A sustainability condition for stochastic forest model

A stochastic forest model of young and old age class trees is studied. First, we prove existence, uniqueness and boundedness of global nonnegative solutions. Second, we investigate asymptotic behavior of solutions by giving a sufficient condition for sustainability of the forest. Under this condition, we show existence of a Borel invariant measure. Third, we present several sufficient conditions for decline of the forest. Finally, we give some numerical examples.


1.
Introduction. In 1975, Antonovsky [1] introduced a mono-species forest model with two age classes of trees: Here, u and v denote the tree densities of young and old age classes, respectively. The parameters ρ, h and f are a reproduction rate, mortality of old trees, and aging rate of young trees, respectively; while γ(v) = a(v − b) 2 + c is a mortality of young trees, which is allowed to depend on the old-tree density. In addition, a, b, c, ρ, f and h are assumed to be positive constants.
It is not difficult to see that for any pair (u 0 , v 0 ) of nonnegative initial values u 0 and v 0 , the system (1) possesses a nonnegative and global solution. Furthermore, (1) possesses nonnegative stationary solutions given by where h * = ρf ab 2 +c+f , h * = ρf c+f . The stability and instability of these solutions depends strongly on the magnitude of the mortality h of old age class trees (see Table 1). Table 1. Stability and instability of stationary solutions of (1) On the basis of (1), Kuznetsov et al. [11] introduced a mathematical model of mono-species forest with two age classes which takes into account the seed production and dispersion. The third author studied that model with his colleagues (see, e.g., [3,4,5,15] and [17,Chapter 11]). It is shown that h plays a crucial role in the asymptotic behavior of solutions.
In the real world, the parameters in the model may be random variables due to unpredictability resulting from environmental, ecological and biological disturbances. In principle, the deterministic forest model can not handle randomness. Investigating the role of fluctuation of parameters by using stochastic models should be an interesting problem in environmental and ecological sciences.
As mentioned above, the asymptotic behavior of solutions to the deterministic forest model depends strongly on the magnitude of h. Therefore, in this paper we restrict ourselves to consider a stochastic forest model, where h is perturbed by (Gaussian) white noise. Since Gaussian white noises can be expressed as the generalized derivative of a Brownian motion, we make a substitution: (1), where {w t , t ≥ 0} is a one-dimensional Brownian motion defined on a filtered complete probability space (Ω, F, {F t } t≥0 , P), and σ > 0 is the intensity of the white noise. Our stochastic forest model is then formulated by Itô stochastic differential equations in R 2 : In this paper, we study the stochastic forest model (2). We prove existence of unique global solutions to (2) and then study their asymptotic behavior. On one hand, we present a sufficient condition for sustainability of the forest. Under this condition, we also prove existence of a non-trivial Borel invariant measure. On the other hand, we give several sufficient conditions for decline of the forest. The results are illustrated by a few numerical examples.
To prove existence of non-trivial invariant measures to (2), a common method is to find four one-dimensional processes, namely u 1 , u 2 , v 1 and v 2 , which satisfy two conditions: (i) u t and v t are bounded by these processes, i.e. u 1 (t) < u t < u 2 (t) and v 1 (t) < v t < v 2 (t) for 0 < t < ∞ (ii) These four processes do not hit the boundaries in the sense that there exist > 0 and M > 0 such that However, this can not be done because lim inf To overcome this difficulty, we use the semigroup method presented in [6,13]. First, we establish some estimates for the average of integrals of solutions (see (19) and (20)). Then, we construct a strongly continuous semigroup generated by solutions of (2). Using these estimates and a theorem in [13], we show that the semigroup enjoys an invariant measure.
The organization of the present paper is as follows. Section 2 proves existence and boundedness of unique global nonnegative solutions to (2). Section 3 investigates sustainability of the forest and existence of a Borel invariant measure. To the contrary, Section 4 presents some sufficient conditions for decline of the forest. Finally, Section 5 gives some numerical examples.
