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An allometric method for measuring leaf growth in eelgrass, Zostera marina, using leaf length data

Héctor Echavarría-Heras, Elena Solana-Arellano, Cecilia Leal-Ramírez and Ernesto Franco Vizcaino
From the journal Botanica Marina


The rate of production of leaf biomass in eelgrass (Zostera marina L.) is an indicator variable for environmental influences on the growth of this important seagrass species. The efforts to restore eelgrass meadows from the harmful human influences make the use of nondestructive evaluations essential. We present, here, an indirect procedure for the estimation of the growth rates of eelgrass leaves by using easily obtained measurements of leaf length and increases in leaf length, and allometric parameters linked to the scaling of leaf biomass and leaf length. This allometric method includes criteria that allow the estimation of leaf growth rates even when the sizes of some of the leaves cannot be determined because of herbivory or other environmental factors. To validate the proposed method, we performed simulation studies and analyzed data from two natural eelgrass populations in the East Pacific (México). These allometric projections of leaf growth rates displayed a high level of correspondence with observed values. We show that whenever the allometric parameters for the scaling of eelgrass leaf dry weight in terms of leaf length have been previously fitted, the method proposed here can provide an alternative for estimating biomass production that is both accurate and nondestructive and uses easily obtained data on leaf length and increases in leaf length between the sampling periods.

Corresponding author: Héctor Echavarría-Heras, Centro de Investigación Científica y de Educación Superior de Ensenada, carretera Ensenada-Tijuana No. 3918, Zona Playitas, Código Postal 22860. Apdo. Postal 360. Ensenada, B.C. Mexico


In this appendix, we derive the results that sustain the method of the allometric approximation presented here. Let us consider a leaf in a retrieved shoot such that its length, l(tt), from its base, at the top of the sheath, to its tip, can be plainly identified and measured. We also assume that the individual leaf biomass can be allometrically scaled (Duarte 1991, Harris et al. 2006) in terms of matching length, i.e.,

where α and β are the fixed parameters. Then, we can obtain an allometric surrogate, Δwla(t, Δt), for the individual leaf dry weight increment, Δwl(t, Δt), produced over the interval, [t, tt]; this is,

Factoring βl(tt)α in equation (A2), we obtain

and by defining

we get

where λl(t, Δt) is the ratio of the leaf length increment to the total length,

Noticing that, at the marking time, the previous longitudinal growth aggregated by l(t) and the length increase, Δl, occurring over the interval [t, tl] are formally related through

Hence, from equation (A3), we get,

then, factoring Δl and rearranging leads to

such that

and with

Now, as Δl is positive and bounded above by l(tt), we get from equation (A6),

Then, equation (A10) shows that Rla(t, Δt) vanishes at both the minimum and maximum values attained by λl(t, Δt). In fact, for leaves emerging from the sheath during the interval [t, tt], we have l(tt)=Δl, this will set λl(t, Δt)=1, Rla(t, Δt)=0, and Δwla(t, Δt)=βl)α. Hence, the term representing new tissue is the fundamental contribution to the growth for these leaves.

Because Solana-Arellano et al. (1997) demonstrated that the assumption of a restricted, asymptotic leaf growth in eelgrass is consistent with observations, the leaves having negligible sizes at marking time can be expected to grow most of their size measured at time tt during the observation period [t, tt]. That is, Δl would be expected to be similar to l(tt), thus, making λl(t, Δt)≅1, and again rendering Rla(t, Δt); and therefore, the new leaf material, represented by βl)α, dominates the growth once more. For the older leaves that were almost fully grown at the marking time, the asymptotic growth assumption implies that l(t) and l(tt) could be expected to be similar, thus, rendering the values of both Δl and λl(t, Δt) negligible, and correspondingly, Rla(t, Δt) will also become negligible making βl)α approximate Δwl(t, Δt) in a reasonable way. Therefore, for leaves leading to values of λl(t, Δt) at the extremes of its variation range set by inequality (A12), Rla(t, Δt) will vanish, and we will have Δwl(t, Δt)=βl)α. Meanwhile, for leaves leading to values close to the extreme values of λl(t, Δt), the continuity of Rla(t, Δt) implies that the bias term, Rla(t, Δt), will become negligible, making βl)α dominant in equation (A9).

Let us now assume that a marked leaf is retrieved undamaged, so its length, l(tt), can be actually measured. Then, for such a leaf, the associated value of the ratio λl(t, Δt) will depend only on Δl, and correspondingly, the sign of Rda(t, Δt) depends only on the factor øla(λ, t, Δt). Furthermore, for fixed l(tt), the value of βl(tt)α will also be fixed, and from equation (A10), we can see that the behavior of Rda(t, Δt) depends fundamentally on the variation of the factor øla(λ, t, Δt). Now, as we have


whenever α>1, Rda(t, Δt) is positive, its maximum value will be attained at λl(t, Δt)=0.5, and from equation (A10), we have,

As by fitting equation (A1) we have verified that α>1, then, Rla(t, Δt) remains positive for all the values of λl(t, Δt) in the interval (0, 1), and βl)α will systematically underestimate Δwl(t, Δt). But for suitable values of λl(t, Δt), we can expect the value of the bias term, Rla(t, Δt), to diminish significantly, thus, making the approximation of Δwl(t, Δt) through βl)α consistent.

In fact, letting θ(t, Δt)=λl(t, Δt)-1, we have Δl=θ(t, Δt) l(tt), then, from equation (A10), we have


Moreover, as

then, whenever 1≤α≤2 and θ satisfies 1≤θ<∞, the factor Ωla(θ, t, Δt) increases.

Now, letting lmax=max{l(tt)}, and Δlmin=min{Δl}, then, for

we have 1≤θ(t, Δt)≤θ, and from equations (A9) and (A16), we have

Then, the smaller the value of Δl becomes, the lesser the difference between βl)α and Δwla(t, Δt) will be. We have observed that most of the leaves that were damaged or had missing tips were to be found among the older and longer leaves. As the asymptotic growth assumption implies that Δl can be expected to be smaller for the older and larger leaves than for the newly produced ones, then inequality (A20) suggests the following criteria for calculating Δwla(t, Δt),


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Received: 2012-10-16
Accepted: 2013-4-17
Published Online: 2013-05-31
Published in Print: 2013-06-01

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