A Statistical Methodology for Evaluating the Potential for Poleward Expansion of Warm Temperate and Subtropical Plants Under Climate Change: A Case Study of South Korean Islands

Abstract

Many studies have examined how species are shifting their ranges poleward in response to climate change, using statistical approaches such as graphical analyses, t-tests, correlation analyses, and circular data methods. However, these methods are often constrained by assumptions of linearity or reliance on a single explanatory variable, which limits their ecological applicability. This study introduces a new statistical methodology to evaluate the significance of poleward range expansion, aiming to overcome these limitations and improve the robustness of ecological inference. We developed four parameterized nonlinear models—simple, multivariable, fixed, and transformed—to characterize the relationship between latitude and species richness across 1253 islands. Model parameters were estimated using the Gauss–Newton algorithm, and residuals were calculated as the difference between observed and predicted values. To test for distributional shifts, likelihood ratio tests were applied to the residuals, with statistical significance assessed using chi-square statistics and p-values derived from the −2 log-likelihood ratio. Finally, an intuitive indicator based on the fitted models was introduced to evaluate the direction of range shifts, thereby providing a direct means of identifying northward expansion trends under climate change. Applying this framework revealed significant poleward shifts of warm temperate and subtropical species (χ² = 52.4–61.3; p < 0.001). Among the four models, the multivariable model incorporating island area provided the best fit (AIC, BIC), reflecting its ability to account for collinearity. Taken together, these results underscore the robustness and ecological relevance of the methodology, demonstrating its utility for detecting species-specific range shifts and comparing alternative models under climate change.

1. Introduction

Many studies have investigated the poleward expansion of selected warm temperate and subtropical (WTS) plant species under climate change. Most previous studies have either concluded or implicitly assumed that the WTS plant zones have shifted poleward in response to climate change. Broadly, prior approaches fall into two types. First, studies that rely on linear-assumption frameworks—including linear trends/regressions, linearized boundary or centroid shifts, and auxiliary mean-comparison tests. Second, studies that employ circular data analysis for directional responses (e.g., bearings in phenology or movement), typically using von Mises models and circular ANOVA.

In the first type, prior empirical work on poleward (northward) range change can be grouped into three strands: (i) Studies that estimate linear trends, correlations, or regressions between temporal or climatic predictors (e.g., year, temperature) and biotic responses (occurrence, richness, centroid latitude, areal extent), often using linearized indicators of boundary or centroid shift [1,2,3,4,5]. (ii) Analyses that adopt linear mixed-effects models to estimate movement rates (e.g., km yr⁻¹) while accounting for hierarchical random effects across regions, species, or periods [6]. (iii) Works that use mean-comparison tests (e.g., one-/two-sample t-tests) as auxiliary inference to evaluate differences in mean latitude or displacement between periods or groups, or to test whether an average shift differs from zero [6,7]. For completeness, circular statistics constitute a distinct, non-Euclidean framework for analyzing directional responses and are not treated as linear in this taxonomy [4].

In the second category, circular data analysis has been widely applied to quantify directional ecological responses under environmental change. Group differences in mean direction are commonly tested with the Watson–Williams procedure (one-way circular ANOVA; equal concentration assumed) and, where two factors are involved, with the Harrison–Kanji two-way circular ANOVA [8,9,10,11]. Likelihood-based modeling typically adopts the von Mises/von Mises–Fisher family as the baseline distribution of angles; when multimodality is expected, finite mixtures (EM or variational inference) can capture heterogeneous directional clusters [12,13,14,15]. For uncertainty quantification and prior structure, Bayesian circular methods—including Laplace-approximate posteriors, Bayesian vMF models, and wrapped distributions—are also used [15,16,17,18]. Applications to climate-related range reorientation, such as directional shifts in avian density, illustrate the utility of these tools for detecting coherent movement directions at macroecological scales [10]. Because our goal is to model nonlinear latitude–richness relationships rather than angular bearings, circular methods are not benchmarked as comparators in this study.

While linear approaches have yielded valuable insights, they can be restrictive when responses are nonlinear in latitude or when multiple predictors interact. This motivates a complementary framework that explicitly accommodates nonlinearity and multivariable effects, enabling inference on range-shift direction and magnitude while permitting formal model comparison.

Many ecological responses to climate are not well described by straight lines; they often exhibit thresholds, saturation effects, and interactions among predictors. The ANOVA framework that works for linear models relies on a clean, additive partition of sums of squares under the usual assumptions of independent, normally distributed errors with constant variance. In nonlinear regression, these conditions typically break down: orthogonality is lost, sums of squares no longer admit a unique additive decomposition, and the familiar F-tests lack a straightforward theoretical justification. Put simply, residual additivity and residual homogeneity (constant variance) are not satisfied in general, so the standard ANOVA table and F-tests are not reliable tools for comparing nonlinear models [19,20,21]. As a result, model comparison must be based on principles other than linear-model ANOVA.

We provide a practical, likelihood-based alternative for formal comparison among nonlinear models. We fit nonlinear relationships between latitude and species richness using the Gauss–Newton algorithm, and we compare competing specifications with likelihood-ratio tests using χ² approximations. We also report AIC/BIC to summarize relative support across models. This framework explicitly accommodates nonlinearity and multiple predictors, and it delivers interpretable statistical tests for both the direction (e.g., poleward expansion) and the magnitude of species-specific range shifts.

We apply the framework to WTS plants across 1253 South Korean islands. We model the nonlinear relationship between latitude and species richness with a set of parameterized curves and estimate parameters by Gauss–Newton. Data are split into two periods (pre/post a benchmark year), and the chosen model is fitted separately to each period to allow formal comparison.

The purpose of this case study is to investigate whether the WTS plant zones have expanded northward in the forests of South Korean islands, using the statistical significance testing methodology proposed in this study. The vegetation of Korean islands is known to exhibit an earlier ecological response to climate change compared to inland regions, largely due to the influence of warm ocean currents [22,23,24]. This heightened sensitivity is not unique to Korea; insular ecosystems worldwide have been shown to be particularly vulnerable to climate-driven shifts in temperature, precipitation, and sea-level rise [25,26]. In addition, island ecosystems possess biogeographical distinctiveness, as their spatial isolation leads to higher variability in species immigration, establishment, and extinction compared to the mainland. According to the Theory of Island Biogeography proposed by MacArthur and Wilson [27], island species richness is determined by a dynamic equilibrium between colonization from external sources (typically mainland) and local extinction. Species diversity tends to increase with island size and proximity to the mainland, which serves as the principal gene pool [28].

To avoid confusion, we clearly separate model selection from formal inference. We use AIC/BIC to compare the four candidate specifications (simple, multivariable, fixed, transformed) and to identify the model with the best relative fit (reporting ΔAIC/ΔBIC and, where useful, Akaike weights). Because AIC/BIC do not provide significance tests, we then assess between-period differences within the selected model using a likelihood-ratio test for nested hypotheses (null: a common parameter vector across periods; alternative: period-specific parameters). The test uses the standard χ² approximation and yields a formal decision on whether the fitted nonlinear relationship differs between periods.

2. Materials and Methods

2.1. Study Area

In this study, the study area is South Korea’s island system, which spans approximately 33.1–38.0° N and 124.6–131.9° E, encompassing the transitional boundary between temperate and subtropical East Asia. Official records enumerate over 3300 islands nationwide, of which nearly 3000 remain uninhabited, underscoring the exceptionally dense insular landscape along the Korean coastline [22]. This transitional zone provides a natural setting to examine potential northward shifts of subtropical plant taxa under ongoing climate change. Recent modeling studies on Korean endemic flora have already demonstrated climate-induced distributional dynamics and highlighted the heightened vulnerability of species with narrow ecological niches across the peninsula [29]. Our island-focused analysis complements these peninsula-wide insights by testing such northward tendencies within a densely insular system, thereby offering a spatially explicit perspective on biogeographical responses to climate change.

2.2. Data

This study employs a two-tiered nested dataset structure to evaluate the poleward expansion of WTS plant species under climate change. Each dataset contains the number of such species recorded at comparable large-scale survey sites. The second tier comprises two temporally distinct datasets—one collected prior to, and the other following, a designated reference year. The first tier represents the pooled dataset, integrating both subsets from the second tier. Ideally, the two temporal datasets in the second tier should be separated by at least 10 years to allow detection of potential range shifts attributable to climate change.

The data used in this study were utilized from vegetation surveys conducted by the National Institute of Ecology under the Ministry of Environment of the Republic of Korea between 1999 and 2023, as well as from peer-reviewed academic publications focusing on island and forest vegetation in South Korea. These surveys covered a total of 1253 islands, the scientific names were reorganized according to the Checklist of Vascular Plants in Korea [30] and the compiled data were organized into a comprehensive database.

