Desert and Vegetation States and Asymptotic Conservation Laws in a Biomass-Water Ecosystem Tianze Wang1 and Yisong Yang2,3 1 2

School of Mathematics, Henan University, Kaifeng, Henan 475000, PR China

Department of Mathematics, Yeshiva University, New York, New York 10033, USA 3

Polytechnic Institute of New York University, Brooklyn, New York 11201, USA

Abstract A fundamental subject in ecology is to understand how an ecosystem responds to its environmental changes. The purpose of this paper is to study the desertification and vegetation pattern formation phenomena and understand the dependence of the biomass density B of vegetation on the level of available environmental water resources, controlled by a water supply rate parameter R, which is governed by a coupled system of nonlinear parabolic equations in a mathematical model proposed recently by Shnerb, Sara, Lavee, and Solomon. It is shown that, when R is below the death rate µ of the vegetation in the absence of water, the solution evolving from any initial state approaches exponentially fast the desert state characterized by B = 0; when R is above µ, the solution evolves into a green vegetation state characterized by B 6→ 0 as time t → ∞. In the flower-pot limit where the system becomes a system of ordinary differential equations, it is shown that nontrivial periodic vegetation states exist provided that the water supply rate R is a periodic function and maintains a suitable average level. Furthermore, some conservation laws relating the asymptotic values of the vegetation biomass B and available water density W are also obtained. Key words and phrases. Desertification, vegetation pattern formation, ecosystems, parabolic equations, global stability, periodic solutions, asymptotic conservation laws.

1

Introduction

It is well known that mathematical modelling [1, 8] plays an important role in the quantitative and qualitative understanding of vegetation pattern formation and, in particular, the disappearance of vegetation, often referred to as desertification or

1

desertization, when there is an adverse change in the living environment of the vegetation under consideration. Such a change may mainly be described in terms of one or several biogeochemical cycles including those of water, carbon, and nitrogen. In return, vegetation also regulates those biogeochemical cycles given as environmental resource parameters, thus, resulting in a highly coupled dynamical ecosystem governing the interaction of the vegetation biomass and the environmental resource parameters. How ecosystems respond to environmental resource changes is one of the main frontiers in ecology [15]. Although environmental resource change may be a slow and gradual process, an ecosystem may undergo a catastrophic change attributed to the existence of two alternative stable or attracting states generically referred to as the onset of bistability [15] in the ecosystem. Mathematical modelling aims at pursuing predicative power for these catastrophic responses to changing environments [5, 6, 7, 10, 11, 14, 15, 17]. Recently, Shnerb, Sara, Lavee, and Solomon [18] proposed a general and simple model describing an interacting water-shrubs ecosystem that illustrates the expected bistable phase transition in terms of desertification and vegetation pattern formation picture with respect to the water resource change controlled by a water supply rate parameter. In their model, the density of shrubs biomass, B, and the available water density, W , are governed by a system of nonlinear parabolic equations for which the evolution of B and W are related to the death rate, µ, of the vegetation in the absence of water, and the water supply rate parameter, R. Shnerb et al [18] obtained numerical results showing vegetation pattern formation in their model when R is sufficiently large and described linear stability of the two equilibrium solutions representing the desert and vegetation states. In this paper, we present a mathematical analysis of the desertification and vegetation pattern formation model of Shnerb et al [18]. One of our main results establishes that the global phase transition point between the desert state characterized by B → 0 as t → ∞ and the green vegetation state characterized by B 6→ 0 as t → ∞ is sharply and explicitly rendered at the threshold level R = Rc = µ. In other words, whether a solution starting from any initial state approaches the desert state is dictated by whether the explicit meager-water-supply condition R < µ holds true. An outline of the rest of the paper is as follows. In the next section, we review the model of Shnerb et al [18] and establish the well-posedness of the model. That is, we prove the global existence, uniqueness, and positivity of a solution to the governing system of equations of [18]. In Section 3, we show that the desert state characterized by a vanishing biomass density, B = 0, is globally stable when R < µ. In Section 4, we consider the situation when R > µ. We first establish the occurrence of a vegetation state characterized by characterized by lim supt→∞ {B} = B∞ > 0 when R > µ. We then present a complete analysis of the homogeneous or zerodimensional limit of the model describing for example a “flower-pot” (in the words of [18]) situation. More precisely, based on the Poincar´e–Bendixon theorem, we will prove that the steady green vegetation state, which is the unique stable equilibrium point of the governing ordinary differential equations, is globally stable, meaning 2

that a solution starting from any nontrivial initial state evolves into the steady green vegetation state, B → B1 > 0, exponentially fast as time t → ∞. Besides, we also obtain an explicit formula for all solutions when µ = 1, which gives useful information about what happens in the critical situation, R = µ. Our results for the vegetation states, combined with the global stability result for desert state obtained in Section 3, implies that the bistability transition point expressed in terms of the water supply rate R is indeed rendered at the critical value R = Rc = µ. In Section 5, we consider the situation when R is a periodic function of t. We prove by a fixed-point theory argument that the homogeneous system allows the existence of a periodic solution when the average value of R over its period is adequately given. Such a result is naturally expected for the onset of a “seasonal” response of the vegetation biomass and available water resource with respect to the change of water supply rate. In Section 6, we consider the full spatially dependent system of equations and study the asymptotic behavior of the solutions under the high-water-supply condition R > µ. We will derive some uniform bounds for B and W and obtain some pointwise and asymptotic “conservation laws” relating or confining B and W . A detailed explanation of these relations will also be seen there. We summarize our results and state some comments in the last section.

