Theoretical ecology
Encyclopedia
Theoretical ecology is the scientific discipline devoted to the study of ecological systems
using theoretical methods such as simple conceptual model
s, mathematical model
s, computational simulation
s, 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 observation, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.
The field is broad and includes foundations in applied mathematics, computer science, biology, statistical physics, genetics, chemistry, evolution, and conservation biology. Theoretical ecology aims to explain a diverse range of phenomena in the life sciences, such as population growth and dynamics
, fisheries, competition, evolutionary theory, epidemiology, animal behavior and group dynamics, food webs, ecosystems, spatial ecology, and the effects of climate change.
Theoretical ecology has further benefited from the advent of fast computing power, allowing the analysis and visualization of large-scale computational simulations of ecological phenomena. Importantly, these modern tools provide quantitative predictions about the effects of human induced environmental change on a diverse variety of ecological phenomena, such as: species invasions, climate change, the effect of fishing and hunting on food network stability, and the global carbon cycle.
Ecological models can be deterministic or stochastic
.
Species can be modelled in continuous or discrete
time.
Models are often used to describe real ecological reproduction processes of single or multiple species.
These can be modelled using stochastic branching process
es. Examples are the dynamics of interacting populations (predation competition and mutualism), which, depending on the species of interest, may best be modeled over either continuous or discrete time. Other examples of such models may be found in the field of mathematical epidemiology
where the dynamic relationships that are to be modeled are host-pathogen interactions.
Bifurcation theory
is used to illustrate how small changes in parameter values can give rise to dramatically different long run outcomes, a mathematical fact that may be used to explain drastic ecological differences that come about in qualitatively very similar systems. Logistic map
s are polynomial mappings, and are often cited as providing archetypal examples of how chaotic behaviour
can arise from very simple non-linear dynamical equations. The maps were popularized in a seminal 1976 paper by the theoretical ecologist Robert May
. The difference equation is intended to capture the two effects of reproduction and starvation.
In 1930, R.A. Fisher published his classic The Genetical Theory of Natural Selection
, which introduced the idea that frequency-dependent fitness brings a strategic aspect to evolution
, where the payoffs to a particular organism, arising from the interplay of all of the relevant organisms, are the number of this organism' s viable offspring. In 1961, Richard Lewontin
applied game theory to evolutionary biology in his Evolution and the Theory of Games,
followed closely by John Maynard Smith
, who in his seminal 1972 paper, “Game Theory and the Evolution of Fighting", defined the concept of the evolutionarily stable strategy
.
Because ecological systems are typically nonlinear, they often cannot be solved analytically and in order to obtain sensible results, nonlinear, stochastic and computational techniques must be used. One class of computational models that is becoming increasingly popular are the agent-based models. These models can simulate the actions and interactions of multiple, heterogeneous, organisms where more traditional, analytical techniques are inadequate. Applied theoretical ecology yields results which are used in the real world. For example, optimal harvesting theory draws on optimization techniques developed in economics, computer science and operations research, and is widely used in fisheries.
where r is the per capita growth rate, or the intrinsic growth rate of the organism. It can also be described as r = b-d, where b and d are the per capita time-invariant birth and death rates, respectively. This first order linear differential equation
can be solved to yield the solution
.
This describes how the population grows exponentially in time, where N(0) is the initial population size, and is applicable in cases where a few organisms have begun a colony and are rapidly growing without any limitations or restrictions impeding their growth (e.g. bacteria inoculated in rich media).
). Both a and c will depend on other environmental factors which, we can for now, assume to be constant in this approximated model. The differential equation is now:
This can be rewritten as:
where r = b-d and K = (b-d)/(a+c).
The biological significance of K becomes apparent when stabilities of the equilibria of the system are considered. It is the carrying capacity
of the population. The equilibria of the system are N = 0 and N = K. If the system is linearized, it can be seen that N = 0 is an unstable equilibrium while K is a stable equilibrium.
The general form of this model is
where Nt is a vector
of the number of individuals in each class at time t and L is a matrix
that contains the survival probability and fecundity for each class. The matrix L is referred to as the Leslie matrix
for age-structured models, and as the Lefkovitch matrix for stage-structured models.
If parameter values in L are estimated from demographic data on a specific population, a structured model can then be used to predict whether this population is expected to grow or decline in the long-term, and what the expected age distribution
within the population will be. This has been done for a number of species including loggerhead sea turtle
s and right whale
s.
Taken in its most general sense, predation comprises predator-prey, host-pathogen, and host parasitoid interactions.
interactions exhibit natural oscillations in the populations of both predator and the prey. In 1925, the US mathematician Alfred J. Lotka
developed simple equations for predator-prey interactions in his book on biomathematics. The following year, the Italian mathematician Vito Volterra
, made a statistical analysis of fish catches in the Adriatic and independently developed the same equations. It is one of the earliest and most recognised ecological models, known as the Lotka-Volterra model:
where N is the prey and P is the predator population sizes, r is the rate for prey growth, taken to be exponential in the absence of any predators, α is the prey mortality rate for per-capita predation (also called ‘attack rate’), c is the efficiency of conversion from prey to predator, and d is the exponential death rate for predators in the absence of any prey.
Volterra originally used the model to explain fluctuations in fish and shark populations after fishing
was curtailed during the First World War. However, the equations have subsequently been applied more generally. Other examples of these models include the Lotka-Volterra model of the snowshoe hare
and Canadian lynx in North America, any infectious disease modeling such as the recent outbreak of SARS
and biological control of California red scale by the introduction of its parasitoid
, Aphytis melinus
.
, differs from predator-prey interactions in that pathogens are much smaller, have much faster generation times, and require a host to reproduce. Therefore, only the host population is tracked in host-pathogen models. Compartmental models that categorize host population into groups such as susceptible, infected, and recovered (SIR) are commonly used.
, can be analyzed by the Nicholson-Bailey model
, which differs from Lotka-Volterra and SIR models in that it is discrete in time. This model, like that of Lotka-Volterra, tracks both populations explicitly. Typically, in its general form, it states:
where f(Nt, Pt) describes the probability of infection (typically, Poisson distribution
), λ is the per-capita growth rate of hosts in the absence of parasitoids, and c is the conversion efficiency, as in the Lotka-Volterra model.
and E. coli,
as well as theoretical analysis of the behavior of the system.
The r coefficients give a “base” growth rate to each species, while K coefficients correspond to the carrying capacity. What can really change the dynamics of a system, however are the α terms. These describe the nature of the relationship between the two species. When α12 is negative, it means that N2 has a negative effect on N1, by competing with it, preying on it, or any number of other possibilities. When α12 is positive, however, it means that N2 has a positive effect on N1, through some kind of mutualistic interaction between the two.
When both α12 and α21 are negative, the relationship is described as competitive. In this case, each species detracts from the other, potentially over competition for scarce resources.
When both α12 and α21 are positive, the relationship becomes one of mutualism
. In this case, each species provides a benefit to the other, such that the presence of one aids the population growth of the other.
is the study of the distribution of species in space and time. It aims to reveal where organisms live, at what abundance, and why they are (or are not) found in a certain geographical area.
Biogeography is most keenly observed on islands, which has led to the development of the subdiscipline of island biogeography
. These habitats are often a more manageable areas of study because they are more condensed than larger ecosystems on the mainland. In 1967, Robert MacArthur
and E.O. Wilson published The Theory of Island Biogeography
. This showed that the species richness in an area could be predicted in terms of factors such as habitat area, immigration rate and extinction rate. The theory is considered one of the fundamentals of ecological theory. The application of island biogeography theory to habitat fragments
spurred the development of the fields of conservation biology
and landscape ecology
.
is a hypothesis proposed by Stephen Hubbell in 2001. The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories
in ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. Neutrality means that at a given trophic level
in a food web
, species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis. This implies that biodiversity arises at random, as each species follows a random walk
. This can be considered a null hypothesis
to niche theory
. The hypothesis has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.
