Competition Models for Plant Stems

The models introduced in this paper describe a uniform distribution of plant stems competing for sunlight. The shape of each stem, and the density of leaves, are designed in order to maximize the captured sunlight, subject to a cost for transporting water and nutrients from the root to all the leaves. Given the intensity of light, depending on the height above ground, we first solve the optimization problem determining the best possible shape for a single stem. We then study a competitive equilibrium among a large number of similar plants, where the shape of each stem is optimal given the shade produced by all others. Uniqueness of equilibria is proved by analyzing the two-point boundary value problem for a system of ODEs derived from the necessary conditions for optimality.

Introduction
Optimization problems for tree branches have recently been studied in [3,5]. In these models, optimal shapes maximize the total amount of sunlight gathered by the leaves, subject to a cost for building a network of branches that will bring water and nutrients from the root to all the leaves. Following [2,8,11,13,14], this cost is defined in terms of a ramified transport.
In the present paper we consider a competition model, where a large number of similar plants compete for sunlight. To make the problem tractable, instead of a tree-like structure we assume that each plant consists of a single stem. As a first step, assuming that the intensity of light I(·) depends only on the height above ground, we determine the corresponding optimal In Section 3, given a light intensity function I(·), we study the optimization problem for Model 1, proving the existence of an optimal solution and deriving necessary conditions for optimality. We also give a condition which guarantees the uniqueness of the optimal solution. A counterexample shows that, in general, if this condition is not satisfied multiple solutions can exist. In Section 4 we consider the competition of a large number of stems, and prove the existence of an equilibrium solution. In this case, the common shape of the plant stems can be explicitly determined by solving a particular ODE.
The subsequent sections extend the analysis to a more general setting (Model 2), where both the length and the thickness of the stems are to be optimized. In Section 5 we prove the existence of optimal stem configurations, and derive necessary conditions for optimality, while in Section 6 we establish the existence of a unique equilibrium solution for the competitive game, assuming that the density (i.e., the average number of stems growing per unit area) is sufficiently small. The key step in the proof is the analysis of a two-point boundary value problem, for a system of ODEs derived from the necessary conditions.
In the above models, the density of stems was assumed to be uniform on the whole space. As a consequence, the light intensity I(h) depends only of the height h above ground. Section 7, on the other hand, is concerned with a family of stems growing only on the positive half line. In this case the light intensity I = I(h, x) depends also on the spatial location x, and the analysis becomes considerably more difficult. Here we only derive a set of equations describing the competitive equilibrium, and sketch what we conjecture should be the corresponding shape of stems.
The final section contains some concluding remarks. In particular, we discuss the issue of phototropism, i.e. the tendency of plant stems to bend in the direction of the light source. Devising a mathematical model, which demonstrates phototropism as an advantageous trait, remains a challenging open problem. For a biological perspective on plant growth we refer to [9]. A recent mathematical study of the stabilization problem for growing stems can be found in [1].

Optimization problems for a single stem
We shall consider plant stems in the x-y plane, where y is the vertical coordinate. We assume that sunlight comes from the direction of the unit vector n = (n 1 , n 2 ), n 2 < 0 < n 1 .
As in Fig. 1, we denote by θ 0 ∈ ]0, π/2[ the angle such that (−n 2 , n 1 ) = (cos θ 0 , sin θ 0 ). (2.1) Moreover, we assume that the light intensity I(y) ∈ [0, 1] is a non-decreasing function of the height y. This is due to the presence of competing vegetation: close to the ground, less light can get through.
Model 1 (a stem with fixed length and constant thickness). We begin by studying a simple model, where each stem has a fixed length . Let s → γ(s) = (x(s), y(s)), s ∈ [0, ], be an arc-length parameterization of the stem. As a first approximation, we assume that the leaves are uniformly distributed along the stem, with density κ. The total distribution of leaves in space is thus described by a measure µ, with for every Borel set A ⊆ IR 2 .
Among all stems with given length , we seek the shape which will collect the most sunlight. This can be formulated as an optimal control problem. Indeed, by the Lipschitz continuity of γ(·), the tangent vector t(s) =γ(s) = (cos θ(s), sin θ(s)) is well defined for a.e. s ∈ [0, ]. The map s → θ(s) will be regarded as a control function. According to the model in [5], calling Φ(·) the density of the projection of µ on the space E ⊥ n orthogonal to n, the total sunlight captured by the stem is S(γ) = 1−exp{−Φ(z)} dz = 0 I(y(s))· 1 − exp −κ cos(θ(s) − θ 0 ) cos(θ(s)−θ 0 ) ds.
As shown in Fig. 2, left, we call z the coordinate along the space E ⊥ n perpendicular to n, and let y be the vertical coordinate. Hence dz(s) = cos(θ(s) − θ 0 ) ds, dy(s) = sin(θ(s)) ds. (2.7) In view of (2.4), one can express both γ and θ as functions of the variable y. Introducing the function the problem can be equivalently formulated as follows.
(OP1) Given a length > 0, find h > 0 and a control function y → θ(y) ∈ [θ 0 , π/2] which maximizes the integral h 0 I(y) g(θ(y)) dy (2.9) subject to h 0 1 sin θ(y) dy = . (2.10) Model 2 (stems with variable length and thickness). Here we still assume that the plant consists of a single stem, parameterized by arc-length: s → γ(s), s ∈ [0, ]. However, now we give no constraint on the length of the stem, and we allow the density of leaves to be variable along the stem.
Call u(s) the density of leaves at the point γ(s). In other words, µ is now the measure which is absolutely continuous w.r.t. arc-length measure on γ, with density u. Instead of (2.2) we thus have Calling I(y) the intensity of light at height y, the total sunlight gathered by the stem is now computed by As in [5], we consider a cost for transporting water and nutrients from the root to the leaves. This is measured by for some 0 < α < 1. Notice that, in Model 1, this cost was the same for all stems and hence it did not play a role in the optimization.
The maximum is sought over all controls θ : IR + → [0, π] and u : IR + → IR + . Calling this leads to an optimal control problem in a more standard form.
Let I(y) be the light intensity, which we assume is a non-decreasing function of the vertical component y. For a given κ > 0 (the thickness of the stem), we seek a curve s → γ(s), starting at the origin and with a fixed length , which maximizes the sunlight functional defined at (2.12).
Hence there exists a maximizing sequence of control functions θ n : [0, h n ] → [θ 0 , π/2], so that hn 0 1 sin θ n (y) dy = for all n ≥ 1, 2. For each n, let θ n be the non-increasing rearrangement of the function θ n . Namely, θ n is the unique (up to a set of zero measure) non-increasing function such that, for every c ∈ IR meas {s ; θ n (s) < c} = meas {s ; θ n (s) < c} .
For every n ≥ 1 we claim that hn 0 1 sin θ n (y)
To prove (3.5) we observe that the function g introduced at (2.8) is smooth and satisfies Therefore, the map y → g(θ n (y)) coincides with the non-decreasing rearrangement of y → g(θ n (y)). On the other hand, since I(·) is non-decreasing, it trivially coincides with the non-decreasing rearrangement of itself. Therefore, (3.5) is an immediate consequence of the Hardy-Littlewood inequality [10].
3. Since all functions θ n are non-increasing, they have bounded variation. Using Helly's compactness theorem, by possibly extracting a subsequence, we can find h > 0 and a non-
The optimal control θ * (·) determined by the necessary condition (3.18) is thus recovered by Next, we need to determine h so that the constraint is satisfied. As shown by Example 3.4 below, the solution of (3.21)-(3.22) may not be unique.

Hence (3.23) is satisfied provided that
(3.24) From the above analysis, we conclude where z → ϕ(z) = θ is the function implicitly defined by (3.19).
The following example shows that, without the bound (3.24) on the sunlight intensity function I(·), the conclusion of Theorem 3.3 can fail. I changes abruptly as a function of the hight, the optimal shape may not be unique, as shown in Example 3.4.
with ε > 0 small. By Theorem 3.1 at least one optimal solution exists. By the previous analysis, any optimal solution θ * : [0, h * ] → [θ 0 , π/2] satisfies the necessary conditions (3.25). In this particular case, this implies that θ * (y) is constant separately for y < 1 and for y > 1. As shown in Fig. 2, right, these necessary conditions can have two solutions.
As ε → ε 1 +, we have Comparing (3.28) with (3.29), by continuity we conclude that there exists some . Hence for ε = ε the optimization problem has two distinct solutions.
We remark that in this example the light intensity I(y) is discontinuous at y = 1. However, by a mollification one can still construct a similar example with two optimal configurations, also for I(·) smooth. Of course, in this case the derivative I (h) will be extremely large for h ≈ 1, so that the assumption (3.24) fails.