2

Governing Equations and Positive Solutions

Consider the interaction of water resource and vegetation biomass in an ecosystem so that the presence of water supply enhances vegetation and the latter, in turn, consumes the former. The evolution equations proposed by Shnerb et al [18] governing such an interaction are given in the form ∂B = W B − µB, ∂t ∂W = D∆W + V · ∇W + R − W − λW B, ∂t B(x, 0) = B0 (x), W (x, 0) = W0 (x),

(2.1) (2.2) (2.3)

where B is the density of shrubs biomass and W is the available water density, the parameters µ, λ, D, R are positive constants, the initial states B0 (x), W0(x) in (2.3) are nonnegative functions of the spatial variable x, the term V · ∇W takes account of the downhill water loss in which V is a constant “terrain slope” vector, and the initial value problem consisting of the equations (2.1)–(2.3) are considered for time t > 0 and subject to a spatially periodic boundary condition over a planar spatial domain Ω, which may be viewed as a flat (closed) torus. For convenience, we assume that B0 and W0 are all smooth. Define the quantities W 0 = max{W0(x) | x ∈ Ω}

(2.4)

K = max{W 0 , R}.

(2.5)

and

3

Naturally, as a preliminary preparation for later development, we need first to establish the well-posedness of the problem (2.1)–(2.3). We have Theorem 2.1. The initial value problem consisting of the equations (2.1)–(2.3) has a unique global smooth solution (B(x, t), W (x, t)) which satisfies 0 ≤ B(x, t) ≤ B0(x)e(K−µ)t , 0 ≤ W (x, t) ≤ K,

(2.6) (2.7)

for all t > 0 and x ∈ Ω, provided that the initial data B0 , W0 given in (2.3) are all nonnegative functions. Proof. We divide our proof into two simple steps: (i) we first obtain local existence of a solution; (ii) we then establish global existence and positivity of the solution. For (i), we will follow the standard approach to reformulate the problem in the framework of a contractive semigroup. To this end, we consider the Banach space X = C(Ω) × C(Ω) and set u = (B, W ) ∈ X, u0 = (B0, W0 ) ∈ X, so that (2.1)–(2.3) are recast into ut = Lu + f(u), x ∈ Ω, t > 0, u = u0 , x ∈ Ω, t = 0,

(2.8) (2.9)

where 

−µ 0 L = 0 D∆ + V · ∇ − 1   WB f(u) = . R − λW B



≡



L1 0 0 L2



,

(2.10) (2.11)

Since L1 and L2 both generate contractive semigroups, say S1(t) and S2 (t), on C(Ω), their direct sum S(t) = diag(S1 (t), S2(t)) is the contractive semigroup generated by L over X. Consequently, the existence and uniqueness of a local solution of (2.8) and (2.9), hence of the initial value problem (2.1)–(2.3), is established. In order to show that the solution is global, it suffices to establish the bounds stated in (2.6) and (2.7). From (2.1), we obtain Rt

B(x, t) = B0 (x)e

0 (W (x,s)−µ) ds

.

(2.12)

In particular, B(x, t) ≥ 0. On the other hand, (2.2) may be rewritten as ∂W = D∆W + V · ∇W + R − (1 + λB)W, ∂t

t > 0,

x ∈ Ω,

(2.13)

so that the coefficient of W (in the last term in the equation above) is of a good sign, 1 + λB > 0, and we may proceed using a maximum principle argument: If there is a 4

point (x0 , t0) such that W (x0, t0) < 0, we may assume without loss of generality that W (x, t0), as a function of x ∈ Ω, attains its local minimum at x0 and Wt (x0, t0 ) ≤ 0. Hence (∆W )(x0, t0) ≥ 0 and (∇W )(x0, t0) = 0. Inserting these into (2.13), we arrive at a contradiction. Thus W (x, t) ≥ 0 everywhere. Setting u = W − K where K is defined in (2.5), we get from (2.13) the inequality ∂u ≤ D∆u + V · ∇u − (1 + λB)u. ∂t

(2.14)

We know in view of (2.4) and (2.5) that u(x, 0) = W0 (x) − K ≤ 0. If there is a point (x0 , t0) such that u(x0 , t0) > 0, we may assume without loss of generality that u(x, t0), as a function of x ∈ Ω, attains its local maximum at x0 and ut(x0 , t0) ≥ 0. Hence (∆u)(x0, t0) ≤ 0 and (∇u)(x0, t0) = 0. Inserting these into (2.14), we get a contradiction. Thus u(x, t) ≤ 0 or W (x, t) ≤ K everywhere. Therefore (2.7) is established and the theorem follows. Note that the simple expression (2.12) gives the clear interpretation that the biomass density at the spatial location x at time t is unambiguously enhanced by the available water density at the point x over the entire time interval [0, t]. In the next section, we consider the situation of the presence of a globally attracting desert state characterized by B = 0 when the water supply rate R is low, resulting in a low level of available water, W , which is insufficient for a sustainable vegetation.

