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Theoretical ecology is the scientific discipline devoted to the study of
ecological systems An ecosystem (or ecological system) consists of all the organisms and the physical environment with which they interact. These biotic and abiotic components are linked together through nutrient cycles and energy flows. Energy enters the syste ...
using theoretical methods such as simple
conceptual model A conceptual model is a representation of a system. It consists of concepts used to help people know, understand, or simulate a subject the model represents. In contrast, physical models are physical object such as a toy model that may be asse ...
s,
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 in the natural sciences (such as physics, ...
s,
computational simulation Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be dete ...
s, and advanced
data analysis Data analysis is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, informing conclusions, and supporting decision-making. Data analysis has multiple facets and approaches, enc ...
. 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. 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 Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, ind ...
, evolutionary theory, epidemiology, animal behavior and group dynamics,
food webs A food web is the natural interconnection of food chains and a graphical representation of what-eats-what in an ecological community. Another name for food web is consumer-resource system. Ecologists can broadly lump all life forms into one ...
, ecosystems, spatial ecology, and the
effects of climate change The effects of climate change impact the physical environment, ecosystems and human societies. The environmental effects of climate change are broad and far-reaching. They affect the water cycle, oceans, sea and land ice ( glaciers), sea le ...
. 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 The carbon cycle is the biogeochemical cycle by which carbon is exchanged among the biosphere, pedosphere, geosphere, hydrosphere, and atmosphere of the Earth. Carbon is the main component of biological compounds as well as a major compon ...
.


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.Bolker BM (2008
''Ecological models and data in R''
Princeton University Press, pages 6–9. .
* 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 Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
or
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
. * 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 dynamics System dynamics (SD) is an approach to understanding the nonlinear behaviour of complex systems over time using stocks, flows, internal feedback loops, table functions and time delays. Overview System dynamics is a methodology and mathematica ...
models are deterministic. * Stochastic models allow for the direct modeling of the random perturbations that underlie real world ecological systems. Markov chain models are stochastic. Species can be modelled in continuous or
discrete time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "po ...
.Soetaert K and Herman PMJ (2009
''A practical guide to ecological modelling''
Springer. .
* Continuous time is modelled using
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, ...
s. * Discrete time is modelled using
difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
s. These model ecological processes that can be described as occurring over discrete time steps.
Matrix algebra In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, '' ...
is often used to investigate the evolution of age-structured or stage-structured populations. The
Leslie matrix The Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology named after Patrick H. Leslie. The Leslie matrix (also called the Leslie model) is one of the most well-known ways to describe ...
, 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 In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables. The random variables of a stochastic process are indexed by the natural numbers. The origi ...
es. Examples are the dynamics of interacting populations (
predation Predation is a biological interaction where one organism, the predator, kills and eats another organism, its prey. It is one of a family of common feeding behaviours that includes parasitism and micropredation (which usually do not kill ...
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 Mathematical models can project how infectious diseases progress to show the likely outcome of an epidemic (including in plants) and help inform public health and plant health interventions. Models use basic assumptions or collected statistics ...
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 of curves, 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 The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often referred to as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popula ...
s are
polynomial map In algebra, a polynomial map or polynomial mapping P: V \to W between vector spaces over an infinite field ''k'' is a polynomial in linear functionals with coefficients in ''k''; i.e., it can be written as :P(v) = \sum_ \lambda_(v) \cdots \lambda_ ...
pings, and are often cited as providing archetypal examples of how chaotic behaviour can arise from very simple
non-linear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...
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 ''The Genetical Theory of Natural Selection'' is a book by Ronald Fisher which combines Mendelian genetics with Charles Darwin's theory of natural selection, with Fisher being the first to argue that "Mendelism therefore validates Darwinism" and ...
'', which introduced the idea that frequency-dependent fitness brings a strategic aspect to
evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
, 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 Charles Lewontin (March 29, 1929 – July 4, 2021) was an American evolutionary biologist, mathematician, geneticist, and social commentator. A leader in developing the mathematical basis of population genetics and evolutionary theory, ...
applied game theory to evolutionary biology in his ''Evolution and the Theory of Games'', followed closely by
John Maynard Smith John Maynard Smith (6 January 1920 – 19 April 2004) was a British theoretical and mathematical evolutionary biologist and geneticist. Originally an aeronautical engineer during the Second World War, he took a second degree in genetics un ...
, who in his seminal 1972 paper, “Game Theory and the Evolution of Fighting", defined the concept of the
evolutionarily stable strategy An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is ''impermeable'' when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set o ...
. Because ecological systems are typically
nonlinear In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many oth ...
, 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 model An agent-based model (ABM) is a computational model for simulating the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) in order to understand the behavior of a system and wh ...
s. 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 Fishery can mean either the enterprise of raising or harvesting fish and other aquatic life; or more commonly, the site where such enterprise takes place ( a.k.a. fishing ground). Commercial fisheries include wild fisheries and fish farms, ...
.


