Theoretical ecology

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

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

.

Theoretical ecology has further benefited from the advent of fast computing power, allowing the analysis and visualization of large-scale computational simulations of ecological phenomena. Importantly, these modern tools provide quantitative predictions about the effects of human induced environmental change on a diverse variety of ecological phenomena, such as: species invasions, climate change, the effect of fishing and hunting on food network stability, and the global carbon cycle.

Modelling approaches

As in most other sciences, mathematical models form the foundation of modern ecological theory.

• Phenomenological models: distill the functional and distributional shapes from observed patterns in the data, or researchers decide on functions and distribution that are flexible enough to match the patterns they or others (field or experimental ecologists) have found in the field or through experimentation.^[3]
• Mechanistic models: model the underlying processes directly, with functions and distributions that are based on theoretical reasoning about ecological processes of interest.^[3]

Ecological models can be

• Deterministic models always evolve in the same way from a given starting point.
  random variation. Many system dynamics
  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

• Continuous time is modelled using differential equations.
• Discrete time is modelled using
  difference equations. These model ecological processes that can be described as occurring over discrete time steps. Matrix algebra is often used to investigate the evolution of age-structured or stage-structured populations. The Leslie matrix, for example, mathematically represents the discrete time change of an age structured population.^[6][7][8]

Models are often used to describe real ecological reproduction processes of single or multiple species. These can be modelled using stochastic

The difference equation is intended to capture the two effects of reproduction and starvation.

In 1930,

followed closely by John Maynard Smith, who in his seminal 1972 paper, “Game Theory and the Evolution of Fighting",^[13] defined the concept of the evolutionarily stable strategy.

Because ecological systems are typically

Population ecology

Population ecology is a sub-field of ecology that deals with the dynamics of species populations and how these populations interact with the environment.^[15] It is the study of how the population sizes 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:

${\displaystyle {\frac {dN(t)}{dt}}=rN(t)}$

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 linear differential equation can be solved to yield the solution

${\displaystyle N(t)=N(0)\ e^{rt}}$,

a trajectory known as

, 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). 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:^[16]

${\displaystyle {\frac {dN(t)}{dt}}=((b-aN(t))-(d-cN(t)))N(t)}$

This can be rewritten as:^[16]

${\displaystyle {\frac {dN(t)}{dt}}=rN(t)\left(1-{\frac {N}{K}}\right)}$

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 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.^[16]

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

${\displaystyle \mathbf {N} _{t+1}=\mathbf {L} \mathbf {N} _{t}}$

where N_t is a vector of the number of individuals in each class at time t and L is a matrix that contains the survival probability and fecundity for each class. The matrix L is referred to as the Leslie matrix for age-structured models, and as the Lefkovitch matrix for stage-structured models.^[17]

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 turtles and right whales.^[18][19]

Community ecology

An ecological community is a group of trophically similar,

Taken in its most general sense, predation comprises predator–prey, host–pathogen, and host–parasitoid interactions.

Predator–prey interaction

:

${\displaystyle {\frac {dN(t)}{dt}}=N(t)(r-\alpha P(t))}$

${\displaystyle {\frac {dP(t)}{dt}}=P(t)(c\alpha N(t)-d)}$

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

[27] and biological control of .[28]

A credible, simple alternative to the Lotka-Volterra predator–prey model and their common prey dependent generalizations is the ratio dependent or

Host–pathogen interaction

The second interaction, that of host and pathogen, 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.^[31]

Host–parasitoid interaction

The third interaction, that of host and parasitoid, can be analyzed by the Nicholson–Bailey model, which differs from Lotka-Volterra and SIR models in that it is discrete in time. This model, like that of Lotka-Volterra, tracks both populations explicitly. Typically, in its general form, it states:

${\displaystyle N_{t+1}=\lambda \ N_{t}\ [1-f(N_{t},P_{t})]}$

${\displaystyle P_{t+1}=c\ N_{t}\ f(N_{t},p_{t})}$

where f(N_t, P_t) describes the probability of infection (typically, Poisson distribution), λ is the per-capita growth rate of hosts in the absence of parasitoids, and c is the conversion efficiency, as in the Lotka-Volterra model.^[21]