3

Desert State and its Global Stability

First, we observe that the inequality (2.6) implies that, when the initial available water density W0 and the water supply rate R are both low so that K < µ,

(3.1)

we have B(x, t) → 0 exponentially fast. This clearly indicates a desert state. Next, we show that we will arrive at the desert state whenever the water supply rate R is low, regardless of the level of initial available water density W0 . This is naturally expected because vegetation consumes water and a low level of R will drive the water level down. For comparison, we use two different methods to approach the desertification problem: an energy method based integral inequalities and a pointwise method based on the maximum principle. Energy Method. For this purpose, we consider the functionals Z I1(W ) = (W − R)2 dx ZΩ I2(W ) = |∇W |2 dx. Ω

5

(3.2)

Using (2.2), we have for any δ, ε > 0 the bound Z Z d 2 I1 (W ) = −2D |∇W | dx + 2 (W − R)V · ∇W dx dt Ω Z Ω Z 2 −2 (W − R) dx − 2λ (W − R)W B dx Ω Ω Z  Z 1 2 2 ≤ −(2D − δ|V | ) |∇W | dx − 2 − − ε (W − R)2 dx δ Ω Ω Z  λ2 2  + K B02 dx e−2(µ−K)t . (3.3) ε Ω

Suppose that

|V |2 < 4D.

(3.4)

If V = 0, we can take δ = 1; if V 6= 0, then choose δ = 2D/|V |2 in (3.3). We see that (3.4) implies that 2 − 1/δ = 2 − 1/(2D/|V |2 ) > 0. Hence we can choose ε > 0 sufficiently small so that 2 − 1/δ − ε > 0. Thus we have d I1(W ) ≤ −γI1(W ) + Ce−2(µ−K)t , (3.5) dt where γ > 0 is a constant. Similarly, we have Z Z d 2 I2 (W ) = −2D (∆W ) dx − 2 ∆W (V · ∇W ) dx dt Z Ω ZΩ −2 |∇W |2 dx + 2λ (∆W )W B dx Ω Ω Z  Z 1 2 2 ≤ − 2D − − ε (∆W ) dx − (2 − δ|V | ) |∇W |2 dx δ Ω Ω Z  λ2 2  B02 dx e−2(µ−K)t . (3.6) + K ε Ω Assume ε > 0 to be small so that 2D − ε > 0. Take δ = 1/(2D − ε) so that 2D − 1/δ − ε = 0. Hence, in view of (3.4), we have 2 − δ|V |2 =

1 (2[2D − ε] − |V |2 ) > 0 (2D − ε)

(3.7)

when ε > 0 is sufficiently small. In other words, under the same condition (3.4), we see that I2(W ) satisfies (3.5) as well for some suitable γ, C > 0. Integrating (3.5) (for I1(W ) and I2(W )), we obtain (I1(W ) + I2(W ))(t) ≤ Ce−αt ,

t > 0,

(3.8)

where C > 0 is a constant and α = min{γ, 2(µ − K)} > 0. Therefore, we arrive at the following intermediate result. 6

(3.9)

Proposition 3.1. If the water supply rate R and the initial water density W0 are both sufficiently low so that R < µ,

W 0 = max{W0(x) | x ∈ Ω} < µ,

(3.10)

then the global solution (B(x, t), W (x, t)) approaches the desert state characterized by B = 0 exponentially fast. Furthermore, when the terrain slope vector V satisfies the bound (3.4), the available land water density W (x, t) approaches the simple water supply level R exponentially fast according to the decay law Z {(W (x, t) − R)2 + |∇W (x, t)|2} dx ≤ Ce−αt , (3.11) Ω

where C is a positive constant, α is given by (3.9), and t > 0. Thus we have obtained a rough description of the “desertification” phenomenon in case of meager water resources in the sense that both initial available water density W0 and the water supply rate R are low. However, since it is imaginable that a low water supply rate makes the ecosystem unlikely to maintain an adequate level of available water density, we are inclined to predict that the onset of the desert state should not depend on the initial available water density but only on the water supply rate. In other words, we may hope to establish the occurrence of the desert state as t → ∞ when R is lower than a critical threshold, say Rc , regardless the level of W0 at t = 0. In the following, we will carry out a precise study of the above-mentioned desertification phenomenon without assuming a low initial water density W0 . From (2.13) and the property B, W ≥ 0, we have ∂W ≤ D∆W + V · ∇W + R − W, ∂t

t > 0,

x ∈ Ω.

(3.12)

Therefore, the function v(x, t) = et W (x, t) − Ret satisfies ∂v ≤ D∆v + V · ∇v, ∂t

t > 0,

x ∈ Ω.

(3.13)

Using the maximum principle in (3.13), we have v(x, t) ≤ max{v(x, 0) | x ∈ Ω} = max{W0 (x) | x ∈ Ω} − R,

t > 0,

x ∈ Ω, (3.14)

which gives W (x, t) ≤ e−t (max{W0 (x) | x ∈ Ω} − R) + R,

t > 0,

x ∈ Ω.