2. Global solutions. In this section, we prove existence of unique global nonnegative solutions to (2) and show boundedness of solutions. Put Then, For biological reasons, throughout this paper, initial values for (2) are taken from Let us first prove existence of unique global nonnegative solutions to (2). We use the following lemma.
Since the proof of the lemma is quite easy, we omit it.
In addition, if u 0 + v 0 > 0, then Proof. Since all the functions on the right-hand side of (2) are locally Lipschitz continuous, there is a unique local solution (u t , v t ) defined on an interval [0, τ ), where τ is a stopping time having the following property (see, e.g., [2,7]). If P{τ < ∞} > 0, then τ is an explosion time on {τ < ∞}, i.e. lim Therefore, it suffices to show that τ = ∞ a.s. and that u t ≥ 0, v t ≥ 0 a.s. for 0 < t < ∞. To prove this, we use the method in [14,16]. Consider the four cases of initial values. Case 1: u 0 = v 0 = 0. This is a trivial case, since u t = v t = 0 a.s. for 0 ≤ t < ∞. Case 2: (u 0 , v 0 ) ∈ R 2 + . Let k 0 > 0 be a positive integer such that u 0 and v 0 lie in the interval [ 1 k0 , k 0 ]. Denote with the convention inf ∅ = ∞. It is obvious that {τ k } ∞ k=k0 is nondecreasing. Hence, there exists a limit τ ∞ of this sequence as k → ∞: Let us prove that τ ∞ = ∞ a.s. Indeed, suppose the contrary, then there would exist T > 0 and 0 < < 1 such that Consider a positive function V on R 2 + , which is defined by The Itô formula gives where the infinitesimal operator L is given by It is possibly seen that In addition, there exist M i > 0 (i = 1, 2) such that Taking the expectations of the two sides of the latter inequality, we obtain that The Gronwall inequality then provides that Hence, On the other hand, (5) gives Thanks to (6), (9) and (10), we observe that Letting k → ∞, we arrive at a contradiction: and Clearly, the sequence {τ k } ∞ k=k0 has a limit τ ∞ as k → ∞: Indeed, due to the comparison theorem (see [8] Obviously, v t = 0. Thus, (11) follows.
Let us now verify that there exists α > 0 such that Ev t∧τ k ≤ α for 0 ≤ t < ∞. In addition, for Indeed, by virtue of the first equation of (2), where M 0 is defined in (3). Solving this differential inequality, we obtain that Hence, Using (14) and applying the comparison theorem for the second equation of (2), we observe that v t ≤v t a.s., 0 ≤ t < τ, wherev t is the solution of the one-dimensional stochastic differential equation: Thanks to Lemma 2.1-(ii) and (15), it is easily seen that (12) holds true. Let us finally observe that In view of (11), it suffices to show that τ ∞ = ∞. Indeed, suppose the contrary, then there would exist T > 0 and 0 < < 1 such that The Itô formula then gives Taking the expectation of the two sides of this inequality and using (12) and (14), we observe that By using (17) instead of (8), we repeat the same argument as in Case 2 to conclude that τ ∞ = ∞ a.s.
The proof for this case is similar to one for Case 3. By the above arguments, the proof of the theorem is complete.
Let us now show boundedness for the density u of young age class trees and for moments of the density v of old age class trees.
Proof. Clearly, (i) and (ii) follow from (13) and (14). Meanwhile, applying Lemma 2.1 to the equation (16) and using the fact that v t ≤v t , (iii) and (iv) follow.
3. Sustainability of forest. In this section, we present a sufficient condition for sustainability of the forest. Under this condition, we also show existence of a Borel invariant measure on R 2 + for the system (2).
3.1. Sustainability condition. Let us show that if the intensity of noise and mortality of old age class trees are small enough, then the forest is sustainable.