Under unified neutral theory, complex ecological interactions are permitted among individuals of an ecological community (such as competition and cooperation), providing all individuals obey the same rules. Asymmetric phenomena such as parasitism and predation are ruled out by the terms of reference; but cooperative strategies such as swarming, and negative interaction such as competing for limited food or light are allowed, so long as all individuals behave the same way. The theory makes predictions that have implications for the management of biodiversity
, especially the management of rare species. It predicts the existence of a fundamental biodiversity constant, conventionally written θ, that appears to govern species richness on a wide variety of spatial and temporal scales.
Hubbell built on earlier neutral concepts, including MacArthur
& Wilson's theory of island biogeography
and Gould's concepts of symmetry and null models.
where m is the rate of colonization
, and e is the rate of extinction
. In this model, if e < m, the steady state value of p is 1 – (e/m) while in the other case, all the patches will eventually be left empty. This model may be made more complex by addition of another species in several different ways, including but not limited to game theoretic approaches, predator-prey interactions, etc. We will consider here an extension of the previous one-species system for simplicity. Let us denote the proportion of patches occupied by the first population as p1, and that by the second as p2. Then,
In this case, if e is too high, p1 and p2 will be zero at steady state. However, when the rate of extinction is moderate, p1 and p2 can stably coexist. The steady state value of p2 is given by
(p*1 may be inferred by symmetry).
It is interesting to note that if e is zero, the dynamics of the system favor the species that is better at colonizing (i.e. has the higher m value). This leads to a very important result in theoretical ecology known as the Intermediate Disturbance Hypothesis
, where the biodiversity
(the number of species that coexist in the population) is maximized when the disturbance (of which e is a proxy here) is not too high or too low, but at intermediate levels.
The form of the differential equations used in this simplistic modelling approach can be modified. For example:
The model can also be extended to combinations of the four possible linear or non-linear dependencies of colonization and extinction on p are described in more detail in.
or abiotic, into ecosystem
s can be disruptive. In some cases, it leads to ecological collapse
, trophic cascade
s and the death of many species within the ecosystem. The abstract notion of ecological health
attempts to measure the robustness and recovery capacity for an ecosystem; i.e. how far the ecosystem is away from its steady state. Often, however, ecosystems rebound from a disruptive agent. The difference between collapse or rebound depends on the toxicity
of the introduced element and the resiliency
of the original ecosystem.
If ecosystems are governed primarily by stochastic
processes, through which its subsequent state would be determined by both predictable and random actions, they may be more resilient to sudden change than each species individually. In the absence of a balance of nature
, the species composition of ecosystems would undergo shifts that would depend on the nature of the change, but entire ecological collapse would probably be infrequent events. In 1997, Robert Ulanowicz
used information theory
tools to describe the structure of ecosystems, emphasizing mutual information (correlations) in studied systems. Drawing on this methodology and prior observations of complex ecosystems, Ulanowicz depicts approaches to determining the stress levels on ecosystems and predicting system reactions to defined types of alteration in their settings (such as increased or reduced energy flow, and eutrophication
.
Ecopath
is a free ecosystem modelling software suite, initially developed by NOAA, and widely used in fisheries management as a tool for modelling and visualising the complex relationships that exist in real world marine ecosystems.
s provide a framework within which a complex network of predator–prey interactions can be organised. A food web model is a network of food chain
s. Each food chain starts with a primary producer or autotroph
, an organism, such as a plant, which is able to manufacture its own food. Next in the chain is an organism that feeds on the primary producer, and the chain continues in this way as a string of successive predators. The organisms in each chain are grouped into trophic level
s, based on how many links they are removed from the primary producers. The length of the chain, or trophic level, is a measure of the number of species encountered as energy or nutrients move from plants to top predators. Food energy
flows from one organism to the next and to the next and so on, with some energy being lost at each level. At a given trophic level there may be one species or a group of species with the same predators and prey.
In 1927, Charles Elton
published an influential synthesis on the use of food webs, which resulted in them becoming a central concept in ecology. In 1966, interest in food webs increased after Robert Paine's
experimental and descriptive study of intertidal shores, suggesting that food web complexity was key to maintaining species diversity and ecological stability. Many theoretical ecologists, including Sir Robert May
and Stuart Pimm
, were prompted by this discovery and others to examine the mathematical properties of food webs. According to their analyses, complex food webs should be less stable than simple food webs. The apparent paradox between the complexity of food webs observed in nature and the mathematical fragility of food web models is currently an area of intensive study and debate. The paradox may be due partially to conceptual differences between persistence of a food web and equilibrial stability
of a food web.
can be seen as an application of general systems theory to ecology. It takes a holistic and interdisciplinary approach to the study of ecological systems, and particularly ecosystems. Systems ecology is especially concerned with the way the functioning of ecosystems can be influenced by human interventions. Like other fields in theoretical ecology, it uses and extends concepts from thermodynamics
and develops other macroscopic descriptions of complex systems. It also takes account of the energy flows
through the different trophic level
s in the ecological networks. In systems ecology the principles of ecosystem energy flows are considered formally analogous to the principles of energetics. Systems ecology also considers the external influence of ecological economics
, which usually is not otherwise considered in ecosystem ecology. For the most part, systems ecology is a subfield of ecosystem ecology.
in some direction. Swarm behaviour is commonly exhibited by insects, but it also occurs in the flocking of birds, the schooling
of fish and the herd behaviour of quadrupeds. It is a complex emergent
behaviour that occurs when individual agents
follow simple behavioral rules. Recently, a number of mathematical models have been discovered which explain many aspects of the emergent behaviour.
Swarm algorithms follow a Lagrangian
approach or an Eulerian approach. The Eulerian approach views the swarm as a field
, working with the density of the swarm and deriving mean field properties. It is a hydrodynamic approach, and can be useful for modelling the overall dynamics of large swarms. However, most models work with the Lagrangian approach, which is an agent-based model following the individual agents (points or particles) that make up the swarm. Individual particle models can follow information on heading and spacing that is lost in the Eulerian approach. Examples include ant colony optimization
, self-propelled particles
and particle swarm optimization
is best known for independently proposing a theory of evolution
due to natural selection
that prompted Charles Darwin
to publish his own theory. In his famous 1858 paper, Wallace proposed natural selection as a kind of feedback mechanism which keeps species and varieties adapted to their environment.
The cybernetician
and anthropologist Gregory Bateson
observed in the 1970s that, though writing it only as an example, Wallace had "probably said the most powerful thing that’d been said in the 19th Century". Subsequently, the connection between natural selection and systems theory
has become an area of active research.
argues that the annual flood pulse is the most important aspect and the most biologically productive feature of a river's ecosystem
.
and his students. Brothers H.T. Odum
and E.P. Odum
are generally recognised as the founders of modern theoretical ecology. Robert MacArthur
brought theory to community ecology. Daniel Simberloff
was the student of E.O. Wilson, with whom MacArthur collaborated on The Theory of Island Biogeography
, a seminal work in the development of theoretical ecology.
Simberloff added statistical rigour to experimental ecology and was a key figure in the SLOSS debate
, about whether it is preferable to protect a single large or several small reserves. This resulted in the supporters of Jared Diamond
's community assembly rules defending their ideas through Neutral Model Analysis. Simberloff also played a key role in the (still ongoing) debate on the utility of corridors for connecting isolated reserves.
Stephen Hubbell and Michael Rosenzweig
combined theoretical and practical elements into works that extended MacArthur and Wilson's Island Biogeography Theory - Hubbell with his Unified Neutral Theory of Biodiversity and Biogeography
and Rosenzweig with his Species Diversity in Space and Time.