Population ecology

Population ecology is a sub-field of
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...
that deals with the dynamics of
species 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 appropriat ...
population Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction usi ...
s and how these populations interact with the
environment Environment most often refers to: __NOTOC__ * Natural environment, all living and non-living things occurring naturally * Biophysical environment, the physical and biological factors along with their chemical interactions that affect an organism or ...
. It is the study of how the
population size In population genetics and population ecology, population size (usually denoted ''N'') is the number of individual organisms in a population. Population size is directly associated with amount of genetic drift, and is the underlying cause of effect ...
s of species living together in groups change over time and space, and was one of the first aspects of ecology to be studied and modelled mathematically.


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: : \frac=rN(t) 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 In mathematics and other formal sciences, first-order or first order most often means either: * "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hi ...
linear differential equation In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y + a_1(x)y' + a_2(x)y'' \cdots + a_n(x)y^ = b ...
can be solved to yield the solution : N(t) = N(0) \ e^ , a trajectory known as Malthusian growth, after
Thomas Malthus Thomas Robert Malthus (; 13/14 February 1766 – 29 December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography. In his 1798 book ''An Essay on the Principle of Population'', Mal ...
, who first described its dynamics in 1798. A population experiencing Malthusian growth follows an exponential curve, where N(0) is the initial population size. The population grows when r > 0, and declines when r < 0. The model is most 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 competition Intraspecific competition is an interaction in population ecology, whereby members of the same species compete for limited resources. This leads to a reduction in fitness for both individuals, but the more fit individual survives and is able to r ...
). 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:Moss R, Watson A and Ollason J (1982
''Animal population dynamics''
Springer, page 52–54. .
: \frac=((b-aN(t))-(d-cN(t)))N(t) This can be rewritten as: : \frac=rN(t) \left(1-\frac\right) 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. The constant K is the
carrying capacity The carrying capacity of an environment is the maximum population size of a biological species that can be sustained by that specific environment, given the food, habitat, water, and other resources available. The carrying capacity is defined as ...
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 population 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 :\mathbf_ = \mathbf\mathbf_t where Nt is a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
of the number of individuals in each class at time ''t'' and L is a
matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** '' The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
that contains the survival probability and fecundity for each class. The matrix L is referred to as the
Leslie matrix The Leslie matrix is a discrete, age-structured model of population growth that is very popular in population ecology named after Patrick H. Leslie. The Leslie matrix (also called the Leslie model) is one of the most well-known ways to describe ...
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 The loggerhead sea turtle (''Caretta caretta'') is a species of oceanic turtle distributed throughout the world. It is a marine reptile, belonging to the family Cheloniidae. The average loggerhead measures around in carapace length when fully ...
s and
right whale Right whales are three species of large baleen whales of the genus ''Eubalaena'': the North Atlantic right whale (''E. glacialis''), the North Pacific right whale (''E. japonica'') and the Southern right whale (''E. australis''). They are c ...
s.


Community ecology

An ecological community is a group of trophically similar,
sympatric In biology, two related species or populations are considered sympatric when they exist in the same geographic area and thus frequently encounter one another. An initially interbreeding population that splits into two or more distinct species s ...
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 interaction