A competition model
In the previous analysis, the light intensity function I(·) was a priori given. We now consider a continuous distribution of stems, and determine the average sunlight I(y) available at height y above ground, depending on the density of vegetation above y.
Let the constants , κ > 0 be given, specifying the length and thickness of each stem. We now introduce another constant ρ > 0 describing the density of stems, i.e. how many stems grow per unit area. Assume that all stems have the same height and shape, described the the function θ : [0, h] → [θ 0 , π/2]. For any y ∈ [0, h], the total amount of vegetation at height ≥ y, per unit length, is then measured by The corresponding light intensity function is defined as while I(y) = 1 for y ≥ h. We are interested in equilibrium configurations, where the shape of the stems is optimal for the light intensity I(·). We recall that θ 0 is the angle of incoming light rays, as in (2.1), while the constants , κ > 0 denote the length and thickness of the stems.
(ii) For all y ≥ 0, the light intensity at height y satisfies If the density of vegetation is sufficiently small, we now show that an equilibrium configuration exists.
For all y ≥ 0, define the light intensity For ρ > 0 small enough, we claim that this function satisfies the assumption of Theorem 3.3. Indeed, for a.e. h ∈ [0,h] the left hand side of (3.24) is estimated by and it clearly approaches zero as ρ → 0. On the other hand, for h >h we have I (h) = 0, hence the inequality (3.24) is trivially satisfied.
By Theorem 3.3, the optimization problem (OP1) has a unique solution θ * : For convenience, we extend this map to the entire interval [0, ] by setting The above definition yields a mapping 2. We claim that the map Λ in (4.5) is continuous.
Otherwise, there would exist a sequence (h n , θ n ) → (h, θ), such that Λ(h n , θ n ) does not converge to Λ(h,θ). By compactness we can choose a subsequence (h n k , θ n k ) such that But then (h , θ ) would be a second optimal solution to the optimization problem with light intensity function (4.3), in contradiction with the uniqueness stated in Theorem 3.3.
By Schauder's theorem, the continuous map Λ from the compact convex set K into itself has a fixed point, which provides the desired equilibrium configuration.

Uniqueness and representation of equilibrium solutions.
By (3.21) and (4.2), this equilibrium configuration (h * , θ * ) must satisfy the necessary condition where ϕ is the function defined in Lemma 3.2. Here the constant h * must be determined so that Based on (4.6), one obtains a simple representation of all equilibrium configurations, for any length > 0. Indeed, for t ∈ ] − ∞, 0], let t → ζ(t) be the solution of the Cauchy problem For any length of the stem, choose h * = h * ( ) so that (4.8) The shape of the stem that achieves the competitive equilibrium is then provided by Since the backward Cauchy problem has a unique solution, we conclude that, if an equilibrium solution exists, by the representation (4.9) it must be unique.

Stems with variable length and thickness
We now consider the optimization problem (OP2), allowing for stems of different lengths and with variable density of leaves.
Proof. 1. Consider a maximizing sequence of couples (θ k , u k ) : be the arc-length parameterization of the stem γ k . Call µ k the Radon measure on IR 2 describing the distribution of leaves along γ k . For every Borel set A ⊆ IR n , we thus have For a given radius ρ > 0, we have the decomposition where µ k is the restriction of µ k to the ball B(0, ρ), while µ k the restriction of µ k to the complement IR 2 \B(0, ρ). By the same arguments used in steps 1-2 of the proof of Theorem 3.1 in [3], if the radius ρ is sufficiently large, then for all k ≥ 1. Here S and I α are the functionals defined at (2.12)-(2.13). According to (5.2), we can replace the measure µ k with µ k without decreasing the objective functional.
Without loss of generality we can thus choose > 0 sufficiently large and assume that u k (s) = 0 for all s > , k ≥ 1.
In turn, since S(µ k ) − cI α (µ k ) ≥ 0, we obtain the uniform bound In this step we show that the measures µ k can be taken with uniformly bounded mass. Consider a measure µ k for which (5.3) holds. By (2.13), for every r ∈ [0, ] one has In view of (5.3), this implies It thus remains to prove that, in our maximizing sequence, the functions u k can be replaced with functionsũ k having a uniformly bounded integral over [0, r], for some fixed r > 0.
Toward this goal we fix 0 < ε < β < 1, and, for j ≥ 1, we define r j = 2 −j , and the interval We then let µ j = u u µ andμ j =ũ j µ be the measures supported on V j , corresponding to these densities.
For a fixed integer j * , whose precise value will be chosen later, consider the set of indices and the modified densityũ Moreover, callμ the measure obtained by replacing u withũ in (2.11). By (5.4) and (5.5) the total mass ofμ is bounded. Indeed We now claim that S(μ) − cI α (μ) ≥ S(µ) − cI α (µ). (5.10) Toward a proof of (5.10), we estimate (5.11) To estimate the difference in the irrigation cost, we first observe that the inequality Sinceũ(s) ≤ u(s) for every s ∈ [0, ], using (5.12) we now obtain By choosing the integer j * large enough in (5.7), for j ≥ j * all terms in the summation on the right hand side of (5.14) are ≥ 0. This implies (5.10).
3. By the two previous steps, w.l.o.g. we can assume that the measures µ k have uniformly bounded support and uniformly bounded total mass. Otherwise, we can replace the sequence (u k ) k≥1 with a new maximizing sequence (ũ k ) k≥1 having these properties.
By taking a subsequence, we can thus assume the weak convergence µ k µ. The upper semicontinuity of the functional S, proved in [5], yields S(µ) ≥ lim sup k→∞ S(µ k ). (5.15) In addition, since all maps s → γ k (s) are 1-Lipschitz, by taking a further subsequence we can assume the convergence γ k (s) → γ(s) (5.16) for some limit function γ, uniformly for s ∈ [0, ].
Since each measure µ k is supported on γ k , the weak limit µ is a measure supported on the curve γ.

4.
Since θ k (s) ∈ [θ 0 , π/2], we can re-parameterize each stem γ k in terms of the vertical variable Calling s = s k (y) the inverse function, we thus obtain a maximizing sequence of couples Moreover, the stem γ k can be described as the graph of the Lipschitz function Since all functions x k (·) satisfy x k (0) = 0 and are non-decreasing, uniformly continuous with Lipschitz constant L = cos θ 0 / sin θ 0 , by possibly extracting a further subsequence, we obtain the convergence h k →h and x k (·) →x(·). Herex : [0,h] → IR is a nondecreasing continuous function with Lipschitz constant L, such thatx(0) = 0. More precisely, the convergence x k →x is uniform on every compact subinterval [0, h] with h <h.

5.
We claim that the irrigation cost of µ is no greater that the lim-inf of the irrigation costs for µ k . Let σ → γ(σ) be an arc-length parameterization of γ. Since s → γ(s) if 1-Lipschitz, one has dσ/ds ≤ 1. We now compute (5.17) 6. Combining (5.15) with (5.17) we conclude that the measure µ, supported on the stem γ, is optimal.
Letū be the density of the absolutely continuous part of µ w.r.t. the arc-length measure on γ, and call µ * the measure that has densityū w.r.t. arc-length measure. Since S(µ * ) = S(µ), it follows that µ * = µ. Otherwise I α (µ * ) < I α (µ) and µ is not optimal. This argument shows that the optimal measure µ is absolutely continuous w.r.t. the arc-length measure on γ.

Necessary conditions for optimality.
Let t → (θ * (t), u * (t)) be an optimal solution to the problem (OP2). The necessary conditions for optimality [4,6,7] yield the existence of dual variables p, q satisfying and such that the maximality condition We recall that G(θ, u) is the function defined at (2.17). An intuitive interpretation of the quantities on the right had side of (5.19) goes as follows: • p(t) is the rate of increase in the gathered sunlight, if the upper portion of stem {γ(s) ; s > t} is raised higher.
• q(t) is the rate at which the irrigation cost increases, adding mass at the point γ(t).
• I(y(t)) G(θ, u) is the sunlight captured by the leaves at the point γ(t).

Uniqueness of the optimal stem configuration
Aim of this section is to show that, if the light intensity I(y) remains sufficiently close to 1 for all y ≥ 0, then the shape of the optimal stem is uniquely determined. This models a case where the density of external vegetation is small. for some constants C > 0 and 0 < β < 1. If for some δ > 0 sufficiently small, then the optimal solution to (OP2) is unique.
Proof. We will show that the necessary conditions for optimality have a unique solution. This will be achieved in several steps.

1.
Given I, p, q, define the functions Θ, U by setting Θ(I, p, q), U (I, p, q) . = arg max We recall that G is the function defined at (2.17). Notice that one can write Denote by H(θ, u) . = p · sin θ − q u + I(y) G(θ, u) − cz α (6.5) the quantity to be maximized in (6.3). Differentiating H w.r.t. θ and imposing that the derivative is zero, we obtain Similarly, differentiating w.r.t. u we find This yields u = − ln q I cos (θ − θ 0 ). (6.7) A lengthy but elementary computation shows that the Hessian matrix of second derivatives of H w.r.t. θ, u is negative definite, and the critical point is indeed the point where the global maximum is attained. By (6.7) it follows U (I, p, q) = − ln q I cos Θ(I, p, q) − θ 0 .  are the functions introduced at (6.3), or more explicitly at (6.8)-(6.9). Notice that the length T of the stem is a quantity to be determined, using the boundary conditions in (6.13). p · sin θ − q u + I(y) G(θ, u) − cz α (6.15) is constant along trajectories of (6.13). Observing that the terminal conditions in (6.

