ahead is to integrate niche and neutral theories, that is to add more processes in neutral theories and more stochasticity in niche theories (Alonso & McKane, 2004; Chave, 2004). This demonstrates that there is a strong wish for the ecologists to search for a general principle about the mechanisms of biodiversity (Tilman, 1999). Nevertheless such integration only within the community ecology may not solve the existing problems and identify the general principle embraced. Ricklefs (2006) pointed out that to assess the relative roles of local ecological constraint compared to regional and historical unfolding of diversity-environment relationships, we must abandon localized concepts of the community and adopt historical (particularly phylogenetic) and geographic methods to evaluate the evolution of diversity within large regions and its influence on diversity at local scales. Although many hypotheses have been proposed and ecologists have amassed a wealth of detail to explain global patterns in species richness, there is no integrated hypothesis of how the ecosystems work as a whole evolved entity. For example, local determinism explains the biodiversity gradients from a physical environment perspective, and the theories of community consider more about ecological processes or population dynamics. Moreover, it is inappropriate to run a regression of species richness against environmental factors by treating the species richness as a dependent variable and the environmental factors as independent variables. This is because from the aspect of ecosystems, both species richness and physical environments are independent variables and their interactions contribute to the properties of the ecosystems. Therefore species richness itself through interactions among species also contributes to further species richness. In addition, we all agree that biodiversity is the emergent property of ecosystems through interactions of physical environments and organisms after a long evolutionary history. In contrast, there have been few studies on such indicators to reflect the ecosystem's potential of emerging properties, which undoubtedly influences our effort to understand the mechanisms of biodiversity thoroughly. Obviously, it is necessary to establish an integrated theory to study the causes of biodiversity by embracing *system thinking*, and regarding the ecosystem as an entity and

treating biodiversity as the emergent property of the ecosystems.

environmental factors from 27 provinces of China.

**2. Creativity Theory of Ecosystems (CTE)**

The objective of this paper is to study the mechanisms of biodiversity by establishing a new theory from the aspect of ecosystem creativity. Here we develop an integrated theory, which we call Creativity Theory of Ecosystems (CTE), to study the mechanisms of biodiversity with a different perspective compared to more conventional approaches. Basing our theory on system thinking, the CTE establishes a model according to four concepts of creativity, energy, environment diversity and adaptability and the relationships among them. Chiefly by introducing the adaptability as one of the independent variables, the CTE model not only integrates biotic and abiotic factors but also combines spatial and temporal scales needed to predict plant species richness. This new approach is a very general theory and can be applied to any ecosystem because it is premised on system thinking, and is not tied to any specific scale or particular experimental design. A quantitative test of CTE was also conducted with statistical methods according to data obtained for species richness and

An ecosystem is one of the complex systems whose properties are not fully explained by an understanding of its component parts (Gallagher & Appenzeller, 1999). It is an essential approach for an ecologist to view the ecosystem as an evolved entity, by doing so we find that biodiversity patterns actually are the self-creations of the global ecosystems after a long history of interactions among organisms and physical environments in different scales. It is clear that biodiversity is one of the most prominent emergent properties for the ecosystems, and creativity can naturally be employed as an indicator for the potential of yielding emergent features. Thus, the CTE is based on four main concepts:

can limit local adaptation in sinks and restrict the use of alternate niches (Urban, 2006). Based on the above definitions we find that ecosystem creativity is positively correlated with energy (*e*) and environmental diversity (*d*), but negatively correlated with adaptability (*a*) of the ecosystems. We then introduce creativity index (*CI*) as an indicator to reflect the creativity of ecosystems. We can write *CI* as

A Study on Biodiversity Mechanism by the Creativity Theory of Ecosystem 29

variables because the variables of *d* and *a* are spatial and temporal dependent, respectively. Thus to calculate the *CI* value quantitatively we need to specify ecosystems in spatial and temporal scales and find the relationship among variables for certain scale ecosystems. We then test the CTE with the methods of Principal Component Analysis (PCA) and regression

China, with a vast area of 9.6 million km2, is an ideal region to test CTE. Its territorial distance from south to north is 5500km, including tropic, subtropic, temperate warm and temperate cool zones. From China's east to west is 5200km, including a great expanse of land from the Pacific Ocean to Mt. Everest with the terrain rising gradually. Due to the various geographic and climatic conditions, China is one of the regions with the most

We collected the data from 27 provinces (22 provinces and 5 autonomous regions) in China (Table 1 in Appendix 1) Three municipalities directly under the Central Government (Beijing, Tianjing, Shanghai) were excluded because their relative small areas have been strongly influenced by urbanization. The data from the Chongqing municipality was included in Sichuan province. The areas of Hong Kong, Macao, and Taiwan were not

1. Animal species richness. The number of terrestrial vertebrate species (including birds, reptiles, amphibians, and mammals) was used to indicate the animal species richness. Data were taken from Editorial board for series of natural resources in China (1995) and

2. Plant species richness. The number of vascular plant species (including pteridophyte, gymnosperm, and angiosperm) was used to indicate the plant species richness. Data were taken from publications related to the flora and vegetation for every selected

3. Energy (*e*). Energy input was estimated by annual mean temperature. Data were available online from the Scientific Database of the Chinese Academy of Science

4. Environmental diversity (*d*). We used annual precipitation, altitude difference (maximum minus minimum altitudes), and land area to estimate environmental diversity. Water availability is a critical factor to constrain species distribution and altitude difference is the most prominent geographic feature in China. Area is also employed as a variable to influence environmental diversity because the areas were greatly different among provinces. The combination of these three factors can predict the *d* variable in regional scale very well. Data were taken from the publication Editorial board for series of natural resources in China (1995) and Editorial board for the complete series of Chinese agriculture (1998) and online from

5. Reciprocal of adaptability (*a-1*). According to Eq. (2): *a-1 = 1/a = 1/ n/(N-n) = (N–n) / n*, this actually is the changed rate of ecosystem properties during evolutionary time period △t. Because we treated every province as a consolidated entity, and it is impossible to

Editorial board for the complete series of Chinese agriculture (1998).

for ecosystems on a regional scale.

abundant displays of biodiversity in the world (Shi, 1991).

included because we were unable to get relevant data. The variables were collected and determined as follows:

province (see Appendix 1).

(http://www.sdb.ac.cn/).

(http://www.sdb.ac.cn/).

**3. Test of the CTE**

**3.1 Material and methods**

$$\text{CI} \equiv f \text{ ( $e$ , d, a)}\tag{1}$$

Generally speaking, in this equation energy can be easily understood and measured, however environmental diversity and adaptability are problematic. Because there is no single parameter to completely express environmental diversity, it is probably wise to consider it according to spatial scales. Freestone & Inouye (2006) found that the mechanisms driving species coexistence and diversity in serpentine seeps appear scale-dependent. Willis & Whittaker (2002) classified the spatial scales in five categories as local, landscape, regional, continental, and global. It is becoming increasingly apparent that the factor best accounting for patterns of biodiversity seems to be delimited by scale. Consequently, *d* is a variable of scale-dependent in Eq. (1).

According to the definitions, an ecosystem is completely adaptable to its environmental conditions if there are no changes occurring, but the ecosystem displays some nonadaptable quality if there are emergent properties occurring after a certain period of evolutionary time. Thus the more emergent properties that appear within the ecosystem the bigger the changes become and the lower the level of adaptability. We then can calculate the *a* value by the reciprocal of the ecosystem's changed rate with the following equation:

$$a = 1/(\text{N-n})/\text{n} = \text{n/(N-n)}\tag{2}$$

Where, *n* is the number of original properties at time t1, *N* is the number of properties at time t2, △t (△t = t2- t1) is the evolutionary time of an ecosystem. Thus, *N–n* is the number of emergent properties and *(N–n)/n* is the changed rate of ecosystem properties during evolutionary time period △t.

Apparently we measure the adaptability of the ecosystem from historical and evolutionary aspects, because in the biological sense current adaptations are the result of selection that was in progress at some time in the past (Ridley, 2004). We actually judge the adaptability from differences between current properties of the ecosystem and its properties in the past. This then implies that our *a* value is also a variable that is time-scale-dependent. In addition, though the ecological processes of competition and predation (Bush, 2003; Fine et al., 2006; Schmitz, 2006; Straub, 2006) are greatly different within evolutionary time periods, the Bible teaches us to judge a tree by its fruit. Correspondingly, we judge the perfection of an organism by its power to survive and multiply (Egbert Giles Leigb, 1971) or in our hypotheses by the final emergent properties of ecosystems.

If the *a* variable in Eq. (1) is replaced by Eq. (2), the model yields: *CI = f(e, d, a) = f[e, d, n/(N– n)]*, Since *a* is negatively correlated with creativity, the above equation should be:

$$\text{CI} \equiv f[\text{e, d, (N-n)/n}] \tag{3}$$

Hence, CTE treats the ecosystems as a consolidated entity, and biodiversity is its emergent property through interactions among organisms and environments after evolving through a certain spatial and temporal scale. For an ecosystem the higher the energy input the more diversified environments become and the less adaptability. Subsequently, the greater the creativity also means the higher the biodiversity.

However, due to the combinations among the variables of *e, d, and a* could be various forms such as plus, multiplication, power, etc., and the *CI* model (3) is only a functional equation that cannot be calculated directly. This is a reflection of complexity of the ecosystems, i.e., one cannot predict the creativity of every ecosystem with a single combination of these variables because the variables of *d* and *a* are spatial and temporal dependent, respectively. Thus to calculate the *CI* value quantitatively we need to specify ecosystems in spatial and temporal scales and find the relationship among variables for certain scale ecosystems. We then test the CTE with the methods of Principal Component Analysis (PCA) and regression for ecosystems on a regional scale.

## **3. Test of the CTE**

28 Research in Biodiversity – Models and Applications

Generally speaking, in this equation energy can be easily understood and measured, however environmental diversity and adaptability are problematic. Because there is no single parameter to completely express environmental diversity, it is probably wise to consider it according to spatial scales. Freestone & Inouye (2006) found that the mechanisms driving species coexistence and diversity in serpentine seeps appear scale-dependent. Willis & Whittaker (2002) classified the spatial scales in five categories as local, landscape, regional, continental, and global. It is becoming increasingly apparent that the factor best accounting for patterns of biodiversity seems to be delimited by scale. Consequently, *d* is a variable of

According to the definitions, an ecosystem is completely adaptable to its environmental conditions if there are no changes occurring, but the ecosystem displays some nonadaptable quality if there are emergent properties occurring after a certain period of evolutionary time. Thus the more emergent properties that appear within the ecosystem the bigger the changes become and the lower the level of adaptability. We then can calculate the *a* value by the reciprocal of the ecosystem's changed rate with the following equation:

Where, *n* is the number of original properties at time t1, *N* is the number of properties at time t2, △t (△t = t2- t1) is the evolutionary time of an ecosystem. Thus, *N–n* is the number of emergent properties and *(N–n)/n* is the changed rate of ecosystem properties during

Apparently we measure the adaptability of the ecosystem from historical and evolutionary aspects, because in the biological sense current adaptations are the result of selection that was in progress at some time in the past (Ridley, 2004). We actually judge the adaptability from differences between current properties of the ecosystem and its properties in the past. This then implies that our *a* value is also a variable that is time-scale-dependent. In addition, though the ecological processes of competition and predation (Bush, 2003; Fine et al., 2006; Schmitz, 2006; Straub, 2006) are greatly different within evolutionary time periods, the Bible teaches us to judge a tree by its fruit. Correspondingly, we judge the perfection of an organism by its power to survive and multiply (Egbert Giles Leigb, 1971) or in our

If the *a* variable in Eq. (1) is replaced by Eq. (2), the model yields: *CI = f(e, d, a) = f[e, d, n/(N–*

Hence, CTE treats the ecosystems as a consolidated entity, and biodiversity is its emergent property through interactions among organisms and environments after evolving through a certain spatial and temporal scale. For an ecosystem the higher the energy input the more diversified environments become and the less adaptability. Subsequently, the greater the

However, due to the combinations among the variables of *e, d, and a* could be various forms such as plus, multiplication, power, etc., and the *CI* model (3) is only a functional equation that cannot be calculated directly. This is a reflection of complexity of the ecosystems, i.e., one cannot predict the creativity of every ecosystem with a single combination of these

*n)]*, Since *a* is negatively correlated with creativity, the above equation should be:

scale-dependent in Eq. (1).

evolutionary time period △t.

hypotheses by the final emergent properties of ecosystems.

creativity also means the higher the biodiversity.

