Limiting similarity

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Limiting similarity (informally "limsim") is a concept in theoretical ecology and community ecology that proposes the existence of a maximum level of niche overlap between two given species that will allow continued coexistence.

Theoretical ecology

Theoretical ecology is the scientific discipline devoted to the study of ecological systems using theoretical methods such as simple conceptual models, mathematical models, computational simulations, and advanced data analysis. Effective models improve understanding of the natural world by revealing how the dynamics of species populations are often based on fundamental biological conditions and processes. Further, the field aims to unify a diverse range of empirical observations by assuming that common, mechanistic processes generate observable phenomena across species and ecological environments. Based on biologically realistic assumptions, theoretical ecologists are able to uncover novel, non-intuitive insights about natural processes. Theoretical results are often verified by empirical and observational studies, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.

Ecological niche The fit of a species living under specific environmental conditions.

In ecology, a niche is the match of a species to a specific environmental condition. It describes how an organism or population responds to the distribution of resources and competitors and how it in turn alters those same factors. "The type and number of variables comprising the dimensions of an environmental niche vary from one species to another [and] the relative importance of particular environmental variables for a species may vary according to the geographic and biotic contexts".

In biology, a species ( ) is the basic unit of classification and a taxonomic rank of an organism, as well as a unit of biodiversity. A species is often defined as the largest group of organisms in which any two individuals of the appropriate sexes or mating types can produce fertile offspring, typically by sexual reproduction. Other ways of defining species include their karyotype, DNA sequence, morphology, behaviour or ecological niche. In addition, paleontologists use the concept of the chronospecies since fossil reproduction cannot be examined. While these definitions may seem adequate, when looked at more closely they represent problematic species concepts. For example, the boundaries between closely related species become unclear with hybridisation, in a species complex of hundreds of similar microspecies, and in a ring species. Also, among organisms that reproduce only asexually, the concept of a reproductive species breaks down, and each clone is potentially a microspecies.

Contents

This concept is a corollary of the competitive exclusion principle, which states that, controlling for all else, two species competing for exactly the same resources cannot stably coexist. It assumes normally-distributed resource utilization curves ordered linearly along a resource axis, and as such, it is often considered to be an oversimplified model of species interactions. Moreover, it has theoretical weakness, and it is poor at generating real-world predictions or falsifiable hypotheses. Thus, the concept has fallen somewhat out of favor except in didactic settings (where it is commonly referenced), and has largely been replaced by more complex and inclusive theories.

Competitive exclusion principle A proposition that two species competing for the same limiting resource cannot coexist at constant population values

In ecology, the competitive exclusion principle, sometimes referred to as Gause's law, is a proposition named for Georgy Gause that two species competing for the same limiting resource cannot coexist at constant population values. When one species has even the slightest advantage over another, the one with the advantage will dominate in the long term. This leads either to the extinction of the weaker competitor or to an evolutionary or behavioral shift toward a different ecological niche. The principle has been paraphrased in the maxim "complete competitors cannot coexist".

History

In 1932, Georgii Gause created the competitive exclusion principle based on experiments with cultures of yeast and paramecium. [1] The principle maintains that two species with the same ecological niches cannot stably coexist. That is to say, when two species compete for identical resource access, one will be competitively superior and it will ultimately supplant the other. Over the next half century, limiting similarity slowly emerged as a natural outgrowth of this principle, aiming (but not necessarily succeeding) to be more quantitative and specific.

Yeast informal group of fungi

Yeasts are eukaryotic single-celled microorganisms classified as members of the fungus kingdom. The first yeast originated hundreds of millions of years ago, and 1,500 species are currently identified. They are estimated to constitute 1% of all described fungal species. Yeasts are unicellular organisms which evolved from multicellular ancestors, with some species having the ability to develop multicellular characteristics by forming strings of connected budding cells known as pseudohyphae or false hyphae. Yeast sizes vary greatly, depending on species and environment, typically measuring 3–4 µm in diameter, although some yeasts can grow to 40 µm in size. Most yeasts reproduce asexually by mitosis, and many do so by the asymmetric division process known as budding.

<i>Paramecium</i> genus of unicellular ciliates, commonly studied as a representative of the ciliate group

Paramecium is a genus of unicellular ciliates, commonly studied as a representative of the ciliate group. Paramecia are widespread in freshwater, brackish, and marine environments and are often very abundant in stagnant basins and ponds. Because some species are readily cultivated and easily induced to conjugate and divide, it has been widely used in classrooms and laboratories to study biological processes. Its usefulness as a model organism has caused one ciliate researcher to characterize it as the "white rat" of the phylum Ciliophora.