(3.15)

The inequality (3.15) and Proposition 3.1 immediately lead us to the following sharpened, precise, statement concerning the desertification phenomenon. 7

Theorem 3.2. When the water supply rate R is sufficiently low so that R < µ,

(3.16)

then the global solution (B(x, t), W (x, t)) of (2.1)–(2.3) approaches the desert state characterized by B = 0 exponentially fast, regardless how high the initial states for the shrubs biomass density and available water density are. Furthermore, when the terrain slope vector V satisfies the bound (3.4), the available water density W (x, t) approaches the water supply rate R exponentially fast according to the decay law (3.11) valid for t > t0 where t0 > 0 is some sufficiently large time moment. Proof. From (3.15), we see that there is some t0 > 0 so that W (x, t0) < µ since R < µ. Now consider the solution of (2.13) for t > t0 with the initial condition given at t = t0 and apply Proposition 3.1. Obviously, the above study suggests that the water supply rate parameter R can actually be replaced by a bounded function of time t (for example, a period function of t) and all the foregoing discussion about the desertification remains valid. In fact, in the following, we will use a more effective method – a pointwise method based on the maximum principle – to study the desertification problem in such a generalized setting. Pointwise Method. We now consider the occurrence of the desert state using the maximum principle. As a consequence, we will derive some pointwise asymptotic estimates for a solution to approach the desert state as t → ∞. In particular, we see that there is no need to have any restricted condition on the terrain slope vector V as stated in Theorem 3.2. More precisely, we can extend Theorem 3.2 into Theorem 3.3. If the water supply rate R is a function of t but stays uniformly low such that R(t) ≤ R0 < µ, (3.17) where R0 > 0 is some constant, then the desert state appears exponentially fast in the t → ∞ limit. Furthermore, if lim R(t) = R0 < µ

t→∞

(3.18)

for some constant R0 , then for any ε > 0 satisfying R0 + ε < µ there is a large moment t0 > 0 such that (R0 − ε) − (R0 − ε)e−(t−t0 ) < W (x, t) < (R0 + ε) + (K − (R0 + ε))e−(t−t0 ) ,

∀t ≥ t0. (3.19)

In particular, B(x, t) → 0 as t → ∞ exponentially fast and W (x, t) → R0 uniformly for x ∈ Ω. 8

as t → ∞

(3.20)

Proof. In fact, the condition (3.17) implies that W satisfies the inequality ∂W ≤ D∆W + V · ∇W + R0 − W, ∂t

(3.21)

which gives us the bound W (x, t) ≤ e−t (W0 − R0 ) + R0 ,

x ∈ Ω,

t > 0,

(3.22)

as before (cf. (3.15)). Therefore, we have W (x, t) < R0 + ε < µ,

x ∈ Ω,

t ≥ t0

(3.23)

for some sufficiently small constant ε > 0 and large moment t0 > 0. In view of (3.23) and the expression Rt

B(x, t) = B(x, t0)e

t0 (W (x,s)−µ) ds

,

x ∈ Ω,

t > t0 ,

(3.24)

we that B(x, t) → 0 exponentially fast as t → ∞, as claimed. To prove the second part of the theorem, we assume that ε > 0 satisfies R0 +ε < µ. The condition (3.18) implies that there is a t1 > 0 such that R(t) < R0 + ε,

∀t ≥ t1.

(3.25)

As before, we have W (x, t) ≤ e−(t−t1 ) (max{W (x, t1) | x ∈ Ω} − (R0 + ε)) + (R0 + ε) ≤ e−(t−t1 ) (K − (R0 + ε)) + (R0 + ε), t ≥ t1.

(3.26)

Inserting (3.26) into (3.24) and with t0 = t1, we have 0 ≤ B(x, t) = B(x, t1)e(1−e

−(t−t1 ) )(K−(R

0 +ε))

e−(µ−(R0 +ε))(t−t1 ) ,

t ≥ t1 .

(3.27)

In particular, B(x, t) = O(e−(µ−(R0 +ε))(t−t1 )),

t ≥ t1 .

(3.28)

Let t2 > 0 be large enough so that R(t) − λW (x, t)B(x, t) > R0 − ε,

t ≥ t2 ,

(3.29)

which is ensured by (2.7) and (3.28). Hence we have ∂W > D∆W + V · ∇W − (W − (R0 − ε)), t ≥ t2 . (3.30) ∂t As a consequence of the maximum principle, we obtain the lower estimate for W given by W (x, t) ≥ e−(t−t2 ) (min{W (x, t2) | x ∈ Ω} − (R0 − ε)) + (R0 − ε) ≥ (R0 − ε) − (R0 − ε)e−(t−t2 ) , x ∈ Ω, t ≥ t2 ,

(3.31)

as anticipated. Setting t0 = max{t1, t2}, we arrive at (3.19) from (3.26) and (3.31). 9

4

Vegetation State and its Global Stability

It will be interesting to study what happens when (3.16) is violated. In particular, it may be desirable to achieve a green vegetation state when the water supply rate R is large enough. In fact, we shall aim at showing that such a state will appear when R > µ.