Predator–prey Predation is a biological interaction where one organism, the predator, kills and eats another organism, its prey. It is one of a family of common feeding behaviours that includes parasitism and micropredation (which usually do not kill the ...
interactions exhibit natural oscillations in the populations of both predator and the prey. In 1925, the American mathematician
Alfred J. Lotka Alfred James Lotka (March 2, 1880 – December 5, 1949) 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 proposa ...
developed simple equations for predator–prey interactions in his book on biomathematics. The following year, the Italian mathematician
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in An ...
, made a statistical analysis of fish catches in the AdriaticGoel, N.S. et al., "On the Volterra and Other Non-Linear Models of Interacting Populations", ''Academic Press Inc.'', (1971) and independently developed the same equations. It is one of the earliest and most recognised ecological models, known as the Lotka-Volterra model: : \frac = N(t)(r-\alpha P(t)) : \frac = P(t)(c \alpha N(t) -d) 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 Fishing is the activity of trying to catch fish. Fish are often caught as wildlife from the natural environment, but may also be caught from fish stocking, stocked bodies of water such as fish pond, ponds, canals, park wetlands and reservoirs. ...
was curtailed during the
First World War World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was List of wars and anthropogenic disasters by death toll, one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, ...
. However, the equations have subsequently been applied more generally. Other examples of these models include the Lotka-Volterra model of the
snowshoe hare The snowshoe hare (''Lepus americanus''), 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. The animal's feet prevent it from sin ...
and Canadian lynx in North America, any infectious disease modeling such as the recent outbreak of
SARS Severe acute respiratory syndrome (SARS) is a viral respiratory disease of zoonotic origin caused by the severe acute respiratory syndrome coronavirus (SARS-CoV or SARS-CoV-1), the first identified strain of the SARS coronavirus species, ''seve ...
and biological control of California red scale by the introduction of its
parasitoid In evolutionary ecology, a parasitoid is an organism that lives in close association with its host (biology), host at the host's expense, eventually resulting in the death of the host. Parasitoidism is one of six major evolutionarily stable str ...
, '' Aphytis melinus'' . A credible, simple alternative to the Lotka-Volterra predator–prey model and their common prey dependent generalizations is the ratio dependent or Arditi-Ginzburg model. The two are the extremes of the spectrum of predator interference models. According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka–Volterra extreme on the interference spectrum that the model can simply be discounted as wrong. They are much closer to the ratio-dependent extreme, so if a simple model is needed one can use the Arditi–Ginzburg model as the first approximation.


Host–pathogen interaction

The second interaction, that of host and
pathogen In biology, a pathogen ( el, πάθος, "suffering", "passion" and , "producer of") in the oldest and broadest sense, is any organism or agent that can produce disease. A pathogen may also be referred to as an infectious agent, or simply a g ...
, 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 interaction

The third interaction, that of host and
parasitoid In evolutionary ecology, a parasitoid is an organism that lives in close association with its host (biology), host at the host's expense, eventually resulting in the death of the host. Parasitoidism is one of six major evolutionarily stable str ...
, can be analyzed by the
Nicholson–Bailey model The Nicholson–Bailey model was developed in the 1930s to describe the population dynamics of a coupled host-parasitoid system. It is named after Alexander John Nicholson and Victor Albert Bailey. Host-parasite and prey-predator systems can als ...
, 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: : N_ = \lambda \ N_t \ 1 - f(N_t, P_t) /math> : P_ = c \ N_t \ f(N_t, p_t) where f(Nt, Pt) describes the probability of infection (typically,
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 or space if these events occur with a known co ...
), 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. discoiderum'' and '' E. coli'', as well as theoretical analysis of the behavior of the system. : \frac = \frac\left( K_1 - N_1 + \alpha_N_2 \right) : \frac = \frac\left( K_2 - N_2 + \alpha_N_1 \right) 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 Competition is a rivalry where two or more parties strive for a common goal which cannot be shared: where one's gain is the other's loss (an example of which is a zero-sum game). Competition can arise between entities such as organisms, indivi ...
. 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. :''See
Competitive Lotka–Volterra equations 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 the Generalized Lotka–Volterra equation to include trophic interactions ...
for further extensions of this model.''