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,^[32] as well as theoretical analysis of the behavior of the system.^[33]

${\displaystyle {\frac {dN_{1}}{dt}}={\frac {r_{1}N_{1}}{K_{1}}}\left(K_{1}-N_{1}+\alpha _{12}N_{2}\right)}$

${\displaystyle {\frac {dN_{2}}{dt}}={\frac {r_{2}N_{2}}{K_{2}}}\left(K_{2}-N_{2}+\alpha _{21}N_{1}\right)}$

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

. 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 for further extensions of this model.

Neutral theory

Unified neutral theory is a hypothesis proposed by Stephen P. Hubbell in 2001.^[20] The hypothesis aims to explain the diversity and relative abundance of species in ecological communities, although like other neutral theories in ecology, Hubbell's hypothesis assumes that the differences between members of an ecological community of trophically similar species are "neutral," or irrelevant to their success. Neutrality means that at a given trophic level in a food web, species are equivalent in birth rates, death rates, dispersal rates and speciation rates, when measured on a per-capita basis.^[34] This implies that biodiversity arises at random, as each species follows a random walk.^[35] This can be considered a null hypothesis to niche theory. The hypothesis has sparked controversy, and some authors consider it a more complex version of other null models that fit the data better.

Under unified neutral theory, complex ecological interactions are permitted among individuals of an

, 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

Spatial ecology

Biogeography

Biogeography 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

r/K-selection theory

Main article:

r/K selection

A population ecology concept is r/K selection theory, one of the first predictive models in ecology used to explain

neutral theories.^[41]

Niche theory

Main article:

Niche models

Metapopulations

Spatial analysis of ecological systems often reveals that assumptions that are valid for spatially homogenous populations – and indeed, intuitive – may no longer be valid when migratory subpopulations moving from one patch to another are considered.^[42] 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

${\displaystyle {\frac {dp}{dt}}=mp(1-p)-ep}$

where m is the rate of colonization, and e is the rate of extinction.^[43] 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,

${\displaystyle {\frac {dp_{1}}{dt}}=m_{1}p_{1}(1-p_{1})-ep_{1}}$

${\displaystyle {\frac {dp_{2}}{dt}}=m_{2}p_{2}(1-p_{1}-p_{2})-ep_{2}-mp_{1}p_{2}}$

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

${\displaystyle p_{2}^{*}={\frac {e}{m_{1}}}-{\frac {m_{1}}{m_{2}}}}$

(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

The form of the differential equations used in this simplistic modelling approach can be modified. For example:

1. 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.[45]
2. Extinction may depend non-linearly on p (e*p*(1-p)) as opposed to the linear (e*p) regime described above. This is referred to as the “rescue effect” and it is again harder to drive a population extinct under this regime.^[45]

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.^[46]

Ecosystem ecology

Introducing new elements, whether

of the original ecosystem.

If ecosystems are governed primarily by

NOAA

, and widely used in fisheries management as a tool for modelling and visualising the complex relationships that exist in real world marine ecosystems.

Food webs

In 1927, Charles Elton published an influential synthesis on the use of food webs, which resulted in them becoming a central concept in ecology.^[50] 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.^[51] Many theoretical ecologists, including Sir Robert May and Stuart Pimm, were prompted by this discovery and others to examine the mathematical properties of food webs. According to their analyses, complex food webs should be less stable than simple food webs.^{[1]: 75–77 [2]: 64} The apparent paradox between the complexity of food webs observed in nature and the mathematical fragility of food web models is currently an area of intensive study and debate. The paradox may be due partially to conceptual differences between persistence of a food web and equilibrial stability of a food web.^[1][2]

Systems 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".^[53][54]

Behavioral ecology

Swarm behaviour

to land.^[55]

See also:

Swarm models

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

.