(4.1)

The Full Model. We first consider the occurrence of a green vegetation state in the full system (2.1)–(2.3). For this purpose, define n o B∞ = lim sup max{B(x, t) | x ∈ Ω} . (4.2) t→∞

It is clear that the desert and vegetation states have the following characterizations: B∞ = 0, B∞ > 0,

desert state; vegetation state.

(4.3)

We show next that the vegetation state appears when (4.1) holds. In fact, we have the following general result: Theorem 4.1. If the water supply rate R is a function of time t and stays uniformly high such that R(t) ≥ R0 > µ, (4.4) then the vegetation state appears ultimately. That is, B∞ > 0, for any solution (B, W ) of (2.1)–(2.3) with B0 6≡ 0. Moreover, if B0 ≡ 0, then B ≡ 0 such that W (x, t) → R0 as t → ∞ uniformly for x ∈ Ω provided that R(t) → R0 as t → ∞, regardless of the value of R0 . Proof. For any ε > 0 sufficiently small so that R0 − ε > µ. Let (B, W ) be a solution of (2.1)–(2.3). Suppose otherwise that B∞ = 0. Using the boundedness of W stated in (2.7), we can find some t0 > 0 such that λW (x, t)B(x, t) < ε,

t ≥ t0 .

(4.5)

Inserting (4.5) into (2.2), we have ∂W > D∆W + V · ∇W + (R0 − ε) − W, ∂t

t ≥ t0 ,

(4.6)

which is of the same form as the inequality (3.30). Therefore (3.31) holds with t2 = t0. Inserting this into (3.24) and using R0 − ε > µ, we see that B∞ = ∞ unless B0 ≡ 0. The last statement of the theorem is obvious because B ≡ 0 reduces (2.2) into a linear equation. 10

The Flower-Pot Model. We next study the ordinary differential equation limit of (2.1)–(2.3) and aim at achieving a complete understanding of the system. Such a situation may be motivated from several considerations. For example, we may assume that the vegetation biomass density B and the available water density W are uniformly distributed so that they are independent of the spatial coordinates (a macroscopic viewpoint). We may also view B as to represent the total or average biomass of the vegetation over a typical spatial domain Ω. Besides, we may also follow Shnerb et al [18] to consider a flowerpot model confined within a local region which may simply be regarded effectively as a spatial point so that we arrive at the zero-dimensional reduced form of (2.1)–(2.3). Of course, such situation occurs so far as the initial states, B0 and W0 , are uniformly distributed at t = 0. In view of all the possible situations described above, we see that the equations (2.1)–(2.3) become the following initial value problem of two ordinary differential equations dB = B(W − µ), dt dW = R − W (1 + λB), t > 0; dt B(0) = B0, W (0) = W0.

(4.7) (4.8) (4.9)

Of course, for the initial data in (4.9), only the nontrivial case, B0 > 0, W0 ≥ 0, will be interesting for a nondesert state. Thus we are led to considering the steady state of the equations (4.7) and (4.8): W1 = µ,

B1 =

 1R −1 . λ µ

(4.10)

The condition (4.1) gives us B1 > 0, which unambiguously defines a steady green vegetation state. First, we consider the local stability of the equilibrium (4.10). Setting B = B1 + u and W = W1 + v in (4.7) and (4.8), we obtain  du 1R = − 1 v + uv, dt λ µ dv R = −λµu − v − λuv. dt µ

(4.11) (4.12)

The matrix of the linear part of the system of equations (4.11) and (4.12) has the characteristic roots given by s  R 2 R − ± − (R − µ), (4.13) 2µ 2µ 11

whose real parts are always negative in view of the condition (4.1). Therefore, the steady state (4.10) is asymptotically stable. In other words, if the initial state (B0 , W0) is sufficiently close to (B1 , W1), then the solution (B(t), W (t)) of (4.7)–(4.9) approaches (B1 , W1) exponentially fast as t → ∞. In fact, here, we are able to establish the following most general result: Any solution of (4.7)–(4.8) will approach (B1 , W1) exponentially fast as t → ∞. In other words, we can show that (B1 , W1) is globally stable. Theorem 4.2. Starting from any initial state (B0 , W0 ) satisfying B0 > 0, W0 ≥ 0, the solution of (4.7)–(4.9) approaches the steady green vegetation state (B1 , W1 ) defined in (4.10) exponentially fast as t → ∞ provided that the water supply rate R is sufficiently high so that (4.1) holds. In other words, in this situation, we observe the unique vegetation state (4.10) asymptotically. Proof. By the Poincare–Bendixon theorem, a bounded solution of (4.7)–(4.9) approaches a nontrivial periodic solution or a critical point, of the equations (4.7) and (4.8) as t → ∞. In order to prove the statement of our theorem, it suffices to show that the system of equations (4.7) and (4.8) has no nontrivial periodic solution and that all solutions of (4.7) and (4.8) satisfying the given initial conditions are bounded. Since B0 > 0, we see that B(t) > 0 for all t > 0. Suppose that (B(t), W (t)) is a periodic solution of the equations (4.7) and (4.8) with period T > 0. We can rewrite (4.7) as d ln B = W − µ. (4.14) dt Inserting (4.14) into (4.8) and setting β = ln B, we obtain  dβ  d2 β =R− + µ (1 + λeβ ). dt2 dt