Ecosystem
An ecosystem is a biological environment consisting of all the organisms living in a particular area, as well as all the nonliving , physical components of the environment with which the organisms interact, such as air, soil, water and sunlight....
using theoretical methods such as simple conceptual model
Conceptual model
In the most general sense, a model is anything used in any way to represent anything else. Some models are physical objects, for instance, a toy model which may be assembled, and may even be made to work like the object it represents. They are used to help us know and understand the subject matter...
s, mathematical model
Mathematical model
A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used not only in the natural sciences and engineering disciplines A mathematical model is a...
s, computational simulation
Computer simulation
A computer simulation, a computer model, or a computational model is a computer program, or network of computers, that attempts to simulate an abstract model of a particular system...
s, and advanced data analysis
Data analysis
Analysis of data is a process of inspecting, cleaning, transforming, and modeling data with the goal of highlighting useful information, suggesting conclusions, and supporting decision making...
. 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 observation, revealing the power of theoretical methods in both predicting and understanding the noisy, diverse biological world.
The field is broad and includes foundations in applied mathematics, computer science, biology, statistical physics, genetics, chemistry, evolution, and conservation biology. Theoretical ecology aims to explain a diverse range of phenomena in the life sciences, such as population growth and dynamics
Population dynamics
Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...
, fisheries, competition, evolutionary theory, epidemiology, animal behavior and group dynamics, food webs, ecosystems, spatial ecology, and the effects of climate change.
Theoretical ecology has further benefited from the advent of fast computing power, allowing the analysis and visualization of large-scale computational simulations of ecological phenomena. Importantly, these modern tools provide quantitative predictions about the effects of human induced environmental change on a diverse variety of ecological phenomena, such as: species invasions, climate change, the effect of fishing and hunting on food network stability, and the global carbon cycle.
Modelling approaches
As in most other sciences, mathematical models form the foundation of modern ecological theory.- Phenomenological models: distill the functional and distributional shapes from observed patterns in the data, or researchers decide on functions and distribution that are flexible enough to match the patterns they or others (field or experimental ecologists) have found in the field or through experimentation.
- Mechanistic models: model the underlying processes directly, with functions and distributions that are based on theoretical reasoning about ecological processes of interest.
Ecological models can be deterministic or stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
.
- Deterministic models always evolve in the same way from a given starting point. They represent the average, expected behavior of a system, but lack random variation. Many system dynamicsSystem dynamicsSystem dynamics is an approach to understanding the behaviour of complex systems over time. It deals with internal feedback loops and time delays that affect the behaviour of the entire system. What makes using system dynamics different from other approaches to studying complex systems is the use...
models are deterministic. - Stochastic models allow for the direct modeling of the random perturbations that underlie real world ecological systems. Markov chain modelsMarkov processIn probability theory and statistics, a Markov process, named after the Russian mathematician Andrey Markov, is a time-varying random phenomenon for which a specific property holds...
are stochastic.
Species can be modelled in continuous or discrete
Discrete time
Discrete time is the discontinuity of a function's time domain that results from sampling a variable at a finite interval. For example, consider a newspaper that reports the price of crude oil once every day at 6:00AM. The newspaper is described as sampling the cost at a frequency of once per 24...
time.
- Continuous time is modelled using differential equationDifferential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s. - Discrete time is modelled using difference equations. These model ecological processes that can be described as occurring over discrete time steps. Matrix algebraMatrix algebraMatrix algebra may refer to:*Matrix theory, is the branch of mathematics that studies matrices*Matrix ring, thought of as an algebra over a field or a commutative ring...
is often used to investigate the evolution of age-structured or stage-structured populations. The Leslie matrixLeslie matrixIn applied mathematics, the Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology. It was invented by and named after Patrick H. Leslie...
, for example, mathematically represents the discrete time change of an age structured population.
Models are often used to describe real ecological reproduction processes of single or multiple species.
These can be modelled using stochastic branching process
Branching process
In probability theory, a branching process is a Markov process that models a population in which each individual in generation n produces some random number of individuals in generation n + 1, according to a fixed probability distribution that does not vary from individual to...
es. Examples are the dynamics of interacting populations (predation competition and mutualism), which, depending on the species of interest, may best be modeled over either continuous or discrete time. Other examples of such models may be found in the field of mathematical epidemiology
Epidemic model
An Epidemic model is a simplified means of describing the transmission of communicable disease through individuals.-Introduction:The outbreak and spread of disease has been questioned and studied for many years...
where the dynamic relationships that are to be modeled are host-pathogen interactions.
Bifurcation theory
Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations...
is used to illustrate how small changes in parameter values can give rise to dramatically different long run outcomes, a mathematical fact that may be used to explain drastic ecological differences that come about in qualitatively very similar systems. Logistic map
Logistic map
The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations...
s are polynomial mappings, and are often cited as providing archetypal examples of how chaotic behaviour
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
can arise from very simple non-linear dynamical equations. The maps were popularized in a seminal 1976 paper by the theoretical ecologist Robert May
Robert May, Baron May of Oxford
Robert McCredie May, Baron May of Oxford, OM, AC, PRS is an Australian scientist who has been Chief Scientific Adviser to the UK Government, President of the Royal Society, and a Professor at Sydney and Princeton. He now holds joint professorships at Oxford, and Imperial College London...
. The difference equation is intended to capture the two effects of reproduction and starvation.
In 1930, R.A. Fisher published his classic The Genetical Theory of Natural Selection
The Genetical Theory of Natural Selection
The Genetical Theory of Natural Selection is a book by R.A. Fisher first published in 1930 by Clarendon. It is one of the most important books of the modern evolutionary synthesis and is commonly cited in biology books.-Editions:...
, which introduced the idea that frequency-dependent fitness brings a strategic aspect to evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
, where the payoffs to a particular organism, arising from the interplay of all of the relevant organisms, are the number of this organism' s viable offspring. In 1961, Richard Lewontin
Richard Lewontin
Richard Charles "Dick" Lewontin is an American evolutionary biologist, geneticist and social commentator. A leader in developing the mathematical basis of population genetics and evolutionary theory, he pioneered the notion of using techniques from molecular biology such as gel electrophoresis to...
applied game theory to evolutionary biology in his Evolution and the Theory of Games,
followed closely by John Maynard Smith
John Maynard Smith
John Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
, who in his seminal 1972 paper, “Game Theory and the Evolution of Fighting", defined the concept of the evolutionarily stable strategy
Evolutionarily stable strategy
In game theory and behavioural ecology, an evolutionarily stable strategy , which is sometimes also called an evolutionary stable strategy, is a strategy which, if adopted by a population of players, cannot be invaded by any alternative strategy that is initially rare. An ESS is an equilibrium...
.
Because ecological systems are typically nonlinear, they often cannot be solved analytically and in order to obtain sensible results, nonlinear, stochastic and computational techniques must be used. One class of computational models that is becoming increasingly popular are the agent-based models. These models can simulate the actions and interactions of multiple, heterogeneous, organisms where more traditional, analytical techniques are inadequate. Applied theoretical ecology yields results which are used in the real world. For example, optimal harvesting theory draws on optimization techniques developed in economics, computer science and operations research, and is widely used in fisheries.
Exponential growth
The most basic way of modeling population dynamics is to assume that the rate of growth of a population depends only upon the population size at that time and the per capita growth rate of the organism. In other words, if the number of individuals in a population at a time t, is N(t), then the rate of population growth is given by:where r is the per capita growth rate, or the intrinsic growth rate of the organism. It can also be described as r = b-d, where b and d are the per capita time-invariant birth and death rates, respectively. This first order linear differential equation
Linear differential equation
Linear differential equations are of the formwhere the differential operator L is a linear operator, y is the unknown function , and the right hand side ƒ is a given function of the same nature as y...
can be solved to yield the solution
.This describes how the population grows exponentially in time, where N(0) is the initial population size, and is applicable in cases where a few organisms have begun a colony and are rapidly growing without any limitations or restrictions impeding their growth (e.g. bacteria inoculated in rich media).