3.
We can use this identity to express z as a function of the other variables: 4. Since I is given as a function of the height y, it is convenient to rewrite the equations (6.13) using y as an independent variable. Using the identity (6.17), we obtain a system of two equations for the variables p, q: where w = w(I, p, q) is the function introduced at (6.10). Note that under our assumptions, f 1 remains bounded, while f 2 diverges as q(y) → I(y). The system (6.13) can now be equivalently formulated as

5.
To prove uniqueness of the solution to the boundary value problem (6.13), it thus suffices to prove the following (see Fig. 4, right). The figure on the right shows the case where I(·) is not constant. As before, h must be determined so that q(0, h) = 0.
In the case I(y) ≡ 1, recalling (6.9) we obtain (see Fig. 4) The system (6.13) can now be written as 6. We will show that a strict inequality as in (6.30) remains valid for a more general function I(·), provided that the assumptions (6.1)-(6.2) hold.
Toward this goal, we need to determine how p and q vary w.r.t. the parameter h. Denoting by their partial derivatives, by (6.20) one obtains the linear system The boundary conditions at y = h require some careful consideration. As y → h−, we expect f 2 (I(y), p(y), q(y)) → +∞ and Q(y) → −∞. To cope with this singularity we introduce the new variable Q(y) . = Q(y) f 2 I(y), p(y), q(y) . (6.33) The system (6.32), together with the new boundary conditions for P, Q can now be written as (6.34) To analyze this system we must compute the partial derivatives of f 1 and f 2 . From the definition (6.10) it follows Using (6.35), from (6.18), (6.19) we obtain z I(y), p, q , z I(y), p, q .
We need to show that the same strict inequality holds when δ > 0 in (6.2) is small enough. Notice that, if the right hand sides of the equations in (6.34) were bounded, letting I L ∞ → 0 a continuity argument would imply the uniform convergence P (y) → 0 and Q(y) → −1. The same conclusion can be achieved provided that the right hand sides in (6.34) are uniformly integrable. This is precisely what will be proved in the next two steps, relying on the identities (6.36).
The lower bound Θ ≥ θ 0 is an immediate consequence of (6.9). To obtain an upper bound on Θ, we set q . = q(y) I(y) .
A similar argument yields Using I (y) ≤ δ and (6.42) in the equation (6.18) we obtain a bound of the form which yields Since α < 1, using (6.41) and (6.44) in (6.9) we obtain the limit Θ(y) → θ 0 as y → h.
On the other hand, when y is bounded away from h, the denominator in (6.10) is strictly positive and the quantity w = w(I, p, q) remains uniformly bounded. By (6.9), we obtain the upper bound Θ ≤ θ + , for some θ + < π/2.

8.
Relying on (6.36), in this step we prove that all terms on the right hand sides of the ODEs in (6.34) are uniformly integrable.
(i) We first consider the terms appearing in the ODE for P (y). Concerning f 1,p , as y → h− one has because of (6.41). Since α < 1, this implies that f 1,p is an integrable function of y.
(ii) By the second equation in (6.36), as y → h− one has iii) The term f 2 blows up as y → h−, due to the factor z α−1 . However, this factor is integrable in y because, by (6.38), (6.41) and (6.42) showing that f 2 is integrable, because α > 0.
(iv) We now solve the linear ODE for P in (6.34) with terminal condition P (h) = 0. By the estimates (6.45)-(6.46) and (6.48) one obtains a bound of the form (v) In a neighborhood of the origin, the function f 1,q contains a logarithm which blows up as y → 0+. However, this is integrable because, for y ≈ 0, we have and ln y is integrable in y. Recalling (6.1), as y > 0 ranges in a neighborhood of the origin, we conclude (vi) It remains to consider the terms appearing in the ODE for Q(y). We first observe that − 2α z −α I(y), p(y), q(y) .

9.
The proof can now be accomplished by a contradiction argument. If the conclusion of the theorem were not true, one could find a sequence of absolutely continuous, non-decreasing functions I n : IR + → [0, 1], all satisfying (6.1), with I n (0) → 1, and such that, for each n ≥ 1, the optimization problem (OP2) has two distinct solutions, say (θ n ,ǔ n ) and (θ n ,û n ). As a consequence, for each n ≥ 1 the system (6.24) has two solutions. To fix the ideas, let the first solution be defined on [0,ȟ n ] and the second on [0,ĥ n ], withȟ n <ĥ n . These two solutions will be denoted by (p n ,q n ,ž n ) and (p n ,q n ,ẑ n ). They both satisfy the boundary conditionš p n (ȟ n ) =p n (ĥ n ) = 0,q n (ȟ n ) = I(ȟ n ),q n (ĥ n ) = I(ĥ n ),q n (0) =q n (0) = 0. (6.54) Thanks to the last identity, by the mean value theorem there exists some intermediate point k n ∈ [ȟ n ,ĥ n ] such that, with the notation introduced at (6.21), ∂q n ∂h (0, k n ) = 0. (6.55) For each n ≥ 1 consider the corresponding system (6.56) Since f 2 I n (0), p n (0, k n ), 0 > 0, by (6.55) it follows Q n (0) = 1 f 2 I n (0), p n (0, k n ), 0 · ∂q n ∂h (0, k n ) = 0. (6.57) Let P n (y) . = ∂p(y, k n ) ∂h , Q n (y) . = 1 f 2 I n (y), p n (y, k n ), q n (y, k n ) · ∂q(y, k n ) ∂h , be the solutions to (6.56). By the previous steps, their derivatives P n , Q n n≥1 form a sequence of uniformly integrable functions defined on the intervals [0, k n ]. Note that the existence of an upper bound sup n k n . = h + < +∞ follows from the existence proof.
Thanks to the uniform integrability, by possibly taking a subsequence, we can assume the convergence k n → h ∈ [0, h + ], the weak convergence of derivatives P n P , Q n Q in L 1 , and the convergence P n → P, Q n → Q, uniformly on every subinterval [0, h] with h <h.
Recalling that every I n satisfies the uniform bounds (6.1), since I n (y) → I(y) ≡ 1 uniformly for all y ≥ 0, we conclude that (P, Q) provides a solution to the linear system (6.34) on [0,h], corresponding to the constant function I(y) ≡ 1. We now observe that, when I(y) ≡ 1, the solution to (6.34) is P (y) ≡ 0 and Q(y) ≡ −1. On the other hand, our construction yields This contradiction achieves the proof of Theorem 6.1.

Existence of an equilibrium solution
Given a nondecreasing light intensity function I : IR + → [0, 1], in the previous section we proved the existence of an optimal solution (θ * , u * ) for the maximization problem (OP2).
Conversely, let ρ 0 > 0 be the constant density of stems, i.e. the number of stems growing per unit area. If all stems have the same configuration, described by the couple of functions y → (θ(y), u(y)) as in (2.18), then the corresponding intensity of light at height y above ground is computed as The main goal of this section is to find a competitive equilibrium, i.e. a fixed point of the composition of the two maps I → (θ * , u * ) and (θ, u) → I (θ,u) .
(i) The couple (θ * , u * ) provides an optimal solution to the optimization problem (OP2), with light intensity function I = I * .
The main result of this section provides the existence of a competitive equilibrium, assuming that the density ρ 0 of stems is sufficiently small.

2.
For each I ∈ F, let (θ (I) , u (I) ) describe the corresponding optimal stem. Calling In order to use Schauder's theorem, we need to check the continuity of these map, in a suitable topology.
We start by observing that From the estimate (6.43) it follows that the functions p (I) are equicontinuous on [0, h + ]. Recalling that q = q · I, by (6.39) we conclude that the functions q (I) are equicontinuous as well.

5.
To prove the continuity of the map Λ 1 , consider a sequence of functions I n ∈ F, with I n → I uniformly on [0, h + ]. Let (θ n , u n ) : [0, h + ] → IR 2 be the corresponding unique optimal solutions.
To prove the claim, let (p n , q n ) be the corresponding solutions of the system (6.20). By the estimates on p , q proved in Section 6, the functions (p n , q n ) are equicontinuous. From any subsequence we can thus extract a further subsequence and obtain the convergence for some functions p, q, uniformly on [0, h + ].
For every j ≥ 1 we now have θ n j (y) = Θ I n (y), p n (y), q n (y) , u n j (y) = U I n (y), p n (y), q n (y) , where U and Θ are the functions in (6.8)-(6.9). By the dominated convergence theorem, the convergence (7.7) together with the uniform integrability of θ n j and u n j yields the L 1 convergence θ n j − θ L 1 → 0, u n j − u L 1 → 0. (7.8) In turn this implies that ( p, q) provide a solution to the problem (6.20), in connection with the light intensity I. By uniqueness, p = p and q = q. Therefore, θ = θ and u = u as well.
The above argument shows that, from any subsequence, one can extract a further subsequence so that the L 1 -convergence (7.8) holds. Therefore, the entire sequence (θ n , u n ) n≥1 converges to (θ, u) in L 1 ([0, h + ]). This establishes the continuity of the map Λ 1 .
6. The map Λ 2 • Λ 1 is now a continuous map of the compact, convex domain F ⊂ C 0 ([0, h + ]) into itself. By Schauder's theorem it admits a fixed point I * (·). By construction, the optimal stem configuration θ (I * ) , u (I * ) yields a competitive equilibrium, in the sense of Definition 7.1.

