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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 knowledge, know, understanding, understand, or simulation, simulate a subject the model represents. In contrast, physical models are physical object su ...
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, enco ...
. 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, indivi ...
, 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 level ...
.
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 Earth's atmosphere, atmosphere of the Earth. Carbon is the main component of biological compounds as well as ...
.
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 consi ...
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
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
. 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 mathematical ...
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, an ...
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, ''U ...
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 the ...
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
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. Mo ...
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 popular ...
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
Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
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, h ...
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 und ...
, 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 of ...
.
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 other ...
, 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 wha ...
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, both ...
.
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 overlaps wi ...
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 appropriate s ...
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 using a ...
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:
:
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 high ...
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
:
,
a trajectory known as
Malthusian 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 ...
, 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. .
:
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. 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 t ...
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
:
where N
t 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'' (franchis ...
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
A population pyramid (age structure diagram) or "age-sex pyramid" is a graphical illustration of the distribution of a population (typically that of a country or region of the world) by age groups and sex; it typically takes the shape of a pyramid ...
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 clas ...
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 sh ...
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 th ...
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 propos ...
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 Anc ...
, made a statistical analysis of fish catches in the Adriatic[Goel, 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:
:
:
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 stocked bodies of water such as ponds, canals, park wetlands and reservoirs. Fishing techniques inclu ...
was curtailed during the First World War
World War I (28 July 1914 11 November 1918), often abbreviated as WWI, was one of the deadliest global conflicts in history. Belligerents included much of Europe, the Russian Empire, the United States, and the Ottoman Empire, with fightin ...
. 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 sink ...
and Canadian lynx
The Canada lynx (''Lynx canadensis''), or Canadian lynx, is a medium-sized North American lynx that ranges across Alaska, Canada, and northern areas of the contiguous United States. It is characterized by its long, dense fur, triangular ears w ...
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, ''sever ...
and biological control of California red scale
''Aonidiella aurantii'' or red scale is an armored scale insect and a major pest of citrus. It is thought to be a native of South China but has been widely dispersed by the agency of man through the movement of infected plant material. In the ...
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
''Aphytis melinus'' is an internal parasite of the California red scale, ''Aonidiella aurantii'', which is a pest of citrus in California and elsewhere. This wasp drums its antennae against the scale insect to find out if it is healthy, if it is ...
''
.
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 germ ...
, 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 also ...
, 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:
: