Computer simulations of ecosystems are useful for revealing in detail the ecological processes concerned with major bioevents. Unlike empirical approaches, theoretical models allow us to see virtual evolutionary changes in interspecific interactions in a much shorter time as compared with evolutionary experiments. Some recent ecosystem models succeeded in constructing hypothetical ecosystems with high species diversity by introducing low connectance and gradual evolution. Some of these hypothetical ecosystems had similar properties to real ecosystems. Some models incorporated biologically meaningful rules for constructing interspecific interactions, whereas others did not. Both types of model are complementary to each other: which to choose depends on the purpose of the study. Most of the former type of model focused on incorporating biologically realistic processes. They do not aim to mimic the topological features of real ecosystems. It is quite natural that such models did not reproduce ecosystems similar to real ecosystems. Therefore, the reality of such models should be tested not only by using the topology of the ecosystem but also by using parameters which are suitable for the purpose of the models.
Introduction
Why did the dinosaurs become extinct? How did ecosystems recover from mass extinction events? What are the factors that have controlled temporal diversity changes over geologic time? These questions viewed on an evolutionary time scale have intrigued many biologists and paleontologists. Studies of these subjects give hints to the future of the earth and human beings.
Researches on major bioevents in the Earth's history have been based mainly on empirical evidence such as the fossil record and sedimentary evidence indicating environmental changes. These approaches have revealed what happened in the past: for example, it is now well known that a large meteorite impact probably contributed to the mass extinction at the end of the Cretaceous (e.g., Alvarez et al., 1980).
However, empirical studies do not always clarify the mechanisms of such bioevents, because the geological evidence of ancient bioevents is generally incomplete. Most species are not preserved in the fossil record, and there is scant evidence of interspecific interactions (Raup and Stanley, 1971). Particularly, ancient ecosystems are hardly reconstructable from empirical evidence. Therefore, we do not as yet understand the detailed ecological processes of bioevents, e.g., how the decline of primary production derived from the impact winter propagated through ecosystems at the end of the Cretaceous.
To advance research on past bioevents, we should attempt not only empirical approaches but also theoretical ones (Raup et al., 1973). For revealing the detailed ecological processes concerned with major bioevents, computer simulations of ecosystem models are very useful. We can trace evolutionary changes in interspecific interactions through a hypothetical ecosystem in a shorter time as compared with evolutionary experiments. In addition, hypotheses proposed by theoretical studies may encourage empirical studies. These advantages of theoretical studies may complement studies on ancient bioevents.
The prototype ecosystem model was developed by Lotka (1920a, b, 1922a, b) and Volterra (1926 and Volterra (1928). Since then, a huge number of studies using the Lotka-Volterra equation have been conducted. However, although real ecosystems often consist of a great number of species, most of the studies have focused on the behavior of very small systems (e.g., Pimm and Lawton, 1978; Namba, 1984; Pimm, 1991; Gilpin, 1994). This is partly because the behavior of a system with low species diversity is more easily understood. A more serious problem is that a system with high species diversity is hard to construct because of its instability (the diversity-stability debate: Gardner and Ashby, 1970; May, 1972; Pimm, 1991; Tokita and Yasutomi, 1999; McCann, 2000).
In recent years, several authors proposed advanced models, some of which overcame the instability of a large system. These models can be roughly classified into two types: one incorporates population dynamics and the other not. The former type of model focuses on effects of ecological processes on the behavior of systems, and some of the latter aim at mimicking real ecosystems. These models are promising tools for studies on an evolutionary time scale. This paper focuses mainly on food web models and briefly discusses the usefulness and perspective of these models.
Models without population dynamics
Niche Model (Williams and Martinez, 2000)
In this model, each species was randomly assigned a niche value, feeding range, and center of feeding range (Figure 1). A given predatory species fed on all species with a niche value that fell in the feeding range of the predatory species. By changing the size of the feeding range, the connectance of the model food web was adjusted to that of a real target food web. Although the model was very simple, it successfully mimicked real food webs.
Figure 1.
Schematic figure of the Niche Model (after Williams and Martinez, 2000). Species (▾) are distributed on a niche axis. For details see the text.
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While this model was not an evolutionary model, it provided the basis for the Nested-Hierarchy Model, which introduced an evolutionary aspect in the definition of interspecific interaction.
Nested-Hierarchy Model (Cattin et al., 2004)
Cattin et al. (2004) modified the Niche Model to introduce a phylogenetic constraint in food web construction. In their model, interspecific interaction was determined as follows; a prey species of predator species i was randomly chosen among species with niche value smaller than i. Here the chosen prey species was named j. If prey j had no predator other than predator i, the next prey of predator i was randomly chosen again. If prey j already had one or more predators, predator i was admitted to the group of predators of prey j, and then the next prey of predator i was randomly chosen among the set of prey of the predator group. These processes were begun with the species having the smallest niche value in the system. By the last process, a phylogenetic constraint was incorporated in the model.
Using the model, Cattin et al. (2004) were able to reproduce model food webs similar to real ones. Although their manner of introducing a phylogenetic constraint was artificial and did not represent phylogenetic evolution, there was no doubt that they tried to represent an evolutionary aspect of food web construction.
Speciation Model (Rossberg et al., 2005)
The name of the model derived from the manner of construction of interspecific interactions via speciation. In this model, species diversity in a food web system was changed by speciation, extinction, and immigration of new species from outside the system. These events occur randomly.
When a new species immigrated into the system, predators of the immigrants were randomly chosen with a certain probability (C0) from possible predators, which were defined as all species whose body size was not less than h times that of the new species. When a new species was born by speciation, predatory species of the new species were determined in the following manner. Candidate predators of the new species were derived from two groups of species in a food web. One was all predators of the ancestor. The other was randomly chosen with C0 from all possible predators of the new species other than the first group. From the candidates, predators of the new species were randomly chosen with a certain probability so as to adjust the connectance of the system to C0. Prey species of the new species were determined in the same manner.
The Speciation Model could well reproduce some features of real food webs. Then, Rossberg et al. (2005) concluded that evolutionary dynamics had an important role in the establishment of a food web system. The detailed behavior of this model was analytically investigated by Rossberg et al. (2006a).
However, this result was obtained only under the unrealistic assumption that the speciation rate of species with smaller body size is higher than that of species with larger body size (Rossberg et al., 2006a). In addition, the model had difficulties with mimicking real food webs consisting of not less than 50 species (Rossberg et al., 2006a). In order to overcome these problems, they have proposed a new model which they call the Matching Model (Rossberg et al., 2006b).
Matching Model (Rossberg et al., 2006b)
This model described the evolution of an abstract species pool. Unlike other models without population dynamics, interspecific interactions were determined with reference to properties of the species. This procedural approach was similar to the models of Caldarelli et al. (1998), Drossel et al. (2001, 2004) and Yoshida (2002, 2003a), which incorporated population dynamics.
In this model, each species had a size parameter and two sequences of 1 and 0 representing foraging and vulnerability traits. (In Rossberg et al. (2006b), the length of the sequences is 256.) If the number of foraging traits of a consumer matching the corresponding vulnerability traits of a resource species exceeded a certain threshold, and the size of the resource did not exceed the upper limit of the resource that the consumer could feed on, the consumer fed on the resource. In the model, extinctions, invasions of new species, and speciation events occurred randomly. The traits and size parameter of an invading species were determined randomly. The logarithm of the size of a new species appearing via speciation was determined by adding a zero-mean Gaussian random number to that of its ancestor. The traits of the ancestor were inherited, but mutations occurred at a certain rate.
Although population dynamics were not incorporated, this model successfully mimicked real food webs by the adjusting parameters mentioned above. Therefore, Rossberg et al. (2006b) concluded that population dynamics were not required to understand food-web topology.
Amaral and Meyer's (1999) model
In this model, each species occupied an ecological niche at each trophic level (Figure 2). Species at the lowest trophic level were assumed to be autotrophic, and those at upper levels were heterotrophic. Heterotrophic species fed on other species in the next lower trophic level. Interspecific interactions were defined randomly. Every existing species was given a chance of speciation at a certain rate. A new species occupied a randomly selected vacant niche at the same trophic level or in one of the two neighboring niches. Inter-specific interactions of new species were defined randomly. At every time step, a certain fraction of the species at the lowest trophic level were randomly selected and became extinct. As a result, if all prey species of a predatory species in the next higher level became extinct, the predatory species became extinct. This procedure was repeated up to the highest level.
Figure 2.