7.
To prove uniqueness, we derive a set of necessary conditions satisfied by the equilibrium solution, and show that this system has a unique solution.
Using (6.8) and (6.11), we can rewrite the light intensity function (7.1) as where w = w(I, p, q) is the function introduced at (6.10). Differentiating w.r.t. y one obtains Here the common height of the stems h > 0 is a constant to be determined.
8. The uniqueness of solutions to (7.10) will be achieved by a contradiction argument. Since this is very similar to the one used in the proof of Theorem 6.1, we only sketch the main steps.
In analogy with (6.31), (6.33), denote by p(y, h), q(y, h), I(y, h) the unique solution to the Cauchy problem (7.10), with terminal conditions given at y = h. Consider the functions By the same arguments used in step 8 of the proof of Theorem 6.1, we conclude that the right-hand side of (7.12) is uniformly integrable.
To reach a contradiction, we observe that J n (y) = − kn y J n (z) dz and the right-hand side of J n in (7.12) consists of uniformly integrable terms which are multiplied by ρ 0,n . This implies J(y) ≡ 0. This corresponds to the case of an intensity function I(y) ≡ 1. But in this case we know that Q(y) ≡ −1, contradicting the fact that, by (7.13), Q(0) = lim n→∞ Q n (0) = 0.

Stem competition on a domain with boundary
We consider here the same model introduced in Section 2, where all stems have fixed length and constant thickness κ. But we now allow the sunlight intensity I = I(x, y) to vary w.r.t. both variables x, y. As shown in Fig. 5, left, we denote by s → γ(s, ξ) = (x(s), y(s)), s ∈ [0, ], (8.1) the arc-length parameterization of the stem whose root is located at (ξ, 0), and write g for the function introduced at (2.8). This leads to the optimization problem 2 ) and stems are distributed along the positive half line, with density as in (8.9). In this case, stems originating from points close to the origin have no incentive to grow upward, because they already receive a nearly maximum light intensity. Hence they bend to the right, almost perpendicularly to the light rays.
An interesting case is where stems grow only on the half line {ξ ≥ 0}. For example, one can takeρ if ξ > b. (8.9) In this case, we conjecture that the competitive equilibrium has the form illustrated in Fig. 5, right.

Concluding remarks
A motivation for the present study was to understand whether competition for sunlight could explain phototropism, i.e. the tendency of plant stems to bend toward the light source. A naive approach may suggest that, if a stem bends in the direction of the light rays, the leaves will be closer to the sun and hence gather more light. However, since the average distance of the earth from the sun is approximately 90 million miles, getting a few inches closer cannot make a difference.
As shown in Fig. 6, if a single stem were present, to maximize the collected sunlight it should be perpendicular to the light rays, not parallel. In the presence of competition among several plant stems, our analysis shows that the best configuration is no longer perpendicular to light rays: the lower part of the stems should grow in a nearly vertical direction, while the upper part bends away from the sun.
Still, our competition models do not predict the tilting of stems in the direction of the sun rays. This may be due to the fact that these models are "static", i.e., they do not describe how plants grow in time. This leaves open the possibility of introducing further models that can explain phototropism in a time-dependent framework. As suggested in [12], the preemptive conquering of space, in the direction of the light rays, can be an advantageous strategy. We leave these issues for future investigation.  Figure 6: The stem γ 1 , oriented perpendicularly to the sun rays, collects much more sunlight than γ 2 . Indeed, γ 1 would give the best orientation for solar panels. Notice that γ 2 minimizes the sunlight gathered because the upper leaves put the lower ones in shade.


THE CHALLENGE OF DEVELOPING HIGH-SPEED RAIL PROJECTS: RECENT EVIDENCE FROM DEVELOPING COUNTRIES

The plan by Indonesian government to build a high-speed rail (HSR) has previously existed for years. Both the Japanese and Chinese government had both showed interest in the project, and both of them have the technological capacity to build a railway that would connect Jakarta and Bandung. This project was however awarded to China because they made a provision of soft loans for the project, while Japan on the other hand wanted the Indonesian government to completely provide the fund required to execute the project from the beginning to the end of the project. The proposed construction of the railway lines will cut travel time between Jakarta and Bandung from about three hours by car to just 45 minutes only. The project will include integration of the HSR with developments along its corridor through transit oriented developments. The line would attract around 10 million passengers per year in first year of operation, this is because of Jakarta’s huge population size in addition to the number of pairs of destinations that the HSR would connect. However, even the KL – SG HSR project signed an agreement on February 2013, it is noteworthy to mention that the surprising decision arrived not long after the newly elected Malaysian prime minister raised the possibility of dropping the project because of its cost implications.

INTRODUCTION
A brief history into the railway system in Indonesia showed that the railway transport system was created in mid-19th century when Indonesia was still under the colonial rule of the Dutch. The railway system was created in 1939 so as to facilitate the movement of cargoes and passengers. As at that time, the length of the railway line was 6,324 km long on Java and 1,833 km long on the Sumatera Island. In 2009, the total railway length had fallen substantially from 6,324 km to 3,464 km on Java and 1,833km to 1,350 km on Sumatera Island [1]. The main reason for this decline was the competition of railway transport with road transport, thus more funds were utilized for building more roads at the expense of the railway lines. Hence, Law No. 23/2007 was approved to make railway transport an important means of transportation within Indonesia. This clearly maps out the development guide for the national railway system.

METHODOLOGY:CASE COMPARISONS
Author compare the proposed Jakarta-Bandung HSR corridor with the some of HSR corridors in Asia and Europe regions and to identify key factors that have contributed to its successful. International comparison is especially important in HSR because the research shows important differences across countries due to topography, demographics, nature of transit demand and government investment schemes [2, 3].

EARLY JAPANESE'S PROPOSAL
Japan had already indicated their interest to replicate their Shinkansen HSR technology in Indonesia since 2008. That same year, Japan had displayed their Shinkansen technology and also done a feasibility study. Considering that the Island of Java is identical to the pre-HSR Honshu in Japan, the Japanese government under the Japanese International Cooperation Agency (JICA) therefore proposed the construction of HSR for the Indonesian island of Java, backed by soft loans from the Japanese government, connecting the densely populated capital of Indonesia, Jakarta which suffers from freight and passenger congestion to Surabaya which is about 730 km apart. A new proposal that focus on building 150 km of HSR from Jakarta to Bandung, which will result in the drastic reduction of the time spent to cover the 150 km from 3 hours to just 45 minutes as shown in Fig.1. Japan which is widely known for the manufacturing of world-class trains was the most favorable to win the contract for construction of HSR from Indonesian government. However, change in government after the 2014 Indonesian gubernatorial election which saw the swearing in of Joko Widodo in October 2014 put an end to this project. The Joko led administration basically the HSR project in January 2015, reason for this action was that the HSR project was quite too expensive for the government to execute and there were several other more important and significant infrastructural projects that were required in the underdeveloped islands outside of Java.   [5]. China's HSR strategy has become part an integral belt-and-road strategy and core of China's foreign policy, all these have occurred just within the past decade or thereabout [6]. The 'belt' component comprises of many land routes and the 'road' component on the other hand comprises of many sea routes. This also arise from the fact that both systems of land and sea routes link China to Europe. This initiative also led to the development of special funding and investments [7].

HISTORY AND RECENT DEVELOPMENT
The HSR project was forecasted to cover a distance of about 143 km, linking Jakarta and Bandung, thus becoming Indonesia's first ever HSR project. As highlighted earlier, the Japanese and Chinese government made known their intention. Although the Japanese and Chinese governments had carried out previous comprehensive studies, it was only Japan that went the extra length of issuing a study for a project extending to Surabaya, which was estimate to be about 730 km. This was followed by a counter bid which was submitted by the Chinese government in April 2015, must to the chagrin of the Japanese. This was followed by a state visit by Joko Widodo, who had a successful meeting with the Chinese president, Xi Jinping on March 26, 2015. After the meeting Xi Jinping publicly announced the support of the Chinese government for the development of Indonesian high-speed project, this was followed by the signing of a memorandum of understanding between the two governments. In July 2015, Indonesia led by Jodo Widodo officially announced their plan to commence the construction of the HSR, this would connect Jakarta and Bandung, the bidding process was then made public, therefore the contest was between the potential bidders which included both the Japanese and Chinese train-makers had. The contest became interesting as the Chinese train-maker organized a Chinese HSR Technology exhibition in August 2015. This was a brilliant move which was not new to either contenders since both the Chinese and Japanese have been engaged in fierce business competition in the past, often using intense lobbying to outdo each other. It was widely acclaimed that the primary reason behind the high display of domineering campaign displayed goes further. The Chinese government played another card in mid-September 2015, they submitted a new proposal that offered to fully meet the Indonesian government's demands while also eliminating funding required to commence. This was followed closely by several months of bidding, negotiations, between the two countries. It temporarily led to the cancellation of the project, however, the Indonesian government made a decision to select China as the preferred bidder for the US$5 billion project because of its financial structure which was so hard to ignore, considering the fact that this was completely different from what the Japanese plan had proposed. China's triumph can be alluded to China's willingness fund the project. This would in fact lead to a waiver, which was different from what the Japanese had offered, and the Japanese had also refused to shift their grounds. China had even gone further ahead alleviate its deal by including its commitment to institute a program for the manufacture of light and electric rail system in partnership with local businesses. These products would be targeted at the Indonesian market, and exported neighboring markets, thus creating a seamless technology transfer system that would lead to growth of local technology for renovating and train stations. Therefore, it looked like Indonesia had benefitted immensely from the Japan-China competition, as the Indonesian HSR bid earmarked continuous competition between Japan and China in their bid for other infrastructure projects with Asia. The HSR system has 71.63 km of the track on the ground level, while 53.54 km of the track will be raised, and the remaining 15.63 km will be below the surface. This was followed by the groundbreaking ceremony, held on January 21, 2016 to commence the construction of the HSR. The HSR system would be open to the public after its official launch in 2021.