Noted ecologist and evolutionary biologist David Lack said retrospectively that he had already begun to mull around with the ideas of limiting similarity as early as the 1940s, but it wasn't until the end of the 1950s that the theory began to be built up and articulated. [2] G. Evelyn Hutchinson's famous "Homage to Santa Rosalia" was the next foundational paper in the history of the theory. Its subtitle famously asks, "Why are there so many kinds of animals?", and the address attempts to answer this question by suggesting theoretical bounds to speciation and niche overlap. For the purposes of understanding limiting similarity, the key portion of Hutchinson's address is the end where he presents the observation that a seemingly ubiquitous ratio (1.3:1) defines the upper bound of morphological character similarity between closely related species. [3] While this so-called Hutchinson ratio and the idea of a universal limit have been overturned by later research, the address was still foundational to the theory of limiting similarity.

David Lack British ornithologist and biologist

David Lambert Lack FRS was a British evolutionary biologist who made contributions to ornithology, ecology, and ethology. His 1947 book, Darwin's Finches, on the finches of the Galapagos Islands was a landmark work as were his other popular science books on Life of the Robin and Swifts in a Tower. He developed what is now known as Lack's Principle which explained the evolution of avian clutch sizes in terms of individual selection as opposed to the competing contemporary idea that they had evolved for the benefit of species. His pioneering life-history studies of the living bird helped in changing the nature of ornithology from what was then a collection-oriented field. He was a longtime director of the Edward Grey Institute of Field Ornithology at the University of Oxford.

G. Evelyn Hutchinson British ecologist (1903–1991).

George Evelyn Hutchinson, was a British ecologist sometimes described as the "father of modern ecology." He contributed for more than sixty years to the fields of limnology, systems ecology, radiation ecology, entomology, genetics, biogeochemistry, a mathematical theory of population growth, art history, philosophy, religion, and anthropology. He worked on the passage of phosphorus through lakes, the chemistry and biology of lakes, the theory of interspecific competition, and on insect taxonomy and genetics, zoo-geography and African water bugs. He is known as one of the first to combine ecology with mathematics. He became an international expert on lakes and wrote the four-volume Treatise on Limnology in 1957.

MacArthur and Levins were the first to introduce the term 'limiting similarity' in their 1967 paper. They attempted to lay out a rigorous quantitative basis for the theory using probability theory and the Lotka–Volterra competition equations. [4] In doing so, they provided the ultimate theoretical framework on which many subsequent studies were based.

Richard Levins Mathematical ecologist, university professor at Harvard School of Public Health, and political activist, who is best known for his work on evolution in changing environments and on metapopulations.

Richard "Dick" Levins was an ex-tropical farmer turned ecologist, a population geneticist, biomathematician, mathematical ecologist, and philosopher of science who had researched diversity in human populations. Until his death, Levins was a university professor at the Harvard T.H. Chan School of Public Health and a long-time political activist. He was best known for his work on evolution and complexity in changing environments and on metapopulations.

Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space. Any specified subset of these outcomes is called an event.

The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource. They can be further generalised to include trophic interactions.

Theory

As proposed by MacArthur and Levins in 1967, the theory of limiting similarity is rooted in the Lotka–Volterra competition model. This model describes two or more populations with logistic dynamics, adding in an additional term to account for their biological interactions. Thus for two populations, x1 and x2:

where

MacArthur and Levins examine this system applied to three populations, also visualized as resource utilization curves, depicted below. In this model, at some upper limit of competition α, between two species x1 and x3, the survival of a third species x2 between the other two is not possible. This phenomenon is termed limiting similarity. Evolutionary, if two species are more similar than some limit L, a third species will converge towards the nearer of the two competitors. If the two species are less similar than some limit L, a third species will evolve an intermediate phenotype.

[embedded graph: U v R. x1, x2, x3 curves.]

For each resource R, U represents the probability of utilization per unit time by an individual. At some level of overlap between species x1 and x3, the survival of a third species x2 is no longer possible.

May [5] extended this theory when considering species with different carrying capacities, concluding that coexistence was unlikely if the distance between the modes of competing resource utilization curves d was less than the standard deviation of the curves w.

