The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order nonlinear differential equations, frequently used to describe the dynamics of biological systems in which two species interact, one as a predator and the other as prey. The populations change through time according to the pair of equations:
The Lotka–Volterra system of equations is an example of a Kolmogorov model, which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions, competition, disease, and mutualism.
The Lotka–Volterra predator–prey model was initially proposed by Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910. This was effectively the logistic equation, originally derived by Pierre François Verhulst. In 1920 Lotka extended the model, via Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example and in 1925 he used the equations to analyse predator–prey interactions in his book on biomathematics. The same set of equations was published in 1926 by Vito Volterra, a mathematician and physicist, who had become interested in mathematical biology. Volterra's enquiry was inspired through his interactions with the marine biologist Umberto D'Ancona, who was courting his daughter at the time and later was to become his son-in-law. D'Ancona studied the fish catches in the Adriatic Sea and had noticed that the percentage of predatory fish caught had increased during the years of World War I (1914–18). This puzzled him, as the fishing effort had been very much reduced during the war years. Volterra developed his model independently from Lotka and used it to explain d'Ancona's observation.
The model was later extended to include density-dependent prey growth and a functional response of the form developed by C. S. Holling; a model that has become known as the Rosenzweig–McArthur model. Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey, such as the lynx and snowshoe hare data of the Hudson's Bay Company and the moose and wolf populations in Isle Royale National Park.
In the late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or Arditi–Ginzburg model. The validity of prey- or ratio-dependent models has been much debated.
The Lotka–Volterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:
When multiplied out, the prey equation becomes
The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; this exponential growth is represented in the equation above by the term αx. The rate of predation upon the prey is assumed to be proportional to the rate at which the predators and the prey meet, this is represented above by βxy. If either x or y is zero, then there can be no predation.
With these two terms the equation above can be interpreted as follows: the rate of change of the prey's population is given by its own growth rate minus the rate at which it is preyed upon.
The predator equation becomes
In this equation, δxy represents the growth of the predator population. (Note the similarity to the predation rate; however, a different constant is used, as the rate at which the predator population grows is not necessarily equal to the rate at which it consumes the prey). γy represents the loss rate of the predators due to either natural death or emigration, it leads to an exponential decay in the absence of prey.
Hence the equation expresses that the rate of change of the predator's population depends upon the rate at which it consumes prey, minus its intrinsic death rate.
If none of the non-negative parameters α, β, γ, δ vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous in x, and the second one in y, the parameters β/α and δ/γ are absorbable in the normalizations of y and x respectively, and γ into the normalization of t, so that only α/γ remains arbitrary. It is the only parameter affecting the nature of the solutions.
Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 10 baboons and 10 cheetahs, one can plot the progression of the two species over time; given the parameters that the growth and death rates of baboon are 1.1 and 0.4 while that of cheetahs are 0.1 and 0.4 respectively. The choice of time interval is arbitrary.
One may also plot solutions parametrically as orbits in phase space, without representing time, but with one axis representing the number of prey and the other axis representing the number of predators for all times.
This corresponds to eliminating time from the two differential equations above to produce a single differential equation
relating the variables x and y. The solutions of this equation are closed curves. It is amenable to separation of variables: integrating
yields the implicit relationship
where V is a constant quantity depending on the initial conditions and conserved on each curve.
An aside: These graphs illustrate a serious potential problem with this as a biological model: For this specific choice of parameters, in each cycle, the baboon population is reduced to extremely low numbers, yet recovers (while the cheetah population remains sizeable at the lowest baboon density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, might cause the baboons to actually go extinct, and, by consequence, the cheetahs as well. This modelling problem has been called the "atto-fox problem", an atto-fox being a notional 10−18 of a fox.
A less extreme example covers:
α = 2/3, β = 4/3, γ = 1 = δ. Assume x, y quantify thousands each. Circles represent prey and predator initial conditions from x = y = 0.9 to 1.8, in steps of 0.1. The fixed point is at (1, 1/2).
In the model system, the predators thrive when there are plentiful prey but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in a cycle of growth and decline.