Based on this database, the number of WTS plant species present on each of the 1253 islands was recorded. The dataset was divided into three subsets: Data_A_2013 (data collected after 2013), Data_B_2013 (data collected before 2013), and Data_AB_2013 (a pooled dataset including both time periods). These datasets form a nested structure, in which Data_AB_2013 constitutes the first tier, encompassing the entire set of records, while Data_A_2013 and Data_B_2013 represent the second tier as temporally distinct subsets (Figure 1). Specifically, Data_A_2013 includes records for 857 islands, Data_B_2013 for 832 islands, and Data_AB_2013 comprises a total of 1689 records corresponding to 1253 unique islands, of which 436 entries are duplicated and 809 are unduplicated (Figure 2). The datasets A_2013 and B_2013 include both overlapping and non-overlapping islands. For the analysis, all islands from both datasets were used regardless of overlap.

WTS plant species, in particular, prefer maritime climates and specific soil types, and their northern distribution limits are generally determined by minimum winter temperatures [31]. The IPCC [32] projected that global mean temperatures could increase by up to 6.4 °C by the end of the 21st century. In response to such climate change, most regions of South Korea—except for some high-altitude inland areas—are expected to shift toward a warm-temperate climate zone, while the southern coastal and insular areas may transition into a subtropical regime. Accordingly, research on the northward expansion of WTS plant species in island ecosystems of the Korean Peninsula provides valuable insights for anticipating and managing future vegetation shifts in inland areas under climate change.

The list of WTS plant species was synthesized with reference to previous studies [33,34,35], and the scientific names were reorganized according to the Checklist of Vascular Plants in Korea [30]. The synthesized list is presented in Appendix A.

2.3. Method

The statistical significance testing methodology proceeds in four steps. (Step 1: Modeling) A parameterized mathematical model is constructed to characterize the relationship between latitude and species richness. The model includes parameters such as slope, intercept shift, and bias, which influence the shape and fit of the response curve. (Step 2: Estimation) Parameter values are estimated using the Gauss–Newton algorithm, and the predicted number of species at each site is computed by applying the estimated parameters to the model equation. (Step 3: Testing) The two datasets are compared via a likelihood ratio test (LRT) based on model residuals. The test statistic, calculated as −2 log-likelihood ratio (LR), is evaluated using the chi-square distribution with appropriate degrees of freedom, following Wilks’ theorem [36]. (Step 4: Determining) An intuitive indicator is proposed to quantify the direction of species distributional changes driven by climate change.

All statistical analyses, including the Gauss–Newton estimation, LRTs, chi-square tests, and correlation analyses, were conducted using R 4.5.0 in RStudio 2025.05.0 (Build 496) [37]. Geospatial visualizations were created using QGIS 3.34.9 ‘Prizren’ [38].

3. Methodology: Statistical Analysis

3.1. Mathematical Model

From the perspective of Island Biogeography, the relationship between distance from the mainland (X) and the number of species (Y) shows a strong first-order linear correlation when both variables are log-transformed in Equation (1) [27].

$$\mathrm{Log}Y= {\beta }_1\log X+{\beta }_2$$

(1)

In the case of the relationship between latitude (X) (rather than distance) and the number of species (Y), a linear model is more appropriate when only the number of species variable is log-transformed in Equation (2). It is generally more suitable to apply a log transformation to causative factors like distance, but not to environmental factors like latitude [39,40].

$$\mathrm{Log}Y={\beta }_{1}X+{\beta }_2$$

(2)

For greater intuitive understanding and explanatory power, parameters ${\beta }_1$ and ${\beta }_2$ are modified in a proper, and a new parameter ${\beta }_3$ is added, resulting in the basic model shown in Equation (3). The parameters are designed to represent the slope (${\beta }_1$), the horizontal shift along the latitude (${\beta }_2$), and the vertical shift representing the overall species richness (${\beta }_3$). Under the assumption that the two random variables X and Y are independent and identically distributed, we construct the simple mathematical parameterized model (SM) as shown in Equation (4).

We built another one mathematical parameterized model, called the multivariable model (MM), added an area of island term (A) taking log to better fit according to Island Biogeography with the parameter ${\beta }_{area}$ to the Simple Model (SM) in Equation (4) as Equation (5) under the assumption that three random variables X, Y and A are independent and identical distributed. We construct a fixed model (FM) in Equation (6), specifically designed to test northward expansion. In this model, ${\beta }_2$ is allowed to vary over time (before and after a specific year), while ${\beta }_1$ and ${\beta }_3$ are held constant. Finally, to address the tendency of SM to underestimate species richness north of 35° N and MM to overestimate it south of 34° N, a complementary model, a transformed model from SM (TM), was developed in Equation (7).

$$\mathrm{Log}\left(Y-{\beta }_3\right)={-\beta }_1\left(X-{\beta }_2\right)+\mathsf{\epsilon }$$

(3)

$$\mathrm{SM}: Y=e^{-{\beta }_1\left(X-{\beta }_2\right)}+{\beta }_3+{\mathsf{\epsilon }}_{SM}$$

(4)

$$\mathrm{MM}: Y=e^{-{\beta }_1\left(X-{\beta }_2\right)}+{\beta }_3+{\beta }_{area}\log A+{\mathsf{\epsilon }}_{MM}$$

(5)

$$\mathrm{FM}: Y=e^{-{\beta }_{1 Fixed}\left(X-{\beta }_2\right)}+{\beta }_{3 Fixed}+{\mathsf{\epsilon }}_{FM}$$

(6)

$$\mathrm{TM}: Y={\beta }_0{e}^{-{\beta }_{1 }{\left(X-{\beta }_2\right)}^2}+{\beta }_3+{\mathsf{\epsilon }}_{TM}$$

(7)

3.2. Parameter Estimation

The parameters $\beta$ are estimated using Gauss-Newton method in Equation (8). Parameter estimation is performed three times: separately for the two datasets (before and after a specific year) and for the pooled dataset. Although various methods can be used—such as gradient descent, Newton–Raphson, Levenberg–Marquardt, and least squares—the Gauss–Newton method is selected for this study. The Gauss–Newton method is widely employed for nonlinear least squares problems, particularly because it avoids the need for second derivative calculations, making it more computationally efficient [41].

$${\beta }_{new}=\beta -{\left(J^{T}J \right)}^{-1}{J}^{T}R$$

(8)

where $\beta$ is the parameter vector, $J$ is the Jacobian matrix, and $R$ is the residual matrix.

Additionally, parameter estimates can be obtained using the nls() function in R. Estimate the parameters using R function nls() in R (version 4.5.0). Estimates obtained using nls() closely match those derived from the Gauss–Newton method.

3.3. Significance Test

3.3.1. Likelihood Ratio Test for the Residuals

The residual (r), which is the difference between observed and estimated values, should satisfy normality and homoscedasticity If these assumptions are violated, appropriate transformations or alternative parameter estimates are required. Given sample sizes over 50, residuals can generally be assumed to follow a normal distribution. In this study, we assume that the residuals satisfy normality and follow a normal distribution with mean ${\mu }_r$ and variation ${\sigma }_r^2$ in Equation (9), estimated by Maximum Likelihood Estimation (MLE) method [42]. In nonlinear models, residuals are generally unlikely to follow a normal distribution. This tendency becomes more pronounced when the model incorporates a wide range of environmental heterogeneity, as is often the case in large-scale ecological modeling. However, when the dataset is derived from island-based surveys grounded in the principles of island biogeography, the residuals tend to exhibit an approximately normal distribution.

$$\mathrm{Residuals} \hspace{1em}{r}_i=Y_i-\widehat{Y_i}, i=1, 2, \cdots ,n ~ N ({\mu }_r, {\sigma }_r^2)$$

(9)

The LRT, including its restricted form (RLRT), provides a statistically rigorous framework for evaluating the goodness-of-fit in nonlinear regression models. Particularly in cases where a parametric null model is compared against a broader semiparametric alternative, such as those incorporating penalized splines, LRTs are well-established and theoretically justified tools [43]. In the present study, the null hypothesis is that all groups share the same set of model parameters, whereas the alternative hypothesis allows at least one parameter to differ across groups.

In contrast, the use of residual-based F-tests in nonlinear regression is generally inappropriate. This is primarily because the decomposition of total variation—central to the definition and interpretation of the F-statistic—is not valid in nonlinear settings. Specifically, the additive identity SST = SSR + RSS, which underpins the F-test in linear models, does not hold in nonlinear models. As a result, the F-statistic lacks both interpretability and theoretical support in this context [44], and it is therefore not adopted in the current analysis.

