数学模型在生态与进化中的应用：时间和空间概论

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"《Wiley.An.Introduction.To.Mathematical.Models.In.Ecology.And.Evolution.Time.And.Space.2nd.Edition.May.2009.eBook-ELOHiM》是一本面向生态学和进化论学生的数学模型入门教材，旨在帮助学生理解生态学和进化论中的基本概念，同时提升他们在应用理论时所需的数学能力。本书由Michael Gillman撰写，适用于本科和研究生，介绍了一系列关键的数学建模方法和基础概念，涵盖了从种群动态到物种进化的广泛主题，并强调了建模在实际问题中的应用，如可持续收获和生物控制。" 在生态学和进化论的研究中，数学模型扮演着至关重要的角色，因为许多核心概念都具有数学性质，这使得初学者往往感到难以理解。这本书的目的就是解决这一难题，它假设读者具备最少的数学和统计学知识，然后逐步引导他们掌握各种前沿的建模技术。书中的内容不仅包括经典的生态学模型，如种群增长模型、竞争排斥原理等，还涉及了最新的进化动力学模型，如基于基因频率的进化模型和空间扩散模型。在生态学部分，书中可能会讲解种群动态模型，如逻辑斯蒂增长模型和Ricker模型，这些模型用于描述种群数量随时间的变化。还会涉及食物网模型，以分析物种间的相互作用和能量流动。在进化论方面，书里可能涵盖哈密顿规则、适应度景观和遗传漂变等概念，以及基于选择和突变的种群遗传学模型。此外，书中特别强调了数学模型在解决实际问题中的应用。例如，通过数学模型可以研究如何实现可持续的渔业管理，预测物种入侵的影响，或者评估生物防治策略的有效性。这些应用实例有助于读者将理论知识与现实世界的问题相结合，提高他们分析和解决问题的能力。在统计方法上，本书可能介绍参数估计、假设检验和回归分析等基础统计技术，以及如何利用这些技术来分析生态数据。对于空间和时间尺度上的生态过程，可能涉及到时空统计模型和地理信息系统（GIS）的应用。《An Introduction to Mathematical Models in Ecology and Evolution: Time and Space》是生态学和进化论领域的宝贵教材，它为那些希望深入理解这两个领域并提升数学建模技能的学生提供了坚实的基础。通过阅读这本书，读者不仅可以掌握一系列重要的数学模型，还能学会如何将这些模型应用于解决生态学和进化论的实际挑战。

INTRODUCTION 7

that some of the problems may lie in the large amount of variation in the

data around the regression. We can consider this by examining the residuals;

that is, the size of the difference of the observed values from the predicted

value on the regression line (Fig. 1.5)

The pattern of residuals of below-ground NPP with respect to the rainfall

shows a large scatter of values (positive and negative) between approximately

200 and 700 mm of rainfall, with small residuals for the higher values there-

after. This is a common distribution of residuals and a source of concern as

regression analyses require that the residuals follow the same distribution

pattern across the full range of independent values. A consequence of this

deviant pattern is that the larger values of rainfall have a large effect on the

regression. Indeed, it is possible to have a completely non-signiﬁ cant scatter

of points at low values and for one point with a high value (for both variables)

to generate a signiﬁ cant positive regression. Generally this means that more

samples are needed in the intermediate range. In this case there is one large

value with a mean temperature above 25°C and rainfall of nearly 1600 mm

(you can see this clearly on the plot of rainfall against temperature, Fig. 1.4).

Not surprisingly this occurs in a tropical region (Thailand). If we remove this

value the regressions are even further away from signiﬁ cance.

Whereas the NPP above or below ground did not show a signiﬁ cant

response to the climate variables, the fraction of below-ground NPP was sig-

niﬁ cant (Fig. 1.6).

This study illustrates the importance of exploring your data. Although

above-ground and below-ground NPP show no signiﬁ cant effects, the fraction

of NPP below ground declines signiﬁ cantly with increasing temperature.

Alternatively, we could also use the fraction of NPP above ground (what

slope would you expect for this regression?). The same effect is seen with

rainfall, which is expected as rainfall and temperature are positively corre-

lated. The relationship of fraction of below-ground NPP with temperature is

given by the equation:

–1000

–500

500

1000

1500

0 200 400 600 800 1000 1200 1400 1600 1800

Rainfall (mm)

Residuals (observed value minus

predicted value) for below-ground NPP

Fig. 1.5 Residuals of below-ground NPP from regression against rainfall.

INTRODUCTION 9

perspective this may be sufﬁ cient to provide a useful model of the effects of

climate change on grassland ecosystems. Conversely, about one-third of the

variation remains unexplained. We will see in the next section how to

improve these models by using more than one explanatory variable.

1.3 Multiple regression

In the previous section we had two possible independent variables which

were correlated. This meant that either could be used and, indeed, it was

only appropriate to use one. However, it is often the case that there are two

or more independent variables that are not correlated and we wish to under-

stand their overall contribution to the regression. This is the domain of mul-

tiple regression. With two independent variables, instead of ﬁ tting a line

through a set of points as in linear regression, we are now ﬁ tting a two-

dimensional plane through a set of points. With three independent variables

it becomes difﬁ cult to visualize the process, but the mathematical principle

of reducing the unexplained variation still applies.

We will now consider the application of multiple regression to an ecosys-

tem analysis. The hypothesis was that methane production from wetlands

was being suppressed by acid rain. As methane is a powerful greenhouse gas

the reduction may have important consequences for global climate change.

An experiment was undertaken involving treatments simulating acid-rain

deposition on a peat bog with the levels of methane ﬂ ux recorded (Gauci

et al. 2002). A further analysis considered the relationship of the extent of

methane ﬂ ux with peat temperature and water-table depth; that is, two

naturally occurring variables. This required use of multiple regression and is

considered here.

The dependent variable is methane ﬂ ux and the independent variables are

peat temperature and peat water table. The data can be plotted as two sepa-

rate graphs (Fig. 1.7) or combined in a three-dimensional graph in which the

two independent axes are at right angles to each other (these may look

impressive but it is often very difﬁ cult to read the values and so they are not

included here).

The plots of the dependent against the independent values separately

show that the percentage of methane ﬂ ux increases with temperature and

with water-table depth. Notice that the r

values of these two are quite low

(30 and 24% of variance explained respectively) although the regressions are

signiﬁ cant (P = 0.0045 and 0.012). There is also the suggestion that there is

a problem with the distribution of the residuals (Fig. 1.8). There are ways of

dealing with these issues without further sampling, such as transformations

of data, which were undertaken by the authors of the paper: we will consider

this later. Here we will focus on the raw data and ignore the odd patterns of

the residuals.

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