Applied examples

It is of note that the theory of limiting similarity does not easily generate falsifiable predictions about natural phenomenon. However, many studies have tried to test the theory by making the highly suspect assumption that character displacement can be used as a close proxy for niche incongruence. [6] One recent paleoecological study, for example, used fossil proxies of gastropod body size to determine levels of character displacement over 42,500 years during the Quaternary. They found little evidence of character displacement, and they concluded that "limiting similarity, as seen in both ecological character displacement and community-wide character displacement, is a transient ecological phenomenon rather than a long-term evolutionary process". [7] Other theoretical and empirical studies tend to find results that similarly play down the strength and role of limiting similarity in ecology and evolution. For example, Abrams (who is prolific on the subject of limiting similarity) and Rueffler find in 2009 that "there is no absolute limit to similarity; there is always some range of mortality rates of one species allowing coexistence, given a fixed mortality of the other species". [8]

What a lot of studies examining limiting similarity find are the weaknesses in the original theory that are addressed below.

Criticism

The key weakness of the theory of limiting similarity is that it is highly system specific and thus difficult to test in practice. In actual environments, one resource axis is inadequate and a specific analysis must be done for each given pair of species. In practice it is necessary to take into account:

While these complications don't invalidate the concept, they render limiting similarity exceedingly difficult to test in practice and useful for little more than didacticism.

Furthermore, Hubbell and Foster point out that extinction via competition can take an extremely long time and the importance of limiting similarity in extinction may even be superseded by speciation. [9] Also, from a theoretical standpoint, small changes in carrying capacities can allow for nearly completely overlapping resource utilization curves and in practice carrying capacity can be difficult to determine. Many studies that attempt to explore limiting similarity (including Huntley et al. 2007) resort to examining character displacement as a proxy for niche overlap, which is suspect at best. While a useful-if simple-model, limiting similarity is nearly untestable in reality.

See also

Related Research Articles

The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:

Alfred J. Lotka American mathematician

Alfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist, Lotka is best known for his proposal of the predator–prey model, developed simultaneously but independently of Vito Volterra. The Lotka–Volterra model is still the basis of many models used in the analysis of population dynamics in ecology.

Population ecology Study of the dynamics of species populations and how these populations interact with the environment

Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment. It is the study of how the population sizes of species change over time and space. The term population ecology is often used interchangeably with population biology or population dynamics.

Metapopulation

A metapopulation consists of a group of spatially separated populations of the same species which interact at some level. The term metapopulation was coined by Richard Levins in 1969 to describe a model of population dynamics of insect pests in agricultural fields, but the idea has been most broadly applied to species in naturally or artificially fragmented habitats. In Levins' own words, it consists of "a population of populations".

Character displacement

Character displacement is the phenomenon where differences among similar species whose distributions overlap geographically are accentuated in regions where the species co-occur, but are minimized or lost where the species’ distributions do not overlap. This pattern results from evolutionary change driven by biological competition among species for a limited resource. The rationale for character displacement stems from the competitive exclusion principle, also called Gause's Law, which contends that to coexist in a stable environment two competing species must differ in their respective ecological niche; without differentiation, one species will eliminate or exclude the other through competition.

The term niche differentiation, as it applies to the field of ecology, refers to the process by which competing species use the environment differently in a way that helps them to coexist. The competitive exclusion principle states that if two species with identical niches compete, then one will inevitably drive the other to extinction. When two species differentiate their niches, they tend to compete less strongly, and are thus more likely to coexist. Species can differentiate their niches in many ways, such as by consuming different foods, or using different parts of the environment.

The paradox of enrichment is a term from population ecology coined by Michael Rosenzweig in 1971. He described an effect in six predator–prey models where increasing the food available to the prey caused the predator's population to destabilize. A common example is that if the food supply of a prey such as a rabbit is overabundant, its population will grow unbounded and cause the predator population to grow unsustainably large. That may result in a crash in the population of the predators and possibly lead to local eradication or even species extinction.

Competition (biology) Interaction where the fitness of one organism is lowered by the presence of another organism

Competition is an interaction between organisms or species in which both the organisms or species are harmed. Limited supply of at least one resource used by both can be a factor. Competition both within and between species is an important topic in ecology, especially community ecology. Competition is one of many interacting biotic and abiotic factors that affect community structure. Competition among members of the same species is known as intraspecific competition, while competition between individuals of different species is known as interspecific competition. Competition is not always straightforward, and can occur in both a direct and indirect fashion.