The hypotheses tested via the LR framework are formally stated in Equation (10). The null hypothesis (H₀) posits that model parameters are identical across datasets, indicating no structural shift. In contrast, the alternative hypothesis (H₁) allows for differences in parameters between datasets, implying a potential shift, such as one driven by climate change

$$\begin{array}{lll}\mathrm{SM}:& H_0: {\beta }_{\left(1,2,3\right) A}={\beta }_{\left(1,2,3\right) B}& \mathrm{vs}. H_1: {\beta }_{\left(1,2,3\right) A}\ne {\beta }_{\left(1,2,3\right) B}\\ \mathrm{MM}:& H_0: {\beta }_{\left(1,2,3,area\right) A}={\beta }_{\left(1,2,3,area\right) B}& \mathrm{vs}. H_1: {\beta }_{\left(1,2,3,area\right) A}\ne {\beta }_{\left(1,2,3,area\right) B }\\ \mathrm{FM}:& H_0: {\beta }_{2 A}={\beta }_{2 B}& \mathrm{vs}. H_1: {\beta }_{2 A}\ne {\beta }_{2 B}\\ \mathrm{TM}:& H_0:{\beta }_{\left(0,1,2,3\right) A}={\beta }_{\left(0,1,2,3\right) B}& \mathrm{vs}. H_1: {\beta }_{\left(0,1,2,3\right) A}\ne {\beta }_{\left(0,1,2,3\right) B}\end{array}$$

(10)

The likelihood function $L_{H0}\left({\beta }_{H0}| r_{AB}\right)$ that can occur under the assumption the null hypothesis is true in Equation (11), the likelihood function $L_{H1}\left({\beta }_{H1}| r_A, r_B\right)$ that can occur under the assumption the alternative hypothesis is true in Equation (12), are as follows. The likelihood function $L_{H0}\left({\beta }_{H0}|{ r}_{AB}\right)$ is for the data pooled across two data groups

$$L_{H0}\left({\beta }_{H0}| r_{AB}\right)=\prod _{i=1}^{n_{AB}}f(r_{AB i} ; {{\mu }_{r AB},{\sigma }_{r AB}^2, \beta }_{ \left(1,2,3\right) AB })$$

(11)

$$\begin{array}{cc}\hfill L_{H1}\left({\beta }_{H1}|{ r}_A, r_B\right)& ={\displaystyle \prod _{i=1}^{n_A}}f (r_{A i}; {{\mu }_{r A},{\sigma }_{r A}^2, \beta }_{ \left(1,2,3\right) A})\hfill \\ & \times {\displaystyle \prod _{j=1}^{n_B}}f(r_{B i}; {{\mu }_{r B},{\sigma }_{r B}^2, \beta }_{ \left(1,2,3\right) B})\hfill \end{array}$$

(12)

The ratio of the likelihood function $L_{H0}\left({\beta }_{H0}|r_{AB}\right)$ and $L_{H1}\left({\beta }_{H1}|r_A,r_B\right)$, called λ, is as follows in Equation (13).

$$\mathsf{\lambda }={\displaystyle \raisebox{1ex}{$L_{H0}\left({\beta }_{H0}|{ r}_{AB}\right)$}\!\left/ \!\raisebox{-1ex}{$L_{H1}\left({\beta }_{H1}|{ r}_A, r_B\right)$}\right.}$$

(13)

3.3.2. Chi-Square Test for Two Times Negative Log-Likelihood Ratio

The statistic −2 log λ, two times negative log-LR, follows an asymptotically chi-squared distributed with degrees of freedom df, where df is the difference between the dimension of $L_{H1}$ under H₁ and the dimension of $L_{H0}$ under H₀ by Wilks’ theorem [36]. Simply speaking, the dimension means the number of the parameters, and the degrees of freedom means the difference in the number of free parameters between the null and alternative hypotheses, likelihood function $L_{H0}\left({\beta }_{H0}|r_{AB}\right)$, $L_{H1}\left({\beta }_{H1}|r_A,r_B\right)$, respectively [45].

In case of SM, the degrees of freedom are 5. The Likelihood function $L_{H0}\left({\beta }_{H0}|r_{AB}\right)$ has five parameters, ${\mu }_{rAB},{\sigma }_{rAB}^2,{\beta }_{1AB},{\beta }_{2AB}$ and ${\beta }_{3AB}$. So $L_{H0}$ has 5 dimensions. The Likelihood function $L_{H1}\left({\beta }_{H1}|r_A,r_B\right)$ has ten parameters ${\mu }_{rA},{\sigma }_{rA}^2,{\beta }_{1A},{\beta }_{2A},{\beta }_{3A}$ and ${\mu }_{rB},{\sigma }_{rB}^2,{\beta }_{1B},{\beta }_{2B},{\beta }_{3B}$. So $L_{H1}$ has 10 dimensions. Thus, the degrees of freedom are 5, calculated as 10 dimensions of $L_{H1}$ minus 5 dimensions of $L_{H0}$.

In case of MM, the degrees of freedom are 6. The Likelihood function $L_{H0}\left({\beta }_{H0}|r_{AB}\right)$ has six parameters, ${\mu }_{rAB},{\sigma }_{rAB}^2,{\beta }_{1AB},{\beta }_{2AB}{,\beta }_{3AB}$ and ${\beta }_{areaAB}$. So $L_{H0}$ has 6 dimensions. The Likelihood function $L_{H1}\left({\beta }_{H1}|r_A,r_B\right)$ has twelve parameters ${\mu }_{rA},{\sigma }_{rA}^2,{\beta }_{1A},{\beta }_{2A},{\beta }_{3A},{\beta }_{areaA}$ and ${\mu }_{rB},{\sigma }_{rB}^2,{\beta }_{1B},{\beta }_{2B},{\beta }_{3B},{\beta }_{areaB}$. So $L_{H1}$ has 12 dimensions. Thus, the degrees of freedom are 6, calculated as 12 dimensions of $L_{H1}$ minus 6 dimensions of $L_{H0}$.

In case of FM, the degree of freedom is 1. The Likelihood function $L_{H0}\left({\beta }_{H0}|r_{AB}\right)$ has five parameters, ${\mu }_{rAB},{\sigma }_{rAB}^2,{\beta }_{1AB},{\beta }_{2AB}$ and ${\beta }_{3AB}$. So $L_{H0}$ has 5 dimensions. The Likelihood function $L_{H1}\left({\beta }_{H1}|r_A,r_B\right)$ has six parameters ${\mu }_{rA},{\sigma }_{rA}^2,{\beta }_{2A}$ and ${\mu }_{rB},{\sigma }_{rB}^2,{\beta }_{2B}$. So $L_{H1}$ has 6 dimensions. Thus, the degree of freedom is 1, calculated as 6 dimensions of $L_{H1}$ minus 5 dimensions of $L_{H0}$.

In case of TM, the degrees of freedom are 6. The Likelihood function $L_{H0}\left({\beta }_{H0}|r_{AB}\right)$ has six parameters, ${\mu }_{rAB},{\sigma }_{rAB}^2,{\beta }_{0AB},{\beta }_{1AB}{,\beta }_{2AB}$ and ${\beta }_{3AB}$. So $L_{H0}$ has 6 dimensions. The Likelihood function $L_{H1}\left({\beta }_{H1}|r_A,r_B\right)$ has twelve parameters ${\mu }_{rA},{\sigma }_{rA}^2,{\beta }_{0A},{\beta }_{1A},{\beta }_{2A},{\beta }_{3A}$ and ${\mu }_{rB},{\sigma }_{rB}^2,{\beta }_{0B},{\beta }_{1B},{\beta }_{2B},{\beta }_{3B}$. So $L_{H1}$ has 12 dimensions. Thus, the degrees of freedom are 6, calculated as 12 dimensions of $L_{H1}$ minus 6 dimensions of $L_{H0}$.

If the p-value < α (0.05, 0.01, or 0.001), the null hypothesis is rejected, indicating significant differences attributable to climate change effects. A summary of the above is provided in

Table 1. Degrees of freedom (DF) for chi-square distribution by model.

Model	${\mathit{H}}_1$			${\mathit{H}}_0$			DF (C–F)
	A. Likelihood Function	B. Parameters	C. Dimension	D. Likelihood Function	E. Parameters	F. Dimension
SM	$L_{H1}\left({\beta }_{H1}|r_A\right)$	${{\mu }_{rA},\sigma }_{rA}^2,{\beta }_{1A},{\beta }_{2A},{\beta }_{3A}$	5	$L_{H0}\left({\beta }_{H0}|r_{AB}\right)$	${\mu }_{rAB},{\sigma }_{rAB}^2,{\beta }_{1AB},{\beta }_{2AB,}$${\beta }_{3AB}$	5	5
	$L_{H1}\left({\beta }_{H1}|r_B\right)$	${{\mu }_{rB},\sigma }_{rB}^2,{\beta }_{1B},{\beta }_{2B},{\beta }_{3B}$	5
MM	$L_{H1}\left({\beta }_{H1}|r_A\right)$	${{\mu }_{rA},\sigma }_{rA}^2,{\beta }_{1A},{\beta }_{2A},{\beta }_{3A},{\beta }_{areaA}$	6	$L_{H0}\left({\beta }_{H0}|r_{AB}\right)$	${{\mu }_{rAB},\sigma }_{rAB}^2,{\beta }_{1AB},{\beta }_{2AB}{,\beta }_{3AB,}$${\beta }_{areaAB}$	6	6
	$L_{H1}\left({\beta }_{H1}|r_B\right)$	${{\mu }_{rB},\sigma }_{rB}^2,{\beta }_{1B},{\beta }_{2B},{\beta }_{3B},{\beta }_{areaB}$	6
FM	$L_{H1}\left({\beta }_{H1}|r_A\right)$	${{\mu }_{rA},\sigma }_{rA}^2,{\beta }_{2A}$	3	$L_{H0}\left({\beta }_{H0}|r_{AB}\right)$	${{\mu }_{rAB},\sigma }_{rAB}^2,{\beta }_{1AB},{\beta }_{2AB}$$, {\beta }_{3AB}$	5	1
	$L_{H1}\left({\beta }_{H1}|r_B\right)$	${{\mu }_{rB},\sigma }_{rB}^2,{\beta }_{2B}$	3
TM	$L_{H1}\left({\beta }_{H1}|r_A\right)$	${{\mu }_{rA},\sigma }_{rA}^2,{\beta }_{0A},{\beta }_{1A},{\beta }_{2A},{\beta }_{3A}$	6	$L_{H0}\left({\beta }_{H0}|r_{AB}\right)$	${{\mu }_{rB},\sigma }_{rAB}^2,{\beta }_{0AB},{\beta }_{1AB},{\beta }_{2AB},{\beta }_{3AB}$	6	6
	$L_{H1}\left({\beta }_{H1}|r_B\right)$	${{\mu }_{rB},\sigma }_{rB}^2,{\beta }_{0B},{\beta }_{1B},{\beta }_{2B},{\beta }_{3B}$	6

.

In likelihood-based inference following nonlinear model fitting, it is generally assumed that the residual mean does not require explicit estimation, as the model specification itself accounts for the mean structure of the data. Accordingly, the residual mean is not typically included as a parameter in the likelihood function. By contrast, the residual variance—representing unexplained variability—is treated as a model parameter and plays a central role in constructing the likelihood function [19,44].

Pawitan [46] further proposed simplifying the LRT by treating the residual variance as a common nuisance parameter across both the null and alternative models. Under this assumption, the shared variance component cancels out in the log-LR, allowing for analytical simplification. This formulation enhances numerical stability and directs inferential focus toward structural model parameters. It is especially advantageous when residuals are approximately normally distributed and when the equal-variance assumption across models is justified.

However, in the present study, we adopted a different approach in both the treatment of residual mean and variance. First, the residual mean was explicitly included as a parameter in the likelihood function. This decision was motivated by the estimation procedure, which relied not on analytical derivation (e.g., via differentiation) but on iterative numerical optimization. Under such estimation frameworks, residuals are not guaranteed to have a mean of zero. Assuming so could lead to bias in the likelihood formulation. Therefore, the residual mean was estimated jointly with other parameters to more accurately reflect the empirical error structure.

Second, instead of assuming a common residual variance across models, we estimated the residual variances under the null and alternative hypotheses separately. This modeling choice was based on the premise that allowing heterogeneous variances provides a more realistic representation of model-specific dispersion patterns, particularly in the presence of potential heteroscedasticity. By treating both variances as estimable parameters, we aimed to avoid oversimplification and to ensure the robustness and flexibility of the LRT in capturing the true data-generating process.

3.4. Evaluation

3.4.1. Meaning of the Parameters

As the value of parameter ${\beta }_1$ increases, the slope of the curve becomes steeper, particularly in its central region. This change reflects a concentration or dispersion effect, resulting in a curve that appears fatter or thinner in the middle. In contrast, increases in ${\beta }_2$ shift the entire curve horizontally along the x-axis, indicating a rightward or leftward displacement without altering the overall shape. Similarly, changes in ${\beta }_3$ shift the curve vertically along the y-axis, reflecting upward or downward movement while preserving the curve’s shape (Figure 3).

Biologically, ${\beta }_1$ can be interpreted as representing the richness or diversity of species within the survey area. Parameters ${\beta }_2$ and ${\beta }_3$ indicate latitudinal shifts and the vertical extent of migration, respectively. An increase in ${\beta }_2$, particularly for WTS plant species, may suggest an expansion of species diversity and richness northward, implying possible range shifts in response to environmental gradients across the surveyed region.

3.4.2. Direction of Expansion

If there is a significant difference in −2 log λ at significance level α, it can be seen the direction of the shift or expansion by the difference in values of parameters, Δ${\beta }_1$, Δ${\beta }_2$ and Δ${\beta }_3$ between two datasets in the second tier. When all the differences of Δ${\beta }_1$, Δ${\beta }_2$ and Δ${\beta }_3$ become larger, it expands (shifts to the north); when smaller, it contracts (shifts to the south). The parameter with the largest differences among Δ${\beta }_1$, Δ${\beta }_2$ and Δ${\beta }_3$ has the greatest influence.

When all the differences in parameters Δ${\beta }_1$, Δ${\beta }_2,$ and Δ${\beta }_3$ are not the same direction (positive or negative), we proposed the intuitive criteria for determining direction as the sum of the standardized marginals (SoSM) for the values of parameters between two models as follow in Equation (14). If the value of SoSM is positive, it means that it shifted to the north, and if negative, it shifted to the south.

$$\mathrm{SoSM}={\displaystyle \frac{∆{\beta }_1}{\frac{{\beta }_{1A}+{\beta }_{1B}}{2}}}+{\displaystyle \frac{∆{\beta }_2}{\frac{{\beta }_{2A}+{\beta }_{2B}}{2}}}+{\displaystyle \frac{∆{\beta }_3}{\frac{{\beta }_{3A}+{\beta }_{3B}}{2}}}$$

(14)

The methodology described in Section 3 can be summarized as follows, with the statistical significance of each step outlined in

Table 2. Summary of the proposed methodology and its significance across four analytical phases.

Step	Description	Method	Application in This Study	Its Significance
1. Modeling	Develop	island	• Based on MacArthur’s theory,	1. Theoretical foundation:
	parameterized	biogeography	log-transformed species richness and distance	Models grounded in island biogeography
	model	Theory	yield a linear relationship	2. Transformability:
			• A baseline model is constructed and refined	Adaptable to research objectives
			to include parameters for northward expansion	3. Model flexibility:
			• Allows nonlinear forms with no constraints	Allows nonlinear or multivariable structures
2. Fitting	Estimate	Gauss-Newton,	• Estimates parameters of the nonlinear form	1. Residuals of nonlinear models typically deviate from
	model	nls() in R	• Evaluates normality of residuals using Kernel	normality, but island-based model shows near-normal
	parameters		Density Estimation (KDE)	residuals.
				2. Islands are suitable for evaluating the northward
				expansion of WTS plant species
3. Testing	Significance	Likelihood	• Conducts LRT	1. LRT is appropriate for nonlinear model.
	test	Ratio Test	• Applies chi-square test on the −2 log-LR	The F-test, based on the ratio of variances, is not suitable
		(LRT),		2. Degrees of freedom challenges in chi-square distribution
		−2 log-LR,		are addressed flexibly
		Chi-square test		3. Statistical significance of differences is confirmed
4. Evaluating	Determine	Intuitive	• Assess directional shift	1. Provides a simple indicator to detect directional changes
	direction of	indicator	in response to significant model differences	2. Directional logic is embedded in the initial model,
	Shifts, if any	(proposed)		supporting interpretation.

Note: Stepwise summary of (1) Modeling, (2) Fitting, (3) Testing, and (4) Evaluating, showing the methods used (Gauss–Newton estimation; likelihood-ratio [deviance] tests; AIC/BIC), their application to WTS plant richness across 1253 South Korean islands, and the rationale for each step.

.

4. Results

4.1. Mathematical Model

Four parameterized mathematical models were established: the Simple Model (SM), the Multivariable Model (MM), the Fixed Model (FM), and the Transformed Model (TM) as follows in Equations (15)–(18), and ${\beta }_{\left(1,2,3\right) A}$, ${\beta }_{\left(1,2,3\right) B}$, ${\beta }_{\left(1,2,3\right) AB}$ (the fourth model additionally includes the parameter ${\beta }_0$) are the parameters to be estimated for the three datasets (Data_A_2013, Data_B_2013, and Data_AB_2013), respectively. Each model is defined to explain the relationship between latitude and the number of WTS plant species, with specific adjustments to parameters to account for changes over time. The models cover the latitude range from 33.2° N to 38.0° N.

$$\begin{array}{lll}\mathrm{SM}:& Y_A=e^{-{\beta }_{1 A}\left(X_A-{\beta }_{2 A}\right)}+{\beta }_{3 A}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{A}\text{\_}2013\\ & Y_B=e^{{-\beta }_{1 B}\left(X_B-{\beta }_{2 B}\right)}+{\beta }_{3 B}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{B}\text{\_}2013\\ & Y_{AB}=e^{{-\beta }_{1 AB}\left(X_{AB}-{\beta }_{2 AB}\right)}+{\beta }_{3 AB}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{AB}\text{\_}2013\end{array}$$

(15)

$$\begin{array}{lll}\mathrm{MM}:& Y_A=e^{-{\beta }_{1 A}\left(X_A-{\beta }_{2 A}\right)}+{\beta }_{3 A}+{\beta }_{area A}\log A& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{A}\text{\_}2013\\ & Y_B=e^{{-\beta }_{1 B}\left(X_B-{\beta }_{2 B}\right)}+{\beta }_{3 B }+{\beta }_{area B}\log A& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{B}\text{\_}2013\\ & Y_{AB}=e^{{-\beta }_{1 AB}\left(X_{AB}-{\beta }_{2 AB}\right)}+{\beta }_{3 AB}+{\beta }_{area AB}\log A& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{AB}\text{\_}2013\end{array}$$

(16)

$$\begin{array}{lll}\mathrm{F}\mathrm{M}:& Y_A=e^{-{\beta }_1\left(X_A-{\beta }_{2 A}\right)}+{\beta }_{3 A}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{A}\text{\_}2013\\ & Y_B=e^{{-\beta }_1\left(X_B-{\beta }_{2 B}\right)}+{\beta }_{3 B}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{B}\text{\_}2013\\ & Y_{AB}=e^{{-\beta }_{1 }\left(X_{AB}-{\beta }_{2 AB}\right)}+{\beta }_{3 AB}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{AB}\text{\_}2013\end{array}$$

(17)

$$\begin{array}{lll}\mathrm{T}\mathrm{M}:& Y_A={\beta }_0{ e}^{-{\beta }_{1 A}\left(X_A-{\beta }_{2 A}\right)}+{\beta }_{3 A}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{A}\text{\_}2013\\ & Y_B={\beta }_0{ e}^{{-\beta }_{1 B}\left(X_B-{\beta }_{2 B}\right)}+{\beta }_{3 B}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{B}\text{\_}2013\\ & Y_{AB}={\beta }_0{ e}^{{-\beta }_{1 AB}\left(X_{AB}-{\beta }_{2 AB}\right)}+{\beta }_{3 AB}& \mathrm{for} \mathrm{Data}\text{\_}\mathrm{AB}\text{\_}2013\end{array}$$

(18)

4.2. Parameter Estimation

The parameters of the models were estimated using the Gauss–Newton method or alternatively using the nls() function in R. The results were as follows in Equations (19)–(22) refer to

Table 3. The results of computation on the estimates of the parameters of SM using nls(), chi-square statistic and p-value for the residuals through −2 log-LR test in R.

Dataset	Parameter	Estimate	Std. Error	t Value	Pr (>|t|)
A_2013	${\beta }_1$	1.185348	0.1406	8.428	<2 × 10−16	***
	${\beta }_2$	36.252680	0.2661	136.261	<2 × 10−16	***
	${\beta }_3$	1.758833	0.5959	2.951	0.00325	**
B_2013	${\beta }_1$	0.845830	0.1507	5.612	2.73 × 10−8	***
	${\beta }_2$	36.544797	0.5583	65.452	<2 × 10−16	***
	${\beta }_3$	1.798106	0.9074	1.982	0.0479	*
AB_2013	${\beta }_1$	0.840041	0.0978	8.596	<2 × 10−16	***
	${\beta }_2$	36.798868	0.3690	99.729	<2 × 10−16	***
	${\beta }_3$	1.431216	0.6020	2.377	0.0175	*
−2 log LR	chi-square statistic	52.35363
	degree of freedom	5
	p-value	0.00000				***

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05.

,

Table 4. The results of computation on the estimates of the parameters of MM using nls(), chi-square statistic and p-value for the residuals through −2 log-LR test in R.

Dataset	Parameter	Estimate	Std. Error	t Value	Pr (>|t|)
A_2013	${\beta }_1$	1.172858	0.12952	9.055	<2 × 10−16	***
	${\beta }_2$	36.275370	0.25013	145.028	<2 × 10−16	***
	${\beta }_3$	1.401977	0.55445	2.529	0.0116	*
	${\beta }_{area}$	0.000993	0.00008	12.480	<2 × 10−16	***
B_2013	${\beta }_1$	0.852618	0.14701	5.800	9.46 × 10−9	***
	${\beta }_2$	36.526810	0.53675	68.052	<2 × 10−16	***
	${\beta }_3$	1.661198	0.88093	1.886	0.597
	${\beta }_{area}$	0.000579	0.00010	5.949	3.99 × 10−9	***
AB_2013	${\beta }_1$	0.835776	0.09241	9.045	<2 × 10−16	***
	${\beta }_2$	36.822240	0.35343	104.186	<2 × 10−16	***
	${\beta }_3$	1.135764	0.57676	1.969	0.0491	*
	${\beta }_{area}$	0.000836	0.00006	13.343	<2 × 10−16	***
−2 log LR	chi-square statistic	61.27253
	degree of freedom	6
	p-value	0.00000				***

Signif. codes: 0 ‘***’ 0.001 ‘*’ 0.05.

,

Table 5. The results of computation on the estimates of the parameters of FM using nls(), chi-square statistic and p-value for the residuals through −2 log-LR test in R.

Dataset	Parameter	Estimate	Std. Error	t Value	Pr (>|t|)
A_2013	${\beta }_2$	36.995720	0.0416	889.4	<2 × 10−16	***
B_2013	${\beta }_2$	36.633600	0.0444	825.6	<2 × 10−16	***
AB_2013	${\beta }_1$	0.840041	0.0978	8.596	<2 × 10−16	***
	${\beta }_2$	36.798868	0.3690	99.729	<2 × 10−16	***
	${\beta }_3$	1.431216	0.6020	2.377	0.0175	*
−2 log LR	chi-square statistic	39.72759
	degree of freedom	1
	p-value	0.00000				***

Signif. codes: 0 ‘***’ 0.001 ‘*’ 0.05.

and

Table 6. The results of computation on the estimates of the parameters of TM using nls(), chi-square statistic and p-value for the residuals through −2 log-LR test in R.

Dataset	Parameter	Estimate	Std. Error	t Value	Pr (>|t|)
A_2013	${\beta }_0$	12.4652	1.4306	8.713	<2 × 10−16	***
	${\beta }_1$	1.8252	0.5009	3.644	0.000285	***
	${\beta }_2$	34.0050	0.1253	271.390	<2 × 10−16	***
	${\beta }_3$	2.9779	0.4143	7.189	1.43 × 10−12	***
B_2013	${\beta }_0$	13.6278	3.8415	3.548	0.000411	***
	${\beta }_1$	0.4444	0.2370	1.875	0.061146	.
	${\beta }_2$	32.9164	0.5576	59.028	<2 × 10−16	***
	${\beta }_3$	3.1046	0.5955	5.214	2.34 × 10−7	***
AB_2013	${\beta }_0$	12.1126	1.1688	10.364	<2 × 10−16	***
	${\beta }_1$	0.8372	0.1777	4.712	2.66 × 10−6	***
	${\beta }_2$	33.5619	0.1340	250.402	<2 × 10−16	***
	${\beta }_3$	3.0300	0.3434	8.824	<2 × 10−16	***
−2 log LR	chi-square statistic	60.78024
	degree of freedom	6
	p-value	0.00000				***

Signif. codes: 0 ‘***’ 0.001 ‘.’ 0.1.

. Note: Initial values were carefully chosen to ensure convergence during the estimation process. For the SM, the initial values of ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ were set to 1, 40, and 2, respectively. For the MM, the initial values of ${\beta }_1$, ${\beta }_2$, ${\beta }_3$ and ${\beta }_{area}$ were set to 1, 40, 2 and 1, respectively.

To determine appropriate initial values, we considered the roles of each parameter. Since ${\beta }_1$ represents the slope, it was assumed not to be excessively steep; thus, values around 1 were used as a reference. ${\beta }_2$ was assumed to approximate the mean number of WTS plant species and was initialized accordingly. As ${\beta }_3$ appeared to be relatively insensitive to initial values, an arbitrary starting value could be used without noticeably affecting convergence. To evaluate parameter sensitivity, we systematically varied the initial values as follows: ${\beta }_1$ from −9 to 11, ${\beta }_2$ from 30 to 50, and ${\beta }_3$ from −8 to 12, each in increments of 0.1. Although some combinations resulted in divergence, all converging cases appeared to yield consistent parameter estimates.

The estimated parameters for each dataset are as follows: For the SM, ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ were estimated at (1.185, 36.252, 1.759) for Data_A_2013, (0.846, 36.545, 1.798) for Data_B_2013, and (0.840, 36.799, 1.431) for Data_AB_2013 in Equation (19) (Table 3). For the MM, the corresponding estimates including the area term were (1.173, 36.275, 1.402, 0.001), (0.853, 36.527, 1.661, 0.001), and (0.836, 36.822, 1.136, 0.001), respectively, in Equation (20) (Table 4). For the FM, the ${\beta }_1$ and ${\beta }_3$ parameters remained constant, while ${\beta }_2$ was estimated as 36.996 for Data_A_2013 and 36.634 for Data_B_2013, with Data_AB_2013 yielding 36.799 in Equation (21) (Table 5). For the TM, the corresponding estimates including the area term were (12.465, 1.825, 34.005, 2.978), (13.628, 0.444, 32.916, 3.105), and (12.113, 0.837, 33.562, 3.030), respectively, in Equation (22) (Table 6).

$$\begin{array}{lll}\mathrm{SM}:& {\beta }_{(1,2,3) A}& =(1.185, 36.252, 1.759)\\ & {\beta }_{\left(1,2,3\right) B }& =(0.846, 36.545, 1.798)\\ & {\beta }_{(1,2,3) AB}& =(0.840, 36.799, 1.431)\end{array}$$

(19)

$$\begin{array}{lll}\mathrm{MM}:& {\beta }_{(1,2,3,area) A}& =(1.173, 36.275, 1.402, 0.001)\\ & {\beta }_{(1,2,3,area) B}& =(0.853, 36.527, 1.661, 0.001)\\ & {\beta }_{(1,2,3,area) AB}& =(0.836, 36.822, 1.136, 0.001)\end{array}$$

(20)

$$\begin{array}{lll}\mathrm{F}\mathrm{M}:& {\beta }_{(1,2,3) A}& =(0.840, 36.996, 1.431)\\ & {\beta }_{\left(1,2,3\right) B }& =(0.840, 36.634, 1.431)\\ & {\beta }_{(1,2,3) AB}& =(0.840, 36.799, 1.431)\end{array}$$

(21)

$$\begin{array}{lll}\mathrm{T}\mathrm{M}:& {\beta }_{(0,1,2,3) A}& =(12.465, 1.825, 34.005, 2.978)\\ & {\beta }_{\left(0,1,2,3\right) B }& =(13.628, 0.444, 32.916, 3.105)\\ & {\beta }_{(0,1,2,3) AB}& =(12.113, 0.837, 33.562, 3.030)\end{array}$$

(22)

After applying the parameter values to the parameterized model, calculating and plotting, the scatter graph with fitting curve for each three datasets is shown below (Figure 4). The upper red curve is for Data_A_2013, the lower blue is for Data_B_2013, and the middle green is for Data_AB_2013.

The fitted curves appear visually similar across models. However, statistical comparisons based on the method described in the main text revealed significant differences in model performance, highlighting the importance of applying formal model evaluation beyond visual inspection.

4.3. Significance Test

The normality of residuals from the four fitted models was evaluated using Kernel Density Estimation (KDE), as shown in Figure 5. All four models—SM, MM, FM, and TM—produced residuals that approximately follow a normal distribution, with slight differences in skewness and kurtosis. Notably, the MM, which includes island area as a predictor, exhibited the most symmetric distribution, suggesting improved residual normality. This is plausible given the strong, well-established relationship between island area and species richness in island biogeography. In contrast, SM and FM show minor deviations, with slightly right- and left-skewed residuals, respectively. These results imply that incorporating island area helps stabilize model performance. Nonetheless, residual asymmetry in some models may reflect additional unaccounted factors, such as anthropogenic influence on species diversity in small islands. Future improvements could include the addition of explanatory variables like human disturbance to further enhance model fit.

In our analysis, the kernel density estimate (KDE) of the residuals for SM and FM displayed a unimodal, symmetric shape that closely resembles a Gaussian distribution. However, the Shapiro–Wilk test for normality yielded a p-value close to zero, suggesting a statistically significant deviation from normality for all four models. This apparent contradiction highlights a well-known limitation of the Shapiro–Wilk test in large samples: its excessive sensitivity to minor deviations from normality. Previous studies have shown that the Shapiro–Wilk test tends to reject the null hypothesis of normality even when the underlying distribution is nearly normal, particularly when the sample size exceeds 100–200 observations [47,48]. In such cases, visual and non-parametric approaches such as KDE and Q-Q plots can offer a more practical and informative assessment of the distributional characteristics of residuals. When the KDE of the residuals demonstrates a bell-shaped curve consistent with a normal distribution, the normality assumption may be considered reasonable for inferential purposes, despite a low p-value from formal tests. This consideration is especially relevant in nonlinear regression models, where inferential procedures such as the LRT rely more on the asymptotic properties of estimators than on strict adherence to normality [44,49]. Thus, in the presence of large samples and visually normal residuals, it may be more appropriate to interpret model adequacy based on distributional shape rather than hypothesis test statistics alone.

Based on this consideration, the LRT was conducted despite the significant result of the Shapiro–Wilk test, as the residuals exhibited a distributional shape closely resembling normality when assessed using KDE. This approach is justified by the asymptotic nature of the LRT, which relies more on distributional convergence than on strict adherence to normality assumptions. For the SM, the LR statistic (−2 log λ) was calculated as 52.4 with 5 degrees of freedom, resulting in a p-value close to 0. Thus, we reject the null hypothesis and conclude that a statistically significant change has occurred (Table 3). For the MM, the LR statistic was 61.3 with 6 degrees of freedom, and again the p-value was approximately 0. Therefore, we reject the null hypothesis, indicating significant changes in parameters (Table 4). For the FM, the LR statistic was 39.7 with 1 degree of freedom, also yielding a p-value close to 0. Consequently, we reject the null hypothesis and recognize a statistically significant shift in the latitude parameter (Table 5). For the TM, the LR statistic was 60.78 with 6 degree of freedom, also yielding a p-value close to 0. We reject the null hypothesis and recognize a statistically significant shift in the latitude parameter (Table 6).

As shown in

Table 7. Model comparison results for predictors A, B, and AB across four modeling approaches (SM, MM, FM, and TM).

Model	${\mathit{R}}^2$			$\mathbf{Adjusted} {\mathit{R}}^2$			AIC			BIC
	A	B	AB	A	B	AB	A	B	AB	A	B	AB
SM	0.212	0.130	0.153	0.210	0.128	0.152	5598	5319	10,961	5617	5338	10,983
MM	0.446	0.285	0.356	0.444	0.283	0.355	5298	5158	10,500	5322	5181	10,527
FM	0.212	0.130	0.153	0.210	0.128	0.152	5598	5319	10,961	5617	5338	10,983
TM	0.241	0.133	0.166	0.239	0.130	0.165	5567	5318	10,936	5591	5342	10,963

Note: The table presents the comparative performance of four models (SM, MM, FM, and TM) based on different predictor sets (A, B, and AB). Performance metrics include the coefficient of determination ($R^2$), adjusted coefficient of determination (adjusted $R^2$), Akaike Information Criterion (AIC), and Bayesian Information Criterion (BIC). Higher $R^2$ and adjusted $R^2$ values indicate better explanatory power, while lower AIC and BIC values indicate better model fit after accounting for model complexity.

, the reported statistics are presented for relative comparison among the models rather than for assessing absolute validity. In nonlinear frameworks, both $R^2$ and adjusted $R^2$ should be interpreted as relative indicators, while AIC and BIC provide measures of model fit penalized for complexity. Across the four models, the values are generally comparable; however, the MM shows consistently higher $R^2$ and adjusted $R^2$ values, along with lower AIC and BIC scores, relative to the other models. It should be noted that the overall $R^2$ and adjusted $R^2$ values remain relatively low, reflecting the complexity of ecological processes and the limited proportion of variance explained by the current set of predictors. Nevertheless, such values are generally considered acceptable within ecological and biological studies, where inherent variability and complex interactions often constrain explanatory power [50,51]. Despite this limitation, the consistent advantage of the MM suggests that the inclusion of an additional predictor helps to mitigate multicollinearity and improve relative model performance.

4.4. Evaluation

Here we determine the direction of the range shift by computing a simple, model-based indicator from the fitted pre–post latitude–richness curves (positive = poleward), and use it to summarize and interpret the observed change.

In the SM, the estimate for ${\beta }_1$ increased by 0.340, while ${\beta }_2$ decreased by 0.292, and ${\beta }_3$ decreased by 0.039 between the two periods. Because the directions of change are not consistent, we calculated the Sum of Standardized Marginals (SoSM), which resulted in 0.304, a positive value in Equation (23). This indicates a northward expansion of the WTS plants. In the MM, the estimate for ${\beta }_1$ increased by 0.316, ${\beta }_2$ decreased by 0.007, and ${\beta }_3$ decreased by 0.169. The SoSM value was 0.140, also positive in Equation (23), suggesting a northward shift. In the FM, only ${\beta }_2$ changed significantly, increasing by 0.362. The calculated SoSM value was 0.010, which is positive in Equation (23), supporting the conclusion of a northward expansion. In all four models, the results indicate that the WTS plant zones have expanded northward since 2013, with very highly statistical significance (p-value < 0.001). The calculated SoSM value was 1.208, which is positive in Equation (23), supporting the conclusion of a northward expansion. In all four models, the results indicate that the WTS plant zones have expanded northward since 2013, with very highly statistical significance (p-value < 0.001).

$$\begin{array}{l}{\mathrm{S}\mathrm{o}\mathrm{S}\mathrm{M}}_{SM}={\displaystyle \frac{∆{\beta }_1}{\frac{{\beta }_{1A}+{\beta }_{1B}}{2}}}+{\displaystyle \frac{∆{\beta }_2}{\frac{{\beta }_{2A}+{\beta }_{2B}}{2}}}+{\displaystyle \frac{∆{\beta }_3}{\frac{{\beta }_{3A}+{\beta }_{3B}}{2}}}={\displaystyle \frac{1.185-0.846}{\frac{1.185+0.846}{2}}}+{\displaystyle \frac{36.253-36.545}{\frac{36.253+36.545}{2}}}+{\displaystyle \frac{1.759-1.798}{\frac{1.759+1.798}{2}}}=0.304\\ {\mathrm{S}\mathrm{o}\mathrm{S}\mathrm{M}}_{MM}=0.140, {\mathrm{S}\mathrm{o}\mathrm{S}\mathrm{M}}_{FM}=0.010, {\mathrm{S}\mathrm{o}\mathrm{S}\mathrm{M}}_{TM}=1.208\end{array}$$

(23)

5. Discussion

Our results suggest that by the time northward shifts become visible in spatial analyses, the underlying ecological changes have already begun and progressed considerably. Relying solely on visual or field-based detection therefore risks underestimating the scale and timing of climate-induced range dynamics. The methodology developed here addresses this limitation by detecting incipient distributional responses before overt shifts appear, effectively functioning as an early-warning system. Applied to warm-temperate and subtropical plant species on Korean islands, the framework revealed clear latitudinal patterns and allowed species to be classified into three functional groups. Importantly, these insights were derived from existing occurrence records rather than long-term monitoring, underscoring the method’s practicality and potential for use in data-limited contexts. By identifying early signals of range shifts, the approach provides timely insight into biodiversity monitoring and conservation under climate change.

5.1. Evaluation of the Proposed Methodology

The proposed four-step framework appeared coherent and statistically consistent, indicating that differences among models can be evaluated even under nonlinear conditions. This is a useful aspect, as ecological and biogeographical processes are rarely linear, and the ability to compare model performance in such settings broadens the applicability of the approach beyond conventional linear analyses.

Among the four models tested, the MM performed best. Its R² and adjusted R² values were more than twice those of the other models, while both AIC and BIC values were slightly lower (Table 7). These results suggest that the MM provided a comparatively better representation of the observed distributional dynamics. At the same time, they indicate that the framework could be further improved by incorporating additional environmental variables and explicitly addressing issues of collinearity. Such adjustments may increase explanatory power and contribute to the development of more refined predictive models.

Therefore, while the present analysis relies on relatively simple nonlinear formulations, this simplicity also provides a basis for further refinement. Future research should extend the framework by integrating a broader suite of ecological and climatic predictors, thereby improving predictive performance and enabling stronger inference on species responses under climate change.

5.2. Functional Range Shifter Stage Analysis for WTS Plant

Fitting the number of WTS plant species to latitude using the method proposed in this study (Figure 4) reveals several important observations.

5.2.1. Similar Patterns Across All Models

All four statistical models showed very similar results across the different datasets, even though the study area spans a wide latitudinal range from 33.2° N to 38.00° N, or about 500 km. While the fitted curves based on the 118 WTS plant species (Appendix A) may appear slightly different when viewed graphically, statistical tests showed that these differences are in fact significant, even at a very low significance level. This finding highlights an important point: what appears visually similar in graphs may still be statistically different. When analyzing large-scale patterns like this, statistical analysis, such as that proposed in this study, is necessary for reliable interpretation.

5.2.2. A Long Tail in the North: The Role of 35° N

From the northern boundary down to about 35° N (roughly 300 km), the number of WTS plant species remains relatively constant. However, south of 35° N, clear differences between the models begin to appear. This suggests that many species have already established stable populations up to 35° N, increasing species richness in that zone. In contrast, species found north of 35° N seem to be in an early stage of expansion, where they are still spreading rather than becoming abundant. In this context, the area between 34° N and 35° N may represent a saturation zone, while the area north of 35° N may still be in the process of colonization. These findings suggest that it could be helpful to divide the study area into two parts—north and south of 35° N—for more detailed future analysis.

The value corresponding to 35° N can be derived mathematically as follows. As we move southward from the northern latitudes, the number of WTS plant species increases gradually, exhibiting incremental differences. At a certain point, however, the differences begin to increase more substantially and become visually discernible. Although there is no universally established threshold to define what constitutes a “significant” change, a practical approach is to define this point as the location where the value exceeds 5% of the total range (i.e., the difference between the maximum and minimum values). This approach yields a latitude of approximately 35° N. Mathematically, this can be derived as follows: we identify the value of x at which the change reaches 5% of the total variation, as formalized in Equation (24). Rearranging this relationship to express x explicitly yields the calculation described in Equation (25). Applying this method to the SM dataset “Data_A_2013” gives a resulting latitude of 34.95475° N. The 5% threshold is an empirical criterion that is commonly used in exploratory analyses to signal early signs of structural change [52].

$$0.05={\displaystyle \frac{f\left(x\right)-f_{min}}{f_{max}-f_{min}}} , f_{min}={\beta }_3 , { f}_{max}=f\left(x_{min}\right)$$

(24)

$$x={\beta }_2-{\displaystyle \frac{1}{{\beta }_1}}\mathrm{l}\mathrm{n}[0.05·(f_{max}-{\beta }_3)]$$

(25)

5.2.3. Latitudinal Patterns of the WTS Plant Species Richness Around the 35° N Threshold

As illustrated in Figure 4, the fitted curves diverge around 35° N across different datasets, suggesting potential latitudinal heterogeneity in WTS plant species richness. To assess whether these visual differences are reflected in actual species counts, we conducted separate t-tests comparing the A_2013 and B_2013 datasets for islands located north and south of 35° N (

Table 8. Comparison of WTS plant species richness before and after 2013 north and south of 35° N.