Logistic growth
The exponential growth model makes a number of assumptions, many of which often do not hold. For example, many factors affect the intrinsic growth rate and is often not time-invariant. A simple modification of the exponential growth is to assume that the intrinsic growth rate varies with population size. This is reasonable: the larger the population size, the fewer resources available, which can result in a lower birth rate and higher death rate. Hence, we can replace the time-invariant r with r’(t) = (b –a*N(t)) – (d + c*N(t)), where a and c are constants that modulate birth and death rates in a population dependent manner (e.g. intraspecific competitionIntraspecific competition
Intraspecific competition is a particular form of competition in which members of the same species vie for the same resource in an ecosystem...
). Both a and c will depend on other environmental factors which, we can for now, assume to be constant in this approximated model. The differential equation is now:
This can be rewritten as:
where r = b-d and K = (b-d)/(a+c).
The biological significance of K becomes apparent when stabilities of the equilibria of the system are considered. It is the carrying capacity
Carrying capacity
The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment...
of the population. The equilibria of the system are N = 0 and N = K. If the system is linearized, it can be seen that N = 0 is an unstable equilibrium while K is a stable equilibrium.
Structured growth
Another assumption of the exponential growth model is that all individuals within a population are identical and have the same probabilities of surviving and of reproducing. This is not a valid assumption for species with complex life histories. The exponential growth model can be modified to account for this, by tracking the number of individuals in different age classes (e.g. one-, two-, and three-year-olds) or different stage classes (juveniles, sub-adults, and adults) separately, and allowing individuals in each group to have their own survival and reproduction rates.The general form of this model is
where Nt is a vector
Vector
Vector, a Latin word meaning "carrier", may refer in English to:-In computer science:*A one-dimensional array**Vector , a data type in the C++ Standard Template Library...
of the number of individuals in each class at time t and L is a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
that contains the survival probability and fecundity for each class. The matrix L is referred to as the Leslie matrix
Leslie matrix
In applied mathematics, the Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology. It was invented by and named after Patrick H. Leslie...
for age-structured models, and as the Lefkovitch matrix for stage-structured models.
If parameter values in L are estimated from demographic data on a specific population, a structured model can then be used to predict whether this population is expected to grow or decline in the long-term, and what the expected age distribution
Population pyramid
A population pyramid, also called an age structure diagram, is a graphical illustration that shows the distribution of various age groups in a population , which forms the shape of a pyramid when the population is growing...
within the population will be. This has been done for a number of species including loggerhead sea turtle
Loggerhead sea turtle
The loggerhead sea turtle , or loggerhead, is an oceanic turtle distributed throughout the world. It is a marine reptile, belonging to the family Cheloniidae. The average loggerhead measures around long when fully grown, although larger specimens of up to have been discovered...
s and right whale
Right whale
Right whales are three species of large baleen whales consisting of two genera in the family Balaenidae of order Cetacea. Their bodies are very dark gray or black and rotund....
s.
Community ecology
An ecological community is a group of trophically similar, sympatric species that actually or potentially compete in a local area for the same or similar resources. Interactions between these species form the first steps in analyzing more complex dynamics of ecosystems. These interactions shape the distribution and dynamics of species. Of these interactions, predation is one of the most widespread population activities.Taken in its most general sense, predation comprises predator-prey, host-pathogen, and host parasitoid interactions.
Predator-prey
Predator-preyPredation
In ecology, predation describes a biological interaction where a predator feeds on its prey . Predators may or may not kill their prey prior to feeding on them, but the act of predation always results in the death of its prey and the eventual absorption of the prey's tissue through consumption...
interactions exhibit natural oscillations in the populations of both predator and the prey. In 1925, the US mathematician Alfred J. Lotka
Alfred J. Lotka
Alfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist best known for his proposal of the predator-prey model, developed simultaneously but independently of Vito Volterra...
developed simple equations for predator-prey interactions in his book on biomathematics. The following year, the Italian mathematician Vito Volterra
Vito Volterra
Vito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....
, made a statistical analysis of fish catches in the Adriatic and independently developed the same equations. It is one of the earliest and most recognised ecological models, known as the Lotka-Volterra model:
where N is the prey and P is the predator population sizes, r is the rate for prey growth, taken to be exponential in the absence of any predators, α is the prey mortality rate for per-capita predation (also called ‘attack rate’), c is the efficiency of conversion from prey to predator, and d is the exponential death rate for predators in the absence of any prey.
Volterra originally used the model to explain fluctuations in fish and shark populations after fishing
Fishery
Generally, a fishery is an entity engaged in raising or harvesting fish which is determined by some authority to be a fishery. According to the FAO, a fishery is typically defined in terms of the "people involved, species or type of fish, area of water or seabed, method of fishing, class of boats,...
was curtailed during the First World War. However, the equations have subsequently been applied more generally. Other examples of these models include the Lotka-Volterra model of the snowshoe hare
Snowshoe Hare
The Snowshoe Hare , also called the Varying Hare, or Snowshoe Rabbit, is a species of hare found in North America. It has the name "snowshoe" because of the large size of its hind feet and the marks its tail leaves. The animal's feet prevent it from sinking into the snow when it hops and walks...
and Canadian lynx in North America, any infectious disease modeling such as the recent outbreak of SARS
and biological control of California red scale by the introduction of its parasitoid
Parasitoid
A parasitoid is an organism that spends a significant portion of its life history attached to or within a single host organism in a relationship that is in essence parasitic; unlike a true parasite, however, it ultimately sterilises or kills, and sometimes consumes, the host...
, Aphytis melinus
Aphytis melinus
Aphytis melinus is an internal parasite of the California Red Scale, Aonidiella aurantii, which is a pest of citrus in California and elsewhere. This chalcid wasp drums its antennae against the scale insect to find out if it is healthy, if it is already parasitized, how large it is, etc., to decide...
.
Host-pathogen
The second interaction, that of host and pathogenPathogen
A pathogen gignomai "I give birth to") or infectious agent — colloquially, a germ — is a microbe or microorganism such as a virus, bacterium, prion, or fungus that causes disease in its animal or plant host...
, differs from predator-prey interactions in that pathogens are much smaller, have much faster generation times, and require a host to reproduce. Therefore, only the host population is tracked in host-pathogen models. Compartmental models that categorize host population into groups such as susceptible, infected, and recovered (SIR) are commonly used.
Host-parasitoid
The third interaction, that of host and parasitoidParasitoid
A parasitoid is an organism that spends a significant portion of its life history attached to or within a single host organism in a relationship that is in essence parasitic; unlike a true parasite, however, it ultimately sterilises or kills, and sometimes consumes, the host...
, can be analyzed by the Nicholson-Bailey model
Nicholson-Bailey model
The Nicholson–Bailey model was developed in the 1930s to describe the population dynamics of a coupled host-parasite system. It is named after Alexander John Nicholson and Victor Albert Bailey....
, which differs from Lotka-Volterra and SIR models in that it is discrete in time. This model, like that of Lotka-Volterra, tracks both populations explicitly. Typically, in its general form, it states:
where f(Nt, Pt) describes the probability of infection (typically, Poisson distribution
Poisson distribution
In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since...
), λ is the per-capita growth rate of hosts in the absence of parasitoids, and c is the conversion efficiency, as in the Lotka-Volterra model.
Competition and mutualism
In studies of the populations of two species, the Lotka-Volterra system of equations has been extensively used to describe dynamics of behavior between two species, N1 and N2. Examples include relations between D. discoiderumDictyostelium discoideum
Dictyostelium discoideum is a species of soil-living amoeba belonging to the phylum Mycetozoa. D. discoideum, commonly referred to as slime mold, is a eukaryote that transitions from a collection of unicellular amoebae into a multicellular slug and then into a fruiting body within its lifetime. D...
and E. coli,
as well as theoretical analysis of the behavior of the system.
The r coefficients give a “base” growth rate to each species, while K coefficients correspond to the carrying capacity. What can really change the dynamics of a system, however are the α terms. These describe the nature of the relationship between the two species. When α12 is negative, it means that N2 has a negative effect on N1, by competing with it, preying on it, or any number of other possibilities. When α12 is positive, however, it means that N2 has a positive effect on N1, through some kind of mutualistic interaction between the two.
When both α12 and α21 are negative, the relationship is described as competitive. In this case, each species detracts from the other, potentially over competition for scarce resources.
When both α12 and α21 are positive, the relationship becomes one of mutualism
Mutualism
Mutualism is the way two organisms of different species biologically interact in a relationship in which each individual derives a fitness benefit . Similar interactions within a species are known as co-operation...
. In this case, each species provides a benefit to the other, such that the presence of one aids the population growth of the other.
- See Competitive Lotka-Volterra equationsCompetitive Lotka-Volterra equationsThe 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.-Overview:...
for further extensions of this model.
Biogeography
BiogeographyBiogeography
Biogeography is the study of the distribution of species , organisms, and ecosystems in space and through geological time. Organisms and biological communities vary in a highly regular fashion along geographic gradients of latitude, elevation, isolation and habitat area...
is the study of the distribution of species in space and time. It aims to reveal where organisms live, at what abundance, and why they are (or are not) found in a certain geographical area.
Biogeography is most keenly observed on islands, which has led to the development of the subdiscipline of island biogeography
Island biogeography
Island biogeography is a field within biogeography that attempts to establish and explain the factors that affect the species richness of natural communities. The theory was developed to explain species richness of actual islands...
. These habitats are often a more manageable areas of study because they are more condensed than larger ecosystems on the mainland. In 1967, Robert MacArthur
Robert MacArthur
Robert Helmer MacArthur was an American ecologist who made a major impact on many areas of community and population ecology....
and E.O. Wilson published The Theory of Island Biogeography
The Theory of Island Biogeography
The Theory of Island Biogeography is a 1967 book by Edward O. Wilson and Robert MacArthur which laid the foundations for the study of island biogeography. An edition with a new preface by Edward O. Wilson was published in 2001 ....
. This showed that the species richness in an area could be predicted in terms of factors such as habitat area, immigration rate and extinction rate. The theory is considered one of the fundamentals of ecological theory. The application of island biogeography theory to habitat fragments
Habitat fragmentation
Habitat fragmentation as the name implies, describes the emergence of discontinuities in an organism's preferred environment , causing population fragmentation...
spurred the development of the fields of conservation biology
Conservation biology
Conservation biology is the scientific study of the nature and status of Earth's biodiversity with the aim of protecting species, their habitats, and ecosystems from excessive rates of extinction...
and landscape ecology
Landscape ecology
Landscape ecology is the science of studying and improving relationships between urban development and ecological processes in the environment and particular ecosystems...
.
Neutral theory
Unified neutral theoryUnified neutral theory of biodiversity
The unified neutral theory of biodiversity and biogeography is a hypothesis and the title of a monograph by ecologist Stephen Hubbell...
is a hypothesis proposed by Stephen Hubbell in 2001. The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories
Neutral theory
Neutral theory may refer to one of these two related theories:* the neutral theory of molecular evolution; or* the unified neutral theory of biodiversity....
in ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. Neutrality means that at a given trophic level
Trophic level
The trophic level of an organism is the position it occupies in a food chain. The word trophic derives from the Greek τροφή referring to food or feeding. A food chain represents a succession of organisms that eat another organism and are, in turn, eaten themselves. The number of steps an organism...
in a food web
Food web
A food web depicts feeding connections in an ecological community. Ecologists can broadly lump all life forms into one of two categories called trophic levels: 1) the autotrophs, and 2) the heterotrophs...
, species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis. This implies that biodiversity arises at random, as each species follows a random walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
. This can be considered a null hypothesis
Null hypothesis
The practice of science involves formulating and testing hypotheses, assertions that are capable of being proven false using a test of observed data. The null hypothesis typically corresponds to a general or default position...
to niche theory
Ecological niche
In ecology, a niche is a term describing the relational position of a species or population in its ecosystem to each other; e.g. a dolphin could potentially be in another ecological niche from one that travels in a different pod if the members of these pods utilize significantly different food...
. The hypothesis has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.
Under unified neutral theory, complex ecological interactions are permitted among individuals of an ecological community (such as competition and cooperation), providing all individuals obey the same rules. Asymmetric phenomena such as parasitism and predation are ruled out by the terms of reference; but cooperative strategies such as swarming, and negative interaction such as competing for limited food or light are allowed, so long as all individuals behave the same way. The theory makes predictions that have implications for the management of biodiversity
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
, especially the management of rare species. It predicts the existence of a fundamental biodiversity constant, conventionally written θ, that appears to govern species richness on a wide variety of spatial and temporal scales.
Hubbell built on earlier neutral concepts, including MacArthur
Robert MacArthur
Robert Helmer MacArthur was an American ecologist who made a major impact on many areas of community and population ecology....
& Wilson's theory of island biogeography
Island biogeography
Island biogeography is a field within biogeography that attempts to establish and explain the factors that affect the species richness of natural communities. The theory was developed to explain species richness of actual islands...
and Gould's concepts of symmetry and null models.
Metapopulations
Spatial analysis of ecological systems often reveals that assumptions that are valid for spatially homogenous populations – and indeed, intuitive – may no longer be valid when migratory subpopulations moving from one patch to another are considered. In a simple one-species formulation, a subpopulation may occupy a patch, move from one patch to another empty patch, or die out leaving an empty patch behind. In such a case, the proportion of occupied patches may be represented aswhere m is the rate of colonization
Colonisation (biology)
Colonisation is the process in biology by which a species spreads into new areas, regions, and continents. It is sometimes also referred to as immigration, but colonisation often refers to successful immigration with integration to a community, having resisted initial local extinction.One classic...
, and e is the rate of extinction
Extinction
In biology and ecology, extinction is the end of an organism or of a group of organisms , normally a species. The moment of extinction is generally considered to be the death of the last individual of the species, although the capacity to breed and recover may have been lost before this point...
. In this model, if e < m, the steady state value of p is 1 – (e/m) while in the other case, all the patches will eventually be left empty. This model may be made more complex by addition of another species in several different ways, including but not limited to game theoretic approaches, predator-prey interactions, etc. We will consider here an extension of the previous one-species system for simplicity. Let us denote the proportion of patches occupied by the first population as p1, and that by the second as p2. Then,
In this case, if e is too high, p1 and p2 will be zero at steady state. However, when the rate of extinction is moderate, p1 and p2 can stably coexist. The steady state value of p2 is given by
(p*1 may be inferred by symmetry).
It is interesting to note that if e is zero, the dynamics of the system favor the species that is better at colonizing (i.e. has the higher m value). This leads to a very important result in theoretical ecology known as the Intermediate Disturbance Hypothesis
Intermediate Disturbance Hypothesis
The Intermediate Disturbance Hypothesis states that local species diversity is maximized when ecological disturbance is neither too rare nor too frequent. At low levels of disturbance, more competitive organisms will push subordinate species to extinction and dominate the ecosystem...
, where the biodiversity
Biodiversity
Biodiversity is the degree of variation of life forms within a given ecosystem, biome, or an entire planet. Biodiversity is a measure of the health of ecosystems. Biodiversity is in part a function of climate. In terrestrial habitats, tropical regions are typically rich whereas polar regions...
(the number of species that coexist in the population) is maximized when the disturbance (of which e is a proxy here) is not too high or too low, but at intermediate levels.
The form of the differential equations used in this simplistic modelling approach can be modified. For example:
- Colonization may be dependent on p linearly (m*(1-p)) as opposed to the non-linear m*p*(1-p) regime described above. This mode of replication of a species is called the “rain of propagules”, where there is an abundance of new individuals entering the population at every generation. In such a scenario, the steady state where the population is zero is usually unstable.
- Extinction may depend non-linearly on p (e*p*(1-p)) as opposed to the linear (e*p) regime described above. This is referred to as the “rescue effect” and it is again harder to drive a population extinct under this regime.