Neutral theory

Unified neutral theory is a hypothesis proposed by Stephen P. 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 The trophic level of an organism is the position it occupies in a food web. A food chain is a succession of organisms that eat other organisms and may, in turn, be eaten themselves. The trophic level of an organism is the number of steps it ...
in a
food web A food web is the natural interconnection of food chains and a graphical representation of what-eats-what in an ecological community. Another name for food web is consumer-resource system. Ecologists can broadly lump all life forms into one o ...
, 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 In mathematics, a random walk is a random process that describes a path that consists of a succession of random steps on some mathematical space. An elementary example of a random walk is the random walk on the integer number line \mathbb Z ...
. This can be considered a
null hypothesis In scientific research, the null hypothesis (often denoted ''H''0) is the claim that no difference or relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is ...
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 In ecology, a community is a group or association (ecology), association of Population ecology, populations of two or more different species occupying the same geographical area at the same time, also known as a biocoenosis, biotic community, ...
(such as competition and cooperation), providing all individuals obey the same rules. Asymmetric phenomena such as
parasitism Parasitism is a close relationship between species, where one organism, the parasite, lives on or inside another organism, the host, causing it some harm, and is adapted structurally to this way of life. The entomologist E. O. Wilson ha ...
and
predation Predation is a biological interaction where one organism, the predator, kills and eats another organism, its prey. It is one of a family of common feeding behaviours that includes parasitism and micropredation (which usually do not kill ...
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 or biological diversity is the variety and variability of life on Earth. Biodiversity is a measure of variation at the genetic ('' genetic variability''), species ('' species diversity''), and ecosystem ('' ecosystem diversity'') ...
, 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 Wilson may refer to: People *Wilson (name) ** List of people with given name Wilson ** List of people with surname Wilson * Wilson (footballer, 1927–1998), Brazilian manager and defender * Wilson (footballer, born 1984), full name Wilson R ...
's theory of
island biogeography Insular biogeography or island biogeography is a field within biogeography that examines the factors that affect the species richness and diversification of isolated natural communities. The theory was originally developed to explain the pattern of ...
and
Gould Gould may refer to: People * Gould (name), a surname Places United States * Gould, Arkansas, a city * Gould, Colorado, an unincorporated community * Gould, Ohio, an unincorporated community * Gould, Oklahoma, a town * Gould, West Virginia, ...
's concepts of symmetry and null models.


Spatial ecology


Biogeography

Biogeography Biogeography is the study of the distribution of species and ecosystems in geographic space and through geological time. Organisms and biological communities often vary in a regular fashion along geographic gradients of latitude, elevation, ...
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 Insular biogeography or island biogeography is a field within biogeography that examines the factors that affect the species richness and diversification of isolated natural communities. The theory was originally developed to explain the pattern of ...
. 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 Helmer MacArthur (April 7, 1930 – November 1, 1972) was a Canadian-born American ecologist who made a major impact on many areas of community and population ecology. Early life and education MacArthur was born in Toronto, Ontario, ...
and E.O. Wilson published ''
The Theory of Island Biogeography ''The Theory of Island Biogeography'' is a 1967 book by the ecologist Robert MacArthur and the biologist Edward O. Wilson. It is widely regarded as a seminal piece in island biogeography and ecology. The Princeton University Press reprinted the ...
''. 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 Conservation biology is the study of the conservation of nature and of Earth's biodiversity with the aim of protecting species, their habitats, and ecosystems from excessive rates of extinction and the erosion of biotic interactions. It is an in ...
and
landscape ecology Landscape ecology is the science of studying and improving relationships between ecological processes in the environment and particular ecosystems. This is done within a variety of landscape scales, development spatial patterns, and organizatio ...
.


r/K-selection theory

A population ecology concept is r/K selection theory, one of the first predictive models in ecology used to explain life-history evolution. The premise behind the r/K selection model is that natural selection pressures change according to
population density Population density (in agriculture: standing stock or plant density) is a measurement of population per unit land area. It is mostly applied to humans, but sometimes to other living organisms too. It is a key geographical term.Matt RosenberPopu ...
. For example, when an island is first colonized, density of individuals is low. The initial increase in population size is not limited by competition, leaving an abundance of available
resources Resource refers to all the materials available in our environment which are technologically accessible, economically feasible and culturally sustainable and help us to satisfy our needs and wants. Resources can broadly be classified upon their av ...
for rapid population growth. These early phases of
population growth Population growth is the increase in the number of people in a population or dispersed group. Actual global human population growth amounts to around 83 million annually, or 1.1% per year. The global population has grown from 1 billion in 1800 to ...
experience ''density-independent'' forces of natural selection, which is called ''r''-selection. As the population becomes more crowded, it approaches the island's carrying capacity, thus forcing individuals to compete more heavily for fewer available resources. Under crowded conditions, the population experiences density-dependent forces of natural selection, called ''K''-selection.