PROJECT'S DESCRIPTION
The HSR is part of the governments' grand ambition of upgrading Indonesia's lagging infrastructure. If successful executed, the project will drastically reduce travel time to just 45 minutes only. The HSR has four stations as seen in Fig. 2. The HSR project will be financed exclusively by China. The project will include integration of the HSR stations with developments along its corridor through transit oriented developments (TOD) [8 -10]. The ambitious proposal for the national railway network comes amid recent news that the cost of the Jakarta-Bandung HSR has swelled to almost US$6 billion, from US$5.  Urban rail lines, including a planned light rail transit (LRT) system in Jakarta, are also expected to help reduce congestion and transport costs, which have become the highest in South-east Asia. It was forecast that the new line would attract around 10 million passengers per year in first year of operation, as shown in Fig. 3.

Fig. 3 Accumulated HSR traffic [12]
For 20 years, Shinkansen services in Japan enjoyed a sustained traffic, it gained 100 billion passengerskilometer. From 1994 to 2004, within the next 20years interval, the demand halved, because only 50 billion additional passengers-kilometer used HSR. When compared with most European HSR projects which are still in their first 20-year period, it is natural to expect high growth rates as expressed by Fig. 3. Hence, Fig. 4 shows accumulated traffic used the HSR services in Asia and Europe based on traffic data from each operator during the 2010 to 2016 period. As confirmed by Fig. 4 the only China is still in it incredible constant growth and gained a huge accumulated traffic around 850 billion passenger-km. Two other Asian countries i.e. South Korea and Taiwan started HSR services in first decade of 21 centuries only gained accumulated traffic of 31.4 and 20.2 billion passenger-km respectively during the same period. Based on long experienced of HSR services in Japan and Europe countries it is easy to predict that most China HSR projects still enjoyed a constant traffic growth for the next two decades most triggering by combined building new dedicated electrified lines and upgrading existing lines. Both South Korea and Taiwan HSR service expressed constant demand growth for the following first decade even the Korea Train Express (KTX) has transported approximately 150 million passengers since the four years after its opening. Taiwan HSR itself has carried about 100,000 passengers per day for fifty first months of commercial service. However,  Based on passenger traffic data obtained from Europe and Asia, the first HSR line constructed in Indonesia is expected to gain considerable amounts of demand. However, it is important to state that in Europe and Asian countries, the construction of HSR lines was as a result of the inability of their conventional lines being unable to meet their demands, thus they needed to add a new capacity to increase rail service. It is also noteworthy to mention that many of these existing rail lines had already been doubled or tripled in a bid to increase the capacity. Therefore, the high demand for conventional rail can be said to have created a market for HSR in these countries. It is inevitable to mention that Indonesia include Jakarta and Bandung lacks of this factor that triggered HSR services successful in Europe and Asian countries as shown in the table above. The Parahyangan conventional line, which began operating in 1971 in the Jakarta -Bandung corridor only carries a total of 560,320 passengers in 2016.

PROPOSAL OF SG -KL HSR PROJECT
The Malaysian and Singaporean governments signed an agreement on February 2013 that would result in the construction of HSR line, connecting Malaysia [15]. It is of utmost important and key to the successful implementation of Malaysia's national development strategy, (one of the main objective is to increase the country's GDP per person by 150%) by the time this HSR line will be completed in 2020, the Malaysian government is also looking forward to a further boost in the country's economics by closely-tied cooperation with the Singaporean government. Cost of this project is estimated at RM 40 billion (USD 12 billion). The source(s) of the funding, however, has not been officially disclosed; although some private sources say that Private-Public-Partnership (PPP) funding will be used, others claim that it will be jointly co-funded by both the Malaysian and Singaporean governments considering the fact 335 km of the line will be in Malaysia while the remaining 15 km will be in Singapore. The HSR will have a total of eight stops, seven of which will be within Malaysia and one will be within Jurong East. The high speed rail line which will consist of bullet trains moving at top speed, above an estimated 300 km/hour is projected to become operational in 2026. This would therefore reduce the land journey between Kuala Lumpur and Singapore from about five hours to a mere 90 minutes' journey as shown in Fig. 5. This project has attracted keen interest from various large corporations within Asia and Europe. However, Mahathir Mohamad who is the current Malaysian Prime Minister had announced on Monday May 28, 2018 that the HSR project will be axed, but he had also explained that it will take some time to execute because the Malaysian government and Singaporean government had also signed a contract on the multi-billion-dollar project. Meanwhile, Dr Mahathir Mohamad, the Malaysian Prime Minister had also assured that the government will remain "business friendly" to all investors involve in the project [16]. The relationship existing between Malaysia and Singapore is considered unique because of certain factors, these include the geography of both countries, economy, politics, Others Europe historical background, cultural heritage, and ethnicity. It is also attributed to the fact that Singapore separated from Malaysia in 1965, hence the similarity. It is also intriguing to know that although both countries have been characterized by healthy competition in economic and social matters, they however enjoy a very high level of economic interdependence as major trading partners. The existing relationship between Singapore and Malaysia has been described as symbiotic.
However, this mutually symbiotic as the relationship has been beneficial to both countries, although the relationship has faced some challenges in the past, it has existed since 1965. The Mahathir administration which was in control in Malaysia from 1997 to 2002, was believed by many to be the most stressful period between both Singapore and Malaysia. However, the situation changed after Abdullah Badawi got into power and became the prime minister of Malaysia in 2003, and since then there has been enhanced contact and cooperation between both governments. History is starting to repeat itself again in the part of the SG -KL HSR project [16].

CONCLUSIONS
In conclusion, the Jakarta-Bandung HSR project, linking Jakarta to Bandung was executed by PT Kereta Cepat Indonesia China (KCIC), through mutually agreed business-to-business plan basis, with the Indonesian government. China, which was not only the contender had been awarded the project because of the provision of soft loans for the project; on the other hand, Japan which had showed interest in the project had requested that the Indonesian government funded the project. Surprisingly, Japan was the first to have indicated interested in the project and had even gone ahead to commence working on a feasibility study for HSR track, connecting Jakarta to the country's second largest city, Surabaya, which is about 730 kilometers apart. Indonesia went further in 2012 to commence another feasibility study focused on the Jakarta-to-Bandung leg, this study was finalized in 2014. The highest operating speed of the trains is estimated at about 350 km/hour, the Jakarta-Bandung high speed rail would result in shortened travel time between the two hubs, reducing the time spent to cover this distance from three hours to 45 minutes only, therefore pushing forward economic development along the line through transit oriented developments. It was projected that the line would also attract about 10 million passengers each year within the first year of operation. This high figure is however a reflection of the densely populated city of Jakarta, and the several number of origin destination pairs that the HSR line would simultaneously serve. The Kuala Lumpur-Singapore HSR is a strategic project between the Malaysian government and Singaporean government, with its primary goal of facilitating seamless travel between the two capital cities of Kuala Lumpur and Singapore, enhancing business linkages, and connecting the citizens of both countries. The high-speed rail link is expected to reduce the travel time between the two cities from about four to five hours by road to 90 minutes only. However, it is also noteworthy to mention that the surprising decision arrived not long after the newly elected Malaysian prime minister raised the possibility of dropping the project because of its cost implications.

REFERENCES
[1] Directorate General of Railway, Ministry of Transportation, Master Plan of National Railway, 2011.
[2] Campos, J., and de Rus., G. Some stylized facts about high-speed rail: A review of HSR


Towards a Universal Pension Points System in France

A specificity of the current French system is that a private sector worker is currently covered by two mandatory public pay-as-you-go (PAYGO) schemes: a general defined benefit (DB) scheme — which is the main component — managed by the state and a points scheme (AgircArrco) managed by social partners, together representing about 70% of paid benefits.

