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Dynamics of Stochastic Coral Reefs Model with Multiplicative Nonlinear Noise
Little seems to be known about the ergodicity of random dynamical systems with multiplicative nonlinear noise. This paper is devoted to discern asymptotic behavior dynamics through the stochastic coral reefs model with multiplicative nonlinear noise. By support theorem and Hörmander theorem, the Markov semigroup corresponding to the solutions is to prove the Foguel alternative. Based on boundary distributions theory, the required conservative operators related to the solutions are further established to ensure the existence a stationary distribution. Meanwhile, the density of the distribution of the solutions either converges to a stationary density or weakly converges to some probability measure.
The coral reefs equation is one of the most famous ecosystem models :where represents the cover of macroalgae; represents the cover of corals.(i)is the rate that corals recruit to and overgrow algal turfs;(ii) is the natural mortality rate of corals;(iii) is the rate that corals are overgrown by macroalgae;(iv) is the rate that macroalgae spread vegetatively over algal turfs;(v) is the grazing rate that parrotfish graze macroalgae without distinction from algal turfs.
By the results in Li et al. , they discuss all kinds of dynamical behaviors. Recently, system (1) was studied extensively that it exhibits complex dynamical phenomena, including chaos, bifurcation, stability, and attractiveness [1–10].
However, ecosystem in the real world is very often subject to environmental noise due to uncertainty and unknown factors [11–16]. From a biological point of view and the generality of the models considered [17–21], these systems can appear very formal. This paper studies a stochastic coral reefs model where the intrinsic growth rate of the cover of macroalgae, , and the one of the cover of corals, , are perturbed stochastically and . In the paper, we only consider that stochastic coral reefs model can be described bywhere is a two-sided canonical Brownian motion. and represent the intensity of random noise and the differential and are to be understood in the sense of multiplicative nonlinear noise. Since the drift and diffusion coefficients of (2) satisfy locally Lipschitz continuous condition, we can apply standard theorems that provide both existence and uniqueness of the positive solution of (2) (see ), for any given initial value. However, since the diffusion term of (2) is not linear but nonlinear, the existing powerful classical results [11–21] fail to work here. Nevertheless, this paper discusses the asymptotic behavior of the stochastic coral reefs model with multiplicative nonlinear noise using Fokker-Planck equations. To overcome the difficulty from the diffusion term, based on boundary distributions theory and conservative operators, we show that the density of the distribution of the solutions either converges to a stationary density or weakly converges to some probability measure.
This paper is organized as follows. In Section 2, we study the global attractiveness of the solution for stochastic coral reefs model with multiplicative nonlinear noise. In Section 3, we discuss the ergodicity of the solution for stochastic coral reefs model with multiplicative nonlinear noise.
2. Global Attractiveness
In the section, we study the global attractiveness of the solution for stochastic coral reefs model with multiplicative nonlinear noise.
Proposition 1. Suppose that , hold.(I)If the cover of macroalgae is absent, then the cover of corals dies with probability one.(II)If the cover of corals is absent, the quantity of the preys oscillates between and , and there exists a unique stationary distribution with the density where is a constant.
Proof. Denoting , , we replace system (2) byLetThen, system (4) becomesor Stratonovich stochastic differential equationLet denote the generator of diffusion (4); that is,Then Fokker-Planck equation (FPE) of (4) can be described by(I) If the cover of macroalgae is absent, the quantity of the cover of corals satisfiesFix It is easy to see that and . Then, we can obtainIt implies that without the cover of macroalgae, the cover of corals dies with probability one.
(II) If the cover of corals is absent, the quantity of the cover of macroalgae satisfiesLet It easily shows that and . We haveIt implies that, without the predators, the quantity of the preys oscillates between and . Furthermore, there exists a stationary distribution of system (13) with the density satisfying the FPESolving (16), then we get where and are real numbers. With the conditions andit means thatIt easily shows that where is standard normal distribution function. Thus, we getBy ergodic theorem, if is a solution of system (13), then we haveFurthermore, converges in probability to when . DefineIt is easy to see that
Theorem 2. Suppose that and hold. Then there exists a constant such thatholds with probability 1, where and are the solution of (2) with the initial condition .