Niche theory


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 as : \frac = m p (1-p) - e p where m is the rate of
colonization Colonization, or colonisation, constitutes large-scale population movements wherein migrants maintain strong links with their, or their ancestors', former country – by such links, gain advantage over other inhabitants of the territory. When ...
, and e is the rate of
extinction Extinction is the termination of a kind of organism or of a group of kinds (taxon), usually a species. The moment of extinction is generally considered to be the death of the Endling, last individual of the species, although the Functional ext ...
. 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, : \frac = m_1 p_1 (1 - p_1) - e p_1 : \frac = m_2 p_2 (1 - p_1 - p_2) - e p_2 - m p_1 p_2 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^*_2 = \frac - \frac (p*1 may be inferred by symmetry). 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 Biodiversity or biological diversity is the variety and variability of life on Earth. Biodiversity is a measure of variation at the genetic ('' genetic variability''), species ('' species diversity''), and ecosystem ('' ecosystem diversity'') ...
(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.Vandermeer JH and Goldberg DE (2003
''Population ecology: first principles''
Princeton University Press, page 175–176. .
# 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 The rescue effect is a phenomenon which was first described by Brown and Kodric-Brown,Brown JH, Kodric-Brown A. 1977 Turnover rates in insular biogeography: effect of immigration on extinction. Ecology 58, 445– 449. (doi:10.2307/ 1935620) and is ...
” 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 biotic or
abiotic In biology and ecology, abiotic components or abiotic factors are non-living chemical and physical parts of the environment that affect living organisms and the functioning of ecosystems. Abiotic factors and the phenomena associated with them under ...
, into
ecosystem An ecosystem (or ecological system) consists of all the organisms and the physical environment with which they interact. These biotic and abiotic components are linked together through nutrient cycles and energy flows. Energy enters the syst ...
s can be disruptive. In some cases, it leads to
ecological collapse Ecological collapse refers to a situation where an ecosystem suffers a drastic, possibly permanent, reduction in carrying capacity for all organisms, often resulting in mass extinction. Usually, an ecological collapse is precipitated by a disastr ...
,
trophic cascade Trophic cascades are powerful indirect interactions that can control entire ecosystems, occurring when a trophic level in a food web is suppressed. For example, a top-down cascade will occur if predators are effective enough in predation to reduce t ...
s and the death of many species within the ecosystem. The abstract notion of
ecological health Ecological health is a term that has been used in relation to both human health and the condition of the environment. *In medicine, ecological health has been used to refer to multiple chemical sensitivity, which results from exposure to synthet ...
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 is the degree to which a chemical substance or a particular mixture of substances can damage an organism. Toxicity can refer to the effect on a whole organism, such as an animal, bacterium, or plant, as well as the effect on a subs ...
of the introduced element and the
resiliency Resilience, resilient, resiliency, or ''variation'', may refer to: Science Ecology * Ecological resilience, the capacity of an ecosystem to recover from perturbations ** Climate resilience, the ability of systems to recover from climate change * ...
of the original ecosystem. If ecosystems are governed primarily by
stochastic Stochastic (, ) refers to the property of being well described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselv ...
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 balance of nature, also known as ecological balance, is a theory that proposes that ecological systems are usually in a stable equilibrium or homeostasis, which is to say that a small change (the size of a particular population, for example) w ...
, 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 Edward Ulanowicz ( ) is an American theoretical ecologist and philosopher of Polish descent who in his search for a ''unified theory of ecology'' has formulated a paradigm he calls ''Process Ecology''. He was born September 17, 1943 in Bal ...
used
information theory Information theory is the scientific study of the quantification, storage, and communication of information. The field was originally established by the works of Harry Nyquist and Ralph Hartley, in the 1920s, and Claude Shannon in the 1940s. ...
tools to describe the structure of ecosystems, emphasizing
mutual information In probability theory and information theory, the mutual information (MI) of two random variables is a measure of the mutual dependence between the two variables. More specifically, it quantifies the " amount of information" (in units such ...
(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 is the process by which an entire body of water, or parts of it, becomes progressively enriched with minerals and nutrients, particularly nitrogen and phosphorus. It has also been defined as "nutrient-induced increase in phyt ...
.
Ecopath Ecopath with Ecosim (EwE) is a free and open source ecosystem modelling software suite, initially started at NOAA by Jeffrey Polovina, but has since primarily been developed at the formerly UBC Fisheries Centre of the University of British Colum ...
is a free ecosystem modelling software suite, initially developed by
NOAA The National Oceanic and Atmospheric Administration (abbreviated as NOAA ) is an United States scientific and regulatory agency within the United States Department of Commerce that forecasts weather, monitors oceanic and atmospheric conditio ...
, 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 web A food web is the natural interconnection of food chains and a graphical representation of what-eats-what in an ecological community. Another name for food web is consumer-resource system. Ecologists can broadly lump all life forms into one o ...
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 A food chain is a linear network of links in a food web starting from producer organisms (such as grass or algae which produce their own food via photosynthesis) and ending at an apex predator species (like grizzly bears or killer whales), de ...
s. Each food chain starts with a
primary producer Primary or primaries may refer to: Arts, entertainment, and media Music Groups and labels * Primary (band), from Australia * Primary (musician), hip hop musician and record producer from South Korea * Primary Music, Israeli record label Works * ...
or
autotroph An autotroph or primary producer is an organism that produces complex organic compounds (such as carbohydrates, fats, and proteins) using carbon from simple substances such as carbon dioxide,Morris, J. et al. (2019). "Biology: How Life Wo ...
, 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 The trophic level of an organism is the position it occupies in a food web. A food chain is a succession of organisms that eat other organisms and may, in turn, be eaten themselves. The trophic level of an organism is the number of steps it ...
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 is chemical energy that animals (including humans) derive from their food to sustain their metabolism, including their muscular activity. Most animals derive most of their energy from aerobic respiration, namely combining the carbohy ...
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 Elton may refer to: *Charles Elton (Born, 1993) Professional Rugby Player for Otago Rugby * Charles Isaac Elton (1839–1900), English lawyer, politician, writer and antiquarian * Charles Sutherland Elton (1900–1991), English biologist ...
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 Stuart Leonard Pimm (born 27 February 1949) is an American-British biologist and theoretical ecologist specializing in scientific research of biodiversity and conservation biology. Education Pimm was born in Derbyshire, United Kingdom. He was e ...
, 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 Stability may refer to: Mathematics *Stability theory, the study of the stability of solutions to differential equations and dynamical systems ** Asymptotic stability ** Linear stability ** Lyapunov stability ** Orbital stability ** Structural sta ...
of a food web.May RM (2001
''Stability and Complexity in Model Ecosystems''
Princeton University Press, reprint of 1973 edition with new foreword. .
Pimm SL (2002
''Food Webs''
University of Chicago Press, reprint of 1982 edition with new foreword. .