The current reference wage for private sector workers is based on the best 25 years of earnings, while for civil servants it is the wage over the last six months. Taking into account the full career is both more transparent and equitable. While taking into account only the best years protects against some forms of career breaks, it also generates perverse, regressive effects by favouring workers experiencing large wage improvements (Aubert and Duc, 2011).
In France, both replacement rates and pension fi nancial balances are highly dependent on real-wage growth. This dependence stems from the price uprating of past wages in the general scheme -formalised by the 1993 reformthe price indexation of the cost and the value of the point, which has often been applied discretionally in the Agirc-Arrco scheme, and the price indexation of pensions in The current French pension system offers good social protection, translating into high average relative incomes and low relative poverty among the elderly. However, it is deeply fragmented and its structure is overly complex. The objective of the planned reform is to simplify the system substantially, improve its management, reduce inequalities in the rules used to compute benefi ts and facilitate labour mobility. The reform aims at introducing a universal points system, thus eventually eliminating special regimes. Several OECD countries have a points system: contributions are used to purchase points, and at retirement the pension is equal to the sum of acquired pension points multiplied by the point value.
The creation of a universal pension regime in France is a major reform. Previous reforms have improved fi nancial sustainability. As there has been suspicion that the planned measures would be used to lower pensions further, the emphasis was on budget-neutral reform. However, it quickly became apparent that such a big overhaul defi ning the way the old-age social protection system will work in the future could not be undertaken without addressing fi nancial sustainability.
Ideally, two reforms were needed, but both dimensions were included in the same draft law. The fi rst reform is the design of the universal system, in which pension rights will be included in the new system from 2022-2025 and the fi rst generation will retire with these points as part of their pension in 2037. The second is the parametric measures aimed at ensuring the fi nancial balance of the current system in 2027. This second component is totally distinct from the implementation of the universal system and Forum apply to survivor pensions -which alone are covered by 13 additional different regimes. These differences also create a suspicion that other workers in other sectors are better treated. In turn, this suspicion fuels the motivation to defend one's own interests and complicates the implementation of policies to improve the system and reduce unequal treatment. Strong opposition to the reform has come from workers covered by schemes that would ultimately disappear, such as those covering employees in the railway sector, energy and subway companies and some self-employed groups.
One historical reason for a preferential treatment in the public sector may be the result of the need to attract civil service workers, a need that has vanished over time. These views also refer to the career being considered a lifelong job; in the absence of mobility, civil service pensions in particular were therefore best organised in a stand-alone system and could have been considered deferred compensation. A job for life, however, is a thing of the past.
France is one of only four OECD countries, along with Belgium, Germany and Korea, that still has totally different schemes covering private and public sector workers. The differences in pension benefi ts between public and private sector workers with similar earnings can be signifi cant in France -although much lower than in the other three countries (OECD, 2016). Moreover, the heterogeneity of rules in France also applies within the public sector, where there are wide differences (Conseil d'orientation des retraites [COR], 2018).
In half of the OECD countries, all employees are covered under the same mandatory schemes at least for new labour market entrants. Ten countries have a fully integrated scheme, but with a top-up component for civil servants above and beyond the mandatory schemes that exist for private sector workers. The difference of benefi ts between public and private sector workers with similar earnings thus depends on the level of the top-up. It is within these countries that the largest OECD differences are found: this is in particular the case in Canada, the United States and the United Kingdom (OECD, 2016).
Teachers are a case in point in France. They benefi t today from a higher pension than private sector workers with similar earnings throughout the career. This might be seen as a compensation for low wages. Indeed, in international comparison, teachers' wages tend to be low in France, especially in the fi rst part of the career. With the reform, teachers would lose in terms of pensions as a result of unifying pension rules. The reform actually reveals serious issues in the wage formation of teachers, which pension schemes are ill equipped to address. The problem thus lies in the wage setting process, and while the govern-payments. Price uprating was chosen in 1993 because it was socially and politically more acceptable given the lack of understanding among the population of its implications for pension benefi ts.

Complexity
There are 42 mandatory pension schemes with very different rules. On average, each individual has currently contributed to more than three regimes, the benefi ts of which do not add up in a straightforward way. Even for private sector workers, France is unusual, as explained above.
Due to the complex interactions between the rules of the various schemes, issues raised by workers affi liated with different regimes often become serious obstacles to the needed reforms that modify some parameters in some schemes. These interactions are also a serious hurdle for the overall management of the system, as the envisaged policy measures only have partial effects. Yet, each of these partial reforms carries some political costs and their needed recurrence gives the impression that pensions require some endless adjustments. Moreover, the overall complexity makes it very diffi cult for contributors to correctly anticipate their retirement income, which in turn makes it hard to take adequate retirement decisions.

Fragmentation
The fragmentation of the system implies substantial differences in the treatment of workers in the private sector, the public sector, those covered by numerous special regimes and the self-employed. Despite recent harmonisation measures, large inequalities result from differences in the rules used to compute benefi ts including those that Forum benefi t (DB), points and notional (non-fi nancial) defi ned contribution (NDC). The 42 mandatory public pension schemes in France mix DB and points schemes, with different rules in each broad category. With the planned reform, France made the clear choice to keep the core of the pension system as a public PAYGO. The new system would be based on a points system for the unifi ed framework.
A generic NDC system is appealing technically because it is designed to ensure fi nancial stability. The counterpart of this fi nancial robustness is twofold. First, cohorts that live longer have, as a rule embedded in the system, to contribute for longer periods to maintain replacement rates at similar levels to older cohorts. Second, for given contribution rates, a generation that relies on a fast growing (declining) working-age population will benefi t from higher (lower) replacement rates. This second implication is inherent in PAYGO systems whereby current workers directly fi nance current pensioners, but it is automatically included in generic NDC schemes. The main shortcoming of NDC might lie in the diffi culty workers have to grasp the concept of notional accounts, how accounts evolve over time and what they concretely imply for the level of benefi ts (Sundén, 2013).
The choice of constraining rules refl ecting trends in wages, employment and life expectancy is therefore critical for the good management of the system. Depending on how uprating and indexation rules are defi ned, there could be some equivalence between points schemes, DB schemes and NDC schemes (Boulhol, 2019). A strict equivalence in the benefi t formula between a points and an NDC scheme holds if: • There is a static condition for a given cohort: the indexation rate of the cost of purchasing a point is equal to the notional interest rate, which in its generic form is equal to the growth rate of total contributions.
• There is a dynamic condition across cohorts: the value of the point at a given (retirement or reference) age has to be indexed to the notional interest rate minus the growth rate of remaining life expectancy at retirement age. On average across OECD countries, remaining life expectancy at 65 is projected to grow by 0.47% per year on average by 2050 and by 0.42% in France.
Nine OECD countries have introduced a points or NDC system at the core of their public pensions since the early 1990s. Table 1 summarises how countries implicitly uprate past wages, index pensions in payments, account for employment trends and changes in longevity within their points or NDC scheme. Uprating past wages are embed-ment intends to increase teachers' wages, the overall effect will be seen in the long term through the combined action of future governments.

Core of the current reform: Simpler and more equitable
The French government created the High Commission for Pension Reform in September 2017 to prepare the reform introducing a universal pension points system in France. In July 2019, the High Commission published its recommendations entitled For a universal pension system, simpler, fairer, for all. In January 2020, a draft law was adopted by the Council of Ministers and is being discussed in parliament as of February 2020. Pension entitlements in the new system would start to accrue gradually from 2022 for those born in 2004 and from 2025 for all individuals born from 1975.
The unifi ed pension system aims to establish common rules for contributions and the calculation of pension entitlements for all workers, drastically simplify the current system, and ensure fi nancial sustainability by reducing the sensitivity of fi nancial balances to economic and demographic trends and short-term shocks.
The reform aims at introducing a universal PAYGO points system based on some key principles: • All 42 regimes will be included based on common rules and replaced by the universal points system; • Financial stability and sustainability should be ensured, and in particular the system should be designed such that it is resistant to economic and demographic shocks; • Pension entitlements will accrue up to a wage ceiling of three times the social security (annual) ceiling (PASS); • The minimum retirement age will be maintained at 62; • Points will be allocated for each child and for career breaks (related to unemployment, maternity leave, sickness, disability, etc.), and specifi c situations will be accounted for (long careers, arduous and hazardous work, handicaps); • The system will include a survivor and a minimum pension scheme.
Points schemes and comparison with countries having a points or NDC scheme Publicly provided earnings-related pension schemes are typically PAYGO and follow three broad types: defi ned Forum payments will be indexed to prices. The cost and value of the point would be indexed to the average income per capita (i.e. accounting for both wages of employees and incomes of self-employed workers) from 2045, after a transition phase starting with price indexation in 2025. It would have been preferable to follow the (long-term) rule from the start and adjust the initial point cost and point value accordingly.

