Litter dynamics and its effects on the survival of Castanopsis sieboldii seedlings in a subtropical forest in southern Japan

Abstract

Based on direct field measurement, this study quantitatively estimated the litter dynamics on the forest floor for a 1-year-period and then investigated its influence on the seedling dynamics of Castanopsis sieboldii, as well as interactions with adults in a subtropical forest in southern Japan. Litter dynamics is composed of three major components: falling litter, transport, and decomposition on the forest floor. Litterfall was measured by litter traps and did not exhibit clear spatial tendency. Lateral input was assessed by newly accumulated litter beneath the traps and showed no spatial variation, either. In contrast, lateral output of litter, which was quantified from disappearance of artificial litter, was correlated with local topography. Consequently, we found considerable spatial variations and seasonal changes in litter dynamics on the forest floor. In addition, we constructed survival models of C. sieboldii seedlings at the individual level. The lateral movement of accumulated litter had an influence on the survival of seedlings, which mostly occurred in periods of typhoons with heavy rain. Meanwhile, the distance from canopy trees, which is assumed to be a spacing mechanism due to seedling/adult interactions, played a lesser role in this subtropical forest. Our results suggest that the stability of accumulated litter on the forest floor was a predominant factor in the spatial dynamics of the early life stage of C. sieboldii.

Appendix: Non-parametric estimation of litter data

When we obtained observations Ob _i (such as litter output and input) at {(x _i , y _i )} = {(x ₁, y ₁), (x ₂, y ₂), ..., (x _M , y _M )} in a rectangular plot, modifying the non-parametric method introduced by Sakamoto and Ishiguro (1985), we can smoothly interpolate the observations and estimate that amount at any (unobserved) point, as follows.

We first fix grid points over the plot, and in principle, estimates are non-parametrically given at these points by the maximum likelihood method. Practicably, it is exceedingly time-consuming to find the best set of parameters simultaneously if their number is large. Here, to save the optimization process, we used a relatively small number of grid points, and other points are interpolated by spline functions.

Let the plot be [A ₀, A ₁] × [B ₀, B ₁], and let the grid points be {(X _j , Y _k )} (j = 0, ..., J − 1 and k = 0, ..., K − 1. In this study, J = K = 8 and X _j , Y _k  = 1, 5, 9, ..., 29.) and denote the estimate at (X _j , Y _k ) by q _j,k . What we want to calculate from given data are these q _j,k .

If we interpolate the estimate at an arbitrary point (x, y) by connecting {q _j,k } by the second-order B-spline functions s _j (x) and s _k (y) (Sakamoto 1991, p. 133), and if the amount of litter at (x, y) follows the normal distribution of average

$$ q(x,y) = {\sum\limits_{j = 0}^{J - 1} {{\sum\limits_{k = 0}^{K - 1} {q_{{j,k}} s_{j} (x)s_{k} (y)} }} } $$

with variance σ², the likelihood of given observations {Ob _i } is given by

$$ L({\mathbf{q}}) = {\prod\limits_{i = 1}^M {\exp ( - (q(x_{i} ,y_{i} ) - Ob_{i} )^{2} /2\sigma ^{2} )/{\sqrt {2\pi } }\sigma } }, $$

(1)

where q is the J × K matrix (q _j,k ).

In general, we want to maximize the (log-) likelihood; however, if litter amounts are spatially smoothly changing, adjacent q _j,k , q _j±1,k , q _j,k±1, q _j±1,k±1 are close to one another to some degree; thus, if we assume linear changes among them,

$$ g({\mathbf{q}}) = {\sum\limits_{j = 1}^{J - 1} {{\sum\limits_{k = 1}^{K - 1} {(q_{{j,k}} - q_{{j - 1,k}} - q_{{j,k - 1}} + q_{{j - 1,k - 1}} )^{2} } }} } + {\sum\limits_{k = 1}^{K - 2} {(q_{{J - 1,k + 1}} - 2q_{{J - 1,k}} + q_{{J - 1,k - 1}} )^{2} } } + {\sum\limits_{j = 1}^{J - 2} {(q_{{j + 1,K - 1}} - 2q_{{j,K - 1}} + q_{{j - 1,K - 1}} )^{2} } } $$

(2)

should be sufficiently small. Unfortunately, the two optimization, log-likelihood (ln(1)) and (2), are not always compatible. Hence, we determine the relative importance between the two by providing weight v as:

$$ \ln {\left({L{\left( {\mathbf{q}} \right)}} \right)}-1/v^{2} \cdot g(\mathbf{q}). $$

(3)