Proof. Define the Lyapunov function on Applying Itô’s formula to function (26), we getThat is,whereis a local martingale with quadratic form:Fix . For any , by martingale inequality, we haveBy using Borel-Cantelli theorem, we can choose a set with and for any there is a such that It implies thatfor and . Substituting (33) into (28), we get for and for almost and . Moreover, there exists a constant satisfyingBy inequality (35), we get it means thatMoreover, there exists a positive constant satisfyingTherefore, we getThen, we havefor any , , and
If with , then we getIt means thatorThe proof is completed.
Theorem 3. Suppose that and hold. Then, with probability 1, we have
Proof. Define -function:Applying Itô’s formula to function (44), we haveThat is,whereis a local martingale with quadratic form:By using Borel-Cantelli theorem and martingale inequality, for , , and , for almost , there is a such that , and we haveBy (47) and (50), we getMoreover, there exists a constant satisfyingBy inequality (52), it easily shows thatMoreover, there exists a positive constant satisfyingCombining (51), (52), and (53), we getfor any .
It is easy to see that there exists a real number independent of satisfyingFrom (55) and (56), we getfor any Then, we haveIf and , we obtainLetting , we getBy (60), for every , , and , then by letting , , and , we obtainThe proof is completed.
Theorem 4. Suppose that , hold. Let denote the solution of the following the equation:with the initial value . Let denote the solution of the following the equation:with the initial value .
Then, with probability 1, there exist and .
Proof. Let , . By Itô’s formula, we have By using comparison theorem, we have for all a.s. It implies that for all . It is easy to see that we show the second assertion by a similar way for all . The proof is completed.
Theorem 5. Suppose that , hold. Then the following assertions are true:
Proof. From Theorem 3, we getTherefore,For any , by martingale inequality, we have By using Borel-Cantelli theorem, for almost all , there is a real constant satisfying, for all and ,It means that for Then, we getCombining (67) and (71), it yieldsMoreover, by Theorem 4, we getTherefore,Letting , we have The first assertion is proven. Using similar way, the second assertion is also proven. The proof is completed.
In the section, we discuss the ergodicity of the solution for stochastic coral reefs model with multiplicative nonlinear noise.
Proof. Let and denote vector fields on ; then the Lie bracket denotes a vector field: Denote Then it is easy to see that where Assume that there is a point and so the vectors , , and can not span the space . Then, the vectors and are parallel; the vectors and are also parallel. Therefore, we getIt is easy to check that equality (80) is impossible.
It implies that the vectors , , span at any point . Therefore, we obtain the Hörmander condition.()For every , the vectors span the space .By Hypothesis () and Hörmander theorem [22–29], then the transition probability function results in a density and . Thus, the first assertion has been proved.
From the first assertion, it easily shows that for any , of system (4) results in the density satisfying the FPE (9). Furthermore, we getDefining the operator isfor any , By using continuation theorem of operator and assertion (I), it easily shows that the operator is an integral Markov semigroup. The proof is completed.
Theorem 7. Suppose that , hold. Then there is no more than three solution curves such that satisfying rank if .
Proof. Let and . We consider the following system:Denote System (84) becomes We show that denote the solution of system (84) with the initial value , and defined as . By using the perturbation method, we get the Frechet derivative of . Let and . denote the fundamental matrix of the following differential equation:That is, and for . Then, we havewhere . Let and if and if By using Taylor formula, we get as . Then, we have Let , , and . By using mean value theorem of integration, we get It is easy to calculate that we show Therefore, we obtain If the two vectors and are not linearly dependent, then we get the rank . Since the between and is linear dependence, thus, it is easy to see that and denote the solution of the following differential equation: