Applicability of Common Algorithms in Species–Area Relationship Model Fitting

Tsung-Jen Shen,Zhidong Liu,Xiaoke Liu

doi:10.3390/d14030212

Abstract

The species–area relationship (SAR) describes a law of species richness changes as the sampling area varies. SAR has been studied for more than 100 years and is of great significance in the fields of biogeography, population ecology, and conservation biology. Accordingly, there are many algorithms available for fitting the SARs, but their applicability is not numerically evaluated yet. Here, we have selected three widely used algorithms, and discuss three aspects of their applicability: the number of iterations, the time consumption, and the sensitivity to the initial parameter setting. Our results showed that, the Gauss–Newton method and the Levenberg–Marquardt method require relatively few iterative steps and take less time. In contrast, the Nelder–Mead method requires relatively more iteration steps and consumes the most time. Regarding the sensitivity of the algorithm to the initial parameters, the Gauss–Newton and the Nelder–Mead methods are more sensitive to the choice of initial parameters, while the Levenberg–Marquardt method is relatively insensitive to the choice of initial parameters. Considering that the Gauss–Newton method and the Levenberg–Marquardt method can only be used to fit smooth SAR models, we concluded that the Levenberg–Marquardt model is the best choice to fit the smooth SARs, while the Nelder–Mead method is the best choice to fit the non-smooth SARs.

Highlights

The species–area relationship (SAR) is one of the most general ecological patterns, depicting the changing process of species richness as the sampling area increases
Note that the first SAR model put forward by Arrhenius was depicted by a power function [8], but many mathematical variants of SAR have been continuously proposed to account for diverse shapes of SAR in practical applications [9,10]
There are many nonlinear optimization algorithms that can fit non-linear functions, this study focuses primarily on three effective classic methods: The Gauss–Newton method, the Levenberg–Marquardt method, and the Nelder–Mead method

Summary

Introduction

The species–area relationship (SAR) is one of the most general ecological patterns, depicting the changing process of species richness as the sampling area increases. Increasing attention is given to exploration of ways in which to use the SAR for conservation purposes, including selection of biodiversity hotspots, determination of the optimal size and shape of natural areas, and the prediction of species extinction [3,4]. Note that the first SAR model put forward by Arrhenius was depicted by a power function [8], but many mathematical variants of SAR have been continuously proposed to account for diverse shapes of SAR in practical applications [9,10]. The mathematical and biological mechanisms behind the SAR have been preliminarily studied, the applicability of the SAR model is controversial. There is no clear biological foundation to give preference to these particular models [13] and the best-fitting model for a particular species–area curve can only be determined empirically [7]

Methods

Results

Discussion

Conclusion