TheLotka–Volterra equations, also known as theLotka–Volterra predator–prey model, are a pair of first-ordernonlineardifferential equations, frequently used to describe thedynamics ofbiological systems in which two species interact, one as apredator and the other as prey. Thepopulations change through time according to the pair of equations:
where
The solution of the differential equations isdeterministic andcontinuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.[1]
The Lotka–Volterra system of equations is an example of aKolmogorov population model (not to be confused with the better knownKolmogorov equations),[2][3][4] which is a more general framework that can model the dynamics of ecological systems with predator–prey interactions,competition, disease, andmutualism.
The prey are assumed to have an unlimited food supply and to reproduce exponentially, unless subject to predation; thisexponential growth is represented in the equation above by the termαx. The rate of predation on the prey is assumed to be proportional to the rate at which the predators and the prey meet; this is represented above byβxy. If eitherx ory is zero, then there can be no predation. With these two terms the prey 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 termδxy 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). The termγy 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.
The Lotka–Volterra predator-prey model makes a number of assumptions about the environment and biology of the predator and prey populations:[5]
None of the assumptions above are likely to hold for natural populations. Nevertheless, the Lotka–Volterra model shows two important properties of predator and prey populations and these properties often extend to variants of the model in which these assumptions are relaxed:
Firstly, the dynamics of predator and prey populations have a tendency to oscillate. Fluctuating numbers of predators and prey have been observed in natural populations, such as thelynx andsnowshoe hare data of theHudson's Bay Company[6] and the moose and wolf populations inIsle Royale National Park.[7]
Secondly, the population equilibrium of this model has the property that the prey equilibrium density (given by) depends on the predator's parameters, and the predator equilibrium density (given by) on the prey's parameters. This has as a consequence that an increase in, for instance, the prey growth rate,, leads to an increase in the predator equilibrium density, but not the prey equilibrium density. Making the environment better for the prey benefits the predator, not the prey (this is related to theparadox of the pesticides and to theparadox of enrichment). A demonstration of this phenomenon is provided by the increased percentage of predatory fish caught had increased during the years ofWorld War I (1914–18), when prey growth rate was increased due to a reduced fishing effort.
A further example is provided by the experimentaliron fertilization of the ocean. In severalexperiments large amounts of iron salts were dissolved in the ocean. The expectation was that iron, which is a limiting nutrient for phytoplankton, would boost growth of phytoplankton and that it would sequester carbon dioxide from the atmosphere. The addition of iron typically leads to a short bloom in phytoplankton, which is quickly consumed by other organisms (such as small fish orzooplankton) and limits the effect of enrichment mainly to increased predator density, which in turn limits thecarbon sequestration. This is as predicted by the equilibrium population densities of the Lotka–Volterra predator-prey model, and is a feature that carries over to more elaborate models in which the restrictive assumptions of the simple model are relaxed.[8]
The Lotka–Volterra model has additional applications to areas such as economics[9] and marketing.[10][11] It can be used to describe the dynamics in a market with several competitors, complementary platforms and products, a sharing economy, and more. There are situations in which one of the competitors drives the other competitors out of the market and other situations in which the market reaches an equilibrium where each firm stabilizes on its market share. It is also possible to describe situations in which there are cyclical changes in the industry or chaotic situations with no equilibrium and changes are frequent and unpredictable.
In economics, thePhillips curve, which shows the statistical relationship between unemployment and the rate of change in nominal wages, has been connected by theGoodwin model. This model reinterprets the dynamics of the biological prey-predator interaction, as described by the Lotka-Volterra model, in economic terms. The way the two species interact in this model led Goodwin to draw parallels with theMarxian classconflict. The Kolmogorov generalization of the prey-predator model, along with further developments of the Goodwin model, has extended these ideas.[12]
The Lotka–Volterra predator–preymodel was initially proposed byAlfred J. Lotka in the theory of autocatalytic chemical reactions in 1910.[13][14] This was effectively thelogistic equation,[15] originally derived byPierre François Verhulst.[16] In 1920 Lotka extended the model, viaAndrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example[17] and in 1925 he used the equations to analyse predator–prey interactions in his book onbiomathematics.[18] The same set of equations was published in 1926 byVito Volterra, a mathematician and physicist, who had become interested inmathematical biology.[14][19][20] Volterra's enquiry was inspired through his interactions with the marine biologistUmberto 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 theAdriatic 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 and, as prey fish the preferred catch, one would intuitively expect this to increase of prey fish percentage. Volterra developed his model to explain D'Ancona's observation and did this independently from Alfred Lotka. He did credit Lotka's earlier work in his publication, after which the model has become known as the "Lotka-Volterra model".[21]
The model was later extended to include density-dependent prey growth and afunctional response of the form developed byC. S. Holling; a model that has become known as the Rosenzweig–MacArthur model.[22] Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to explain the dynamics of natural populations of predators and prey.