On cellular levels, individual organisms also demonstrated swarm behavior.

were indicated to be vital for large multicellular animals in the evolutionary pathway.

Synchronization

Photinus carolinus firefly will synchronize their shining frequencies in a collective setting. Individually, there are no apparent patterns for the flashing. In a group setting, periodicity emerges in the shining pattern.^[62] The coexistence of the synchronization and asynchronization in the flashings in the system composed of multiple fireflies could be characterized by the chimera states. Synchronization could spontaneously occur.^[63] The agent-based model has been useful in describing this unique phenomenon. The flashings of individual fireflies could be viewed as oscillators and the global coupling models were similar to the ones used in condensed matter physics.

Evolutionary ecology

The British biologist Alfred Russel Wallace is best known for independently proposing a theory of evolution due to natural selection that prompted Charles Darwin to publish his own theory. In his famous 1858 paper, Wallace proposed natural selection as a kind of feedback mechanism which keeps species and varieties adapted to their environment.^[64]

The action of this principle is exactly like that of the centrifugal governor 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.^[65]

The cybernetician and anthropologist Gregory Bateson observed in the 1970s that, though writing it only as an example, Wallace had "probably said the most powerful thing that’d been said in the 19th Century".^[66] Subsequently, the connection between natural selection and systems theory has become an area of active research.^[64]

Other theories

In contrast to previous ecological theories which considered

History

Theoretical ecology draws on pioneering work done by

Simberloff added statistical rigour to experimental ecology and was a key figure in the SLOSS debate, about whether it is preferable to protect a single large or several small reserves.^[70] This resulted in the supporters of Jared Diamond's community assembly rules defending their ideas through Neutral Model Analysis.^[70] 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:

• Category:Mathematical ecologists
• Category:Theoretical biologists

Journals

See also

References

Further reading

• The classic text is Theoretical Ecology: Principles and Applications, by
  ISBN 978-0-19-920998-9
  .

---

• Bolker BM (2008) Ecological Models and Data in R Princeton University Press.
  ISBN 978-0-691-12522-0
  .
• Case TJ (2000) An illustrated guide to theoretical ecology Oxford University Press.
  ISBN 978-0-19-508512-9
  .
• Caswell H (2000) Matrix Population Models: Construction, Analysis, and Interpretation, Sinauer, 2nd Ed.
  ISBN 978-0-87893-096-8
  .
• Edelstein-Keshet L (2005) Mathematical Models in Biology Society for Industrial and Applied Mathematics.
  ISBN 978-0-89871-554-5
  .
• Gotelli NJ (2008) A Primer of Ecology Sinauer Associates, 4th Ed.
  ISBN 978-0-87893-318-1
  .
• Gotelli NJ & A Ellison (2005) A Primer Of Ecological Statistics Sinauer Associates Publishers.
  ISBN 978-0-87893-269-6
  .
• Hastings A (1996) Population Biology: Concepts and Models Springer.
  ISBN 978-0-387-94853-9
  .
• 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.
  ISBN 978-0-521-83132-1
  .
• Kot M (2001) Elements of Mathematical Ecology Cambridge University Press.
  ISBN 978-0-521-00150-2
  .
• Lawton JH (1999). "Are there general laws in ecology?" (PDF). Oikos. 84 (2): 177–192.
  JSTOR 3546712. Archived from the original
  (PDF) on 2010-06-11.
• Murray JD (2002) Mathematical Biology, Volume 1 Springer, 3rd Ed.
  ISBN 978-0-387-95223-9
  .
• Murray JD (2003) Mathematical Biology, Volume 2 Springer, 3rd Ed.
  ISBN 978-0-387-95228-4
  .
• Pastor J (2008) Mathematical Ecology of Populations and Ecosystems Wiley-Blackwell.
  ISBN 978-1-4051-8811-1
  .
• Roughgarden J (1998) Primer of Ecological Theory Prentice Hall.
  ISBN 978-0-13-442062-2
  .
• Ulanowicz R (1997) Ecology: The Ascendant Perspective Columbia University Press.