35° N	Period	No of Islands	Mean	SD	Mean Difference (A–B)	t Value	Pr (>|t|)
North	A_2013	312	3.55	3.48	−0.70	−1.245	0.107
	B_2013	229	3.97	4.18
South	A_2013	545	8.61	7.96	1.23	2.888	0.002	**
	B_2013	603	7.35	6.73

Note: Summary of two-sample t-tests comparing the number of WTS plant species recorded on islands before (B_2013) and after (A_2013) 2013, separated by geographic zone (north vs. south of 35° N). The tests assess whether mean richness differs significantly between periods. Asterisks denote levels of significance (** p < 0.01).

). In the northern region, the mean number of species decreased after 2013; however, the difference was not statistically significant at the 0.05 level (p-value = 0.107). In contrast, in the southern region, the mean number of species increased substantially after 2013, and the difference was statistically significant at the 0.01 level (p-value = 0.002). This result is consistent with the signs of the standardized model coefficients (SoSMs): while the coefficient for richness (${\beta }_1$) showed a positive shift, both ${\beta }_2$ and ${\beta }_3$—related to spatial displacement—showed negative shifts.

These findings suggest that, after 2013, WTS plant richness did not change markedly in the north, aside from a slight decrease, but increased significantly in the southern zone. This implies not merely a poleward shift, but rather a localized concentration of richness around the mid-latitudinal band, indicating a regionally intensified diversity rather than uniform latitudinal migration. Although individual WTS species may have exhibited range shifts, the overall distribution of WTS plant species appears to be more strongly influenced by changes in abundance than by directional movement. In particular, the 35° N latitude seems to act as a threshold or barrier, beyond which species richness does not substantially increase, suggesting a latitudinal limit to further expansion.

5.2.4. Key Species That Require Further Observation and the Need for Classification

The changes in WTS plant distribution patterns near 35° N indicate that certain species may require more focused observation in the future. These species show differing responses to climate change and could play key roles in shaping the future structure of temperate forests. Based on these observations, it seems useful to classify WTS plants into three groups according to their current range shift status. The first group, Established Range Shifters, includes species that have already expanded northward and successfully formed stable populations. These species likely have strong dispersal ability, broad ecological tolerance, and rapid responsiveness to changing climatic conditions. The second group, Active Range Expanders, consists of species that are currently moving northward but have not yet become fully established. These may be in a transitional stage, affected by delayed population dynamics or environmental filtering. The third group, Range-Constrained Species, shows little or no evidence of range shift at this time. This may be due to limited dispersal capacity, specialized habitat requirements, or physiological inertia in response to warming (

Table 9. Classification of WTS plant species into three functional range shift groups, with comparisons of occurrence counts and northward abundance ratios between two datasets from 2013.

Group	WTS Plant Species	2013_A		2013_B		Difference
		a. Occurrence	b. Ratio	c. Occurrence	d. Ratio	Occurrence (a–c)	Ratio (b–d)
1. Established	Elaeagnus macrophylla Thunb.	407	0.62	307	0.63	100	−0.02
Range Shifters	Euonymus japonicus Thunb.	370	0.54	382	0.40	−12	0.14
	Pseudosasa japonica Makino ex Nakai	127	0.74	80	0.51	47	0.23
2. Active Range	Machilus thunbergii Siebold & Zucc.	193	0.29	191	0.21	2	0.08
Expanders	Neolitsea sericea Koidz.	147	0.26	78	0.26	69	0.00
	Hedera rhombea Siebold & Zucc. ex Bean	228	0.22	238	0.19	−10	0.03
	Trachelospermum asiaticum Nakai	339	0.22	318	0.25	21	−0.03
	Eurya japonica Thunb.	406	0.18	448	0.14	−42	0.04
	Ardisia japonica (Thunb.) Blume	229	0.16	200	0.16	29	0.00
	Pittosporum tobira (Thunb.) W.T.Aiton	247	0.12	290	0.16	−43	−0.04
	Litsea japonica (Thunb.) Juss.	115	0.12	129	0.09	−14	0.02
	Rhaphiolepis indica var. umbellate	243	0.08	228	0.10	15	−0.02
	Ligustrum japonicum Thunb.	263	0.07	249	0.05	14	0.02
	Ficus oxyphylla Miq. ex Zoll.	141	0.04	75	0.07	66	−0.03
	Eurya emarginata (Thunb.) Makino	166	0.02	169	0.01	−3	0.02
3. Range-Constrained	The remaining WTS plants, not including Groups 1 and 2
Species

Note: Columns a and c indicate the number of plots where each species was recorded (i.e., total occurrences). Columns b and d show the proportion of occurrences north and south of 35° N.

).

This classification not only helps in understanding species-specific responses but also highlights the importance of identifying and monitoring key taxa that may lag or vary in their responses. Continued observation of these groups is necessary to improve future projections of vegetation changes under ongoing climate change.

5.2.5. Relationship Between Island Area and Climate-Driven Distribution Shift of WTS Plants

To evaluate the influence of island area on the northward migration of WTS plant species under climate change, we applied the methodology developed with the SM in this study while sequentially excluding islands below increasing area thresholds, ranging from 0 ha to 7.9 ha in 0.1 ha increments. The results (Figure 6) demonstrate that WTS plant species exhibited statistically significant northward shifts up to an area threshold of approximately 5.9 ha. However, when the analysis was restricted to islands larger than or equal to 6.0 ha, the latitudinal shift was no longer statistically significant (p ≥ 0.05).

These findings indicate that climate-driven range shifts of WTS plant species are predominantly occurring on smaller islands (<6 ha), while larger islands (≥6 ha) show relatively stable species distributions, at least as of the year 2013. Notably, when the model was applied exclusively to islands smaller than 6 ha, p-values approached zero across all thresholds, suggesting strong statistical support for northward migration within this subset.

This pattern suggests that smaller islands may serve as more sensitive indicators of climate-induced biogeographic responses in insular plant communities. The increased detectability of distributional shifts in smaller islands may be attributed to their limited area, greater susceptibility to environmental fluctuations, or higher edge-to-interior ratios, all of which could amplify the impacts of climate change on plant species distributions.

By applying the same analytical framework using the modified MM, we observed statistically significant northward shifts of WTS plant species up to an island size threshold of 11.8 ha (Figure 7). In contrast, the SM detected this signal only up to 5.9 ha, suggesting that the MM—through the inclusion of island area as an explicit covariate—achieves enhanced sensitivity in capturing spatial distributional responses to climate change. This improvement highlights the value of incorporating ecologically relevant landscape metrics into distributional shift models. It also underscores the flexibility and scalability of the proposed methodology: the modeling framework can be iteratively refined by integrating spatial or habitat-specific variables, thereby improving its capacity to detect early biogeographic signals under climate pressure. These results collectively point to a robust and adaptable analytical tool for climate impact detection in fragmented island ecosystems.

5.3. Global Applicability

This study highlights that island vegetation is a more sensitive system for detecting ecological responses to climate change compared to continental environments [25,26]. By examining potential northward shifts, we obtained insights relevant to forest conservation and management, including the early detection of distributional changes and the classification of functional response types. Such an approach is not limited to the Korean context but is applicable to other nations with extensive island systems. Because the methodology developed here is grounded in principles of island biogeography, it can be transferred to island ecosystems worldwide. Thus, the framework presented in this study has international relevance as a transferable tool for monitoring and conserving island biodiversity under climate change.

6. Conclusions

This study developed and applied a statistical approach to evaluate whether WTS plant species in South Korea are shifting their ranges northward under climate change. Analysis of distribution records from 1253 islands revealed significant northward shifts (χ² = 52.4–61.3; p < 0.001), providing quantitative evidence that WTS plants are responding to recent climate warming.

Our findings suggest that nonlinear range-shift models, when assessed with formal statistical tests, can provide a more structured way to evaluate distributional changes compared with descriptive assessments. In this study, model performance indicators (AIC, BIC) were generally supportive of the proposed approach, indicating that such methods can complement existing tools rather than replace them.

Importantly, the results illustrate how climate change is already influencing WTS plants on South Korean islands. Beyond documenting ecological responses, these findings may inform broader discussions on adaptive management of island forests, where shifts in species composition and ecosystem function require forward-looking strategies.

Nevertheless, limitations remain. The choice of initial model structure may restrict the ecological complexity that can be represented, and future work should refine model forms and incorporate additional environmental variables—such as temperature, precipitation, and elevation—with attention to collinearity. These steps would help to improve both the precision and ecological relevance of predictions.

Despite these limitations, the study provides evidence of climate-driven range expansion in WTS species and highlights the usefulness of statistical evaluation in detecting such changes. Continued monitoring and model development will be important for advancing ecological understanding and supporting effective forest management under climate change.