Systems ecology

Systems ecology Systems ecology is an interdisciplinary field of ecology, a subset of Earth system science, that takes a holistic approach to the study of ecological systems, especially ecosystems. Systems ecology can be seen as an application of general syst ...
can be seen as an application of
general systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
to ecology. It takes a
holistic Holism () is the idea that various systems (e.g. physical, biological, social) should be viewed as wholes, not merely as a collection of parts. The term "holism" was coined by Jan Smuts in his 1926 book '' Holism and Evolution''."holism, n." OED On ...
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 is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
and develops other macroscopic descriptions of complex systems. It also takes account of the energy flows through the different
trophic level The trophic level of an organism is the position it occupies in a food web. A food chain is a succession of organisms that eat other organisms and may, in turn, be eaten themselves. The trophic level of an organism is the number of steps it ...
s in the ecological networks. 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.


Ecophysiology

This is the study of how "the environment, both physical and biological, interacts with the physiology of an organism. It includes the effects of climate and nutrients on physiological processes in both plants and animals, and has a particular focus on how physiological processes scale with organism size".


Behavioral ecology


Swarm behaviour

Swarm behaviour Swarm behaviour, or swarming, is a collective animal behaviour, collective behaviour exhibited by entities, particularly animals, of similar size which aggregate together, perhaps milling about the same spot or perhaps moving ''en masse'' or a ...
is a collective behaviour exhibited by animals of similar size which aggregate together, perhaps milling about the same spot or perhaps migrating 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 Herd behavior is the behavior of individuals in a group acting collectively without centralized direction. Herd behavior occurs in animals in herds, packs, bird flocks, fish schools and so on, as well as in humans. Voting, demonstrations, riot ...
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 Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
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 An agent-based model (ABM) is a computational model for simulating the actions and interactions of autonomous agents (both individual or collective entities such as organizations or groups) in order to understand the behavior of a system and wh ...
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 In computer science and operations research, the ant colony optimization algorithm (ACO) is a probabilistic technique for solving computational problems which can be reduced to finding good paths through graphs. Artificial ants stand for multi ...
,
self-propelled particles Self-propelled particles (SPP), also referred to as self-driven particles, are terms used by physicists to describe autonomous agents, which convert energy from the environment into directed or persistent motion. Natural systems which have insp ...
and particle swarm optimization