The model can also be extended to combinations of the four possible linear or non-linear dependencies of colonization and extinction on p are described in more detail in.
Ecosystem ecology
Introducing new elements, whether bioticBiotic component
Biotic components are the living things that shape an ecosystem. A biotic factor is any living component that affects another organism, including animals that consume the organism in question, and the living food that the organism consumes. Each biotic factor needs energy to do work and food for...
or abiotic, into ecosystem
Ecosystem
An ecosystem is a biological environment consisting of all the organisms living in a particular area, as well as all the nonliving , physical components of the environment with which the organisms interact, such as air, soil, water and sunlight....
s can be disruptive. In some cases, it leads to ecological collapse
Ecological collapse
Ecological Collapse refers to a situation where an ecosystem suffers a drastic, if not permanent, reduction in carrying capacity for all organisms, often resulting in mass extinction...
, trophic cascade
Trophic cascade
Trophic cascades occur when predators in a food web suppress the abundance of their prey, thereby releasing the next lower trophic level from predation...
s and the death of many species within the ecosystem. The abstract notion of ecological health
Ecological health
Ecological health or ecological integrity or ecological damage are the symptoms of an ecosystem's pending loss of carrying capacity, its ability to perform ecological services, or a pending ecocide, due to cumulative causes such as pollution. it can also be defined as farming so as to minimize the...
attempts to measure the robustness and recovery capacity for an ecosystem; i.e. how far the ecosystem is away from its steady state. Often, however, ecosystems rebound from a disruptive agent. The difference between collapse or rebound depends on the toxicity
Toxicity
Toxicity is the degree to which a substance can damage a living or non-living organisms. Toxicity can refer to the effect on a whole organism, such as an animal, bacterium, or plant, as well as the effect on a substructure of the organism, such as a cell or an organ , such as the liver...
of the introduced element and the resiliency
Resilience (ecology)
In ecology, resilience is the capacity of an ecosystem to respond to a perturbation or disturbance by resisting damage and recovering quickly. Such perturbations and disturbances can include stochastic events such as fires, flooding, windstorms, insect population explosions, and human activities...
of the original ecosystem.
If ecosystems are governed primarily by stochastic
Stochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
processes, through which its subsequent state would be determined by both predictable and random actions, they may be more resilient to sudden change than each species individually. In the absence of a balance of nature
Balance of nature
The balance of nature is a theory that says that ecological systems are usually in a stable equilibrium , which is to say that a small change in some particular parameter will be corrected by some negative feedback that will bring the parameter back to its original "point of balance" with the rest...
, the species composition of ecosystems would undergo shifts that would depend on the nature of the change, but entire ecological collapse would probably be infrequent events. In 1997, Robert Ulanowicz
Robert Ulanowicz
Robert Edward Ulanowicz is an American theoretical ecologist and philosopher who is best known for his search for a unified theory of ecology. He was born September 17, 1943 in Baltimore, Maryland....
used information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
tools to describe the structure of ecosystems, emphasizing mutual information (correlations) in studied systems. Drawing on this methodology and prior observations of complex ecosystems, Ulanowicz depicts approaches to determining the stress levels on ecosystems and predicting system reactions to defined types of alteration in their settings (such as increased or reduced energy flow, and eutrophication
Eutrophication
Eutrophication or more precisely hypertrophication, is the movement of a body of water′s trophic status in the direction of increasing plant biomass, by the addition of artificial or natural substances, such as nitrates and phosphates, through fertilizers or sewage, to an aquatic system...
.
Ecopath
Ecopath
Ecopath with Ecosim is a free ecosystem modelling software suite. It was initially a NOAA initiative led by Jeffrey Polovina, but since primarily developed at the Fisheries Centre of the University of British Columbia. In 2007, it was named as one of the ten biggest scientific breakthroughs in...
is a free ecosystem modelling software suite, initially developed by NOAA, and widely used in fisheries management as a tool for modelling and visualising the complex relationships that exist in real world marine ecosystems.
Food webs
Food webFood web
A food web depicts feeding connections in an ecological community. Ecologists can broadly lump all life forms into one of two categories called trophic levels: 1) the autotrophs, and 2) the heterotrophs...
s provide a framework within which a complex network of predator–prey interactions can be organised. A food web model is a network of food chain
Food chain
A food web depicts feeding connections in an ecological community. Ecologists can broadly lump all life forms into one of two categories called trophic levels: 1) the autotrophs, and 2) the heterotrophs...
s. Each food chain starts with a primary producer or autotroph
Autotroph
An autotroph, or producer, is an organism that produces complex organic compounds from simple inorganic molecules using energy from light or inorganic chemical reactions . They are the producers in a food chain, such as plants on land or algae in water...
, an organism, such as a plant, which is able to manufacture its own food. Next in the chain is an organism that feeds on the primary producer, and the chain continues in this way as a string of successive predators. The organisms in each chain are grouped into trophic level
Trophic level
The trophic level of an organism is the position it occupies in a food chain. The word trophic derives from the Greek τροφή referring to food or feeding. A food chain represents a succession of organisms that eat another organism and are, in turn, eaten themselves. The number of steps an organism...
s, based on how many links they are removed from the primary producers. The length of the chain, or trophic level, is a measure of the number of species encountered as energy or nutrients move from plants to top predators. Food energy
Food energy
Food energy is the amount of energy obtained from food that is available through cellular respiration.Food energy is expressed in food calories or kilojoules...
flows from one organism to the next and to the next and so on, with some energy being lost at each level. At a given trophic level there may be one species or a group of species with the same predators and prey.
In 1927, Charles Elton
Charles Sutherland Elton
Charles Sutherland Elton FRS was an English zoologist and animal ecologist. His name is associated with the establishment of modern population and community ecology, including studies of invasive organisms.-Personal life:...
published an influential synthesis on the use of food webs, which resulted in them becoming a central concept in ecology. In 1966, interest in food webs increased after Robert Paine's
Robert T. Paine (zoologist)
Robert T. Paine, PhD, is a zoologist and professor emeritus of the University of Washington,who coined the keystone species concept in order to explain the relationship between Pisaster ochraceus, a species of starfish, and Mytilus californianus, a species of mussel.In his classic 1966 paper, Dr...
experimental and descriptive study of intertidal shores, suggesting that food web complexity was key to maintaining species diversity and ecological stability. Many theoretical ecologists, including Sir Robert May
Robert May, Baron May of Oxford
Robert McCredie May, Baron May of Oxford, OM, AC, PRS is an Australian scientist who has been Chief Scientific Adviser to the UK Government, President of the Royal Society, and a Professor at Sydney and Princeton. He now holds joint professorships at Oxford, and Imperial College London...
and Stuart Pimm
Stuart Pimm
Stuart Pimm is an American-British biologist and theoretical ecologist specializing in scientific research of biodiversity and conservation biology.- Professional career :...
, were prompted by this discovery and others to examine the mathematical properties of food webs. According to their analyses, complex food webs should be less stable than simple food webs. The apparent paradox between the complexity of food webs observed in nature and the mathematical fragility of food web models is currently an area of intensive study and debate. The paradox may be due partially to conceptual differences between persistence of a food web and equilibrial stability
Ecological stability
Ecological stability can refer to types of stability in a continuum ranging from resilience to constancy to persistence. The precise definition depends on the ecosystem in question, the variable or variables of interest, and the overall context...
of a food web.