*CI = f (e, d, a)* (1)

*a = 1/(N–n)/n = n/(N-n)* (2)

*CI = f[e, d, (N–n)/n]* (3)

## **3.1 Material and methods**

China, with a vast area of 9.6 million km2, is an ideal region to test CTE. Its territorial distance from south to north is 5500km, including tropic, subtropic, temperate warm and temperate cool zones. From China's east to west is 5200km, including a great expanse of land from the Pacific Ocean to Mt. Everest with the terrain rising gradually. Due to the various geographic and climatic conditions, China is one of the regions with the most abundant displays of biodiversity in the world (Shi, 1991).

We collected the data from 27 provinces (22 provinces and 5 autonomous regions) in China (Table 1 in Appendix 1) Three municipalities directly under the Central Government (Beijing, Tianjing, Shanghai) were excluded because their relative small areas have been strongly influenced by urbanization. The data from the Chongqing municipality was included in Sichuan province. The areas of Hong Kong, Macao, and Taiwan were not included because we were unable to get relevant data.

The variables were collected and determined as follows:

A Study on Biodiversity Mechanism by the Creativity Theory of Ecosystem 31

with a single parameter of temperature, altitude difference, or land area are not ideal and account for 12.3%,17.6%, and 28.0% of the variation in plant species richness respectively, with less F value and P<0.05. The model with multiple parameters of temperature, precipitation, and altitude difference shows much better than that of models with a single parameter, and account for 59.3% of the variation in plant species richness (F=19.915,

The CTE successfully predicts the plant species richness distribution in provinces of China. This success comes mainly from our new system thinking approach of study. First, we treat every province as a consolidated ecosystem (though their land areas may differ greatly), and that plant species richness is one of its emergent properties through the interactions of biotic and abiotic factors. Conventional approaches using equal and small area quadrat as the basic studying unit actually divide the ecosystems into many small component parts. This approach is probably reasonable for a small scale evaluation within a community or landscape, but is inappropriate for a large spatial scale like China with its 9.6 million km2 land area, and distances exceeding 5000 km from south to north and east to west. Adding up all of the component parts is not equal to a whole ecosystem. This also explains why Willis and Whittaker (2002) concluded that variables that best account for species richness on a local spatial scale may not be the same as those accounting for

Secondly, we use an integrated factor from the results of PCA to calculate *CI*, which represents the contribution weights of every independent variable to the whole system. This approach is much better than the single variable model of temperature, precipitation, or area, because species richness is the emergent property of ecosystems through interactions of multiple factors. So a single factor model cannot explain the ecosystem property, especially on a large scale. On the other hand, though model established by multiple variables with stepwise regression fits better than a single variable model, it has a major drawback from the aspect of system thinking. This is because it selects variables only considering the correlations between dependent and independent variables, but neglects the interactions among variables. For instance, the last model in Table 1 only selects altitude difference and precipitation and rejects temperature. But we well know that nothing will happen in the real world without energy input. Thus, Hawkins et al. (2003) concluded that the interaction between water and energy provides a strong explanation for globally extensive plant and animal diversity gradients. Those analyses that do not include waterenergy variables are missing a key component for explaining the broad-scale patterns of diversity. In our theory we not only consider correlation, but also pay great attention to the

Finally, and most importantly, we introduce adaptability into the independent variables. This is estimated by the changed rate of animal species richness in an evolutionary time scale. In this way, we not only integrate biotic and abiotic factors but also combine spatial and temporal scales to predict plant species richness. This is a brand new approach when compared to the conventional hypotheses. Our results show that the changed rate of animal species richness has a great influence on plant species richness. Not only do theoretical works support this, but also empirical studies at the population and community levels have

indispensable components of the emergent properties of ecosystems.

*P*<0.001), and is far behind the CI model.

richness at regional spatial scales.

**3.3 Discussion**

measure the complete properties of the ecosystems, we used the animal species richness at present-day (*N*) to estimate the ecosystem properties. We then assumed that the number of original properties (animal species richness) at the processes occurring was 1, that is *n* = 1. The evolutionary time period (△t) might be over the last 10000 years, i.e., since the end of the last glacial period (Willis 2002).

*CI* values for every province were calculated from Eq. (3) by the following steps and methods: First, the data (from Table 1 in Appendix 1) were normalized using the standard deviation method (Xu, 2002) in order to eliminate influences caused by different units and dimensions (Table 2 in Appendix 1). Second, a Principal Component Analysis (PCA) was performed on the *e*, *d* and *a-1* in an effort to reduce the dimensionality of the data sets. The varimax rotation was used to simplify the interpretation of the results. Two components accounted for 87.63% of variance in *e*, *d* and *a-1* depending on the eigenvalues, percent of variance, and cumulative percent (Table 3 in Appendix 1). The major components of the first factor are temperature, precipitation, and the reciprocal of adaptability (Table 4 in Appendix 1). This is consistent with the natural situation in China where there are two most prominent climatic features of temperature increasing from north to south and precipitation decreasing from east to west. Factor 2 is composed of altitude difference and area (Table 4 in Appendix 1), each of which reflects the geographic characteristics in China. From the rotated component matrix (Table 4 in Appendix 1) and component score coefficient matrix (Table 5 in Appendix 1), we find the component score coefficient rotated matrix for all the provinces, which were F1 and F2 (Table 6 in Appendix 1). Finally, the *CI* value (Table 6 in Appendix 1) was estimated by the integrated factor (∑F) according to the percent of variance for F1 and F2:

#### *CI =*∑F = % of variance for component 1×F1 + % of variance for component 2×F2 = 0.55569×F1 + 0.32065×F2

Therefore, *CI* represents integrated levels of energy, reciprocal of adaptability, and environmental diversity in terms of annual precipitation, altitude difference, and area. *CI* value is above average if it is positive, and below average if negative.

In order to test the CTE we took the plant species richness as the emergent property of the ecosystem for each province and as the dependent variable, where CI was the independent variable. We ran regressions on the normalized data using the equation: plant species richness = a + b(*CI*) + c(*CI*)2, in linear, quadratic and cubic models. Only the best model was shown in Table 1. The results were compared with that of conventional methods, i.e., *energyhypothesis*, *spatial heterogeneity hypothesis, area hypothesis*, and *water-energy dynamic hypothesis,* with a multiple regression equation of: plant species richness = a + b (energy or area…) + c (energy or area…)2 also in linear, quadratic and cubic models. In order to be as liberal as possible in discovering patterns, relationships were considered significant if *P* < 0.05. The majority of relationships considered significant had *P* < 0.01.

#### **3.2 Results**

Regression analysis shows (Table 1) relationships that are almost exactly those predicted by the CTE hypotheses. We find the CI value is the best predictor with the cubic model and accounted for 94.0% of the variation in plant species richness (F=137.516, *P*<0.0001) , whereas model with an integrated environmental factor excluding adaptability can only account for 42.3% of the variation in plant species richness (F=20.054, *P*<0.0001). The models with a single parameter of temperature, altitude difference, or land area are not ideal and account for 12.3%,17.6%, and 28.0% of the variation in plant species richness respectively, with less F value and P<0.05. The model with multiple parameters of temperature, precipitation, and altitude difference shows much better than that of models with a single parameter, and account for 59.3% of the variation in plant species richness (F=19.915, *P*<0.001), and is far behind the CI model.

#### **3.3 Discussion**

30 Research in Biodiversity – Models and Applications

*CI* values for every province were calculated from Eq. (3) by the following steps and methods: First, the data (from Table 1 in Appendix 1) were normalized using the standard deviation method (Xu, 2002) in order to eliminate influences caused by different units and dimensions (Table 2 in Appendix 1). Second, a Principal Component Analysis (PCA) was performed on the *e*, *d* and *a-1* in an effort to reduce the dimensionality of the data sets. The varimax rotation was used to simplify the interpretation of the results. Two components accounted for 87.63% of variance in *e*, *d* and *a-1* depending on the eigenvalues, percent of variance, and cumulative percent (Table 3 in Appendix 1). The major components of the first factor are temperature, precipitation, and the reciprocal of adaptability (Table 4 in Appendix 1). This is consistent with the natural situation in China where there are two most prominent climatic features of temperature increasing from north to south and precipitation decreasing from east to west. Factor 2 is composed of altitude difference and area (Table 4 in Appendix 1), each of which reflects the geographic characteristics in China. From the rotated component matrix (Table 4 in Appendix 1) and component score coefficient matrix (Table 5 in Appendix 1), we find the component score coefficient rotated matrix for all the provinces, which were F1 and F2 (Table 6 in Appendix 1). Finally, the *CI* value (Table 6 in Appendix 1) was estimated by the integrated factor (∑F) according to the percent of variance for F1 and

*CI =*∑F = % of variance for component 1×F1 + % of variance for component 2×F2 = 0.55569×F1 + 0.32065×F2 Therefore, *CI* represents integrated levels of energy, reciprocal of adaptability, and environmental diversity in terms of annual precipitation, altitude difference, and area. *CI*

In order to test the CTE we took the plant species richness as the emergent property of the ecosystem for each province and as the dependent variable, where CI was the independent variable. We ran regressions on the normalized data using the equation: plant species richness = a + b(*CI*) + c(*CI*)2, in linear, quadratic and cubic models. Only the best model was shown in Table 1. The results were compared with that of conventional methods, i.e., *energyhypothesis*, *spatial heterogeneity hypothesis, area hypothesis*, and *water-energy dynamic hypothesis,* with a multiple regression equation of: plant species richness = a + b (energy or area…) + c (energy or area…)2 also in linear, quadratic and cubic models. In order to be as liberal as possible in discovering patterns, relationships were considered significant if *P* < 0.05. The

Regression analysis shows (Table 1) relationships that are almost exactly those predicted by the CTE hypotheses. We find the CI value is the best predictor with the cubic model and accounted for 94.0% of the variation in plant species richness (F=137.516, *P*<0.0001) , whereas model with an integrated environmental factor excluding adaptability can only account for 42.3% of the variation in plant species richness (F=20.054, *P*<0.0001). The models

value is above average if it is positive, and below average if negative.

majority of relationships considered significant had *P* < 0.01.

since the end of the last glacial period (Willis 2002).