Transition
The impetus for the systemic reform in these nine countries often stemmed from the need to make pensions financially sustainable given ageing prospects. These countries already had a fairly integrated system, except Italy, which unifi ed the system with the introduction of the NDC scheme, and Germany, which did not tackle that aspect.
In terms of transition, Germany is a special case because the system basically functioned like a points-based scheme before the reform. The reform was very fast in the Baltic countries and in the Slovak Republic. In Sweden and Norway, the implementation started about fi ve years after being legislated with a transition of 10 to 15 years. In Italy, the new NDC system gradually applied across 20-25 yearly cohorts. In France, it will be across 30 cohorts, as the individuals born in 1975 would be the fi rst generation accruing some entitlements in the new system, and those born in 2004 would be the fi rst to have accrued all their pension rights in the new scheme.
ded in rules governing points and NDC schemes through the indexation of point cost and the chosen NDC interest rate, respectively.
On one extreme, rules can be closely related to actuarial principles and sustainable returns. On another extreme, there can be a total discretion left to the managers, such as in the Agirc-Arrco scheme, which limits transparency and perhaps confi dence in the system.
Changes in life expectancy are automatically embedded in the computations of the initial pension in countries with NDC systems. Changes in the size of the labour force are not refl ected in many countries covered in Table 1, which makes their system less stable (Auerbach and Lee, 2009), generating the need for another instrument to adjust benefi ts and/or fi nances. Within points schemes, Germany accounts for employment and demographic trends in the point value through the sustainability factor, which depends on the ratio of pensioners to contributors.
In France, while the minimum retirement age is maintained at 62 years, a reference age (âge d'équilibre) will be introduced to serve as the retirement age at which the point value is computed for a given birth cohort. People of the same cohort but retiring earlier or later would have a penalty or bonus applied to this value, respectively. According to the draft law, the reference retirement age would be linked to life expectancy, with two-thirds of changes in life expectancy being passed on. Pensions in

Forum
proposed measures should focus on increasing effective retirement ages given that the draft law refers to measures that will lead to neither lower pensions nor higher labour costs.

Adjustment mechanisms in the new system
Pension systems cannot be managed on 'autopilot' as there are always unpredictable events that affect pension levels and fi nances. Hence, although it is important to design the rules in a way that best accounts for future trends and that is conditional on the exact (albeit initially uncertain) scope of these trends, additional adjustment mechanisms are needed.
Among the nine countries in Table 1, only Germany and Sweden have implemented additional adjustment rules (Boulhol, 2019). In Germany, the contribution rate must be increased if the pension account balance -taking the sustainability factor into consideration -deteriorates beyond a certain threshold, which in turn automatically lowers the point value (that is indexed to net wages), thereby sharing the burden of the adjustment between current workers and current pensioners. One signifi cant difference from the Swedish mechanism is that the adjustment is triggered based on short-term liquidity imbalances and not on the estimated long-term solvency of the system.
Sweden directly adjusts pension entitlements and benefi ts to ensure solvency, based on current pension liabilities and assets made of contribution assets and (the market value of) the buffer fund. The buffer fund represents around 30% of GDP. An automatic balancing mechanism is applied when the calculations imply a potential defi cit.
In that case, a brake is activated, as it was initially in 2010 based on 2008 values, reducing the notional interest rate below the wage growth rate in order to both limit accumulation in notional accounts and reduce indexation of pensions in payments. By contrast, there is no balancing mechanism in Norway, and the general government budget backed by the national wealth fund remains the shock absorber.
A balancing mechanism is a valuable tool, but even Sweden experienced some diffi culties in applying the brake rule. In that sense, the economic and fi nancial turmoil provided a stress test. The adjustment implied by the rule would have amounted to a real-term reduction of 4.6% in pensions and was considered so large that policymakers decided to spread it out over a longer period (Sundén, 2009). Overall, the Swedish mechanism proved resilient, only requiring a small adjustment. As a result, its broad principles remain largely unchallenged even though the experience shows Another factor infl uencing the length of the transition refers to the number of years until the fi rst affected cohort retires. In Sweden, individuals who were age 60 at the time of the voted reform were affected as 20% of their pension was computed based on entitlements converted in the new scheme. In Norway, this applied to those who were 58 years of age, while in Italy it applied to those who were about 38 years old. In France, it will be 45 years old, i.e. the age of those born in 1975 at the expected time of the vote of the reform (2020).
Finally, the question of the conversion of acquired rights is critical. In Sweden and Norway, there has been a conversion of entitlements in the new scheme based on a weighting depending on the birth year, hence a mixed system for 15 years in Sweden and ten years in Norway. By contrast, Italy has been going through a very long process as there has not been any conversion, hence a very long transition of 45-50 years after the adoption of the reform. In France, the government decided in February 2020 that there will be no conversion of rights acquired in old schemes. Benefi ts from entitlements accrued before the implementation of the new system (2022-2025) will be computed at the time of retirement based on all the rules of each regime, implying that these rules will continue to apply, albeit with a decreasing weight, until about 2065. This means that the generations born until 2004 will not benefi t from the simplifi ed calculation of what they can expect for their total pension.

Balancing the current system
As part of the reform, the objective of ensuring the fi nancial balance of the overall pension system by 2027 is now included. Given the complexity of the system, assessing the fi nancial situation of the system is in itself a complex task. In November 2019, the Conseil d'orientation des retraites at the government's request published a report about the 2030 pensions outlook in France (COR, 2019). Depending on assumptions related to real-wage growth and, more importantly, to the chosen accounting rules, the current pension system shows a projected defi cit in 2030 between 0.3% and 1% of GDP.
To eliminate the defi cit by 2027, the government intended to introduce a new penalty when retiring before an 'equilibrium age' that would have gradually increased from 62 years and four months in 2022 to 64 years in 2027. However, it eventually decided to open the possibility of alternative measures based on an agreement between social partners. The latter should make proposals by the end of April 2020. Their mandate is constrained though as the Forum the sensitivity of replacement rates and pension fi nances to long-term productivity growth, unintended forms of redistribution due to the complex interaction of rules and the fact that only parts of the career are taken into account to calculate pension entitlements. Actually, it will do this in the long term only, as France opted for a very long transition, which will delay the benefi ts of the reform. The long transition is due to both applying the reform to those born from 1975 on only (instead of 1963 as initially envisaged) and not converting entitlements acquired in the current schemes as Italy had done.
In this context, it is important to put in place clear indexation rules applying to the cost and the value of points as well as pensions in payments. Those rules should be completed by a balancing mechanism that focuses on fi scal discipline, i.e. in a credible way to ensure long-term cumulated fi nancial balances, while preserving a high enough level of pensions.
Given population ageing, it is crucial to raise effective retirement ages, especially in France, where employment of older workers and the average age of the labour market exit are low. The planned reform defi nes an 'equilibrium' retirement age, which will be used as a reference for the point value and increase (decrease) based on two-thirds of life expectancy gains (losses).
that in periods of large volatility, interventions by politicians are still needed (Weaver and Willén, 2014).
In France, the draft law introduces a golden rule, according to which in each year t the fi ve-year projected balance cumulated over t and t+4 cannot be negative. Focusing on a cumulated indicator (stock concept) is welcome as the target is often to reach a yearly balance (fl ow concept), implying that past defi cits are not corrected and therefore increase debt. However, the objective of this type of mechanism should be to ensure the long-term balance, and fi ve years is not a long period to cover a full economic cycle. This might imply the need to take pro-cyclical measures to respect the golden rule, thereby amplifying the impact of short-term shocks.
It seems that the choice of a fairly short period was guided by the fear that adjustments might be deferred for too long. Even with the proposed design, there is always the risk that needed measures are deferred to year t+3 or t+4. These are complex issues when there is a limited track record of fi scal discipline that would provide some initial credibility. Hence, there is no easy answer. The best option would be to focus on the real objective, a sane long-term fi nancial management with two options. Simple rules should be designed and enforced, but it is not clear whether, even with the simpler universal points system, they can be defi ned to meet this objective. Otherwise, this mandate should be given to an independent institution, which would propose a menu of corrective actions as needed.
One central issue for the management of public PAYGO pension systems, and of points and NDC schemes in particular, refers to how pension accounts are clearly separated or even ring-fenced from the other components of general government balances. The credibility of a golden rule would be enhanced by ring fencing pension accounts.

Conclusions
The OECD has long supported the principle of a universal pension system in France. The current system provides effective social protection for current retirees, but it is overly complex and generates signifi cant differences in the treatment of various population groups. This also leads to serious complications in both the coordination of the large number of different schemes and the management of the overall system. The main goal of the reform is to simplify the system and treat everyone equally. On top of systemic changes, the reform aims at ensuring the fi nancial balance of the current system by 2027. This reform will eliminate or substantially reduce some signifi cant shortcomings of the current system, including


Extension Agents Perception of Open Data Usage in Agricultural Communication in Abia State

The study assessed agricultural extension agent’s perception of open data usage in agricultural communication in Abia State. Simple random sampling technique was used to select 83 extension agents for the study. Data for the study was collected with the use of questionnaire and described with frequency counts, percentages and mean. A moderate percentage (68%) of extension agents were aware of open data with only 15% and 7% having high and very high level of awareness. A relatively high percentage (54%) understand what open data is all about with only 49% consenting to be involved in the use of open data mainly at a moderate level (66%). Extension agents mainly perceive open data to facilitate agricultural research ( = 3.2), facilitate access to research findings ( = 3.2), give access to the work of other extension agents in other states ( = 3.2), expose them to innovation in agriculture ( = 3.2) and ensure timely access to all necessary information ( = 3.1). The major challenges to the use of open data among extension agents were lack of fund to purchase internet bundle ( = 2.2), unavailability of internet facilities( = 2.1), lack of basic ICT facilities ( = 2.1) as well as lack of adequate knowledge of the use of ICT( = 2.0) and open data ( = 2.0). Propagation of the use of open data could be done mainly by provision of ICTs in offices ( = 3.6), training of extension agents to understand the concept of open data ( = 3.6) and provision of internet facilities for extension agents ( = 3.5). The study emphasized the need for Government through the Agricultural Development Programme (ADP) to provide basic ICT tools in offices and provision of internet facilities to enable extension agents maximize the use of open data.