Systems ecology
Systems ecologySystems ecology
Systems ecology is an interdisciplinary field of ecology, taking a holistic approach to the study of ecological systems, especially ecosystems. Systems ecology can be seen as an application of general systems theory to ecology. Central to the systems ecology approach is the idea that an ecosystem...
can be seen as an application of general systems theory to ecology. It takes a holistic and interdisciplinary approach to the study of ecological systems, and particularly ecosystems. Systems ecology is especially concerned with the way the functioning of ecosystems can be influenced by human interventions. Like other fields in theoretical ecology, it uses and extends concepts from thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
and develops other macroscopic descriptions of complex systems. It also takes account of the energy flows
Ecological energetics
Ecological energetics is the quantitative study of the flow of energy through ecological systems. It aims to uncover the principles which describe the propensity of such energy flows through the trophic, or 'energy availing' levels of ecological networks. In systems ecology the principles of...
through the different trophic level
Trophic level
The trophic level of an organism is the position it occupies in a food chain. The word trophic derives from the Greek τροφή referring to food or feeding. A food chain represents a succession of organisms that eat another organism and are, in turn, eaten themselves. The number of steps an organism...
s in the ecological networks. In systems ecology the principles of ecosystem energy flows are considered formally analogous to the principles of energetics. Systems ecology also considers the external influence of ecological economics
Ecological economics
Image:Sustainable development.svg|right|The three pillars of sustainability. Clickable.|275px|thumbpoly 138 194 148 219 164 240 182 257 219 277 263 291 261 311 264 331 272 351 283 366 300 383 316 394 287 408 261 417 224 424 182 426 154 423 119 415 87 403 58 385 40 368 24 347 17 328 13 309 16 286 26...
, which usually is not otherwise considered in ecosystem ecology. For the most part, systems ecology is a subfield of ecosystem ecology.
Swarm behaviour
Swarm behaviour is a collective behaviour exhibited by animals of similar size which aggregate together, perhaps milling about the same spot or perhaps migratingAnimal migration
Animal migration is the relatively long-distance movement of individuals, usually on a seasonal basis. It is a ubiquitous phenomenon, found in all major animal groups, including birds, mammals, fish, reptiles, amphibians, insects, and crustaceans. The trigger for the migration may be local...
in some direction. Swarm behaviour is commonly exhibited by insects, but it also occurs in the flocking of birds, the schooling
Shoaling and schooling
In biology, any group of fish that stay together for social reasons are said to be shoaling , and if, in addition, the group is swimming in the same direction in a coordinated manner, they are said to be schooling . In common usage, the terms are sometimes used rather loosely...
of fish and the herd behaviour of quadrupeds. It is a complex emergent
Emergent
It may also mean:* Emergent , Neural Simulation Software* Emergent , a 2003 album by Gordian Knot* emergent plant, a plant which grows in water but which pierces the surface so that it is partially in air...
behaviour that occurs when individual agents
Agent-based model in biology
Agent-based models have many applications in biology, primarily due to the characteristics of the modeling method. Agent-based modeling is a rule-based, computational modeling methodology that focuses on rules and interactions among the individual components or the agents of the system...
follow simple behavioral rules. Recently, a number of mathematical models have been discovered which explain many aspects of the emergent behaviour.
Swarm algorithms follow a Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
approach or an Eulerian approach. The Eulerian approach views the swarm as a field
Field (physics)
In physics, a field is a physical quantity associated with each point of spacetime. A field can be classified as a scalar field, a vector field, a spinor field, or a tensor field according to whether the value of the field at each point is a scalar, a vector, a spinor or, more generally, a tensor,...
, working with the density of the swarm and deriving mean field properties. It is a hydrodynamic approach, and can be useful for modelling the overall dynamics of large swarms. However, most models work with the Lagrangian approach, which is an agent-based model following the individual agents (points or particles) that make up the swarm. Individual particle models can follow information on heading and spacing that is lost in the Eulerian approach. Examples include ant colony optimization
Ant colony optimization
In computer science and operations research, the ant colony optimization algorithm ' is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs....
, self-propelled particles
Self-propelled particles
Self-propelled particles , also referred to as self-driven particles or as the Couzin–Vicsek algorithm, is a concept used to model swarm behaviour. The concept was introduced in 1995 by Vicsek and Couzin et al. as a special case of the Boids model introduced in 1986 by Reynolds...
and particle swarm optimization
Particle swarm optimization
In computer science, particle swarm optimization is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality...
Evolutionary ecology
The British biologist Alfred Russel WallaceAlfred Russel Wallace
Alfred Russel Wallace, OM, FRS was a British naturalist, explorer, geographer, anthropologist and biologist...
is best known for independently proposing a theory of evolution
Evolution
Evolution is any change across successive generations in the heritable characteristics of biological populations. Evolutionary processes give rise to diversity at every level of biological organisation, including species, individual organisms and molecules such as DNA and proteins.Life on Earth...
due to natural selection
Natural selection
Natural selection is the nonrandom process by which biologic traits become either more or less common in a population as a function of differential reproduction of their bearers. It is a key mechanism of evolution....
that prompted Charles Darwin
Charles Darwin
Charles Robert Darwin FRS was an English naturalist. He established that all species of life have descended over time from common ancestry, and proposed the scientific theory that this branching pattern of evolution resulted from a process that he called natural selection.He published his theory...
to publish his own theory. In his famous 1858 paper, Wallace proposed natural selection as a kind of feedback mechanism which keeps species and varieties adapted to their environment.
The action of this principle is exactly like that of the centrifugal governorCentrifugal governorA centrifugal governor is a specific type of governor that controls the speed of an engine by regulating the amount of fuel admitted, so as to maintain a near constant speed whatever the load or fuel supply conditions...
of the steam engine, which checks and corrects any irregularities almost before they become evident; and in like manner no unbalanced deficiency in the animal kingdom can ever reach any conspicuous magnitude, because it would make itself felt at the very first step, by rendering existence difficult and extinction almost sure soon to follow.
The cybernetician
Cybernetics
Cybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related to information theory, control theory and systems theory, at least in its first-order form...
and anthropologist Gregory Bateson
Gregory Bateson
Gregory Bateson was an English anthropologist, social scientist, linguist, visual anthropologist, semiotician and cyberneticist whose work intersected that of many other fields. He had a natural ability to recognize order and pattern in the universe...
observed in the 1970s that, though writing it only as an example, Wallace had "probably said the most powerful thing that’d been said in the 19th Century". Subsequently, the connection between natural selection and systems theory
Systems theory
Systems theory is the transdisciplinary study of systems in general, with the goal of elucidating principles that can be applied to all types of systems at all nesting levels in all fields of research...
has become an area of active research.
Other theories
In contrast to previous ecological theories which considered floods to be catastrophic events, the river flood pulse conceptFlood pulse concept
In contrast to previous ecological theories which considered floods to be catastrophic events, the river flood pulse concept argues that the annual flood pulse is the most important aspect and the most biologically productive feature of a river's ecosystem. It is a geomorphological concept that...
argues that the annual flood pulse is the most important aspect and the most biologically productive feature of a river's ecosystem
Lotic ecosystem
A lotic ecosystem is the ecosystem of a river, stream or spring. Included in the environment are the biotic interactions as well as the abiotic interactions ....
.
History
Theoretical ecology draws on pioneering work done by G. Evelyn HutchinsonG. Evelyn Hutchinson
George Evelyn Hutchinson FRS was an Anglo-American zoologist known for his studies of freshwater lakes and considered the father of American limnology....
and his students. Brothers H.T. Odum
Howard Odum
Howard Odum may refer to:* Howard W. Odum , American sociologist* Howard T. Odum , his son, ecologist...
and E.P. Odum
Eugene Odum
Eugene Pleasants Odum was an American scientist known for his pioneering work on ecosystem ecology. He wrote the first ecology textbook: Fundamentals of Ecology....
are generally recognised as the founders of modern theoretical ecology. Robert MacArthur
Robert MacArthur
Robert Helmer MacArthur was an American ecologist who made a major impact on many areas of community and population ecology....
brought theory to community ecology. Daniel Simberloff
Daniel Simberloff
Daniel Simberloff is a biologist and ecologist who earned his Ph.D. from Harvard University in 1969.Simberloff started his studies in ecology as a student of the biologist E. O. Wilson, one of the co-authors of the theory of Island biogeography . For his Ph.D...
was the student of E.O. Wilson, with whom MacArthur collaborated on The Theory of Island Biogeography
The Theory of Island Biogeography
The Theory of Island Biogeography is a 1967 book by Edward O. Wilson and Robert MacArthur which laid the foundations for the study of island biogeography. An edition with a new preface by Edward O. Wilson was published in 2001 ....