Population equilibrium occurs in the model when neither of the population levels is changing, i.e. when both of the derivatives are equal to 0:
The above system of equations yields two solutions:
Hence, there are two equilibria.
The first solution effectively represents the extinction of both species. If both populations are at 0, then they will continue to be so indefinitely. The second solution represents a fixed point at which both populations sustain their current, non-zero numbers, and, in the simplified model, do so indefinitely. The levels of population at which this equilibrium is achieved depend on the chosen values of the parameters α, β, γ, and δ.
The Jacobian matrix of the predator–prey model is
and is known as community matrix.
When evaluated at the steady state of (0, 0), the Jacobian matrix J becomes
The eigenvalues of this matrix are
In the model α and γ are always greater than zero, and as such the sign of the eigenvalues above will always differ. Hence the fixed point at the origin is a saddle point.
The stability of this fixed point is of significance. If it were stable, non-zero populations might be attracted towards it, and as such the dynamics of the system might lead towards the extinction of both species for many cases of initial population levels. However, as the fixed point at the origin is a saddle point, and hence unstable, it follows that the extinction of both species is difficult in the model. (In fact, this could only occur if the prey were artificially completely eradicated, causing the predators to die of starvation. If the predators were eradicated, the prey population would grow without bound in this simple model.) The populations of prey and predator can get infinitesimally close to zero and still recover.
Evaluating J at the second fixed point leads to
The eigenvalues of this matrix are
As the eigenvalues are both purely imaginary and conjugate to each others, this fixed point is elliptic, so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a period .
As illustrated in the circulating oscillations in the figure above, the level curves are closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with period .
The value of the constant of motion V, or, equivalently, K = exp(V), , can be found for the closed orbits near the fixed point.
Increasing K moves a closed orbit closer to the fixed point. The largest value of the constant K is obtained by solving the optimization problem
The maximal value of K is thus attained at the stationary (fixed) point and amounts to
where e is Euler's number.
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 proposal of the predator–prey model, developed simultaneously but independently of Vito Volterra. The Lotka–Volterra model is still the basis of many models used in the analysis of population dynamics in ecology.Arditi–Ginzburg equations
The Arditi–Ginzburg equations describe ratio dependent predator–prey dynamics. Where N is the population of a prey species and P that of a predator, the population dynamics are described by the following two equations:
Here f(N) captures any change in the prey population not due to predator activity including inherent birth and death rates. The per capita effect of predators on the prey population (the harvest rate) is modeled by a function g which is a function of the ratio N/P of prey to predators. Predators receive a reproductive payoff, e, for consuming prey, and die at rate u. Making predation pressure a function of the ratio of prey to predators contrasts with the prey dependent Lotka–Volterra equations, where the effect of predators on the prey population is simply a function of the magnitude of the prey population g(N). Because the number of prey harvested by each predator decreases as predators become more dense, ratio dependent predation represents an example of a trophic function. Ratio dependent predation may account for heterogeneity in large-scale natural systems in which predator efficiency decreases when prey is scarce. The merit of ratio dependent versus prey dependent models of predation has been the subject of much controversy, especially between the biologists Lev R. Ginzburg and Peter A. Abrams. Ginzburg purports that ratio dependent models more accurately depict predator-prey interactions while Abrams maintains that these models make unwarranted complicating assumptions.Bacterivore
Bacterivores are free-living, generally heterotrophic organisms, exclusively microscopic, which obtain energy and nutrients primarily or entirely from the consumption of bacteria. Many species of amoeba are bacterivores, as well as other types of protozoans. Commonly, all species of bacteria will be prey, but spores of some species, such as Clostridium perfringens, will never be prey, because of their cellular attributes.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 include trophic interactions.Copiotroph
A copiotroph is an organism found in environments rich in nutrients, particularly carbon. They are the opposite to oligotrophs, which survive in much lower carbon concentrations.
Copiotrophic organisms tend to grow in high organic substrate conditions. For example, copiotrophic organisms grow in Sewage lagoons. They grow in organic substrate conditions up to 100x higher than oligotrophs.Dominance (ecology)
Ecological dominance is the degree to which a taxon is more numerous than its competitors in an ecological community, or makes up more of the biomass.