(4.15)

Multiplying (4.15) by dβ/dt and integrating over the periodic interval [0, T ], we have Z T  Z T  2  dβ d 1 h dβ i2 β (1 + λe ) dt = − + Rβ − µ(β + λeβ ) dt = 0, (4.16) dt 2 dt 0 dt 0

which implies that β(t) ≡constant. Using (4.14) again, we have W ≡ µ, and we arrive at the steady state (4.10). In other words, the equations (4.7) and (4.8) cannot have a nontrivial periodic solution with the stated property. To show that all solutions remain bounded, using B(t) > 0 and W ≥ 0 for all t > 0 and the equations (4.7) and (4.8), we have d (λB + W ) ≤ R − σ(λB + W ), dt

σ ≡ min{1, µ},

(4.17)

with equality to hold only when µ = 1. Integrating (4.17), we obtain the bound λB(t) + W (t) ≤

R (1 − e−σt ) + (λB0 + W0 )e−σt , σ 12

t ≥ 0,

(4.18)

with equality only when µ = 1. Since we know that B(t) > 0 and W (t) ≥ 0, the boundedness of any solution is established. The above method may be used to derive a more explicit upper estimate for the biomass B(t). Indeed, inserting (4.18) into the equation (4.7), we have h R i h dB R i −σt  ≤ −λB 2 + − µ + λB0 + W0 − e B, dt σ σ

(4.19)

which is a Bernoulli type differential inequality and may be integrated to give us B0e( σ −µ)t+ σ (λB0+W0 − σ )(1−e ) , Rt R 1 R −σs 1 + λB0 0 e( σ −µ)s+ σ (λB0+W0 − σ )(1−e ) ds R

B(t) ≤

1

R

−σt

t ≥ 0,

(4.20)

with equality only when µ = 1. Now assume µ = 1. Then in (4.18)–(4.20), all equalities hold and σ = 1 (cf. (4.17)). Therefore we have the explicit solution B(t) =

−t B0e(R−1)t+(λB0+W0 −R)(1−e ) , Rt −s 1 + λB0 0 e(R−1)s+(λB0 +W0 −R)(1−e ) ds

W (t) = (λB0 + W0 )e−t + R(1 − e−t ) − λB(t).

(4.21)

It is clear that this solution approaches (B1, W1 ) exponentially fast as t → ∞. It remains as a curious open question whether one may obtain the solution of (4.7)–(4.9) explicitly when µ 6= 1. Although (4.21) gives the explicit solution only in the special case µ = 1 for the equations (4.7)–(4.9), it may provide useful information to some technical questions. For example, we have seen that R < µ leads to the desert state and R > µ leads to the vegetation state. So it is interesting to ask what happens at the critical situation, R = µ. In (4.21), after setting R = 1, we see that B → 0 and W → 1 as t → ∞, which gives rise to the desert state for R = µ = 1.

5

Existence of Periodic Solutions for the FlowerPot Model

Naturally, it will be interesting to consider the situation when the water supply rate is a function of time t. A relevant assumption is that R is a periodic function modelling the seasonal change of water supply and we ask whether the flower pot ecosystem governed by the coupled equations (4.7) and (4.8) possesses a periodic solution of the same period as R. Such seasonal vegetation pattern formation phenomena have indeed been studied previously. See, for example, the recent work of Guttal and Jayaprakash [4] and references therein.

13

For our purpose, we assume the water supply rate R to be a periodic function of the time t of period T : R(t) = R(t + T ), T > 0, (5.1) and we hope to establish the existence of a periodic solution of the same period for the differential equations (4.7) and (4.8). Integrating (4.7) and (4.8), we have Z T B(T ) = B0 + B(t)(W (t) − µ) dt, (5.2) 0 Z T W (T ) = W0 + (R(t) − W (t)(1 + λB(t))) dt. (5.3) 0

By the continuous dependence theorem of the solution of ordinary differential equations on its initial data, we know that the map (B(T ), W (T )) = F (B0, W0 ),

B0 ≥ 0,

W0 ≥ 0,

(5.4)

defined by (5.2) and (5.3) is continuous. In order to prove the existence of a periodic solution, it suffices to show that F has a fixed point in the first quadrant of R2 . To show that F has a fixed point in the first quadrant of R2 , it will be sufficient to show that F maps a compact convex subset of the first quadrant of R2 into itself. For this purpose, we set SL = {(p, q) ∈ R2 | p ≥ 0, q ≥ 0, λp + q ≤ L}.

(5.5)

Of course, SL is a compact convex subset of R2 for any L > 0. Define R0 = max{R(t) | 0 ≤ t ≤ T }.

(5.6)

Then we can replace (4.18) by λB(t) + W (t) ≤

R0 (1 − e−σt ) + (λB0 + W0 )e−σt , σ

t ≥ 0,

(5.7)

We assume L to be large so that R0 ≤ σL.