F2:

**3.2 Results**

measure the complete properties of the ecosystems, we used the animal species richness at present-day (*N*) to estimate the ecosystem properties. We then assumed that the number of original properties (animal species richness) at the processes occurring was 1, that is *n* = 1. The evolutionary time period (△t) might be over the last 10000 years, i.e.,

> The CTE successfully predicts the plant species richness distribution in provinces of China. This success comes mainly from our new system thinking approach of study. First, we treat every province as a consolidated ecosystem (though their land areas may differ greatly), and that plant species richness is one of its emergent properties through the interactions of biotic and abiotic factors. Conventional approaches using equal and small area quadrat as the basic studying unit actually divide the ecosystems into many small component parts. This approach is probably reasonable for a small scale evaluation within a community or landscape, but is inappropriate for a large spatial scale like China with its 9.6 million km2 land area, and distances exceeding 5000 km from south to north and east to west. Adding up all of the component parts is not equal to a whole ecosystem. This also explains why Willis and Whittaker (2002) concluded that variables that best account for species richness on a local spatial scale may not be the same as those accounting for richness at regional spatial scales.

> Secondly, we use an integrated factor from the results of PCA to calculate *CI*, which represents the contribution weights of every independent variable to the whole system. This approach is much better than the single variable model of temperature, precipitation, or area, because species richness is the emergent property of ecosystems through interactions of multiple factors. So a single factor model cannot explain the ecosystem property, especially on a large scale. On the other hand, though model established by multiple variables with stepwise regression fits better than a single variable model, it has a major drawback from the aspect of system thinking. This is because it selects variables only considering the correlations between dependent and independent variables, but neglects the interactions among variables. For instance, the last model in Table 1 only selects altitude difference and precipitation and rejects temperature. But we well know that nothing will happen in the real world without energy input. Thus, Hawkins et al. (2003) concluded that the interaction between water and energy provides a strong explanation for globally extensive plant and animal diversity gradients. Those analyses that do not include waterenergy variables are missing a key component for explaining the broad-scale patterns of diversity. In our theory we not only consider correlation, but also pay great attention to the indispensable components of the emergent properties of ecosystems.

> Finally, and most importantly, we introduce adaptability into the independent variables. This is estimated by the changed rate of animal species richness in an evolutionary time scale. In this way, we not only integrate biotic and abiotic factors but also combine spatial and temporal scales to predict plant species richness. This is a brand new approach when compared to the conventional hypotheses. Our results show that the changed rate of animal species richness has a great influence on plant species richness. Not only do theoretical works support this, but also empirical studies at the population and community levels have

A Study on Biodiversity Mechanism by the Creativity Theory of Ecosystem 33

richness 0.0648 0.4557 0.3886 0.3962 0.9239

Table 2. Partial Correlation Coefficients between plant species richness and area, altitude

In addition, we find an interesting phenomenon that the geographical area has a very weak partial correlation coefficient with plant species richness, and the best model is an inverse one. This finding is counter to the *area hypothesis* (Table 2, Table 1). We postulate that the *area hypothesis* mainly considers a small scale, and that the pattern of species richness increasing with area will not exist if the area exceeds a critical size. Lyons & Willing (1999) also found that area effects on species richness for bats and marsupials are a minor importance at the area scales of 1000-25000km2. We believe that the smallest area in our study (33900km2 Hainan Province) may have been sufficiently large to have sampled most taxa as a regional

The mechanism of biodiversity is a complex issue that needs additional study from both system-specific models, and a more general theoretical framework that subsumes systemspecific models as special cases (Fox, 2006). By embracing system thinking and regarding an ecosystem as an entity, and by treating biodiversity as the emergent property of the ecosystem, the Creativity Theory of Ecosystems integrates biotic and abiotic factors and combines spatial and temporal scales into a single model. Among the three variables of the model, the introduction of an adaptability variable is a unique and most important innovation. This enables our model to embrace both biotic and temporal factors. Thus we believe that the CTE provides a new approach to the study of the mechanisms of biodiversity from the aspect of a general theoretical framework. In addition, using the methods of PCA, the CI can be quantitatively calculated and will successfully predict plant species richness distribution on a regional scale within China. This demonstrates that the

We thank Z. H. Li for providing climatic data assistance Q. Q. Zhang for data collecting and analyzing assistance, and Mr. Richard R. Faltonson for editing in English. This work was supported by the National Natural Science Foundation of China (NSFC) grant 30972353 and by the Doctoral Discipline Special Foundation of High Educational Universities in China

*P* 0.748 0.017 0.045 0.041 0.000

Annual precipitation Annual Mean air temperature

Animal species richness

difference

difference, precipitation, temperature, and animal species richness

Creativity Theory of Ecosystems is feasible and promising.

Area Altitude

Plant species

species pool within China.

**5. Acknowledgements**

grant<PHONE_NUMBER>0011.

**4. Conclusions**

documented that herbivores can reduce a plants' potential distribution, restricting them to a subset of habitats that they might physiologically tolerate (Harley, 2003). More studies demonstrated that higher trophic-levels can have important effects on plant diversity and ecosystem properties (Fine, 2006; Schmitz, 2006).

Table 1. Regression analysis of plant species richness against *CI* value and other environmental variables

Table 2. Partial Correlation Coefficients between plant species richness and area, altitude difference, precipitation, temperature, and animal species richness

In addition, we find an interesting phenomenon that the geographical area has a very weak partial correlation coefficient with plant species richness, and the best model is an inverse one. This finding is counter to the *area hypothesis* (Table 2, Table 1). We postulate that the *area hypothesis* mainly considers a small scale, and that the pattern of species richness increasing with area will not exist if the area exceeds a critical size. Lyons & Willing (1999) also found that area effects on species richness for bats and marsupials are a minor importance at the area scales of 1000-25000km2. We believe that the smallest area in our study (33900km2 Hainan Province) may have been sufficiently large to have sampled most taxa as a regional species pool within China.

## **4. Conclusions**

32 Research in Biodiversity – Models and Applications

documented that herbivores can reduce a plants' potential distribution, restricting them to a subset of habitats that they might physiologically tolerate (Harley, 2003). More studies demonstrated that higher trophic-levels can have important effects on plant diversity and

Table 1. Regression analysis of plant species richness against *CI* value and other

environmental variables

ecosystem properties (Fine, 2006; Schmitz, 2006).

The mechanism of biodiversity is a complex issue that needs additional study from both system-specific models, and a more general theoretical framework that subsumes systemspecific models as special cases (Fox, 2006). By embracing system thinking and regarding an ecosystem as an entity, and by treating biodiversity as the emergent property of the ecosystem, the Creativity Theory of Ecosystems integrates biotic and abiotic factors and combines spatial and temporal scales into a single model. Among the three variables of the model, the introduction of an adaptability variable is a unique and most important innovation. This enables our model to embrace both biotic and temporal factors. Thus we believe that the CTE provides a new approach to the study of the mechanisms of biodiversity from the aspect of a general theoretical framework. In addition, using the methods of PCA, the CI can be quantitatively calculated and will successfully predict plant species richness distribution on a regional scale within China. This demonstrates that the Creativity Theory of Ecosystems is feasible and promising.

## **5. Acknowledgements**

We thank Z. H. Li for providing climatic data assistance Q. Q. Zhang for data collecting and analyzing assistance, and Mr. Richard R. Faltonson for editing in English. This work was supported by the National Natural Science Foundation of China (NSFC) grant 30972353 and by the Doctoral Discipline Special Foundation of High Educational Universities in China grant<PHONE_NUMBER>0011.

A Study on Biodiversity Mechanism by the Creativity Theory of Ecosystem 35

Anhui -0.547 -0.668 0.571 0.472 -0.398 -0.267 -0.398\* Fujian -0.594 -0.530 1.367 0.991 0.860 0.109 0.860 Gansu 0.256 0.964 -1.236 -0.877 0.915 -0.084 0.915 Guangdong -0.449 -0.638 1.717 1.539 0.952 0.783 0.952 Guangxi -0.300 -0.629 1.384 1.473 1.177 0.969 1.177 Guizhou -0.454 -0.234 0.437 0.494 1.323 0.799 1.323 Hainan -0.818 -0.667 1.532 2.095 -0.278 -0.288 -0.278 Hebei -0.425 -0.179 -0.783 -0.311 -0.375 -0.534 -0.375 Henan -0.478 -0.542 -0.339 0.215 -0.889 -0.219 -0.889 Heilongjiang 0.257 -0.905 -0.782 -1.686 -0.577 -0.902 -0.577 Hubei -0.429 -0.070 0.620 0.536 -0.347 -0.037 -0.347 Hunan -0.363 -0.564 1.066 0.762 -0.200 0.107 -0.200 Jilin -0.425 -0.271 -0.530 -1.297 -0.971 -0.665 -0.971 Jiangsu -0.642 -1.261 0.237 0.446 -0.650 -0.674 -0.650 Jiangxi -0.478 -0.535 1.523 0.881 -0.416 0.053 -0.416 Liaoning -0.531 -0.914 -0.485 -0.596 -0.664 -1.074 -0.664

Mongolia 2.119 0.107 -1.253 -1.213 -0.531 -0.572 -0.531 Ningxia -0.772 -0.238 -1.262 -0.656 -1.091 -0.972 -1.091 Qinghai 0.938 0.941 -1.142 -1.818 -1.026 -0.599 -1.026 Shandong -0.503 -0.827 -0.465 0.025 -0.788 -0.983 -0.788 Shanxi -0.506 -0.179 -0.837 -0.587 -0.994 -0.582 -0.994 Shanxi(Xi'an) -0.379 0.190 -0.556 -0.115 -0.264 -0.208 -0.264 Sichuan 0.544 2.029 0.056 0.080 1.764 1.710 1.764 Tibet 2.164 1.967 -0.806 -1.366 0.497 0.487 0.497 Xinjiang 3.349 2.648 -1.557 -0.814 -0.283 -0.318 -0.283 Yunnan 0.102 1.639 0.452 0.703 3.178 3.896 3.178 Zhejiang -0.635 -0.633 1.072 0.625 0.075 0.063 0.075 *Notes:* § *a-1* = *(N–n) / n*, where N is the animal species richness, n is 1, first calculated it from table 1, then

\* The data are normalized by standard deviation method with the following equations:

*j x x x im <sup>s</sup>*

1

*ij*

1 *m ij i <sup>x</sup> <sup>m</sup> <sup>j</sup> <sup>s</sup>*<sup>=</sup>

*xij* are the original data from table 1, *xij'* are the normalized data

( 1,2,... ; j 1,2,...,n) *ij j*

1 <sup>1</sup> ( ) *m*

*i x x <sup>m</sup>* 2

*ij j*

Annual Mean temperature

Animal species richness

Plant species richness

a-1§

Annual Precipitation

Provinces (or Autonomous region)

Inner

normalized it.

where, *<sup>j</sup> x* =

Table 2. The normalized data from Table 1

Area Altitude

difference

## **6. Appendix 1**

Table 1. The data on climate, environmental diversity, and species richness for the 27 provinces in China.

*Notes:* § *a-1* = *(N–n) / n*, where N is the animal species richness, n is 1, first calculated it from table 1, then normalized it.