Schematic figure of Amaral and Meyer's (1999) model (after Amaral and Meyer, 1999).
At each trophic level, there are a number of niches. Shaded boxes represent niches occupied by a species, and others represent vacant niches. Solid lines represent predator-prey interactions. Species in the upper trophic level feed on those in the neighboring lower level. Species at the trophic level 0 are autotrophic. If species A, which is the only prey of species B, is chosen for a victim of random extinction, species B also becomes extinct because of starvation.
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This model was regarded as a stochastic model covering a trophic hierarchy. Ecological processes were barely involved. However, this model could reproduce diversity fluctuation patterns observed in the fossil record (Amaral and Meyer, 1999). Drossel (1998) analytically investigated this model by focusing on the resultant network structure and showed how the size distribution of extinction events in the model followed a power law.
Chowdhury and Stauffer's (2004) model
Chowdhury and Stauffer tried to incorporate “microevolution” and “macroevolution” into a single model of a predator-prey network with a structure similar to that assumed in Amaral and Meyer's (1999) model. What they called “microevolution” means birth, growth (ageing), and natural death of the individual organisms. “Macroevolution” represents adaptive coevolution of species.
The simulation began with a single species at the lowest trophic level. The state of the system was updated at each discrete time step by six sequential procedures (for details, see Chowdhury and Stauffer, 2004). (1) At each discrete time step, each individual which was older than the “minimum reproduction age” gave birth to a certain number of offspring. (2) Each individual met its natural death with a given probability which increased with growth. (3) At a given probability, each of the species randomly changed its properties. At the same probability, interspecific interactions were readjusted in a manner representing the fact that each species tried to minimize predation pressure but at the same time to its maximize chances to find new food. (4) Some individuals of a given species were killed by predators. The number of victim individuals depended on the number of predatory individuals. If the available food for a species was less than required, some individuals of the species died of starvation. If the number of individuals of the species becomes 0, the species becomes extinct. (5) With a certain probability per unit time, all vacant niches at a given trophic level were refilled by new species, all of which derived from a single common ancestor species at the same trophic level. The properties of new species were defined by adding small mutations to those of the ancestor. Interspecific interactions of new species were equal to those of the ancestor. (6) With a low probability, a new trophic level was added to the system.
This model was regarded as a modification of Amaral and Meyer's (1999) model. Although the manner of constructing interspecific interactions did not refer to full biological processes, the model had the potential to link individual-level processes with macroevolutionary level processes. This model gave an important hint for developing food-web models.
Models incorporating population dynamics
Replicator Equation Model (Happel and Stadler, 1998; Tokita and Yasutomi, 2003)
These models utilized the replicator equation for representing interspecific interactions. The replicator equation is generally used to describe self-replicating entities (replicators) and has been well used in various fields, e.g., sociobiology, mathematical biology, game theory, economics (Hofbauer and Sigmund, 1988; Tokita and Yasutomi, 1999). Interspecific interactions were defined randomly (interaction coefficients in the equation were determined by assigning random values). In Happel and Stadler's (1998) model, interactions of a new species born via speciation were determined by adding slight mutations to all interactions of its ancestor. On the other hand, Tokita and Yasutomi (2003) investigated three manners of determining interspecific interaction of new species: 1) all interspecific interaction coefficients of a new species were determined randomly, 2) interspecific interactions of a new species were set by adding mutations to all interaction coefficients of its ancestor, 3) interspecific interactions of a new species were set by copying those of its ancestor and replacing one of the interaction coefficients with a new one. Tokita and Yasutomi (2003) showed that a very large and complex system could be constructed only by the third manner. This result indicated that gradual evolution was important for constructing a large and complex system.
Web World Model (Caldarelli et al., 1998; Drossel et al., 2001, 2004)
Caldarelli et al. (1998) and Drossel et al. (2001, 2004) tried to introduce real biological bases for the construction and evolution of food webs; that is, inter-specific interactions were constructed based on characters of species, and the model system evolved via the evolution of species.