(5.8)

Therefore, for (B0 , W0 ) ∈ SL , we see in view of (5.7) and (5.8) that λB(T )+W (T ) ≤ L which proves F (B0, W0 ) ∈ SL (in fact, we have shown that λB(t) + W (t) ≤ L for all t). Thus, F has a fixed point in SL which confirms the existence of a desired periodic solution of the ordinary differential equations (4.7) and (4.8). As a by-product of our argument, we have seen from (5.8) that this periodic solution must satisfy λB(t) + W (t) ≤

R0 . σ

(5.9)

It will also be interesting to know whether there is a periodic solution so that the biomass B(t) is nontrivial, B(t) > 0, which represents a periodic vegetation state. 14

Inserting λB(t) ≤ R0 /σ (see (5.9)) into the equation (4.7), we have  dW R0  ≥ R(t) − 1 + W, dt σ

which gives us

0

0

W (t) ≥ W0 e

−(1+ Rσ )t

+e

−(1+ Rσ )t

Z

(5.10)

t

R(s)e(1+

R0 )s σ

ds.

(5.11)

0

Therefore, we have the lower bound B(T ) = B0 exp

Z

T

W (t) dt − µT

0



≥ B0 ,

provided that there holds the condition Z T Z t 0 R0 −(1+ Rσ )t e R(s)e(1+ σ )s ds dt ≥ µT. 0

(5.12)

(5.13)

0

As a consequence, we may modify our convex set SL by defining SLε0 = {(p, q) ∈ R2 | p ≥ ε0, q ≥ 0, λp + q ≤ L},

(5.14)

where L satisfies (5.8) and the positive number ε0 satisfies ε0 < R0 /σ. The above study shows that F maps SLε0 into itself under (5.13). Thus F has a fixed point in SLε0 which leads to the existence of a T -periodic solution satisfying B(t) > 0 for all t. In summary, we can state Theorem 5.1. If the water supply rate R is a T -periodic function of the time t, the equations (4.7) and (4.8) always possess a T -periodic solution. In the situation when the water supply rate satisfies the condition Z   R0 −1 T  R0 1 − e−(1+ σ )(T −t) R(t) dt ≥ µT, 1+ (5.15) σ 0 then the shrubs biomass B(t) of the periodic solution never vanishes which represents a periodic green vegetation state. Proof. We only need to verify that the condition (5.13) is the same as (5.15). To this end, we denote the left-hand side of (5.13) by y(T ). It is straightforward to see that y satisfies the nonhomogeneous differential equation y 00(T ) + (1 +

R0 0 )y (T ) = R(T ). σ

(5.16)

Using the method of variation of parameters, we obtain the form of y(T ) given by the left-hand side of (5.15). 15

The condition (5.15) implies that, in response to the periodic change of the water supply rate R = R(t), a periodic non-desert state exists for the water-biomass ecosystem under consideration when the average value of R(t) is suitably given. On the other hand, we have seen in Section 3 that, when the water supply rate R is low such that (3.17) holds, all solutions approach the desert state characterized by B = 0. However, since Theorem 5.1 ensures the existence of a periodic solution for any periodic water supply rate R = R(t), we see that, under the condition (3.17), such a periodic solution must have a trivial biomass, B ≡ 0. Therefore, we arrive at a periodic desert state defined by the unique T -periodic solution of the reduced equation dW = R(t) − W. (5.17) dt

6

Asymptotic Conservation Laws

We now turn our attention back to the full partial differential equations (2.1)–(2.3) and we wish to reveal a certain close relation between the vegetation biomass density B(x, t) and the available water density W (x, t) over the spatial domain Ω. Notice that the total biomass and total available water over Ω at any time t are respectively given by the quantities Z Z B(t) = B(x, t) dx, W(t) = W (x, t) dx. (6.1) Ω

Ω

On the other hand, using the periodic boundary condition, we have Z Z ∆W dx = 0, V · ∇W dx = 0. Ω

(6.2)

Ω

As a consequence of (6.2), we may integrate (2.1) and (2.2) over Ω to obtain R|Ω| − γ(λB + W ) ≤

d (λB + W) ≤ R|Ω| − σ(λB + W), dt

(6.3)

where γ = max{1, µ} and σ = min{1, µ}. Integrating (6.3), we find R|Ω| (1 − e−γt ) + (λB0 + W0 )e−γt ≤ λB(t) + W(t) γ R|Ω| ≤ (1 − e−σt ) + (λB0 + W0 )e−σt , (6.4) σ where the initial total biomass and total water resource are given by Z Z B0 = B0(x) dx, W0 = W0 (x) dx. Ω

Ω

The relation (6.4) indicates how B(t) and W(t) consume each other. 16

(6.5)

The above method may be explored further to investigate the spots in Ω where the biomass density B(x, t) attains its spatial maxima and minima with respect to those of the water density function W (x, t), although it may be difficult to establish similar pointwise results. When the initial water density W0 depends on the spatial coordinate x, we do not know whether for any fixed t > 0, the biomass B(x, t) would assume its maxima or minima according to the distribution of W (x, t). However, when B0 is constant, the expression (2.12) implies that B(x, t) attains its local maxima or minima where W (x, t) does, which seems to be a naturally expected property and will be observed throughout the following consideration. For any given t > 0, we use x0 (t) (or x0(t)) to denote a local maximum (or minimum) point for the function W (x, t) evaluated over Ω. In view of (2.12), we see that x0 (t) (or x0(t)) is also a local maximum (or minimum) point for the function B(x, t). We have (∆W )(x0(t), t) ≤ 0, (∇W )(x0(t), t) = 0,

(∆W )(x0(t), t) ≥ 0, (∇W )(x0(t), t) = 0.