Interspecific competition where individuals of different species compete for the same resources

Interspecific competition, in ecology, is a form of competition in which individuals of different species compete for the same resources in an ecosystem. This can be contrasted with interspecific cooperation, a type of symbiosis. Competition between members of the same species is called intraspecific competition.

A population model is a type of mathematical model that is applied to the study of population dynamics.

In mathematical biology, the community matrix is the linearization of the Lotka–Volterra equation at an equilibrium point. The eigenvalues of the community matrix determine the stability of the equilibrium point.

Relative species abundance is a component of biodiversity and refers to how common or rare a species is relative to other species in a defined location or community. Relative abundance is the percent composition of an organism of a particular kind relative to the total number of organisms in the area. Relative species abundances tend to conform to specific patterns that are among the best-known and most-studied patterns in macroecology. Different populations in a community exist in relative proportions; this idea is known as relative abundance.

The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator–prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their dynamics can be analysed analytically to some extent. This makes them useful as a theoretical tool for modeling food webs. However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth.

A trophic function was first introduced in the differential equations of the Kolmogorov predator–prey model. It generalizes the linear case of predator–prey interaction firstly described by Volterra and Lotka in the Lotka–Volterra equation. A trophic function represents the consumption of prey assuming a given number of predators. The trophic function was widely applied in chemical kinetics, biophysics, mathematical physics and economics. In economics, "predator" and "prey" become various economic parameters such as prices and outputs of goods in various linked sectors such as processing and supply. These relationships, in turn, were found to behave similarly to the magnitudes in chemical kinetics, where the molecular analogues of predators and prey react chemically with each other.

Coexistence theory

Coexistence theory is a framework to understand how competitor traits can maintain species diversity and stave-off competitive exclusion even among similar species living in ecologically similar environments. Coexistence theory explains the stable coexistence of species as an interaction between two opposing forces: fitness differences between species, which should drive the best-adapted species to exclude others within a particular ecological niche, and stabilizing mechanisms, which maintains diversity via niche differentiation. For many species to be stabilized in a community, population growth must be negative density-dependent, i.e. all participating species have a tendency to increase in density as their populations decline. In such communities, any species that becomes rare will experience positive growth, pushing its population to recover and making local extinction unlikely. As the population of one species declines, individuals of that species tend to compete predominantly with individuals of other species. Thus, the tendency of a population to recover as it declines in density reflects reduced interspecific competition (between-species) relative to intraspecific competition (within-species), the signature of niche differentiation.

The R* rule is a hypothesis in community ecology that attempts to predict which species will become dominant as the result of competition for resources. It predicts that if multiple species are competing for a single limiting resource, then whichever species can survive at the lowest equilibrium resource level can outcompete all other species. If two species are competing for two resources, then coexistence is only possible if each species has a lower R* on one of the resources. For example, two phytoplankton species may be able to coexist if one is more limited by nitrogen, and the other is more limited by phosphorus.

References

  1. ↑ Gause, GF. 1932. Experimental studies on the struggle for existence. Journal of Experimental Biology 9: 389–402.
  2. ↑ Lack, D. 1973. My life as an amateur ornithologist. Ibis 115: 421–434.
  3. ↑ Hutchinson, GE. 1959. Homage to Santa Rosalia, or Why are there so many kinds of animals?. The American Naturalist 93(870): 145–159.
  4. ↑ MacArthur, R and R Levins. 1967. The Limiting Similarity, Convergence, and Divergence of Coexisting Species. The American Naturalist 101(921): 377–385.
  5. ↑ May, R. M. 1973. Stability and Complexity in Model Ecosystems. Princeton: Princeton Univ. Press
  6. ↑ Abrams P. 1983. The Theory of Limiting Similarity. Annual Review of Ecology, Evolution, and Systematics 14: 359–376.
  7. ↑ Huntley JW, Yanes Y, Kowalewski M, Castillo C, Delgado-Huertas A, Ibanez M, Alonso MR, Ortiz JE and T de Torres. 2008. Testing limiting similarity in Quaternary terrestrial gastropods. Paleobiology 34(3): 378–388.
  8. ↑ Abrams PA and C Rueffler. 2009. Coexistence and limiting similarity of consumer species competing for a linear array of resources. Ecology 90(3): 812–822.
  9. ↑ Hubbell, S. P. and Foster, R.B. (1986). Biology, chance, and history and the structure of tropical rain forest tree communities. In: Diamond, J. and Case, T.J. eds. Community ecology. Harper and Row, New York, pp. 314–329.
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