In the late 1980s, an alternative to the Lotka–Volterra predator–prey model (and its common-prey-dependent generalizations) emerged, the ratio dependent orArditi–Ginzburg model.[23] The validity of prey- or ratio-dependent models has been much debated.[24]
The Lotka–Volterra equations have a long history of use ineconomic theory; their initial application is commonly credited toRichard Goodwin in 1965[25] or 1967.[26][27]
The equations haveperiodic solutions. These solutions do not have a simple expression in terms of the usualtrigonometric functions, although they are quite tractable.[28][29][30]
If none of the non-negative parametersα,β,γ,δ vanishes, three can be absorbed into the normalization of variables to leave only one parameter: since the first equation is homogeneous inx, and the second one iny, the parametersβ/α andδ/γ are absorbable in the normalizations ofy andx respectively, andγ into the normalization oft, so that onlyα/γ remains arbitrary. It is the only parameter affecting the nature of the solutions.
Alinearization of the equations yields a solution similar tosimple harmonic motion[31] with the population of predators trailing that of prey by 90° in the cycle.
.svg)
Suppose there are two species of animals, a rabbit (prey) and a fox (predator). If the initial densities are 10 rabbits and 10 foxes per square kilometre, one can plot the progression of the two species over time; given the parameters that the growth and death rates of rabbits are 1.1 and 0.4 while that of foxes are 0.1 and 0.4 respectively. The choice of time interval is arbitrary.
One may also plot solutions parametrically asorbits inphase space, without representing time, but with one axis representing the number of prey and the other axis representing the densities of predators for all times.
This corresponds to eliminating time from the two differential equations above to produce a single differential equation
relating the variablesx (prey) andy (predator). The solutions of this equation are closed curves. It is amenable toseparation of variables: integrating
yields the implicit relationship
whereV is a constant quantity depending on the initial conditions and conserved on each curve.
An aside: These graphs illustrate a serious potential limitation in the application as a biological model: for this specific choice of parameters, in each cycle, the rabbit population is reduced to extremely low numbers, yet recovers (while the fox population remains sizeable at the lowest rabbit density). In real-life situations, however, chance fluctuations of the discrete numbers of individuals might cause the rabbits to actually go extinct, and, by consequence, the foxes as well. This modelling problem has been called the "atto-fox problem", anatto-fox being a notional 10−18 of a fox.[32][33] A density of 10−18 foxes per square kilometre equates to an average of approximately 5×10−10 foxes on the surface of the earth, which in practical terms means that foxes are extinct.
Since the quantity is conserved over time, it plays role of a Hamiltonian function of the system.[34] To see this we can definePoisson bracket as follows. ThenHamilton's equations readThe variables and are not canonical, since. However, using transformations[35] and we came up to a canonical form of theHamilton's equations featuring the Hamiltonian:ThePoisson bracket for the canonical variables now takes the standard form.
Another example covers:
α = 2/3,β = 4/3,γ = 1 =δ. Assumex,y quantify thousands each. Circles represent prey and predator initial conditions fromx =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 prey is plentiful but, ultimately, outstrip their food supply and decline. As the predator population is low, the prey population will increase again. These dynamics continue in apopulation 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:and
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 parametersα,β,γ, andδ.
The stability of the fixed point at the origin can be determined by performing alinearization usingpartial derivatives.
TheJacobian matrix of the predator–prey model isand is known as thecommunity matrix.
When evaluated at the steady state of(0, 0), the Jacobian matrixJ becomes
Theeigenvalues of this matrix are
In the modelα andγ 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 asaddle point.
The instability 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.
EvaluatingJ at the second fixed point leads to
The eigenvalues of this matrix are
As the eigenvalues are both purely imaginary and conjugate to each other, this fixed point must either be a center for closed orbits in the local vicinity or an attractive or repulsive spiral. In conservative systems, there must be closed orbits in the local vicinity of fixed points that exist at the minima and maxima of the conserved quantity. The conserved quantity is derived above to be on orbits. Thus orbits about the fixed point are closed andelliptic, so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency and period.
As illustrated in the circulating oscillations in the figure above, the level curves are closedorbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate withoutdamping around the fixed point with frequency.
The value of theconstant of motionV, or, equivalently,K = exp(−V),, can be found for the closed orbits near the fixed point.
IncreasingK moves a closed orbit closer to the fixed point. The largest value of the constantK is obtained by solving the optimization problemThe maximal value ofK is thus attained at the stationary (fixed) point and amounts towheree isEuler's number.
| International | |
|---|---|
| National | |
| Other | |