A species in the model had a number of characters, which were chosen from a character pool consisting of a total number of K characters (the value of K is set to 500 in the original study). Predator-prey interactions were determined by using a K * K matrix of scores representing the usefulness of any character i of a species for predation against any character j of another species. The score matrix was antisymmetric. Each element of the matrix was determined by using a Gaussian random number. In order to judge whether a predatory species could feed on a prey species, the characters of the predatory species were compared with those of the prey species, and then the sum of scores was obtained by using the score matrix. When the sum of scores was positive, the predatory species could feed on the prey. The biomass of a prey species was distributed among its predators according to their scores for predation. When the score of a given predatory species was lower than a threshold, the predatory species could not feed on the prey species, even though the score of the predatory species was positive. In this way, resource competition among predators was introduced.
At each time step, a species in the system was randomly chosen for speciation. The characters of a new species were set by adding a slight mutation to those of its ancestor; that is, a randomly chosen character of a new species was replaced by another which was randomly chosen from the character pool. By adding a new species, scores of species in the system were changed. Then, interactions in the system were reconstructed.
Drossel et al. (2001, 2004) modified the Web World Model. They used a set of equations based on ratio-dependent functional response (Holling, 1959, 1965, 1966; Arditi and Ginzburg, 1989; Arditi and Michalski, 1996) and introduced the Optimal Foraging Theory (Stephens and Krebs, 1986). (See Appendix for ratio-dependent, functional response, and Optimal Foraging Theory.) Their modification introduced several biologically supported assumptions. This was one of the strong points of this model. Using the new model, Drossel et al. (2004) stated that a large and complex food web with several trophic layers did not arise in a model without any kind of functional response and optimal foraging.
Feeding Preference Model (Yoshida, 2002, 2003a)
As does the Web World Model (Caldarelli et al., 1998; Drossel et al., 2001, 2004), Yoshida's (2002, 2003a) food web model also adopted the concept that interspecific interactions were defined based on the characters of species, and the food web evolved via the evolution of species.
In this model, population dynamics was represented using a simple form of Lotka-Volterra equations in contrast to newly published models (e.g., Drossel et al., 2004; Ito and Ikegami, 2006). A food web consisted of plant and animal species (in the modified Yoshida model (unpublished), animal species were divided into herbivores, carnivores, and omnivores). Plant species conducted primary production. Plant species having similar characters competed with each other. Animal species could not increase their biomass without feeding on other species. A predatory species fed on a prey species if a character set of the prey met the feeding preference of the predator. In predator-prey interactions among animal species, bigger animals fed on smaller ones. The procedure of constructing interspecific interactions was described in detail in Yoshida (2002, 2003a), and a schematic diagram of the procedure was shown in the figure 1 in Yoshida (2003b). At every certain time interval, a randomly chosen species was favored with a chance of speciation. The characters of a new species were defined by adding slight mutations to those of its ancestor. Inter-specific interactions were readjusted at this time.
In this model, values of characters for species did not directly indicate which species was superior, unlike the Niche Model, in which species having a higher niche value fed on others having a lower niche value. Therefore, in Yoshida's food web model, there was no determinant trend in evolution.
In the Web World Model and Replicator Equation Model, characters of a new species that appeared via speciation were determined by replacing one or more values of characters of its ancestor by new values. In contrast, in the Feeding Preference Model, values of characters of a new species that appeared via speciation were determined by adding slight mutations to those of its ancestor. Consequently, characters of phylogenetically related species were similar to one another. In this way, the model could represent phylogenetic constraint in an evolutionary plausible manner. Therefore, this model was very powerful for studying the dynamics of clades (Yoshida 2002, 2003a, 2006a, b).
Ito and Ikegami's (2006) model
In the previous models, a new species suddenly appeared through division of a population by some “external force.” A detailed speciation process was not involved. On the other hand, Ito and Ikegami's (2006) model tried to introduce a detailed speciation process via predator-prey interaction.
This model analyzed the dynamics of the distribution pattern of resource and utilization properties (phenotypes) arranged along two hypothetical axes. Species were defined by clusters formed in this two-dimensional phenotype space. This particular model portrayed the evolution of “trophic species,” that is, species groups sharing the same set of predators and prey.
The distribution patterns of resource and utilization properties changed depending on the interaction between them. The density of utilization properties rapidly increased where the resource properties were concentrated (Figure 3A). Consequently, the density of resource properties under high predation pressure decreased (Figure 3B). New properties were always provided by mutation, which was represented by a diffusion process (Kimura, 1983). In the tail of the resource distribution, the density of utilization properties did not increase, because of the low density of resource properties (Figure 3B). As a result, lateral diffusion of utilization properties was inhibited (Figure 3B). On the other hand, because of low predation pressure in the area, the density of the resource property rapidly increased, and resource properties rapidly diffused (Figure 3B). This process divided the distribution of resource properties, and finally speciation occurred (Figure 3C). Following this event, speciation with respect to the utilization property would occur (Figure 3C).
Figure 3.
Schematic figures of Ito and Ikegami's (2006) model.
Schematic temporal change in the distributions of utilization and resource properties are shown. The distributions change from the state shown in Figure 3A to that shown in 3C. For details see the text.
![i1342-8144-10-4-375-f03.gif](ContentImages/Journals/jpal/10/4/prpsj.10.375/graphic/WebImages/i1342-8144-10-4-375-f03.gif)
This model represented the process of diversification of trophic species very well. However, a problem of the model was that different populations with distinct evolutionary histories were treated as a single species. As Ito and Ikegami (2006) pointed out, if a trophic species consisted of two or more biological species, the evolutionary dynamics of the trophic species might be different from the case in a single biological species. If the model could be extended to trace the evolutionary history of each population, it would dramatically contribute to our understanding of the mechanism of adaptive diversification.
Discussion
High species diversity
One of the main themes of ecosystem modeling, which might derive from the diversity-stability debate (McCann, 2000), has been the construction of systems with high species diversity. For this purpose, each species in a model is assumed to interact with a small number of other species (low connectance), as is well observed in real ecosystems (Yodzis, 1980; Cohen et al., 1990; Schoenly et al., 1991; Havens et al., 1996; Martinez et al., 1999; Thompson and Townsend, 1999; Memmott et al., 2000). This idea originated with Gardner and Ashby (1970) and May (1972, 1973). Under this condition, the effects of various kinds of stimulations are hard to spread over a whole system. For example, a new species can invade the system without expelling other species. Such invasions increase the diversity of the system. Therefore, a system with a low connectance can maintain a high species diversity (May, 1972, 1973).
In some models, the low connectance is realized by suppressing the probability of establishing interactions (Happel and Stadler, 1998; Amaral and Meyer, 1999; Tokita and Yasutomi, 2003; Cattin et al., 2004; Chowdhury and Stauffer, 2004; Rossberg et al., 2005). On the other hand, in the models in which interspecific interactions are determined based on features of species, the range of the feeding preference of species is reduced (Williams and Martinez, 2000; Yoshida, 2002, 2003a). In the Web World Model (Caldarelli et al., 1998; Drossel et al., 2001, 2004), resource competition suppresses the connectance of the system.
Another process that creates a system with high species diversity is gradual evolution. To be favored with a chance of speciation, species must survive for a sufficiently long time. Such species are considered to have adaptive properties, including interaction. So their descendants born through gradual evolution also have adaptive properties and, consequently, may be able to survive for a long time as its ancestor did. In addition, such species, like their ancestors, do not disturb the system severely (Tokita and Yasutomi, 2003). Therefore, a system involving gradual evolution easily maintains a high species diversity.
Thus, suppressing connectance and gradual evolution are key factors for food web models with high species diversity.
The right model for the right job
Each model makes its own assumptions. For example, interspecific interactions are determined randomly in several models (Happel and Stadler, 1998; Amaral and Meyer, 1999; Tokita and Yasutomi, 2003; Cattin et al., 2004; Chowdhury and Stauffer, 2004), whereas they are based on characters of species in other models (Caldarelli et al., 1998; Williams and Martinez, 2000; Drossel et al., 2001, 2004; Yoshida, 2002, 2003a; Rossberg et al., 2006b). However, neither type of model is inferior. Each suits a different purpose.
The former type of model is very simple, so that the behavior of the system can be easily understood. Therefore, such models are suitable for the study of the behavior of a whole system and physical principles governing the system. For example, it is well known that there are various patterns of species-abundance relationship (rank-abundance relationship), and several models were published to explain each pattern (Motomura, 1932; Fisher et al., 1943; Preston, 1948). Tokita (2004) suggested that various patterns of the species-abundance relationship can be generally explained by variation in productivity. In addition, some simple models can produce a system quite similar to real ecosystems, even though they do not involve biologically meaningful rules for the construction of interspecific interactions (Cattin et al., 2004; Rossberg et al., 2005, 2006b). These results indicate that biologically meaningful rules for the construction of interspecific interactions are not indispensable for the purpose of revealing mechanisms controlling the topological features of an ecosystem.