Theorem 2.2 and the Itô formula then provide that where the operator L is defined in (7). After some simple calculations, we obtain that Thereby, by using the estimate (i) of Theorem 2.3, we observe that Let us show that there exists ε 1 > 0 such that for all (u, v) ∈ R 2 + , Since σ 2 < 2(ρ−κh) κ , it is easily seen that there exists a small ε 1 > 0 such that the quadratic equation F (u, v) = 0 in the variable u has two non-positive solutions for every v ≥ 0. Thus, F (u, v) ≥ 0 for all (u, v) ∈ R 2 + . Proof for (19). Due to (23), (24) and the fact that v t ≤v t , wherev t is the solution of (16), we have Put Then, {N t } 0≤t<∞ is a real-valued continuous martingale vanishing at t = 0. Furthermore, {N t } 0≤t<∞ has a quadratic form given by The strong law of large numbers for martingale (see, e.g., [9,12]) then gives In the meantime, applying Lemma 2.1-(i) to the equation (16) and using Theorem 2.3-(ii), we observe that Taking the limit as t → ∞ of the two sides of (25), we hence obtain that lim inf

from which it follows (19).
Proof for (20). Taking the expectation of the two sides of (25), we have Evds , here we used the estimate On account of Lemma 2.1, the solutionv of (16) satisfies the estimate where α 1 is some positive constant. We thus have shown that lim inf Meanwhile, taking the expectation of the two sides of the second equation of (2), it follows that This means that (2) is sustainable. We complete the proof.

3.2.
Existence of Borel invariant measure. Let us show existence of a Borel invariant measure of the Itô process (u t , v t ), which concentrates on some domain of R 2 + under the assumptions in Theorem 3.2. Let P (·, ·, ·, ·) be the transition probability of (u t , v t ): for 0 ≤ t < ∞, (x, y) ∈ R 2 + , and K ∈ B(R 2 + ). It is well known that (see, e.g., [6,13]) (i) P (t, x, y, ·) induces a strongly continuous semigroup {P t } 0≤t<∞ of operators on the space C B (R 2 + ) of bounded continuous functions: x, y, ·) induces a positive contraction [·P t ] on the space M (R 2 + , B(R 2 + )) of finite signed measures: A Borel measure ν on R 2 + (i.e. a positive measure which is finite on any compact set of R 2 + ) is said to be invariant with respect to {P t } 0≤t<∞ if for 0 < t < ∞ and K ∈ B(R 2 + ), The following result is well known. . Let X be a locally compact perfectly normal topological space. Let {Q t } 0≤t<∞ be a strongly continuous semigroup on C B (X) generated by a transition probability on (X, B(X)). If there exists a nonnegative function g in the space C 0 (X) of continuous functions with compact support such that then there exists a Borel invariant measure for {Q t } 0≤t<∞ .
We are now ready to state our theorem.
Theorem 3.5. Let (18) be satisfied. Then, {P t } 0≤t<∞ has a Borel invariant measure which concentrates on some domain of R 2 To prove this theorem, we construct a function g ∈ C 0 (R 2 + ) which satisfies the assumption in Theorem 3.4.
On account of Theorem 3.2, we have lim inf Using Theorem 2.3-(iii) and the Hölder inequality, for any 0 ≤ θ < 2h σ 2 , there exists n θ > 0 such that Thereby, there exists 0 > 0 such that lim inf On the other hand, by Theorem 2.3-(iii), there exists α > 0 such that Ev t ≤ α for 0 ≤ t < ∞. The Markov inequality then provides that Let us show that lim inf Indeed, suppose the contrary, then there would exist an increasing sequence {t n } such that t n → ∞ as n → ∞ and Combining this and (28), we arrive at a contradiction: Therefore, (30) holds true. Let us fix a nonnegative function g ∈ C 0 (R 2 + ) such that g(x, y) = 1, (x, y) ∈ K, 0, (x, y) ∈ R 2 + \K 1 , where K 1 ⊃ K is some bounded open set of R 2 + . In view of (30), we have t 0 P s g(x, y)ds = t 0 R 2 + g(ξ, η)P (s, x, y, dξdη)ds Thanks to Theorem 3.4, we conclude that there exists a Borel invariant measure ν on R 2 + for {P t } 0≤t<∞ such that ν(K) > 0. By Theorems 2.2-2.3, ν concentrates on some domain of R 2 + ∩ {(u, v); u ≤ M 0 }. The proof is now complete.