Evolutionary ecology

The British biologist
Alfred Russel Wallace Alfred Russel Wallace (8 January 1823 – 7 November 1913) was a British natural history, naturalist, explorer, geographer, anthropologist, biologist and illustrator. He is best known for independently conceiving the theory of evolution thro ...
is best known for independently proposing a theory of
evolution Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are the expressions of genes, which are passed on from parent to offspring during reproduction. Variation ...
due to
natural selection Natural selection is the differential survival and reproduction of individuals due to differences in phenotype. It is a key mechanism of evolution, the change in the heritable traits characteristic of a population over generations. Cha ...
that prompted
Charles Darwin Charles Robert Darwin ( ; 12 February 1809 – 19 April 1882) was an English naturalist, geologist, and biologist, widely known for his contributions to evolutionary biology. His proposition that all species of life have descended ...
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 governor A centrifugal governor is a specific type of governor with a feedback system that controls the speed of an engine by regulating the flow of fuel or working fluid, so as to maintain a near-constant speed. It uses the principle of proportional con ...
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 A cyberneticist or a cybernetician is a person who practices cybernetics. Heinz von Foerster once told Stuart Umpleby that Norbert Wiener preferred the term "cybernetician" rather than "cyberneticist", perhaps because Wiener was a mathematician r ...
and anthropologist
Gregory Bateson Gregory Bateson (9 May 1904 – 4 July 1980) was an English anthropologist, social scientist, linguist, visual anthropologist, semiotician, and cyberneticist whose work intersected that of many other fields. His writings include ''Steps to ...
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 is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent components that can be natural or human-made. Every system has causal boundaries, is influenced by its context, defined by its structu ...
has become an area of active research.


Other theories

In contrast to previous ecological theories which considered
floods A flood is an overflow of water ( or rarely other fluids) that submerges land that is usually dry. In the sense of "flowing water", the word may also be applied to the inflow of the tide. Floods are an area of study of the discipline hydrolog ...
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.Thorp, J. H., & Delong, M. D. (1994). The Riverine Productivity Model: An Heuristic View of Carbon Sources and Organic Processing in Large River Ecosystems. Oikos , 305-308Benke, A. C., Chaubey, I., Ward, G. M., & Dunn, E. L. (2000). Flood Pulse Dynamics of an Unregulated River Floodplain in the Southeastern U.S. Coastal Plain. Ecology , 2730-2741.


History

Theoretical ecology draws on pioneering work done by G. Evelyn Hutchinson and his students. Brothers H.T. Odum and E.P. Odum are generally recognised as the founders of modern theoretical ecology.
Robert MacArthur Robert Helmer MacArthur (April 7, 1930 – November 1, 1972) was a Canadian-born American ecologist who made a major impact on many areas of community and population ecology. Early life and education MacArthur was born in Toronto, Ontario, ...
brought theory to
community ecology In ecology, a community is a group or association of populations of two or more different species occupying the same geographical area at the same time, also known as a biocoenosis, biotic community, biological community, ecological community, ...
.
Daniel Simberloff Daniel Simberloff is a biologist and ecologist who earned his Ph.D. from Harvard University in 1969. He is currently Gore Hunger Professor of Environmental Science at the University of Tennessee, editor-in-chief of the journal '' Biological In ...
was the student of E.O. Wilson, with whom MacArthur collaborated on ''
The Theory of Island Biogeography ''The Theory of Island Biogeography'' is a 1967 book by the ecologist Robert MacArthur and the biologist Edward O. Wilson. It is widely regarded as a seminal piece in island biogeography and ecology. The Princeton University Press reprinted the ...
'', a seminal work in the development of theoretical ecology.Cuddington K and Beisner BE (2005
''Ecological paradigms lost: routes of theory change''
Academic Press. .
Simberloff added statistical rigour to experimental ecology and was a key figure in the
SLOSS debate The SLOSS debate was a debate in ecology and conservation biology during the 1970's and 1980's as to whether a single large or several small (SLOSS) reserves were a superior means of conserving biodiversity in a fragmented habitat. Since its incepti ...
, about whether it is preferable to protect a single large or several small reserves. This resulted in the supporters of
Jared Diamond Jared Mason Diamond (born September 10, 1937) is an American geographer, historian, ornithologist, and author best known for his popular science books '' The Third Chimpanzee'' (1991); ''Guns, Germs, and Steel'' (1997, awarded a Pulitzer Priz ...
'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 P. 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.