Biologically meaningful rules for the construction of interspecific interactions sometimes make the model so complicated as to inhibit our understanding of the behavior of a system. However, such rules are indispensable for studies of the evolution of species in a system and the influence of each species on the behavior of the whole system. For example, Yoshida (2002) suggested that the two major features of living fossils—low taxonomic diversity and low evolutionary rate—are closely related via food web dynamics. Yoshida (2003b) suggested that the evolution of species toward mutualism causes a sudden decline of species diversity in a system. Such studies cannot be conducted with models in which interspecific interactions are constructed without biological bases. Therefore, we must choose a model according to our purpose (Iwasa et al., 1987, 1989).
This is the same for population dynamics. Even a model lacking in an evolutionary process and/or a process of population dynamics (extinction of species occurs randomly) sometimes mimics real ecosystems (Williams and Martinez, 2000; Cattin et al., 2004; Rossberg et al., 2005, 2006b) and temporal diversity patterns observed in the fossil record (Amaral and Meyer, 1999). However, such models are inadequate to reveal the cause of extinction of each species. In particular, for investigating the fluctuation of biomass or the number of individuals of each species, population dynamics is indispensable. For example, the effect of a sudden decline in biomass of a species (caused by disturbance) on a system (Yoshida, 2006a) cannot be studied with a model without population dynamics.
This paper does not discuss the individual-based models. Such models are not suitable for studies on a very long time scale, because a huge number of individuals must be simulated. However, this kind of model is very powerful for studies of evolution influenced by individual-level phenomena, such as the bias of distribution of individuals (e.g., Takenaka, 2005) and temporal change in gene frequency in a population (Kawata, 1995).
Must models mimic the topology of real ecosystems?
To test the reality of a model by comparing the model with the real world is a crucial procedure for any theoretical study. Ecosystem models are usually tested by using topological parameters of the web, e.g., number of links per species, the ratios of top predator, intermediate, and basal species (Williams and Martinez, 2000; Cattin et al., 2004; Rossberg et al., 2005, 2006b). Using topological features is adequate to test models aiming at mimicking topological features of real ecosystems. Is it also adequate to test models whose main purpose is not mimicking the topological features of real ecosystems?
Generally speaking, topological features of model webs incorporating population dynamics are not very similar to those of real webs. This is quite natural because models incorporating population dynamics do not mainly aim to mimic the topological features of real webs, whereas it is a main purpose of some models without population dynamics. In addition, models without population dynamics can more directly control the topology of resultant webs by adjusting parameters than those incorporating population dynamics.
Figure 4 represents the result of comparison between Drossel et al.'s (2004) model and the modified Yoshida (2002, 2003a) one, both of which incorporate population dynamics. As mentioned above, Drossel et al.'s (2004) model adopts an improved form of population dynamics and Optimal Foraging Theory. On the other hand, Yoshida's food web model uses the simplest form of Lotka-Volterra equations and does not incorporate Optimal Foraging Theory. However, the modified Yoshida model gives food web properties that are more similar to the average food web than Drossel et al.'s (2004) model does (Figure 4, Table 1). Several causes can be considered: 1) Yoshida is luckier than Drossel et al. (2004) in choice of values of parameters. 2) Ludwig and Walters's (1985) statement that a seemingly unrealistic simple model sometimes better represents a feature of the real world than a complex model which involves biologically adequate processes may be true of food web models. 3) Incorporating Holling's types II and III functional response is not appropriate for a model on an evolutionary time scale, because Holling's types II and III functional response are based on experiments or observation on the time scale of the individual-level life span (Holling, 1959; Begon et al., 1996; See Appendix), which is far shorter than the evolutionary time scale (In addition, the reason why the value of link density in Drossel et al.'s (2004) model is far from the average of real webs is due to an assumption adopted in Drossel et al.'s (2004) model that weak links are not counted, whereas all links are counted in Yoshida's (2002, 2003a) model.). The reason why biologically precise and detailed models do not well mimic real ecosystems must be discussed in further works.
Figure 4.
Comparison between model ecosystems and the mean of real ones.