Most ecological communities are defined by their dominant species.
In many examples of wet woodland in western Europe, the dominant tree is alder (Alnus glutinosa).
In temperate bogs, the dominant vegetation is usually species of Sphagnum moss.
Tidal swamps in the tropics are usually dominated by species of mangrove (Rhizophoraceae)
Some sea floor communities are dominated by brittle stars.
Exposed rocky shorelines are dominated by sessile organisms such as barnacles and limpets.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 real system.Using data gathered from the field, ecological relationships—such as the relation of sunlight and water availability to photosynthetic rate, or that between predator and prey populations—are derived, and these are combined to form ecosystem models. These model systems are then studied in order to make predictions about the dynamics of the real system. Often, the study of inaccuracies in the model (when compared to empirical observations) will lead to the generation of hypotheses about possible ecological relations that are not yet known or well understood. Models enable researchers to simulate large-scale experiments that would be too costly or unethical to perform on a real ecosystem. They also enable the simulation of ecological processes over very long periods of time (i.e. simulating a process that takes centuries in reality, can be done in a matter of minutes in a computer model).Ecosystem models have applications in a wide variety of disciplines, such as natural resource management, ecotoxicology and environmental health, agriculture, and wildlife conservation. Ecological modelling has even been applied to archaeology with varying degrees of success, for example, combining with archaeological models to explain the diversity and mobility of stone tools.Feeding frenzy
In ecology, a feeding frenzy occurs when predators are overwhelmed by the amount of prey available. For example, a large school of fish can cause nearby sharks, such as the lemon shark, to enter into a feeding frenzy. This can cause the sharks to go wild, biting anything that moves, including each other or anything else within biting range. Another functional explanation for feeding frenzy is competition amongst predators. This term is most often used when referring to sharks or piranhas. It has also been used as a term within journalism.Generalized Lotka–Volterra equation
The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator–prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their dynamics can be analysed analytically to some extent. This makes them useful as a theoretical tool for modeling food webs. However, they lack features of other ecological models such as predator preference and nonlinear functional responses, and they cannot be used to model mutualism without allowing indefinite population growth.
The generalised Lotka-Volterra equations model the dynamics of the populations of biological species. Together, these populations can be considered as a vector . They are a set of ordinary differential equations given by
where the vector is given by
where is a vector and A is a matrix known as the community matrix.Kill the Winner hypothesis
The "Kill the Winner" hypothesis (KTW) is a model of population growth involving prokaryotes, viruses and protozoans. It is based on the concept of prokaryotes taking one of two reactions to limited resources: "competition", that is, that priority directed to growth of the population, or a "winner"; and "defense", where the resources are directed to survival against attacks. It is then assumed that the better strategy for a phage, or virus which attacks prokaryotes, is to concentrate on the "winner", the most active population (possibly the most abundant). This tends to moderate the relative populations of the prokaryotes, rather than the "winner take all". The model is related to the Lotka–Volterra equations.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 corollary of the competitive exclusion principle, which states that, controlling for all else, two species competing for exactly the same resources cannot stably coexist. It assumes normally-distributed resource utilization curves ordered linearly along a resource axis, and as such, it is often considered to be an oversimplified model of species interactions. Moreover, it has theoretical weakness, and it is poor at generating real-world predictions or falsifiable hypotheses. Thus, the concept has fallen somewhat out of favor except in didactic settings (where it is commonly referenced), and has largely been replaced by more complex and inclusive theories.Mesotrophic soil
Mesotrophic soils are soils with a moderate inherent fertility. An indicator of soil fertility is its base status, which is expressed as a ratio relating the major nutrient cations (calcium, magnesium, potassium and sodium) found there to the soil's clay percentage. This is commonly expressed in hundredths of a mole of cations per kilogram of clay, i.e. cmol (+) kg−1 clay.Mycotroph
A mycotroph is a plant that gets all or part of its carbon, water, or nutrient supply through symbiotic association with fungi. The term can refer to plants that engage in either of two distinct symbioses with fungi:
Many mycotrophs have a mutualistic association with fungi in any of several forms of mycorrhiza. The majority of plant species are mycotrophic in this sense. Examples include Burmanniaceae.