, a seminal work in the development of theoretical ecology.
Simberloff added statistical rigour to experimental ecology and was a key figure in the SLOSS debate
SLOSS Debate
The SLOSS Debate was a debate in ecology and conservation biology during the 1970s and 1980s as to whether a single large or several small reserves were a superior means of conserving biodiversity in a fragmented habitat....
, about whether it is preferable to protect a single large or several small reserves. This resulted in the supporters of Jared Diamond
Jared Diamond
Jared Mason Diamond is an American scientist and author whose work draws from a variety of fields. He is currently Professor of Geography and Physiology at UCLA...
's community assembly rules defending their ideas through Neutral Model Analysis. Simberloff also played a key role in the (still ongoing) debate on the utility of corridors for connecting isolated reserves.
Stephen Hubbell and Michael Rosenzweig
Michael Rosenzweig
Michael L. Rosenzweig is an ecologist at the University of Arizona who has developed and popularized the concept of Reconciliation ecology. He founded and developed the Department of Ecology and Evolutionary Biology, at UA Tucson, now a major center for the study of Evolutionary Ecology...
combined theoretical and practical elements into works that extended MacArthur and Wilson's Island Biogeography Theory - Hubbell with his Unified Neutral Theory of Biodiversity and Biogeography
Unified neutral theory of biodiversity
The unified neutral theory of biodiversity and biogeography is a hypothesis and the title of a monograph by ecologist Stephen Hubbell...
and Rosenzweig with his Species Diversity in Space and Time.
Theoretical and mathematical ecologists
A distinction can be made between mathematical ecologists, ecologists who apply mathematics to ecological problems, and mathematicians who develop the mathematics itself that arises out of ecological problems.- Robert MacArthurRobert MacArthurRobert Helmer MacArthur was an American ecologist who made a major impact on many areas of community and population ecology....
- Joel CohenJoel CohenJoel Cohen, is an American musician specializing in early music repertoires. Joel graduated Classical High school in Providence, R.I. in 1959. He then graduated from Brown University in 1963. He continued graduate education at Harvard University...
- Donald DeAngelis
- Madhav GadgilMadhav GadgilMadhav Gadgil is an Indian ecologist.-Biography:He was born in Maharashtra studied biology at University of Poona and University of Bombay before doing a Ph.D. thesis in the area of mathematical ecology at Harvard University...
- Alan HastingsAlan HastingsAlan Hastings is a theoretical ecologist and distinguished professor in the Department of Environmental Science and Policy at the University of California, Davis. In 2005 he became a fellow of the American Academy of Arts and Sciences and in 2006 he won the Robert H. MacArthur Award...
- Ray HilbornRay HilbornRay Hilborn is a marine biologist and fisheries scientist, known for his work on conservation and natural resource management in the context of fisheries.He is currently professor of aquatic and fishery science at the University of Washington...
- Henry S. HornHenry S. HornHenry S. Horn is a natural historian and ecologist. He is currently an emeritus professor in the Ecology and Evolutionary Biology Department at Princeton University...
- Cang HuiCang Hui- Biography :Cang Hui is a mathematical ecologist currently working at Stellenbosch University...
- Everett HughesEverett HughesEverett Cherrington Hughes was an American sociologist best known for his work on ethnic relations, work and occupations and the methodology of fieldwork. His take on sociology was, however, very broad...
- Evelyn HutchinsonG. Evelyn HutchinsonGeorge Evelyn Hutchinson FRS was an Anglo-American zoologist known for his studies of freshwater lakes and considered the father of American limnology....
- Hanna KokkoHanna KokkoHanna Kokko is a scientist, working in the field of evolution and ecology. She was a Professor of Evolutionary Biology at Helsinki University, Finland, and has recently taken up a position at the Australian National University as a Professor of Evolutionary Ecology. She has been awarded the 2010...
- Simon Levin
- Richard LevinsRichard LevinsRichard "Dick" Levins is a mathematical ecologist, and political activist. He is best known for his work on evolution in changing environments....
- Richard LewontinRichard LewontinRichard Charles "Dick" Lewontin is an American evolutionary biologist, geneticist and social commentator. A leader in developing the mathematical basis of population genetics and evolutionary theory, he pioneered the notion of using techniques from molecular biology such as gel electrophoresis to...
- Marc Mangel
- Ramon MargalefRamón MargalefRamon Margalef i López was Emeritus Professor of Ecology at the Faculty of Biology of the University of Barcelona. Margalef, unquestionably one of the most important scientists that Spain has produced , worked at the Institute of Applied Biology , and at the Fisheries Research Institute, which he...
- Jacqueline McGladeJacqueline McGladeJacqueline M. McGlade is a leading marine biologist and environmental informatics professor. Her research focusses on the spatial and nonlinear dynamics of ecosystems, climate change and scenario development....
- Angela McLeanAngela McLeanAngela McLean, FRS is Professor of Mathematical Biology in the Department of Zoology at the University of Oxford.She studied Mathematics at Oxford and earned a PhD in Biomathematics from Imperial College, London...
- Robert MayRobert May, Baron May of OxfordRobert McCredie May, Baron May of Oxford, OM, AC, PRS is an Australian scientist who has been Chief Scientific Adviser to the UK Government, President of the Royal Society, and a Professor at Sydney and Princeton. He now holds joint professorships at Oxford, and Imperial College London...
- Robert V. O'Neill
- Howard T. OdumHoward T. OdumHoward Thomas Odum was an American ecologist...
- E. C. PielouE. C. PielouEvelyn Chrystalla Pielou is a statistical ecologist. She began her career as a researcher for the Canadian Department of Forestry and the Canadian Department of Agriculture...
- Stuart PimmStuart PimmStuart Pimm is an American-British biologist and theoretical ecologist specializing in scientific research of biodiversity and conservation biology.- Professional career :...
- Hugh PossinghamHugh PossinghamHugh Phillip Possingham, FAA , is an ARC Federation Fellow in the Department of Mathematics, and the School of Integrative Biology at the University of Queensland...
- Erik RauchErik RauchErik Rauch was a biophysicist and theoretical ecologist who worked at NECSI, MIT, Santa Fe Institute, Yale University, Princeton University, and other institutions....
- Joan (Jonathan) RoughgardenJoan RoughgardenJoan E. Roughgarden is an American evolutionary biologist.- Biography :...
- Graeme RuxtonGraeme RuxtonGraeme Ruxton is Professor of theoretical ecology at the University of Glasgow, Scotland, United Kingdom. His studies focus on the evolutionary pressures on aggregation by animals, and predator-prey aspects of sensory ecology...
- John Maynard SmithJohn Maynard SmithJohn Maynard Smith,His surname was Maynard Smith, not Smith, nor was it hyphenated. F.R.S. was a British theoretical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics under the well-known biologist J.B.S....
- George SugiharaGeorge SugiharaGeorge Sugihara is a theoretical biologist who has worked across a wide variety of fields, including landscape ecology, algebraic topology, algal physiology and paleoecology, neurobiology, atmospheric science, fisheries science, and quantitative finance...
- David TilmanG. David TilmanG. David Tilman born in 1949 in Aurora, Illinois, is a prominent American ecologist who earned his Ph.D. at the University of Michigan in 1976. He is Regent's Professor and McKnight Presidential Chair in Ecology at the University of Minnesota, as well as an instructor in Conservation Biology;...
- Robert UlanowiczRobert UlanowiczRobert Edward Ulanowicz is an American theoretical ecologist and philosopher who is best known for his search for a unified theory of ecology. He was born September 17, 1943 in Baltimore, Maryland....
- E.O. Wilson