\* The data are normalized by standard deviation method with the following equations:

$$\mathbf{x}'\_{ij} = \frac{\mathbf{x}\_{ij} - \overline{\mathbf{x}\_j}}{s\_j} \text{(\$i = 1, 2, \dots m\$; j = 1, 2, \dots, n\$)}$$

34 Research in Biodiversity – Models and Applications

Annual Precipitation (mm)

Anhui 13.98 1860 1192.1 15.06 535 3644 Fujian 12.14 2148 1588.3 18.21 809 4709 Gansu 45.4 5258 292.48 6.88 821 4164 Guangdong 17.8 1922 1762.5 21.53 829 6621 Guangxi 23.67 1941 1596.7 21.13 878 7148 Guizhou 17.61 2763 1125.5 15.19 910 6665 Hainan 3.39 1863.1 1670.5 24.9 561 3585 Hebei 18.77 2879 518.31 10.31 540 2888 Henan 16.7 2123 739.38 13.5 428 3779 Heilongjiang 45.46 1366 518.62 1.97 496 1846 Hubei 18.59 3105.4 1216.3 15.45 546 4295 Hunan 21.17 2076 1438.4 16.82 578 4705 Jilin 18.74 2686 644.19 4.33 410 2516 Jiangsu 10.26 624.7 1025.8 14.9 480 2492 Jiangxi 16.69 2138 1665.8 17.54 531 4552 Liaoning 14.59 1348 666.54 8.58 477 1358

Mongolia 118.34 3474.4 284.13 4.84 506 2781 Ningxia 5.18 2756 279.56 8.22 384 1647 Qinghai 72.12 5210 339.37 1.17 398 2703 Shandong 15.7 1530 676.2 12.35 450 1616 Shanxi 15.6 2878 491.29 8.64 405 2751 Shanxi(Xi'an) 20.56 3647 631.29 11.5 564 3813 Sichuan 56.71 7476 935.72 12.68 1006 9249 Tibet 120.1 7348.13 506.78 3.91 730 5780 Xinjiang 166.49 8765 132.81 7.26 560 3500 Yunnan 39.4 6663.6 1133 16.46 1314 15444 Zhejiang 10.53 1933 1441.7 15.99 638 4579

Table 1. The data on climate, environmental diversity, and species richness for the 27

Annual Mean temperature (℃)

Animal species richness (No. of species)

Plant species richness (No. of species)

**6.1 Tables of basic data and steps for CI value calculation by PCA**

Altitude difference (m)

**6. Appendix 1**

Provinces (or Autonomous region)

Inner

provinces in China.

Area (×104Km2)

$$\text{where,}\\
\begin{aligned}
\overline{\boldsymbol{\alpha}\_{j}} &= \frac{1}{m} \sum\_{i=1}^{m} \boldsymbol{\alpha}\_{ij} \quad \boldsymbol{s}\_{j} = \sqrt{\frac{1}{m} \sum\_{i=1}^{m} (\boldsymbol{\alpha}\_{ij} - \overline{\boldsymbol{\alpha}\_{j}})^2} \end{aligned}$$

*xij* are the original data from table 1, *xij'* are the normalized data

Table 2.  Component score coefficient rotated matrix and *CI* values for all the provinces

% of Variance for component 1

% of Variance for component 2

*CI*﹡

F1§ F2※

Provinces (or Autonomous region)

0.419 × *a-1*

0.32065 × F2

*Note:* Component 1 and 2 accounted for 87.63% of variance in *e*, *d* and *a-1* depending on the initial eigenvalues in percent of variance and cumulative percent, which fulfill the requirements of cumulative percent for the two components larger than 80.00%, and of total initial eigenvalues larger than 1. So, we need only calculate the two components, F1 and F2, and they can represent the integrated level of the total variables.

Table 3. Total Variance Explained

*Note:* The major components of the first factor (F1) include temperature, precipitation, and the reciprocal of adaptability (*a-1*); F2 is composed of altitude difference and area.

#### Table 4. Rotated Component Matrix

Table 5. Component Score Coefficient Matrix

Initial Eigenvalues Extraction Sums of Squared Loadings

1 2.778 55.569 55.569 2.778 55.569 55.569 2 1.603 32.065 87.634 1.603 32.065 87.634

%

*Note:* Component 1 and 2 accounted for 87.63% of variance in *e*, *d* and *a-1* depending on the initial eigenvalues in percent of variance and cumulative percent, which fulfill the requirements of cumulative percent for the two components larger than 80.00%, and of total initial eigenvalues larger than 1. So, we need only calculate the two components, F1 and F2, and they can represent the integrated level of the

> Annual Mean Temperature 0.895 -0.305 Annual Precipitation 0.884 -0.343

Altitude difference -0.113 0.953

 Component 1 2

*a-1* 0.419 0.391

Area -0.0820 0.360

*Note:* The major components of the first factor (F1) include temperature, precipitation, and the reciprocal

Annual Mean Temperature 0.371 -0.0609 Annual Precipitation 0.382 -0.0399

Altitude difference 0.0709 0.474

*a-1* 0.730 0.578

Area -0.396 0.803

% Total % of Variance Cumulative

Component 1 2 %

Comp

Comp

total variables.

Table 3. Total Variance Explained

Table 4. Rotated Component Matrix

Table 5. Component Score Coefficient Matrix

onent Total % of Variance Cumulative

3 0.373 7.453 95.087 4 0.159 3.181 98.269 5 0.087 1.731 100.000 Rotation Sums of Squared Loadings

onent Total % of Variance Cumulative

1 2.284 45.681 45.681 2 2.098 41.953 87.634

of adaptability (*a-1*); F2 is composed of altitude difference and area.

*Notes:* § F1=-0.0820×area+0.0709×altitude difference + 0.371 × precipitation + 0.382 × temperature + 0.419 × *a-1*

※ F2=0.360×area+0.474×altitude difference -0.0609 × precipitation -0.0399 × temperature + 0.391 × *a-1*  ﹡*CI=*∑F = % of variance for component 1 × F1 + % of variance for component 2 × F2 = 0.55569 × F1 + 0.32065 × F2

Table 6. Component score coefficient rotated matrix and *CI* values for all the provinces

A Study on Biodiversity Mechanism by the Creativity Theory of Ecosystem 39

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Why do some places have much higher diversity than others? Explaining patterns of species diversity on the earth is a problem of long-standing for ecologists. Numerous diversity patterns have been documented [1], but the mechanisms behind these patterns remain poorly understood. If we knew the mechanisms, surer decisions in conservation and management would be possible. Understanding can be sought in many different ways. A dominant approach in community ecology is to search for explanations through the study of species interactions [2, 3]. This approach is motivated by the competitive exclusion principle, which suggests that there are limits to how similar species can be in their ecology while coexisting with one another in a stable way [2-6]. However, the competitive exclusion principle is challenged by the existence of many highly diverse communities of seemingly similar species. A response to this challenge is neutral theory, which postulates that many coexisting species are ecologically identical in respects that matter, and do not coexist stably. Instead, they undergo random walks to extinction, with diversity replenished by speciation, and immigration of species from other areas [7]. The neutral theory has been successful in predicting some diversity patterns in nature [8], but fails in other areas [9-12], and is limited

Other responses to the diversity challenge take the competitive exclusion principle seriously, although with a broad definition of stability [5]. These responses seek to understand the various ways that species differ from one another, how these differences structure species interactions and ultimately contribute to diversity maintenance in terms of species coexistence mechanisms [3, 5, 14, 15]. Traditional approaches focus on the differences between similar species in the ways they exploit resources, with some attention to the role of natural enemies in mediating or modifying interactions. New developments emphasize that the relationships that species have with their natural enemies are potentially just as important for diversity maintenance as their relationships with resources, and can have very similar outcomes [16]. Other directions focus on structuring and variation in the physical environment, emphasizing that the abilities of similar species to coexist with one

another may have much to do with physical environmental structure [15, 17-19].

Testing diversity maintenance hypotheses in nature is a vexing challenge [3]. Data are often limited and manipulating systems experimentally for testing diversity maintenance questions poses serious difficulties. Moreover, devising definitive tests of mechanisms has been problematic [20, 21]. However, work on coexistence mechanisms in variable

in the nature of the predictions that it can produce [13].

**1. Introduction**

Peter Chesson

*USA*

*Department of Ecology and Evolutionary Biology,*

*The University of Arizona, Tucson,*

Zhou, S. R. & Zhang, D. Y. (2006). Neutral theory in community ecology. *Journal of Plant Ecology*. 30:868-877 **3**

#### Peter Chesson

*Department of Ecology and Evolutionary Biology, The University of Arizona, Tucson, USA*

#### **1. Introduction**

42 Research in Biodiversity – Models and Applications

Zhou, S. R. & Zhang, D. Y. (2006). Neutral theory in community ecology. *Journal of Plant*

Why do some places have much higher diversity than others? Explaining patterns of species diversity on the earth is a problem of long-standing for ecologists. Numerous diversity patterns have been documented [1], but the mechanisms behind these patterns remain poorly understood. If we knew the mechanisms, surer decisions in conservation and management would be possible. Understanding can be sought in many different ways. A dominant approach in community ecology is to search for explanations through the study of species interactions [2, 3]. This approach is motivated by the competitive exclusion principle, which suggests that there are limits to how similar species can be in their ecology while coexisting with one another in a stable way [2-6]. However, the competitive exclusion principle is challenged by the existence of many highly diverse communities of seemingly similar species. A response to this challenge is neutral theory, which postulates that many coexisting species are ecologically identical in respects that matter, and do not coexist stably. Instead, they undergo random walks to extinction, with diversity replenished by speciation, and immigration of species from other areas [7]. The neutral theory has been successful in predicting some diversity patterns in nature [8], but fails in other areas [9-12], and is limited in the nature of the predictions that it can produce [13].

Other responses to the diversity challenge take the competitive exclusion principle seriously, although with a broad definition of stability [5]. These responses seek to understand the various ways that species differ from one another, how these differences structure species interactions and ultimately contribute to diversity maintenance in terms of species coexistence mechanisms [3, 5, 14, 15]. Traditional approaches focus on the differences between similar species in the ways they exploit resources, with some attention to the role of natural enemies in mediating or modifying interactions. New developments emphasize that the relationships that species have with their natural enemies are potentially just as important for diversity maintenance as their relationships with resources, and can have very similar outcomes [16]. Other directions focus on structuring and variation in the physical environment, emphasizing that the abilities of similar species to coexist with one another may have much to do with physical environmental structure [15, 17-19].

Testing diversity maintenance hypotheses in nature is a vexing challenge [3]. Data are often limited and manipulating systems experimentally for testing diversity maintenance questions poses serious difficulties. Moreover, devising definitive tests of mechanisms has been problematic [20, 21]. However, work on coexistence mechanisms in variable

*c Rv*

1

*l R R*

Sensitivity

*R jl l l kl R j k*

2

*j jl l l l l*

*c vK c*

*r*

*s c vK r*

*l l*

1

<sup>1</sup> <sup>1</sup>

2

<sup>1</sup> <sup>1</sup>

Notation *Nj* : Density of (consumer) species *j*.

species *j* on species *i*

The fitness measure *κ* derived from this model involves three things (Table 1). The first is the ability of a species to harvest resources. This is the total resource intake of a species

*Rl* : Density of resource *l*.

*<sup>l</sup> v* : unit value of resource *l*.

Harvesting ability requirement

( )

*jl l l j*

*R*

*cKv*

/

*R R*

*s s*

*dN r NN*

*dN r NN*

>

*jl c* : Consumption rate of resource *l* by species *j*.

*<sup>j</sup>* : resource maintenance requirement of species

*<sup>l</sup>* : Intraspecific competition for resource *l*

*ij* : Coefficient for competition for the effect of

1 11 1 12 2

2 21 1 22 2

*lr* : Maximum growth rate of resource l.

1 / *<sup>R</sup> Kl* , the carrying capacity of the resource.