If we maximize (3) using a large v, the g(q) term will be almost ignored and the resulting q will fit well to the given data but will be spatially fluctuating. Conversely, if the optimization is conducted for a small v, we will emphasize smoothness but the estimates might exhibit poor fitting. The most appropriate weighting should be between the two extremes. In order to solve this problem, we first rewrite g(q) in the form of the JK-dimensional normal distribution, which can be satisfied if we add other J + K + 1 parameters ({q _j,K }, {q _J,k }, q _J,K ) and extend (2) to

$$ g({\mathbf{q}}) = {\sum\limits_{j = 1}^J {{\sum\limits_{k = 1}^K {(q_{{j,k}} - q_{{j - 1,k}} - q_{{j,k - 1}} + q_{{j - 1,k - 1}} )^{2} } }} } + {\sum\limits_{k = 1}^{K - 1} {(q_{J,{K-1}}-2q_{J,K}+q_{J,{K-1}} )^{2} } } + {\sum\limits_{j = 1}^{J - 1} {(q_{{J+1},K}-2q_{J,K}+q_{{J-1},K})^{2} } } $$

(4)

and express its exponential as:

$$ \pi ({\mathbf{q}}) = (1/{\sqrt {2\pi } }v)^{{(J + 1)(K + 1) - 3}} \cdot \exp ( - 1/2v^{2} \cdot ||{\mathbf{D}}({\mathbf{q}} - {\mathbf{D}}^{{ - 1}} {\mathbf{Eq}}_{0} )||^{2} ), $$

(5)

Here, we consider q = (q _0,0, q _1,0, ..., q _{J – 1, K – 1,} q _0,K , q _1,K , ..., q _J−2,K , q _J,0, q _J,1, ..., q _J,K−2)^t and q ₀ = (q _J−1,K , q _{J, K} , q _{J, K−1})^t as column vectors (these, σ and v are called hyperparameters), and ((J + 1)(K + 1) − 3) × ((J + 1)(K + 1) − 3) matrix D and ((J + 1)(K + 1) − 3) × 3 matrix E are determined so as to satisfy

$$ ||{\mathbf{D}}({\mathbf{q}} - {\mathbf{D}}^{{ - 1}} {\mathbf{Eq}}_{0} )||^{2} = g({\mathbf{q}}) .$$

Then we can say that q follows the ((J + 1)(K + 1)−3)-dimensional normal distribution of mean D ⁻¹ Eq ₀ and covariance matrix (D ^t D/v ²)⁻¹. What we need to conduct is the maximization of L(q)π(q). Because, under a Bayesian framework, the posterior distribution is proportional to the product of the likelihood and the prior distribution, Sakamoto (1991, p. 90) proposes that: “find the mode of the posterior distribution of q when the prior distribution is given by (5).” Utilizing the marginal likelihood

$$ {\int {L({\mathbf{q}})\pi ({\mathbf{q}};v,\sigma ,q_{{J - 1,K}} ,q_{{J,K}} ,q_{{J,K - 1}} )d{\mathbf{q}}} } $$

(6)

which expresses the probability that the data are obtained from a given prior distribution, Akaike (1980a, b) proposed the application of the Akaike Bayesian Information Criterion (ABIC):

$$ {\text{ABIC}} = - 2 \times \ln {\left\{ {{\int {L({\mathbf{q}})\pi ({\mathbf{q}};v,\sigma ,q_{{J - 1,K}} ,q_{{J,K}} ,q_{{J,K - 1}} )d{\mathbf{q}}} }} \right\}} + 2 \times {\left\{ {{\text{number}}\;{\text{of}}\;{\text{hyperparameters}}} \right\}} $$

(7)

and v (and (σ, q _J−1,K , q _J,K , q _J,K−1)) that minimizes ABIC is selected as the most appropriate weighting.

Ishiguro (2004) showed that (7) can be calculated by:

$$ {\text{ABIC}} = n\ln {\left( {2\pi } \right)} + \ln {\left( {\det {\left( {\varvec{\upsigma}_{x} } \right)}} \right)} + {\mathbf{\Delta x}}^{{\text{t}}} \varvec{\upsigma}^{{ - 1}}_{x} {\mathbf{\Delta x}}, $$

(8)

where Σ _x  = A·(1/v ² D ^t D)⁻¹ A ^t + σ² I, Δx = Ob − AD ⁻¹ Eq ₀ (Ob = (Ob ₁, Ob ₂, ..., Ob _n )^t, and ((J + 1)(K + 1) − 3) × n matrix A is determined so as to satisfy {ith component of Aq} = q(x _i , y _i ). Hence, practicably, we first try to find (v, σ, q _J−1,K , q _{J, K} , q _{J, K−1}) that minimizes (8), then, using these hyperparameter values, we need to find q that maximizes (6). Then

$$ q(x,y) = {\sum\limits_{j = 0}^{J - 1} {{\sum\limits_{k = 0}^{K - 1} {q_{{j,k}} s_{j} (x)s_{k} (y)} }} } $$

provides the estimate at arbitrary point (x, y), and we can draw a three-dimensional graph or a contour map. In this paper, we showed q(x, y) at every 2 m grid point in Fig. 5.