Decline of forest.
In this section, we show decline of the forest when either the mortality h of old age class trees or the intensity σ of noise is large. More precisely, if either then the forest falls into the decline. Here, M * is defined in Theorem 2.3-(i).
. Then, as t → ∞, u t and v t converge to 0 in expectation, i.e. In particular, u t and v t converge to 0 in probability: Proof. Let us first prove that u t and v t converge to 0 in expectation. Consider the two cases of the mortality h. Case 1: h ≥ ρf c+f . It follows from (2) that Since the functions 1 and 2 defined by 2 (X, Y ) = f X − hY are non-decreasing with respect to arguments Y and X, respectively, the comparison theorem applied to the latter system provides that where (x t , y t ) is the positive solution to the linear system: with (x t , y t )| t=0 = (u 0 , v 0 ). From this system and a fact that h ≥ ρf c+f , we observe that Hence, hx t + ρy t and f x t + (c + f )y t are non-increasing as t increases. As a consequence, there exist two nonnegative constants β 1 and β 2 such that It is then seen that is a stationary solution of (35). Substituting this solution for (x t , y t ) in (35), we obtain that Solving this system of algebraic equations, we arrive at Case 2: h ≥ f (ρ+2abM * ) ab 2 +c+f . From (2) and Theorem 2.3-(i), we have Using the same argument as in Case 1, we conclude that lim t→∞ Eu t = lim t→∞ Ev t = 0.
Under somewhat stronger assumptions than those of Theorem 4.1, we can show almost sure convergence of u t and v t to 0. Consider two functions F 1 and F 2 defined by and Assume that either inf or 2ρ σ 2 + 2h < c + f f and there exists λ such that 2ρ holds true. Then, the following theorem shows such convergence.
We again use the function Q defined by Q(u, v) = log(u + κv) as in the proof of Theorem 3.2, where κ is a positive constant that will be fixed below.
Let us first show that under (39) or (40), there exists a small > 0 such that where [LQ] is defined in (22). Indeed, it is easily seen that a sufficient condition for this (in fact, it is also a necessary condition) is that there exists > 0 such that If (39) takes place, choose κ such that It is then easily seen that there exists a small > 0 such that the quadratic equation It follows from (21) that On account of (26) and (27), Hence, taking the limit as t → ∞ of both the hand sides of (42), we observe that lim sup This implies that lim t→∞ Q(u t , v t ) = −∞ a.s. Thus, The proof is complete.

Remark 1.
It is possibly seen that if F 1 (1) < 0, then (39) takes place. After some simple calculations on the inequality F 1 (1) < 0, we arrive at this condition: According to Theorem 4.2, we therefore conclude that a noise with large intensity causes decline of the forest. 5.1. Sustainability of forest. In the system (2), set a = 2, b = 1, c = 2.5, f = 4, h = 1, ρ = 5, σ = 0.5, and take initial value (u 0 , v 0 ) = (2, 1). Figure 1 gives sample trajectories of u and v in the phase space and in time. Figure 2 plots points (u T , v T ) of 10 4 sample trajectories of (u, v) at time T = 1000.
By computing 10 3 sample trajectories of (u, v), Figure 3 shows a graph of the expectation of tree densities of young and old age classes. Figure 4 gives a sample trajectory of two processes I and J defined by   Distribution of (u t , v t ) of (2) at t = 10 3 . The parameters and initial value are taken as in the legend of Fig. 1.