<sup>1</sup> <sup>1</sup>

*dR r R Nc*

*l ll j jl <sup>j</sup> <sup>l</sup>*

*jl l l j l*

Maintenance

(consumer equation)

*j*

*dN*

*j*

*R dt*

1

*s*

1

*N dt*

2

*j*. *R*

> *R*

Table 1. MacArthur's consumer-resource equations

when its resources are at their carrying capacities.

*N dt*

*j*

*j l*

*N dt*

(resource equation)

MacArthur Consumerresource Equations

Average fitness measure

Overlap measure

Derived Lotka-Volterra Equations

environments has led to methods of quantifying the strength of coexistence mechanisms, and these quantifications have suggested definitive tests of mechanisms in nature [22]. Implementing these new methods comes with all the usual difficulties of ecological field manipulation, but the ability to focus on surer methods [20] may ultimately lead to the kind of focus in experimental technique that leads to breakthroughs.

In spite of the difficulty of understanding which coexistence mechanisms are active in a given system, enough commonalities exist between different mechanisms of stable coexistence to allow some general advice for conservation and management. Although we are a long way from truly understanding diversity maintenance, we have enough knowledge to suggest a number of areas where caution is needed. Species should not be managed in isolation. Factors improving the situation for one set of species may degrade it for others. Maintaining trophic structure, physical environmental structure, disturbance regimes, and spatial connectivity are all common sense ideas that receive support from existing understanding of diversity maintenance mechanisms.

### **2. Fitness differences, niches and coexistence**

When we consider how species interactions limit diversity, an important concept is the average fitness of a species as a whole, for this average fitness determines a species' ability to dominate other species with which it interacts. This concept is not to be confused with the fitness of a genotype or individual organism. The most common use of the idea of fitness is for genotypes or individual alleles of a gene, where its use is often relative, to predict the survival of a genotype or allele compared with others [23, 24]. Ecologists, however, often give fitness an absolute meaning as the performance of an individual organism, and this individual-level fitness is often measured as the total number of offspring that an individual leaves in its life time [25]. An alternative and more pragmatic meaning is its contribution to the population over a defined period of time through its own survival and reproduction [23]. Fitness at the species level applies in a similar relative context to fitness of a genotype or allele, but in this usage fitness specifies the relative degree of adaptedness of a species compared to others having a similar way of life and living in the same area, i.e. this fitness measure is to relative other species in the same guild, living in the same area [20].

When this species-level fitness measure is applied to problems of species coexistence, it assumes that the area of land in question is large enough for populations to be closed on an ecological timescale [5]. Then the fitness measure, which we denote *κ*, determines the degree of adaptedness of a species to the conditions applicable on that area of land. If the species in a guild cannot coexist with one another, then it is the species with the largest value of *κ* that persists, excluding others. More generally, the *κ* values rank the species in terms of their adaptation to the environment, and in essence rank the species in terms of how secure their persistence is when interacting with the others species in the guild.

#### **2.1 The MacArthur consumer-resource model**

A program to measure average fitness measures in nature has been proposed [26], but at the present time, these measures are easier to define in models. We consider the model of MacArthur [27] where a guild of animal species consume common biological resources in a lower trophic level, as reworked by Chesson [5, 28]. This model has had a key role in the development of ideas on resource partitioning for animal guilds (Table 1). Fig. 1 diagrams the foodweb being modeled, and Table 1 specifies the equations.

environments has led to methods of quantifying the strength of coexistence mechanisms, and these quantifications have suggested definitive tests of mechanisms in nature [22]. Implementing these new methods comes with all the usual difficulties of ecological field manipulation, but the ability to focus on surer methods [20] may ultimately lead to the kind

In spite of the difficulty of understanding which coexistence mechanisms are active in a given system, enough commonalities exist between different mechanisms of stable coexistence to allow some general advice for conservation and management. Although we are a long way from truly understanding diversity maintenance, we have enough knowledge to suggest a number of areas where caution is needed. Species should not be managed in isolation. Factors improving the situation for one set of species may degrade it for others. Maintaining trophic structure, physical environmental structure, disturbance regimes, and spatial connectivity are all common sense ideas that receive support from

When we consider how species interactions limit diversity, an important concept is the average fitness of a species as a whole, for this average fitness determines a species' ability to dominate other species with which it interacts. This concept is not to be confused with the fitness of a genotype or individual organism. The most common use of the idea of fitness is for genotypes or individual alleles of a gene, where its use is often relative, to predict the survival of a genotype or allele compared with others [23, 24]. Ecologists, however, often give fitness an absolute meaning as the performance of an individual organism, and this individual-level fitness is often measured as the total number of offspring that an individual leaves in its life time [25]. An alternative and more pragmatic meaning is its contribution to the population over a defined period of time through its own survival and reproduction [23]. Fitness at the species level applies in a similar relative context to fitness of a genotype or allele, but in this usage fitness specifies the relative degree of adaptedness of a species compared to others having a similar way of life and living in the same area, i.e. this fitness

measure is to relative other species in the same guild, living in the same area [20].

persistence is when interacting with the others species in the guild.

the foodweb being modeled, and Table 1 specifies the equations.

**2.1 The MacArthur consumer-resource model**

When this species-level fitness measure is applied to problems of species coexistence, it assumes that the area of land in question is large enough for populations to be closed on an ecological timescale [5]. Then the fitness measure, which we denote *κ*, determines the degree of adaptedness of a species to the conditions applicable on that area of land. If the species in a guild cannot coexist with one another, then it is the species with the largest value of *κ* that persists, excluding others. More generally, the *κ* values rank the species in terms of their adaptation to the environment, and in essence rank the species in terms of how secure their

A program to measure average fitness measures in nature has been proposed [26], but at the present time, these measures are easier to define in models. We consider the model of MacArthur [27] where a guild of animal species consume common biological resources in a lower trophic level, as reworked by Chesson [5, 28]. This model has had a key role in the development of ideas on resource partitioning for animal guilds (Table 1). Fig. 1 diagrams

of focus in experimental technique that leads to breakthroughs.

existing understanding of diversity maintenance mechanisms.

**2. Fitness differences, niches and coexistence**

Table 1. MacArthur's consumer-resource equations

The fitness measure *κ* derived from this model involves three things (Table 1). The first is the ability of a species to harvest resources. This is the total resource intake of a species when its resources are at their carrying capacities.

*j ij i jj*

where the quantity *ρ* measures overlap in resource use between and species *j* [5]. Fundamentally, we see that the ratio of the fitness measures determines how much one species affects another, compared with how much it affects itself. Understandably, however, relative fitness alone does not do this. It is most important to know how much the species interact with one another also. Thus, the fitness ratio is multiplied by the overlap measure, *ρ*, which is on a scale from 0 to 1, with 1 meaning complete overlap and 0 meaning

Fig. 2. Niche relationships of two species, in terms of the consumption rates on limiting resources. (A) Partial overlap in resource use, *ρ* = 0.33. (B) Complete overlap, *ρ* = 1.

affect the overlap, but is a factor explaining the difference between the fitnesses, *κ*.

Fig. 2 shows the niche relationships of two species as specified by their consumption rate curves for both partial overlap and complete overlap, while Table 1 shows how this overlap measure is calculated. The difference in the heights of the consumption rate curves does not

Interpreting this result is simplest in the case of two species competing with one another. If expression (1) is greater than 1, then species *j* excludes *i* from the system. In terms of competition coefficients, this means that species *j* places less limitation on its own growth as it increases in abundance than it places on the growth of the species *i*. Thus, species *j* continues increasing in abundance beyond the point that the growth of species *i* becomes negative. In terms of the fitness ratio and overlap measure, if the species overlap completely in resource use, then species *j* excludes *i* if it simply has a higher fitness than species *i*. However, with less than complete overlap, this fitness ratio needs to be discounted by the degree of overlap to determine if species *j* can exclude *i*. For example, if *ρ* = 2/3, then the fitness ratio needs to be more than 3/2, rather than just more than 1. Said another way, if one species is fitter than another, it only harms that other species to the extent that they

, (1)

no overlap.

This quantity, therefore, is a maximum resource harvesting rate. Subtracted from it is the resource maintenance requirement, which is the level of resource intake needed for a per capita growth rate of exactly zero, i.e. it is the resource intake that gives an individual organism a fitness of 1, which means that that one individual is replaced by exactly one individual on average in one unit of time. The maximum resource intake, less the maintenance requirement, measures the ability of a species to meet its needs, and is called the net maximum harvesting rate. Finally, the net maximum harvesting rate is divided by a quantity *s* which measures the sensitivity of the growth rate of the species to changes in resource availability, as discussed in detail by Chesson [20]. The resulting quantity is the average fitness measure *κ* (Table 1).

Fig. 1. Foodweb in the MacArthur model. Squares are the focal guild members and circles are resources. Arrows indicate the directions of effects of one species on another, with thickness differences indicating differences in strengths of these effects. The feedback loops for guild members come from links from a square to a circle back to a square. Circular loops for resources indicate resource self-limitation.

Although this fitness measure *κ* may seem intuitively reasonable, the real test is its ability to predict the outcome of the interactions of this species with others in the same guild. The measure *κ* has this desired property because it is related to competition coefficients derived from the MacArthur consumer-resource model. These competition coefficients are denoted *ij* giving the effect of interspecific competition from species *j* onto species *i*, and *jj* for intraspecific competition within species *j*. They measure the feedback loops illustrated in Fig. 1 from one species to another, and from one species back to itself. Moreover, when resource dynamics are fast relative to consumer dynamics, e.g. because resource species have much shorter generation times, the MacArthur consumer-resource model reduces to the Lotka-Volterra competition model defined by these competition coefficients [5] — see Table 1 for the two-species case. Of most importance, the ratio of interspecific competition to intraspecific competition is related to the ratio of the *κ* values:

This quantity, therefore, is a maximum resource harvesting rate. Subtracted from it is the resource maintenance requirement, which is the level of resource intake needed for a per capita growth rate of exactly zero, i.e. it is the resource intake that gives an individual organism a fitness of 1, which means that that one individual is replaced by exactly one individual on average in one unit of time. The maximum resource intake, less the maintenance requirement, measures the ability of a species to meet its needs, and is called the net maximum harvesting rate. Finally, the net maximum harvesting rate is divided by a quantity *s* which measures the sensitivity of the growth rate of the species to changes in resource availability, as discussed in detail by Chesson [20]. The resulting quantity is the

Fig. 1. Foodweb in the MacArthur model. Squares are the focal guild members and circles are resources. Arrows indicate the directions of effects of one species on another, with thickness differences indicating differences in strengths of these effects. The feedback loops for guild members come from links from a square to a circle back to a square. Circular loops

Although this fitness measure *κ* may seem intuitively reasonable, the real test is its ability to predict the outcome of the interactions of this species with others in the same guild. The measure *κ* has this desired property because it is related to competition coefficients derived from the MacArthur consumer-resource model. These competition coefficients are denoted

intraspecific competition within species *j*. They measure the feedback loops illustrated in Fig. 1 from one species to another, and from one species back to itself. Moreover, when resource dynamics are fast relative to consumer dynamics, e.g. because resource species have much shorter generation times, the MacArthur consumer-resource model reduces to the Lotka-Volterra competition model defined by these competition coefficients [5] — see Table 1 for the two-species case. Of most importance, the ratio of interspecific competition to

*jj* for

*ij* giving the effect of interspecific competition from species *j* onto species *i*, and

average fitness measure *κ* (Table 1).

for resources indicate resource self-limitation.

intraspecific competition is related to the ratio of the *κ* values:

$$\frac{\kappa\_j}{\kappa\_i} \, \rho = \frac{\alpha\_{ij}}{\alpha\_{ji}} \, \tag{1}$$

where the quantity *ρ* measures overlap in resource use between and species *j* [5]. Fundamentally, we see that the ratio of the fitness measures determines how much one species affects another, compared with how much it affects itself. Understandably, however, relative fitness alone does not do this. It is most important to know how much the species interact with one another also. Thus, the fitness ratio is multiplied by the overlap measure, *ρ*, which is on a scale from 0 to 1, with 1 meaning complete overlap and 0 meaning no overlap.