Some mycotrophs are parasitic upon fungi in an association known as myco-heterotrophy.Organotroph
An organotroph is an organism that obtains hydrogen or electrons from organic substrates. This term is used in microbiology to classify and describe organisms based on how they obtain electrons for their respiration processes. Some organotrophs such as animals and many bacteria, are also heterotrophs. Organotrophs can be either anaerobic or aerobic.
Antonym: Lithotroph, Adjective: Organotrophic.Phase plane
In applied mathematics, in particular the context of nonlinear system analysis, a phase plane is a visual display of certain characteristics of certain kinds of differential equations; a coordinate plane with axes being the values of the two state variables, say (x, y), or (q, p) etc. (any pair of variables). It is a two-dimensional case of the general n-dimensional phase space.
The phase plane method refers to graphically determining the existence of limit cycles in the solutions of the differential equation.
The solutions to the differential equation are a family of functions. Graphically, this can be plotted in the phase plane like a two-dimensional vector field. Vectors representing the derivatives of the points with respect to a parameter (say time t), that is (dx/dt, dy/dt), at representative points are drawn. With enough of these arrows in place the system behaviour over the regions of plane in analysis can be visualized and limit cycles can be easily identified.
The entire field is the phase portrait, a particular path taken along a flow line (i.e. a path always tangent to the vectors) is a phase path. The flows in the vector field indicate the time-evolution of the system the differential equation describes.
In this way, phase planes are useful in visualizing the behaviour of physical systems; in particular, of oscillatory systems such as predator-prey models (see Lotka–Volterra equations). In these models the phase paths can "spiral in" towards zero, "spiral out" towards infinity, or reach neutrally stable situations called centres where the path traced out can be either circular, elliptical, or ovoid, or some variant thereof. This is useful in determining if the dynamics are stable or not.Other examples of oscillatory systems are certain chemical reactions with multiple steps, some of which involve dynamic equilibria rather than reactions that go to completion. In such cases one can model the rise and fall of reactant and product concentration (or mass, or amount of substance) with the correct differential equations and a good understanding of chemical kinetics.Recruitment (biology)
In biology, especially marine biology, recruitment occurs when a juvenile organism joins a population, whether by birth or immigration, usually at a stage whereby the organisms are settled and able to be detected by an observer.There are two types of recruitment: closed and open.In the study of fisheries, recruitment is "the number of fish surviving to enter the fishery or to some life history stage such as settlement or maturity".Relative abundance distribution
In the field of ecology, the relative abundance distribution (RAD) or species abundance distribution describes the relationship between the number of species observed in a field study as a function of their observed abundance. The graphs obtained in this manner are typically fitted to a Zipf–Mandelbrot law, the exponent of which serves as an index of biodiversity in the ecosystem under study.Replicator equation
In mathematics, the replicator equation is a deterministic monotone non-linear and non-innovative game dynamic used in evolutionary game theory. The replicator equation differs from other equations used to model replication, such as the quasispecies equation, in that it allows the fitness function to incorporate the distribution of the population types rather than setting the fitness of a particular type constant. This important property allows the replicator equation to capture the essence of selection. Unlike the quasispecies equation, the replicator equation does not incorporate mutation and so is not able to innovate new types or pure strategies.Volterra (disambiguation)
Volterra may refer to the following:
Volterra, a town in Italy
Daniele da Volterra (1509–1566), an Italian painter
Francesco da Volterra, a 14th-century Italian painter
Vito Volterra (1860–1940), an Italian mathematician
Volterra Semiconductor, an American semiconductor company
Volterra (crater), a lunar impact crater on the far side of the MoonIn mathematics:
Lotka–Volterra equations, also known as the predator–prey equations
The Smith–Volterra–Cantor set, a Cantor set with measure greater than zero
Volterra's function, a differentiable function whose derivative is not Riemann integrable
Volterra integral equation, a generalization of the indefinite integral
Volterra operator, a bounded linear operator on the space of square integrable functions, the operator corresponding to an indefinite integral
Volterra space, a property of topological spaces