(6.6) (6.7)

Besides, we also have (∇B)(x, t) = 0 for x = x0 or x = x0 . Therefore d ∂ d ∂ B(x, t) = B(x, t), W (x, t) = W (x, t), (6.8) dt ∂t dt ∂t for x = x0(t) or x = x0(t). Inserting (6.6), (6.7), and (6.7) into (2.1) and (2.2) and using the notation b0 (t) = B(x0(t), t), b0(t) = B(x0(t), t), w0 (t) = W (x0(t), t), w0 (t) = W (x0(t), t), we arrive at d (λb0 + w0 ) ≤ R − σ(λb0 + w0 ), (6.9) dt d (λb0 + w0) ≥ R − γ(λb0 + w0 ). (6.10) dt Consequently, similar to (6.4), we can integrate (6.9) and (6.10) to get R (6.11) λb0 (t) + w0 (t) ≤ (1 − e−σt ) + (λB0 + max{W0 (x)})e−σt , x∈Ω σ and R λb0 (t) + w0(t) ≥ (1 − e−γt ) + (λB0 + min{W0 (x)})e−γt . (6.12) x∈Ω γ These estimates give us the pointwise relations between the biomass density and water density at their extreme points. Of practical importance are the asymptotic quantities Bmin ≡ lim inf B(t),

Bmax ≡ lim sup B(t),

t→∞

t→∞

Wmin ≡ lim inf W(t), t→∞

Wmax ≡ lim sup W(t), t→∞

Bmax ≡ lim sup b0 (t),

Bmin ≡ lim inf b0(t), t→∞

t→∞

Wmin ≡ lim inf w0(t), t→∞

Wmax ≡ lim sup w0 (t). t→∞

17

(6.13)

Letting t → ∞ in (6.4), (6.11), (6.12), we have R|Ω| R|Ω| ≤ λBmin + Wmin ≤ λBmax + Wmax ≤ , γ σ R R ≤ λBmin + Wmin ≤ λBmax + Wmax ≤ . γ σ In summary, we can state

(6.14) (6.15)

Theorem 6.1. The total biomass and total water resource given in (6.1) remain bounded uniformly for t > 0 and satisfy the pointwise “conservation” law (6.4) at any t > 0. In the situation where the initial biomass density B0 is homogeneously distributed (so that B0 is a constant), the biomass density attains its local maxima or minima where the water density attains for any given t > 0 and if their respective maxima or minima are denoted by b0 (t), w0(t) or b0 (t), w0(t), then these quantities satisfy the pointwise “conservation” law expressed by (6.11) and (6.12). As t → ∞, all solutions described are absorbed exponentially fast into a universal attractor characterized by (6.14) for the total biomass and total water resource or by (6.15) for the extreme spots of the biomass density and water density functions, which may be interpreted as two “asymptotic conservation laws.” As a by-product of the above stated conservation laws, we also obtain a clear interpretation for the parameter λ > 0 as follows: For a given level of water W , if λ is large, B has to be small; if λ is small, B will be large. This result shows that 1/λ measures the effectiveness of the plants to utilize the water supply in the environment.

7

Conclusions and Remarks

In conclusion, in this paper, a systematic study is carried out for the coupled system of parabolic equations (2.1)–(2.3), proposed by Shnerb, Sarah, Lavee, and Solomon [18], which governs the interaction of vegetation biomass density B and the available water density W in a closed ecosystem. The main results are (i) Desertification when the water supply rate R is below the death rate µ of the vegetation in the absence of water. That is, when R < µ. (ii) The occurrence of vegetation state when R > µ. (iii) The dominance and global stability of the steady green vegetation state within the flower-pot limit of the system when R > µ. (iv) The occurrence of nontrivial periodic states within the flower-pot model when the water supply rate R is a periodic function of time which assumes a suitable average level over its period. (v) Some asymptotic expressions relating or regulating the vegetation biomass density and available water density in the full model.

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It is hoped that our study will provide hints and mathematical methods for the investigation of the models of more extended ecosystems containing vegetation diffusion/dispersion [5, 7, 9, 13, 16, 17, 19, 20] and recognizing the distinction between available soil and surface water resources [4, 14]. See also [2, 3, 6, 10, 11, 12] for reviews on various directions of research on the general subject of pattern formation problems in biological systems which are related to the problem considered here. Acknowledgements. The research of YY was supported in part by NSF and an Othmer senior faculty fellowship at Polytechnic Institute of New York University.

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