Theoretical and mathematical ecologists

A tentative 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. Some notable theoretical ecologists can be found in these categories: * :Mathematical ecologists * :Theoretical biologists


Journals

* ''
The American Naturalist ''The American Naturalist'' is the monthly peer-reviewed scientific journal of the American Society of Naturalists, whose purpose is "to advance and to diffuse knowledge of organic evolution and other broad biological principles so as to enhance t ...
'' * '' Journal of Mathematical Biology'' * ''
Journal of Theoretical Biology The ''Journal of Theoretical Biology'' is a biweekly peer-reviewed scientific journal covering theoretical biology, as well as mathematical, computational, and statistical aspects of biology. Some research areas covered by the journal include cell ...
'' *
Theoretical Ecology
' *
Theoretical Population Biology
' *
Ecological Modelling
'


See also

* Butterfly effect *
Complex system biology Complex systems biology (CSB) is a branch or subfield of mathematical and theoretical biology invented by Robert Rosen concerned with complexity of both structure and function in biological organisms, as well as the emergence and evolution of organ ...
*
Ecological systems theory Ecological systems theory (also called development in context or human ecology theory) was developed by Urie Bronfenbrenner. It offers a framework through which community psychologists examine individuals' relationships within communities and th ...
*
Ecosystem model An ecosystem model is an abstract, usually mathematical, representation of an ecological system (ranging in scale from an individual population, to an ecological community, or even an entire biome), which is studied to better understand the re ...
*
Integrodifference equation In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: : n_(x) = \int_ k(x, y)\, f(n_t(y))\, dy, where \\, is a sequence in the function space and \Omega\, is the domain of those functio ...
– widely used to model the dispersal and growth of populations *
Limiting similarity 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. This concept is a ...
*
Mathematical biology Mathematical and theoretical biology, or biomathematics, is a branch of biology which employs theoretical analysis, mathematical models and abstractions of the living organisms to investigate the principles that govern the structure, development a ...
*
Population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has a ...
*
Population modeling A population model is a type of mathematical model that is applied to the study of population dynamics. Rationale Models allow a better understanding of how complex interactions and processes work. Modeling of dynamic interactions in nature can ...
* Quantitative ecology * Taylor's law * Theoretical biology


References


Further reading

* The classic text is ''Theoretical Ecology: Principles and Applications'', by
Angela McLean Angela McLean (born August 19, 1970) is an American politician and educator who served as the 35th Lieutenant Governor of Montana from February 10, 2014, to January 3, 2016. Governor Steve Bullock, a Democrat, selected McLean in 2014 to replac ...
and Robert May. The 2007 edition is published by the Oxford University Press. . ---- * Bolker BM (2008
''Ecological Models and Data in R''
Princeton University Press. . * Case TJ (2000
''An illustrated guide to theoretical ecology''
Oxford University Press. . * Caswell H (2000) ''Matrix Population Models: Construction, Analysis, and Interpretation'', Sinauer, 2nd Ed. . * Edelstein-Keshet L (2005
''Mathematical Models in Biology''
Society for Industrial and Applied Mathematics. . * Gotelli NJ (2008
''A Primer of Ecology''
Sinauer Associates, 4th Ed. . * Gotelli NJ & A Ellison (2005
''A Primer Of Ecological Statistics''
Sinauer Associates Publishers. . * Hastings A (1996
''Population Biology: Concepts and Models''
Springer. . * Hilborn R & M Clark (1997
''The Ecological Detective: Confronting Models with Data''
Princeton University Press. * Kokko H (2007
''Modelling for field biologists and other interesting people''
Cambridge University Press. . * Kot M (2001
''Elements of Mathematical Ecology''
Cambridge University Press. . * * Murray JD (2002
''Mathematical Biology, Volume 1''
Springer, 3rd Ed. . * Murray JD (2003
''Mathematical Biology, Volume 2''
Springer, 3rd Ed. . * Pastor J (2008
''Mathematical Ecology of Populations and Ecosystems''
Wiley-Blackwell. . * Roughgarden J (1998
''Primer of Ecological Theory''
Prentice Hall. . * Ulanowicz R (1997
''Ecology: The Ascendant Perspective''
Columbia University Press. {{DEFAULTSORT:Theoretical ecology *
Ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overl ...