Introduction
Effective linkage between key players in the agricultural sector is one important way of improving the agricultural sector (Andrason & Van Schalkwyk, 2016). With the advent of mobile phones in many developing countries, agricultural communication is experiencing major changes. Though these changes may not be happening at an appreciable rate, more knowledge and skill in these ICT gadgets and applications may facilitate more use. In the Nigerian contemporary agricultural sector, accessing useful data, processing and managing it, is critical to innovation awareness and acceptance among the farmers. In addition, with increasing environmental and other critical challenges faced by farming households, it is important to embrace a timelier and accessible means of accessing and disseminating agricultural information among the users in the agricultural sector. Verweij, Roosenschoon, Parr, Kamau, Macharia and Msengezi (2017) recognized open data as a driver for economic growth with the potential to contribute to Sustainable Development Goal of zero hunger. As data has become increasingly digital, and thus potentially much more accessible, there is an increased sensitization for the right to access information and knowledge as public goods. Open data simply involves making data available for anyone to access, use and share it while respecting any rights attached to it. When data is available to the people, it enhances institutional responsiveness and better decision-making by both policy-makers and citizens. Open Data by definition is accessible, interoperable, reusable and universal data. Laperrière (2019) emphasized that Open Data do not only allow wider access to historical and usable data that would enable farmers to develop their farming and production practices for the better, it also facilitates efficiency in anticipating and monitoring changes for instance weather variables and aids sharing crucial information across country borders leading to transfer of best practices across countries. The author noted that through the use of satellite data, remote sensing and mapping, stakeholders in agri-business, farmers inclusive can harness the most relevant and useful information to improve and adapt practices, make better decisions and ensure sustainability.
Also, the agriculture sector is currently increasing the combination of technologies, such as geolocation, soil and environmental conditions monitoring, Artificial Intelligence (AI), etc to uphold precision in combination of variables to improve the quantity and quality of agricultural products (Open data in the agriculture system, 2019). To successfully implement and combine these technologies, data particularly open data is very important. In other words, open data facilitates precision agriculture.
Open data is key to information transparency and scientific advancement, especially in Agricultural research. It exposes extension agents to the same raw data that scientists and policy makers use hence allowing them a more in depth understanding of concepts. The use of open data supports the development of a more analytical and collaborative network in the agricultural sector which is very instrumental to generation of new knowledge and innovation. With the increasing emphasis on the use of digital tools in agricultural communication, the concept of open data becomes very important as digital tools will be of no benefit if the data to be disseminated is not accessible. Also, in the face of global covid-19 pandemic, agricultural information dissemination based on physical contacts becomes more limited, researchers in agriculture, governments, and funding agencies are beginning to embrace information dissemination using digital tools, without relevant and useful data being accessible, the use of these digital tools may not be maximally harnessed. Food and Agriculture Organisation (FAO) (2020) noted that the agricultural extension system needs to rapidly innovate to ensure sustenance of timely information dissemination.
The extension agents are the bridge between researchers and farmers. The responsibility of transferring research findings to farmers' rest mainly on the extension agents. Unfortunately, these extension agents are faced with numerous challenges in tackling this important task. Firstly, the agricultural system is a dynamic system with new challenges evolving over time. The extension agents need constant update of their knowledge-base in order to remain valid and effective in their work. Having access to relevant and timely information will help them in updating their knowledge. Secondly, the problem of shortage of extension agents in the system is a major challenge to effective information dissemination. Extension agent-farmer ratio in Nigeria particularly Abia state is poor and cannot support an effective information dissemination. Making data open and accessible to both extension agents and farmers will enhance the use of digital tools, hence ensure more effective information dissemination by the few available extension agents. Improved access to, and use of, open data at grassroots, local, national and global levels holds the potential to transform both long-standing and emerging problems as well as finding solutions that benefit farmers and global food security in timelier and less expensive manner.
Global Open Data for Agriculture and Nutrition (2018) noted that because open data is accessible, usable and sharable by anyone, it is speedily giving solutions to problems that would otherwise be expensive, time intensive or impossible to solve using closed data sources. It explained that making open data work in the agriculture sector requires a shared agenda to increase the supply, quality, and interoperability of data, alongside action to build capacity for the use of data by all stakeholders. Proper understanding of the extension agent's perception of open data usage and the challenges they faced in the use of open data is very important as this will help proffer more effective means of open data usage as well as harnessing more its potentials.

Objectives of the study
The broad objective of the study was to assess agricultural extension agent

Methodology
This study was conducted in Abia State, Nigeria. Abia State occupies about 6320 square kilometers with a projected population of 3,727, 347 (National Bureau of Statistics, 2017) situated between longitude 7° 22' 0.01" E and Latitude 5° 06' 23.69" N. The population of the study consist of all the extension agents in Abia State. Simple random sampling technique was used to select eighty-three (83) extension agents from the total of 106 extension agents in the state. Structured questionnaire was used in the collection of data for the study and the data obtained was analysed with frequency counts, percentages and mean. Agricultural extension agent's perception of open data and ways of propagating open data was captured using a 11 item statements and 9 item statements respectively rated on a Four-point Likert-type scale which was assigned values of 4(strongly agree), 3(Agree), 2(Disagree) and 1(Strongly disagree). A midpoint of 2.50 was obtained, with the decision rule that the extension agents agree with statements with mean score of 2.50 and above and disagree to statements with mean score less than 2.50. Challenges to the use of Open data by the extension agents was captured using an eight item statements rated on a Three-point Likert type scale which was assigned values of 3(major challenge), 2(moderate challenge) and 1 (minor challenge). A midpoint of 2.0 was obtained and decision rule was that statements with mean score of 2.0 and above are major challenges to open data usage while those with mean score less than 2.0 are not major challenges.

Agricultural Extension Agent Perception of Open Data Usage in Agricultural Communication
The result as presented in Table 2 shows that out of the 12 item statements used to capture extension agents' perception of open data usage in agricultural communication, the extension agents agreed to 9. They mainly perceive open data to facilitate: agricultural research ( = 3.2), access to research findings ( = 3.2), access to the work of other extension agents in other states ( = 3.2), timely access to all necessary information ( = 3.1); exposes them to innovation in agriculture ( = 3.2) and is time consuming ( = 2.7).
Results imply that extension agents in the study area have right perception of open data. This is an indication of readiness to adopt open data usage. Ifeanyi-obi (2016) noted that awareness and right understanding of contemporary concepts in agriculture is a prerequisite to farmers' readiness to adopt. People can only adopt what they understand and have proper disposition to. In addition, extension agents' right perception of open data could be attributed to the fact that majority (72%) of them hold at least a bachelor's degree. Level of education is known to be a major determining factor in understanding and adoption of innovation in agriculture. Okoli, Okereke, Onubuogu and Esiobu (2014) noted that high level of education is an added advantage for achieving understanding, huge income and running efficient and sustainable agribusiness enterprise. Use of open data will facilitate marketing information hence enhance agricultural system 3.1** 1.6 Source: Field Survey, 2019: ** ≥ 2.5, * ˂2.5

Challenges to Use of Open Data among Extension Agents
The major challenges as indicated by the extension agents were lack of fund to purchase internet bundle( = 2.2), unavailability of internet facilities( = 2.1), lack of basic ICT facilities ( = 2.1), lack of adequate knowledge of the use of ICT( = 2.0) and open data information is a very vital resource for development in any sector ( = 2.0). This corroborates with Bakker and Addison (2019) which identified poor ICT infrastructure and power supply, (digital) illiteracy and high costs of services as major barriers to adoption and scaling up of ICT for Agriculture initiatives in rural areas. Access to timely relevant information will help farmers in making informed decision. Unfortunately, access to information in the agricultural sector is faced with several challenges and this has reduced the rate of agricultural development. Oyindeinbofa (2017) noted that the lack of access to basic agricultural knowledge and information by rural farmers in Nigeria contributes to poor crop and livestock productivity. He further explained that access to information and knowledge are very vital in agricultural development of any community and where they are poorly disseminated the community's agricultural development becomes highly impeded. Extension agents are the major key players in ensuring timely information access to farmers, improving their access to information through open data usage will contribute in ensuring farmers receive necessary information timely. Akinnagbe, Ezeuzo and Onwubuya (2017) found weak communication and linkage among farmers and extension service providers to be a challenge in reaching these rural women.
The use of open data will help in filling this gap as extension workers will have more access to information and disseminate same to farmers timely. Don't find useful information in my area In the same vein, Man, Saleh, Hassan, Zidane, Nawi, and Umar (2016) highlighted that training results in more motivation, increased knowledge and value of human resources among extension agents in Iraq. Lack and inadequate provision of needed resources and facilities has been a major detrimental factor to effectiveness of extension agents in Nigeria. Most of the ADP offices in the country are poorly furnished with the basic infrastructure needed for effective performance lacking. Having institutional structures that support the use of open data 2.9** 0.9 Source: Field Survey, 2019: ** ≥ 2.5, * ˂2.5