Fig. 2. Niche relationships of two species, in terms of the consumption rates on limiting resources. (A) Partial overlap in resource use, *ρ* = 0.33. (B) Complete overlap, *ρ* = 1.

Fig. 2 shows the niche relationships of two species as specified by their consumption rate curves for both partial overlap and complete overlap, while Table 1 shows how this overlap measure is calculated. The difference in the heights of the consumption rate curves does not affect the overlap, but is a factor explaining the difference between the fitnesses, *κ*.

Interpreting this result is simplest in the case of two species competing with one another. If expression (1) is greater than 1, then species *j* excludes *i* from the system. In terms of competition coefficients, this means that species *j* places less limitation on its own growth as it increases in abundance than it places on the growth of the species *i*. Thus, species *j* continues increasing in abundance beyond the point that the growth of species *i* becomes negative.

In terms of the fitness ratio and overlap measure, if the species overlap completely in resource use, then species *j* excludes *i* if it simply has a higher fitness than species *i*. However, with less than complete overlap, this fitness ratio needs to be discounted by the degree of overlap to determine if species *j* can exclude *i*. For example, if *ρ* = 2/3, then the fitness ratio needs to be more than 3/2, rather than just more than 1. Said another way, if one species is fitter than another, it only harms that other species to the extent that they

degree of separation of those loops for the relative degrees of resource shortage that the two species can tolerate due to their overall adaptation to the environment, to give the

Fig. 3. Simulation showing species coexisting stably according to the lottery competition in a temporally varying environment. Note that each species shows a strong tendency to recover

When the conditions (2) above are satisfied in the MacArthur model, the result is stable coexistence in the sense that if either species is perturbed to low density, it recovers from that low density state. In general, the MacArthur model leads to a stable equilibrium point. However, by stable coexistence, a stable equilibrium is not implied. Instead, the requirement is recovery of each species from low density, which is fully compatible with fluctuating coexistence, as illustrated in Fig. 3. Indeed, Fig. 3 illustrates stable coexistence in the lottery model [31], which requires temporal fluctuations for stable coexistence to occur. i.e. it is an example of fluctuation-dependent coexistence [32], as discussed below under variation in time and space. Neutral models, on the other hand, do not allow stable coexistence. Instead, they are characterized by having average fitnesses the same for all species. Thus, all species have the same *κ* values, and the niche overlap measures *ρ* between all pairs of species are all 1. In general, the key assumption of neutral models is stronger: it is that the species identity of an individual has no bearing on how it is affected by any other individual or how it affects other individuals [33]. However, this stronger assumption is no particular concern here. With equal average fitnesses and complete niche overlap, in the MacArthur model expression (1) is exactly equal to 1, and the inequalities (2) are not satisfied. Stable coexistence does not occur, but the model is neutrally stable. For example,

in the two-species case, population densities approach the line 11 1 22 2

remain at any point on this line, until perturbed. In particular, there is no tendency for either species to increase from low density if perturbed there, and so there is no stable

 *N N*

1 , but can

competition coefficient ratio (1).

from fluctuations to low density.

coexistence.

**2.2 Stable versus neutral coexistence**

overlap in resource use. As we shall see below, this statement generalizes to all forms of overlap associated with interactions between the species.

In the case where neither of two competing species can exclude the other, they coexist. For two species labeled 1 and 2, this means that expression (1) must be less than 1 for (i, j) = (1, 2) and (2, 1), discounting the unlikely boundary case of equality with 1, discussed below. This condition can be rearranged to

$$
\rho < \frac{\kappa\_1}{\kappa\_2} < \frac{1}{\rho} \tag{2}
$$

and shows that overlap, *ρ*, places constraints on how different the average fitness measures for the two species can be and still allow coexistence. Simply put, the more similar the species are in resource use (the closer *ρ* is to 1), the more similar they have to be in average fitness for coexistence to occur. In opposite terms, the more different the species are in resource use, the more different they must be in average fitness before one excludes the other.

This analysis of MacArthur's model yields some important general lessons. The competitive exclusion principle holds that species that are very similar in their ecology should have difficulty coexisting. Similarity in ecology within the MacArthur model can be interpreted as meaning a *ρ* value near 1, i.e. high overlap in resource use. Difficulty in coexisting can now be interpreted as meaning that the species have to be evenly matched in average fitness if they are to coexist, i.e. if one species gets a small edge in average fitness over the other, then exclusion will occur.

These findings for the MacArthur model generalize to others [5, 29], and they highlight two different sorts of similarity between species. The first is similarity in average fitness, i.e. having *κ* values near to each other. As discussed above, the *κ* values allow ranking of overall adaptation to the environment. Almost by definition, having similar *κ* values is favorable to species coexistence. However, similarity in *κ* values is a very different concept from similarity in way of life, or similarity in niches, when niches are defined in terms of the way of life of a species [5, 30]. In the MacArthur model, similarity in way of life reduces to similarity in resource use, which then determines how much the species interact with one another. Thus, similarity in niches constrains differences in average fitness compatible with coexistence, by conditons (2).

Looked at another way, we can think of these similarities and differences as relating to average performance compared with performance under specific conditions. Species 1 and 2 in Fig. 2(A) have very different performance under specific conditions; for example, species 1 derives very little benefit from resources 13 to 17, instead gaining most benefit from resources 5 to 12 while species 2 has an opposite pattern. The average performances of the species, which depend on the heights of the curves in the figure, are nowhere near as different as their performances for most specific resources.

Differences in performance under specific conditions lead to tolerance of average performance differences. For example, in the MacArthur model, specific-condition differences give a small value of *ρ*, and thus wide tolerance in the *κ* ratio. More generally, the nature of these specific differences is very important. They must relate directly or indirectly to separation of feedback loops because it is separation of these feedback loops that makes it possible for intraspecific competition to exceed interspecific competition, the key to coexistence. With the MacArthur model, the measure *ρ* is a specific measure of the degree of separation of those feedback loops. Multiplication by the fitness ratio adjusts the

overlap in resource use. As we shall see below, this statement generalizes to all forms of

In the case where neither of two competing species can exclude the other, they coexist. For two species labeled 1 and 2, this means that expression (1) must be less than 1 for (i, j) = (1, 2) and (2, 1), discounting the unlikely boundary case of equality with 1, discussed below.

> 1 2 1

and shows that overlap, *ρ*, places constraints on how different the average fitness measures for the two species can be and still allow coexistence. Simply put, the more similar the species are in resource use (the closer *ρ* is to 1), the more similar they have to be in average fitness for coexistence to occur. In opposite terms, the more different the species are in resource use, the

This analysis of MacArthur's model yields some important general lessons. The competitive exclusion principle holds that species that are very similar in their ecology should have difficulty coexisting. Similarity in ecology within the MacArthur model can be interpreted as meaning a *ρ* value near 1, i.e. high overlap in resource use. Difficulty in coexisting can now be interpreted as meaning that the species have to be evenly matched in average fitness if they are to coexist, i.e. if one species gets a small edge in average fitness over the other,

These findings for the MacArthur model generalize to others [5, 29], and they highlight two different sorts of similarity between species. The first is similarity in average fitness, i.e. having *κ* values near to each other. As discussed above, the *κ* values allow ranking of overall adaptation to the environment. Almost by definition, having similar *κ* values is favorable to species coexistence. However, similarity in *κ* values is a very different concept from similarity in way of life, or similarity in niches, when niches are defined in terms of the way of life of a species [5, 30]. In the MacArthur model, similarity in way of life reduces to similarity in resource use, which then determines how much the species interact with one another. Thus, similarity in niches constrains differences in average fitness compatible with

Looked at another way, we can think of these similarities and differences as relating to average performance compared with performance under specific conditions. Species 1 and 2 in Fig. 2(A) have very different performance under specific conditions; for example, species 1 derives very little benefit from resources 13 to 17, instead gaining most benefit from resources 5 to 12 while species 2 has an opposite pattern. The average performances of the species, which depend on the heights of the curves in the figure, are nowhere near as

Differences in performance under specific conditions lead to tolerance of average performance differences. For example, in the MacArthur model, specific-condition differences give a small value of *ρ*, and thus wide tolerance in the *κ* ratio. More generally, the nature of these specific differences is very important. They must relate directly or indirectly to separation of feedback loops because it is separation of these feedback loops that makes it possible for intraspecific competition to exceed interspecific competition, the key to coexistence. With the MacArthur model, the measure *ρ* is a specific measure of the degree of separation of those feedback loops. Multiplication by the fitness ratio adjusts the

, (2)

more different they must be in average fitness before one excludes the other.

overlap associated with interactions between the species.

This condition can be rearranged to

then exclusion will occur.

coexistence, by conditons (2).

different as their performances for most specific resources.

degree of separation of those loops for the relative degrees of resource shortage that the two species can tolerate due to their overall adaptation to the environment, to give the competition coefficient ratio (1).

Fig. 3. Simulation showing species coexisting stably according to the lottery competition in a temporally varying environment. Note that each species shows a strong tendency to recover from fluctuations to low density.

#### **2.2 Stable versus neutral coexistence**

When the conditions (2) above are satisfied in the MacArthur model, the result is stable coexistence in the sense that if either species is perturbed to low density, it recovers from that low density state. In general, the MacArthur model leads to a stable equilibrium point. However, by stable coexistence, a stable equilibrium is not implied. Instead, the requirement is recovery of each species from low density, which is fully compatible with fluctuating coexistence, as illustrated in Fig. 3. Indeed, Fig. 3 illustrates stable coexistence in the lottery model [31], which requires temporal fluctuations for stable coexistence to occur. i.e. it is an example of fluctuation-dependent coexistence [32], as discussed below under variation in time and space. Neutral models, on the other hand, do not allow stable coexistence. Instead, they are characterized by having average fitnesses the same for all species. Thus, all species have the same *κ* values, and the niche overlap measures *ρ* between all pairs of species are all 1. In general, the key assumption of neutral models is stronger: it is that the species identity of an individual has no bearing on how it is affected by any other individual or how it affects other individuals [33]. However, this stronger assumption is no particular concern here. With equal average fitnesses and complete niche overlap, in the MacArthur model expression (1) is exactly equal to 1, and the inequalities (2) are not satisfied. Stable coexistence does not occur, but the model is neutrally stable. For example, in the two-species case, population densities approach the line 11 1 22 2 *N N* 1 , but can remain at any point on this line, until perturbed. In particular, there is no tendency for either species to increase from low density if perturbed there, and so there is no stable coexistence.

very similar effects to feedback loops through resources. As consequence Holt coined the term "apparent competition" [41]. While previously we focused on species differences based on their use of resources, equally we can focus on species differences based on their patterns of susceptibility to the various predator species of the guild in question. Thus, the idea of a

Within this expanded concept of a niche, the niche overlap measure *ρ* is extended to predator overlap too. Indeed, the MacArthur consumer-resource model extends to a consumer-resource-predator model, with *ρ* depending on both resource and predator overlap, measuring resource-sensitivity and predator-sensitivity similarities between species [16]. For example, *ρ* would be lowered if one species were more sensitive to changes in resource abundance while the other species were more sensitive to changes in predatorabundance. The extended fitness measure, *κ*, accounts for overall predation susceptibility as

species' niche can be expanded to include susceptibility to predator species [3, 5].

Fig. 4. Foodweb for the consumer-resource-predator model. As for Fig. 1, but with

interspecific and intraspecific density dependence, and thus include both resource competition and apparent competition. Of most importance, the relationship (1) between competition coefficient ratios, fitness ratios and niche overlap still holds, and the coexistence conditions (2) remain valid. Thus, no longer can competition and predation be regarded as having very different effects on species coexistence. We should not think of competition as the primary factor limiting coexistence, with predation modifying what competition does. Instead, the view emerging corresponds to Holt's [38] advice that what is possible for

*ij*, and

*jj*, become coefficients of

well as resource harvesting ability.

predators signified by the diamond symbols.

competition is possible for predation too.

In this fuller concept of the niche, competition coefficients

The significance of neutral models is their ability to predict some patterns of diversity seen in nature, for example in very diverse communities, neutral models have successfully reproduced the observed frequencies of species with different degrees of commonness and rarity [8]. To produce any pattern at all, however, they need to go beyond the strict deterministic description of equations like those in Table 1, and generally include demographic stochasticity. They thus recognize the finiteness of population size, and include independent chance events in the lives of individuals that determine individual fates [7]. For a model like the MacArthur consumer-resource equations, this would mean that the abundances of the species would be constantly randomly perturbed up and down the line 11 1 22 2 *N N* 1 , or with more than two species, over the hyperspace <sup>1</sup> *jj j N* . Extinctions do occur in such models, and so in modern versions they include stochastic speciation and immigration to balance species losses [7].

The predictions of neutral models have often not held up [9-12], and neutral models do not give predictions about the differential effects that environmental change often has on the species in a guild [13], which are critical for conservation and management questions. On the other hand, models in which individuals of different species are not identical in their ecology can reproduce at least some of the patterns of multispecies diversity predicted by neutral models [34]. Neutral models provide a simpler mathematical route to some predictions, and in this way have highlighted some of the processes that lead to some observed patterns in nature, but these same processes are able to produce these patterns in nonneutral models as well.

#### **2.3 Multitrophic diversity maintenance**

Similar species can interact with one another by feedback loops through resources, or by feedback loops through predators. Naturally, they may also interact directly by interfering [35] with or facilitating each other's activities [36], or through intraguild predation [37]. However, the focus here is on the much misunderstood and critical area of feedback loops through predators [38]. Fig. 4 expands the foodweb model of Fig. 1 to include predators of the species in our guild of interest. Much emphasis has focused on idea that competition limits diversity of guild members, while predation modifies what competition does. For example, the keystone species idea was developed around the concept that predation on a competitive dominant would prevent competitive exclusion [39, 40].

The keystone species idea can be view directly within the competition framework above if it is assumed that the predator, in causing mortality, increases the resource maintenance requirement of the dominant species. This is a natural expectation because if a species has higher mortality, to persist it will need higher reproduction, and higher resource consumption to fuel that reproduction. The increased maintenance requirement for the dominant then decreases its *κ* value making it more comparable to that of other species, potentially enabling coexistence. For example, in the two-species case, the coexistence conditions (2) might become satisfied [6].

The above keystone species discussion focuses just on the mortality that the predators cause, and neglects the feedback loops associated with them. In causing mortality, the predator benefits and can build up in density, with the potential then of inflicting greater mortality. The feedback loops from guild members (the prey) to guild members arise because higher densities allow predator numbers to increase, increasing mortality on guild members. As Holt [38, 41] pointed out many years ago, these feedback loops through predators can have

The significance of neutral models is their ability to predict some patterns of diversity seen in nature, for example in very diverse communities, neutral models have successfully reproduced the observed frequencies of species with different degrees of commonness and rarity [8]. To produce any pattern at all, however, they need to go beyond the strict deterministic description of equations like those in Table 1, and generally include demographic stochasticity. They thus recognize the finiteness of population size, and include independent chance events in the lives of individuals that determine individual fates [7]. For a model like the MacArthur consumer-resource equations, this would mean that the abundances of the species would be constantly randomly perturbed up and down

*jj j N* . Extinctions do occur in such models, and so in modern versions they include

The predictions of neutral models have often not held up [9-12], and neutral models do not give predictions about the differential effects that environmental change often has on the species in a guild [13], which are critical for conservation and management questions. On the other hand, models in which individuals of different species are not identical in their ecology can reproduce at least some of the patterns of multispecies diversity predicted by neutral models [34]. Neutral models provide a simpler mathematical route to some predictions, and in this way have highlighted some of the processes that lead to some observed patterns in nature, but these same processes are able to produce these patterns in

Similar species can interact with one another by feedback loops through resources, or by feedback loops through predators. Naturally, they may also interact directly by interfering [35] with or facilitating each other's activities [36], or through intraguild predation [37]. However, the focus here is on the much misunderstood and critical area of feedback loops through predators [38]. Fig. 4 expands the foodweb model of Fig. 1 to include predators of the species in our guild of interest. Much emphasis has focused on idea that competition limits diversity of guild members, while predation modifies what competition does. For example, the keystone species idea was developed around the concept that predation on a

The keystone species idea can be view directly within the competition framework above if it is assumed that the predator, in causing mortality, increases the resource maintenance requirement of the dominant species. This is a natural expectation because if a species has higher mortality, to persist it will need higher reproduction, and higher resource consumption to fuel that reproduction. The increased maintenance requirement for the dominant then decreases its *κ* value making it more comparable to that of other species, potentially enabling coexistence. For example, in the two-species case, the coexistence

The above keystone species discussion focuses just on the mortality that the predators cause, and neglects the feedback loops associated with them. In causing mortality, the predator benefits and can build up in density, with the potential then of inflicting greater mortality. The feedback loops from guild members (the prey) to guild members arise because higher densities allow predator numbers to increase, increasing mortality on guild members. As Holt [38, 41] pointed out many years ago, these feedback loops through predators can have

stochastic speciation and immigration to balance species losses [7].

competitive dominant would prevent competitive exclusion [39, 40].

1 , or with more than two species, over the hyperspace

the line 11 1 22 2

nonneutral models as well.

**2.3 Multitrophic diversity maintenance**

conditions (2) might become satisfied [6].

<sup>1</sup>

 *N N*

very similar effects to feedback loops through resources. As consequence Holt coined the term "apparent competition" [41]. While previously we focused on species differences based on their use of resources, equally we can focus on species differences based on their patterns of susceptibility to the various predator species of the guild in question. Thus, the idea of a species' niche can be expanded to include susceptibility to predator species [3, 5].

Within this expanded concept of a niche, the niche overlap measure *ρ* is extended to predator overlap too. Indeed, the MacArthur consumer-resource model extends to a consumer-resource-predator model, with *ρ* depending on both resource and predator overlap, measuring resource-sensitivity and predator-sensitivity similarities between species [16]. For example, *ρ* would be lowered if one species were more sensitive to changes in resource abundance while the other species were more sensitive to changes in predatorabundance. The extended fitness measure, *κ*, accounts for overall predation susceptibility as well as resource harvesting ability.

Fig. 4. Foodweb for the consumer-resource-predator model. As for Fig. 1, but with predators signified by the diamond symbols.

In this fuller concept of the niche, competition coefficients *ij*, and *jj*, become coefficients of interspecific and intraspecific density dependence, and thus include both resource competition and apparent competition. Of most importance, the relationship (1) between competition coefficient ratios, fitness ratios and niche overlap still holds, and the coexistence conditions (2) remain valid. Thus, no longer can competition and predation be regarded as having very different effects on species coexistence. We should not think of competition as the primary factor limiting coexistence, with predation modifying what competition does. Instead, the view emerging corresponds to Holt's [38] advice that what is possible for competition is possible for predation too.

Some ideas about spatial and temporal variation, especially those under the heading of disturbance, have viewed variation in the same way that has often coloured thinking about predation: competition has been seen as limiting diversity, with variation in time and space as modifying or even nullifying what competition does [57-60]. Many of these views are influenced by the idea that variation in time and space make ecological communities "nonequilibrial," while niche ideas are presumed to depend on the concept of equilibrium

Theoretical studies converge on the conclusion that variation in time and space are best viewed as providing more opportunities for the niches of species to be distinguished, defined by the responses of species to the changing conditions they encounter [3, 5]. This is most easily understood with respect to spatial variation [50, 62-64]. If species favor different habitats, and their resources are consumed and regenerated locally in those habitats, the habitats provide separate feedback loops, just like separate resources. Separate feedback loops with respect to predators are less likely, because predators are often wider ranging than their prey and so prey densities in one habitat can affect predation in other habitats, but this case has been developed theoretically [38]. The same would be true of resources too if resources moved between habitats, or if habitats changed their status over time on a shorter time scale than resource changes, i.e. if there were rapid spatio-temporal

Niche theory for spatial and temporal variation has developed techniques for assessing the extent to which spatial and temporal variation can separate the niches of species [20, 22]. This is the concept of covariance between environment and competition and its generalization to include apparent competition [51, 52]. Such covariances assess whether density-dependent feedback loops change with the environmental conditions under consideration, and can thus separate the niches of species. Based on them, powerful techniques for definitively testing diversity maintenance hypotheses based on spatial and temporal have been proposed [20] and implemented in some cases [22, 66]. Extension of these ideas for other niche-based diversity maintenance mechanisms should be possible, and might well lead to much progress on deciding between the various diversity maintenance

Although it is not possible at the present time to say with confidence which of the many possible diversity maintenance ideas applies in any particular system, these ideas share enough features in common that some general advice is possible. Our principal example has been MacArthur's consumer-resource model, and its extension to a consumer-resourcepredator model, for which average fitness and overlap measures are easy to define. Per capita growth rates in these models are linear functions of the densities of the various species. Models similar in spirit, but having nonlinear relationships, have been studied extensively by Tilman [17, 67], especially for plants limited by nonbiological resources, and compared with others by Chase and Leibold [3]. Although these simple measures of average fitness and niche overlap are not available, the underlying concepts remain applicable [3]. Other models of resource and apparent competition in both constant and variable environments have yielded measures of fitness and niche overlap, when they have been sought explicitly [5, 29, 47, 51, 52, 68], clearly demonstrating how these ideas extend

environmental variation [65]. Similar issues apply to temporal variation [5].

[61]. Nothing could be further from the truth [3, 5].

ideas that have been proposed.

**3. Conservation and management**

Relationships with resources and with predators can equally promote or limit diversity depending on the circumstances. Coexistence is promoted if different guild members have different relationships with their resources or different relationships with their predators, and exclusion is promoted if these relationships tend to be similar. Differences with respect to either resource sensitivity or predator susceptibility lower *ρ*, while similarities in either respect increase *ρ*. As noted above, an extra complication is that complementary relationships between species between sensitivity to resources and susceptibility to predators also lowers *ρ*, and is referred to in the literature as a competition-predation tradeoff [42].

