair of parameters, λ = 5andρ = 3, for the above specified matrix σ which satisfy hypothesis H. As a result, Theorem 6 yields ln xt lim sup 3 a.s. t lnt This means that neither of two species will grow faster than a polynomial of time t of order 3. Figure shows a sample path of ln xt /lnt which supports this theoretical result. 5. Summary Our aim in this paper is to discuss the asymptotic properties of the stochastic Lotka Volterra model in populations dynamics. In our earlier paper 8 we revealed an important fact that even a tiny amount of stochastic noise can suppress an explosion in populations dynamics. Due to the page limit we have not investigated in 8 the asymptotic behaviour of the stochastic populations dynamics but the theory there guarantees the nice property that the solution of the stochastic Lotka Volterra model will remain in the positive cone with probability one. Making use of this property we have in this paper designed various

16 56 X. Mao et al. / J. Math. Anal. Appl Fig.. A sample path of ln xt / lnt produced by generating 6 points with time step = 5 and initial condition x = x = 5. types of Lyapunov functions to discuss the asymptotic behaviour in some detail. Several moment and pathwise asymptotic estimators are obtained. These essentially enhance each other, so that they can be used to reveal better features of the stochastic Lotka Volterra model. Two computer simulations are presented which support the theoretical results. Acknowledgments The authors thank BBSRC and EPSRC for their financial support under Grant 78/MMI97 and, in addition, an anonymous referee for his/her helpful comments on the manuscript. References D.H. Boucher Ed., The Biology of Mutualism, Groom Helm, London, 985. X. He, K. Gopalsamy, Persistence, attractivity, and delay in facultative mutualism, J. Math. Anal. Appl R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, 98. X. Mao, Stability of Stochastic Differential Equations with Respect to Semimartingales, Longman Scientific and Technical, X. Mao, Exponential Stability of Stochastic Differential Equations, Marcel Dekker, X. Mao, Stochastic Differential Equations and Applications, Horwood, G. Marion, X. Mao, E. Renshaw, Convergence of the Euler scheme for a class of stochastic differential equation, Internat. J. Math X. Mao, G. Marion, E. Renshaw, Environmental Brownian noise suppresses explosions in populations dynamics, Stochastic Process. Appl C.L. Wolin, L.R. Lawlor, Models of facultative mutualism: density effects, Amer. Natural