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.