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 dynamics, fisheries, competition, evolutionary theory, epidemiology, animal behavior and group dynamics, food webs, ecosystems, spatial ecology, and the effects of climate change.
Theoretical ecology has further benefited from the advent of fast computing power, allowing the analysis and visualization of large-scale computational simulations of ecological phenomena. Importantly, these modern tools provide quantitative predictions about the effects of human induced environmental change on a diverse variety of ecological phenomena, such as: species invasions, climate change, the effect of fishing and hunting on food network stability, and the global carbon cycle.
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 have found in the field or through experimentation.
- Mechanistic models: model the underlying processes directly, with functions and distributions that are based on theoretical reasoning about ecological processes of interest.
- 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 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.
- 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.
These can be modelled using stochastic branching processes. Examples are the dynamics of interacting populations, which, depending on the species of interest, may best be modeled over either continuous or discrete time. Other examples of such models may be found in the field of mathematical epidemiology where the dynamic relationships that are to be modeled are host–pathogen interactions.
Bifurcation theory is used to illustrate how small changes in parameter values can give rise to dramatically different long run outcomes, a mathematical fact that may be used to explain drastic ecological differences that come about in qualitatively very similar systems. Logistic maps are polynomial mappings, and are often cited as providing archetypal examples of how chaotic behaviour can arise from very simple non-linear dynamical equations. The maps were popularized in a seminal 1976 paper by the theoretical ecologist Robert May. The difference equation is intended to capture the two effects of reproduction and starvation.
In 1930, R.A. Fisher published his classic The Genetical Theory of Natural Selection, which introduced the idea that frequency-dependent fitness brings a strategic aspect to evolution, where the payoffs to a particular organism, arising from the interplay of all of the relevant organisms, are the number of this organism' s viable offspring. In 1961, Richard Lewontin applied game theory to evolutionary biology in his Evolution and the Theory of Games,
followed closely by John Maynard Smith, who in his seminal 1972 paper, "Game Theory and the Evolution of Fighting", defined the concept of the evolutionarily stable strategy.
Because ecological systems are typically nonlinear, they often cannot be solved analytically and in order to obtain sensible results, nonlinear, stochastic and computational techniques must be used. One class of computational models that is becoming increasingly popular are the agent-based models. These models can simulate the actions and interactions of multiple, heterogeneous, organisms where more traditional, analytical techniques are inadequate. Applied theoretical ecology yields results which are used in the real world. For example, optimal harvesting theory draws on optimization techniques developed in economics, computer science and operations research, and is widely used in fisheries.
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. 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, then the rate of population growth is given by:where r is the per capita growth rate, or the intrinsic growth rate of the organism. It can also be described as r = b-d, where b and d are the per capita time-invariant birth and death rates, respectively. This first order linear differential equation can be solved to yield the solution
a trajectory known as Malthusian growth, after Thomas Malthus, who first described its dynamics in 1798. A population experiencing Malthusian growth follows an exponential curve, where N 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.
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' = –, where a and c are constants that modulate birth and death rates in a population dependent manner. Both a and c will depend on other environmental factors which, we can for now, assume to be constant in this approximated model. The differential equation is now:This can be rewritten as:
where r = b-d and K = /.
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.
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 or different stage classes separately, and allowing individuals in each group to have their own survival and reproduction rates.The general form of this model is
where Nt is a vector of the number of individuals in each class at time t and L is a matrix that contains the survival probability and fecundity for each class. The matrix L is referred to as the Leslie matrix for age-structured models, and as the Lefkovitch matrix for stage-structured models.
If parameter values in L are estimated from demographic data on a specific population, a structured model can then be used to predict whether this population is expected to grow or decline in the long-term, and what the expected age distribution within the population will be. This has been done for a number of species including loggerhead sea turtles and right whales.
Community ecology
An ecological community is a group of trophically similar, sympatric species that actually or potentially compete in a local area for the same or similar resources. Interactions between these species form the first steps in analyzing more complex dynamics of ecosystems. These interactions shape the distribution and dynamics of species. Of these interactions, predation is one of the most widespread population activities.Taken in its most general sense, predation comprises predator–prey, host–pathogen, and host–parasitoid interactions.