The degree of difference between a model and the mean of real ecosystems is shown by using the standard deviations of values in each model (vertical axis). Div: species diversity in an ecosystem. %T, %I, %B: ratios of top predator, intermediate species, and basal species, respectively. L/S: link density, calculated by dividing the number of links by the species diversity. Each point represents the mean of simulations. A point with a positive value shows that the mean of the model is overestimated.
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Table 1.
Mean values of topological parameters of real and model ecosystems.
For parameters, see the legend of Figure 4. Values in parentheses represent standard deviations. Data for real webs are obtained from Dunne et al. (2002) in Drossel et al. (2004), those for Drossel et al.'s (2004) model are obtained from their paper, and those of Yoshida's model are based on 20 iterations of simulations.
![i1342-8144-10-4-375-t01.gif](ContentImages/Journals/jpal/10/4/prpsj.10.375/graphic/WebImages/i1342-8144-10-4-375-t01.gif)
To show topological similarity with real ecosystems is useful for models to show the reality of the models. It is possible that such models incorporate some essential mechanism controlling real ecosystems. This may be true especially in models aiming at mimicking the topological features of real ecosystems. In these models, it is significant to show which model is more similar to the topology of real ecosystems (Rossberg et al., 2006b). However, using topological features of ecosystems is not the only way to test the reality of models, because topology is only one of the features of ecosystems. Moreover, most of the models incorporating population dynamics do not mainly aim at mimicking the topological features of real ecosystems. Therefore, it is not always adequate to test such models only by depending on topological parameters. In comparison among such models, a result like Figure 4 is not so significant and does not indicate which model is superior. Therefore, it is nonsense to deny models addressing great challenges only on the grounds that they cannot mimic the topological features of real ecosystems. Then, one of the most important issues for theoretical studies is to establish the manner of testing models suitable for each purpose of the models.
Acknowledgments
This paper is based on an oral presentation given at a symposium commemorating the 70th anniversary of the Palaeontological Society of Japan. I especially thank Hiroshi Kitazato (Japan Agency of Marine-Earth Science and Technology) and Kazushige Tanabe (University of Tokyo) for giving me the opportunity of presenting this work, Takao Ubukata (Shizuoka University), Axel G. Rossberg (Yokohama National University), and an anonymous reviewer for their fruitful comments on the manuscript.
References
Appendices
Appendix
Ratio- and density-dependent models
In ratio-dependent models, the response of populations is proportional to the ratio of prey and predator (or predator and prey) densities, whereas in density-dependent models, the response is proportional to the product, as in the chemical principle of mass action (e.g., Berryman, 1992). Which model should be used was actively debated in the early 90's (e.g., Arditi et al., 1991; Berryman, 1992; Ginzburg and Akçakaya, 1992; Abrams, 1994; Gleeson, 1994; Sarnelle, 1994; Akçakaya et al., 1995; Berryman et al., 1995).
Optimal foraging theory
The optimal foraging theory states that animals feed in a way that maximizes the rate of energy intake per unit time. For the details, see Stephens and Krebs (1986).
Functional and numerical responses
Solomon (1949) classified the response of predator to the density of prey into numerical and functional responses. Numerical response means the change of density of predators by reproduction or migration related to the change of density of prey. Functional response represents the change of consumption rate of predators per capita related to the change of density of prey (Solomon, 1949). Functional response assumes that the consumption rate of predators per capita is influenced by searching and handling time for prey (Holling, 1959).
Holling's classification of functional response
Holling (1959) classified functional response into the following three types. Type I: The consumption rate of predators per capita linearly increases to a plateau, which is scarcely observed. Type II: The rate asymptotically increases to a plateau. Type III: The rate increases following a sigmoid curve to a plateau.
Holling's type II or type III functional response is frequently used because of the following four reasons: 1) the equations represent biologically appropriate processes (e.g., handling time for resources, not feeding on a full stomach), 2) the prototype of these equations was decided by an actual experiment (the dice equation; Holling, 1959), 3) Holling's type II functional response is widely observed in the real world (Holling, 1959; Begon et al., 1996), and 4) Holling's type III is considered to be adequate when there are refugia for prey species or the predation behavior of predators follows optimal foraging (Holling, 1965; Murdoch and Stewart-Oaten, 1975; Hassell et al., 1977).