Feedback loops through resources and through predators naturally do have some differences. For example, predators fitting the most common notion as species that hunt and kill prey species, are often larger than their prey, or of comparable size, and are normally less species rich than the prey they focus on, while the resources of their prey may be more species rich. Opportunities for niche distinctions between species due to their relationships with their predators would therefore seem less than the opportunities due their relationships with their resources. However, predatory behavior can be complex [43], and when predators have frequency-dependent behavior, a single predator has a similar effect in terms of the coexistence of prey species to several predators that do not have this frequency-dependent behavior, but are instead specialists on particular species. Moreover, interpreting "predators" as natural enemies more generally, such as diseases and parasites, makes predators appear every bit as able as resources to define distinct niches for their prey species.

The potential symmetry between resources and predators in promoting coexistence or exclusion depends also on the relative strengths of these processes. The niche overlap measure *ρ* takes these relative strengths into account. This can be seen in the formula in Table 1, where the reciprocal of the renewal rate of a resource weights its importance in the overlap measure. Slow renewal means that the measure is more strongly affected by consumption, and therefore more strongly contributes to resource competition. Similarly, in the extended overlap measure to include predation, the ability of a predator to build up in response to prey consumption weights its importance, and determines its contribution to apparent competition [16]. Of most significance, if resource separation is strong, but resource competition as a whole is weak relative to apparent competition, then that resource separation will have little effect on coexistence. Whether the species coexist depends on whether their niches are distinguished by predation. If their niches are not distinguished by predation, then they will have difficulty coexisting despite substantial separation at the level of resources. Naturally, the opposite conclusion is reached if it is resource competition that is strong relative to apparent competition.

#### **2.4 Variation in time and space**

All natural environments vary substantially in time or space in at least some major ecologically significant ways. Environmental variables and population densities nearly always vary substantially in both time and space. Empirical studies show this, but at great variance with the realities of nature, attention of theory to these facts historically was not a mainstream endeavor [44], and sadly that remains true today. Despite the absence of mainstream attention, considerable theory on the role of variation in time and space has been developed [14, 17, 19, 20, 38, 45-52]. Although the influence of this theory on empirical studies has not yet matched its potential importance, there is now a growing body of related empirical studies [53-56]

Relationships with resources and with predators can equally promote or limit diversity depending on the circumstances. Coexistence is promoted if different guild members have different relationships with their resources or different relationships with their predators, and exclusion is promoted if these relationships tend to be similar. Differences with respect to either resource sensitivity or predator susceptibility lower *ρ*, while similarities in either respect increase *ρ*. As noted above, an extra complication is that complementary relationships between species between sensitivity to resources and susceptibility to predators also lowers *ρ*, and is referred to in the literature as a competition-predation

Feedback loops through resources and through predators naturally do have some differences. For example, predators fitting the most common notion as species that hunt and kill prey species, are often larger than their prey, or of comparable size, and are normally less species rich than the prey they focus on, while the resources of their prey may be more species rich. Opportunities for niche distinctions between species due to their relationships with their predators would therefore seem less than the opportunities due their relationships with their resources. However, predatory behavior can be complex [43], and when predators have frequency-dependent behavior, a single predator has a similar effect in terms of the coexistence of prey species to several predators that do not have this frequency-dependent behavior, but are instead specialists on particular species. Moreover, interpreting "predators" as natural enemies more generally, such as diseases and parasites, makes predators appear

The potential symmetry between resources and predators in promoting coexistence or exclusion depends also on the relative strengths of these processes. The niche overlap measure *ρ* takes these relative strengths into account. This can be seen in the formula in Table 1, where the reciprocal of the renewal rate of a resource weights its importance in the overlap measure. Slow renewal means that the measure is more strongly affected by consumption, and therefore more strongly contributes to resource competition. Similarly, in the extended overlap measure to include predation, the ability of a predator to build up in response to prey consumption weights its importance, and determines its contribution to apparent competition [16]. Of most significance, if resource separation is strong, but resource competition as a whole is weak relative to apparent competition, then that resource separation will have little effect on coexistence. Whether the species coexist depends on whether their niches are distinguished by predation. If their niches are not distinguished by predation, then they will have difficulty coexisting despite substantial separation at the level of resources. Naturally, the opposite conclusion is reached if it is resource competition that

All natural environments vary substantially in time or space in at least some major ecologically significant ways. Environmental variables and population densities nearly always vary substantially in both time and space. Empirical studies show this, but at great variance with the realities of nature, attention of theory to these facts historically was not a mainstream endeavor [44], and sadly that remains true today. Despite the absence of mainstream attention, considerable theory on the role of variation in time and space has been developed [14, 17, 19, 20, 38, 45-52]. Although the influence of this theory on empirical studies has not yet matched its potential importance, there is now a growing body of related

every bit as able as resources to define distinct niches for their prey species.

is strong relative to apparent competition.

**2.4 Variation in time and space**

empirical studies [53-56]

tradeoff [42].

Some ideas about spatial and temporal variation, especially those under the heading of disturbance, have viewed variation in the same way that has often coloured thinking about predation: competition has been seen as limiting diversity, with variation in time and space as modifying or even nullifying what competition does [57-60]. Many of these views are influenced by the idea that variation in time and space make ecological communities "nonequilibrial," while niche ideas are presumed to depend on the concept of equilibrium [61]. Nothing could be further from the truth [3, 5].

Theoretical studies converge on the conclusion that variation in time and space are best viewed as providing more opportunities for the niches of species to be distinguished, defined by the responses of species to the changing conditions they encounter [3, 5]. This is most easily understood with respect to spatial variation [50, 62-64]. If species favor different habitats, and their resources are consumed and regenerated locally in those habitats, the habitats provide separate feedback loops, just like separate resources. Separate feedback loops with respect to predators are less likely, because predators are often wider ranging than their prey and so prey densities in one habitat can affect predation in other habitats, but this case has been developed theoretically [38]. The same would be true of resources too if resources moved between habitats, or if habitats changed their status over time on a shorter time scale than resource changes, i.e. if there were rapid spatio-temporal environmental variation [65]. Similar issues apply to temporal variation [5].

Niche theory for spatial and temporal variation has developed techniques for assessing the extent to which spatial and temporal variation can separate the niches of species [20, 22]. This is the concept of covariance between environment and competition and its generalization to include apparent competition [51, 52]. Such covariances assess whether density-dependent feedback loops change with the environmental conditions under consideration, and can thus separate the niches of species. Based on them, powerful techniques for definitively testing diversity maintenance hypotheses based on spatial and temporal have been proposed [20] and implemented in some cases [22, 66]. Extension of these ideas for other niche-based diversity maintenance mechanisms should be possible, and might well lead to much progress on deciding between the various diversity maintenance ideas that have been proposed.

#### **3. Conservation and management**

Although it is not possible at the present time to say with confidence which of the many possible diversity maintenance ideas applies in any particular system, these ideas share enough features in common that some general advice is possible. Our principal example has been MacArthur's consumer-resource model, and its extension to a consumer-resourcepredator model, for which average fitness and overlap measures are easy to define. Per capita growth rates in these models are linear functions of the densities of the various species. Models similar in spirit, but having nonlinear relationships, have been studied extensively by Tilman [17, 67], especially for plants limited by nonbiological resources, and compared with others by Chase and Leibold [3]. Although these simple measures of average fitness and niche overlap are not available, the underlying concepts remain applicable [3]. Other models of resource and apparent competition in both constant and variable environments have yielded measures of fitness and niche overlap, when they have been sought explicitly [5, 29, 47, 51, 52, 68], clearly demonstrating how these ideas extend

This outcome can be understood intuitively: one species increases in abundance when a valuable habitat type becomes more available, and, due to dispersal between habitat types, its abundance everywhere is increased. It thus has greater competitive effects on other species in all habitats. From the perspective of the theory above, average fitnesses and niche overlap change in the same way as when resource supply is changed. Similar affects apply to temporal environmental variation, an issue of particular concern with climate change [78]. For example, desert annual plants are believed to coexist with one another because different species are favored in different years, depending on the weather [19, 55]. Changes in weather patterns can change the relative *κ* values of the species if the weather becomes on average more favorable for some than for others. Changes in disturbance regimes can have

Although systems where coexistence relies on disturbance are sometimes regarded as nonequilibrial [80], they do not escape these general priniciples. Individual species may be adapted to survive and even take advantage of temporal change, but there is every reason to expect that individual species, and the system as a whole, are sensitive to changes in the average frequency and intensity of environmental events, such as disturbances [81, 82]. Surviving and taking advantage of temporal change may rely on life-history attributes such as dormancy, but may also depend on dispersal to and from refuge habitats that escape change either permanently or at given time (e.g when a fire sweeps through) [83]. Maintaining landscape connectivity, for example through maintenance of dispersal

Finally, we come to the important question of trophic structure. Recognition that predation and competition can have similar roles in diversity maintenance, and that the effects that each has can be undermined by the other, depending on their relative strengths, means that maintenance of trophic structure is vital for conservation [85]. There is much concern about loss of large predators in many systems [86-89]. These predators are often valued for their own sake. Conservation measures focused on them might aim merely for their persistence in nature, rather than for maintenance of the roles that they have in systems [90, 91]. For example, practices in the United States relating to wolf restoration face contentious arguments over the mere persistence of wolves, versus maintaining their numbers so that the ecosystem roles are restored as well [92, 93]. Discussion of trophic cascades, which are increasingly seen in marine systems, as the larger species are eliminated, focus much on the overall abundance of a given trophic level [94, 95]. However, the theory discussed here emphasizes that decimating one trophic level will affect the diversity maintenance roles that it has in other trophic levels for which is either a predator trophic level or a resource trophic

Taken together, these principles and examples reemphasize that species should not be managed in isolation. Factors improving the situation for one set of species may degrade it for others. Maintaining trophic structure, physical environmental structure, disturbance regimes, and spatial connectivity are all common sense ideas that are backed up by the

similar effects through changes in the patterns of spatio-temporal variation [79].

corridors, is especially important in such circumstances [84].

level, as has been found empirically in some systems [96].

This research was supported by NSF grant DEB-0542991.

theory of diversity maintenance mechanisms.

**4. Conclusions**

**5. Acknowledgements**

beyond MacArthur's [27] beginnings. Moreover, Shigesada [69] provides an explicit representation of spatial competition models in the MacArthur framework.

The fact that different species do differ from one another ecologically means that changes in the environment do affect different species differently or to different degrees. Under the individual-species approach to conservation and management, this would simply mean that different species are at risk to different degrees from some change in the environment due to human activities. Under the interactive model given here, changes in the environment need not directly affect a species for it to be affected indirectly through its linkages with other species. The fitness measures *κ* reveal this most simply. A mortality rate of a particular species might be changed by hunting, elimination of a predator, or changes in physical stress in the environment. That change would have a direct effect on that species, either increasing or decreasing its abundance and perhaps putting it at risk, but that change would also have effects on a species' relationships with others. In particular, in the MacArthur model, we see that the ratio of the *κ* values would be affected. An increase in the *κ* ratio in one species favor disadvantages another, potentially leading to its extinction.

One example of this *κ*-ratio analysis is its application to understanding invasive species of large effect [30, 70]. A large *κ* value in a particular system would allow a species to invade that system, but would have also have the effect of depressing native species and potentially driving them extinct. There are a number of ways that an invasive species might achieve a larger *κ* value including low susceptibility to natural enemies in the invaded system [71], advantages in resource harvesting [72], and lower sensitivity to competition than the native species [73]. Alternatively, an invasive species might have a larger *κ* value than native species because habitat degradation has increased stresses on native species, lowering their *κ* values, and rendering the system vulnerable to invasion [30, 74]. Viewed from another perspective, a native species might become endangered not because some environmental change has directly affected it, but because other guild members have had their *κ* values elevated by changes that directly benefited them.