# LotkaāVolterra equations Information

*https://en.wikipedia.org/wiki/LotkaāVolterra_equations*

The **LotkaāVolterra equations**, also known as the **predatorāprey 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:

- x is the number of prey (for example, rabbits);
- y is the number of some predator (for example, foxes);
- and represent the instantaneous growth rates of the two populations;
- t represents time;
- Ī±, Ī², Ī³, Ī“ are positive real parameters describing the interaction of the two species.

The LotkaāVolterra system of equations is an example of a
Kolmogorov model,^{
[1]}^{
[2]}^{
[3]} which is a more general framework that can model the dynamics of ecological systems with predatorāprey interactions,
competition, disease, and
mutualism.

## History

The LotkaāVolterra predatorāprey
model was initially proposed by
Alfred J. Lotka in the theory of autocatalytic chemical reactions in 1910.^{
[4]}^{
[5]} This was effectively the
logistic equation,^{
[6]} originally derived by
Pierre FranĆ§ois Verhulst.^{
[7]} In 1920 Lotka extended the model, via
Andrey Kolmogorov, to "organic systems" using a plant species and a herbivorous animal species as an example^{
[8]} and in 1925 he used the equations to analyse predatorāprey interactions in his book on
biomathematics.^{
[9]} The same set of equations was published in 1926 by
Vito Volterra, a mathematician and physicist, who had become interested in
mathematical biology.^{
[5]}^{
[10]}^{
[11]} 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 (1914ā18). 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.^{
[12]}

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 RosenzweigāMacArthur model.^{
[13]} Both the LotkaāVolterra and RosenzweigāMacArthur 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^{
[14]} and the moose and wolf populations in
Isle Royale National Park.^{
[15]}

In the late 1980s, an alternative to the LotkaāVolterra predatorāprey model (and its common-prey-dependent generalizations) emerged, the ratio dependent or
ArditiāGinzburg model.^{
[16]} The validity of prey- or ratio-dependent models has been much debated.^{
[17]}

The LotkaāVolterra equations have a long history of use in
economic theory; their initial application is commonly credited to
Richard Goodwin in 1965^{
[18]} or 1967.^{
[19]}^{
[20]}

## Physical meaning of the equations

The LotkaāVolterra model makes a number of assumptions, not necessarily realizable in nature, about the environment and evolution of the predator and prey populations:^{
[21]}

- The prey population finds ample food at all times.
- The food supply of the predator population depends entirely on the size of the prey population.
- The rate of change of population is proportional to its size.
- During the process, the environment does not change in favour of one species, and genetic adaptation is inconsequential.
- Predators have limitless appetite.

In this case the solution of the differential equations is
deterministic and
continuous. This, in turn, implies that the generations of both the predator and prey are continually overlapping.^{
[22]}

### Prey

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 *Ī±x*. 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 *Ī²xy*. 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.

### Predators

The predator equation becomes

In this equation, *Ī“xy* 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 term *Ī³y* 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.

## Solutions to the equations

The equations have
periodic solutions. These solutions do not have a simple expression in terms of the usual
trigonometric functions, although they are quite tractable.^{
[23]}^{
[24]}^{
[25]}

If none of the non-negative parameters Ī±, Ī², Ī³, Ī“ 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 *Ī²*/*Ī±* and *Ī“*/*Ī³* are absorbable in the normalizations of y and x respectively, and Ī³ into the normalization of t, so that only *Ī±*/*Ī³* remains arbitrary. It is the only parameter affecting the nature of the solutions.

A
linearization of the equations yields a solution similar to
simple harmonic motion^{
[26]} with the population of predators trailing that of prey by 90Ā° in the cycle.

### A simple example

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^{ā18} of a fox.^{
[27]}^{
[28]}

### Phase-space plot of a further example

A less extreme example covers:

*Ī±* = 2/3, *Ī²* = 4/3, *Ī³* = 1 = *Ī“*. 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).

## Dynamics of the system

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 population cycle of growth and decline.

### Population equilibrium

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 *Ī±*, *Ī²*, *Ī³*, and *Ī“*.

### Stability of the fixed points

The stability of the fixed point at the origin can be determined by performing a linearization using partial derivatives.

The Jacobian matrix of the predatorāprey model is

#### First fixed point (extinction)

When evaluated at the steady state of (0, 0), the Jacobian matrix J becomes

The eigenvalues of this matrix are

In the model Ī± and Ī³ 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 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.

#### Second fixed point (oscillations)

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 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 and elliptic, 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 closed orbits surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without damping around the fixed point with frequency .

The value of the
constant of motion *V*, or, equivalently, *K* = exp(ā*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

*K*is thus attained at the stationary (fixed) point and amounts to

*e*is Euler's number.

## See also

- Competitive LotkaāVolterra equations
- Generalized LotkaāVolterra equation
- Mutualism and the LotkaāVolterra equation
- Community matrix
- Population dynamics
- Population dynamics of fisheries
- Nicholson–Bailey model
- Reactionādiffusion system
- Paradox of enrichment
- Lanchester's laws, a similar system of differential equations for military forces

## Notes

**^**Freedman, H. I. (1980).*Deterministic Mathematical Models in Population Ecology*. Marcel Dekker.**^**Brauer, F.; Castillo-Chavez, C. (2000).*Mathematical Models in Population Biology and Epidemiology*. Springer-Verlag.**^**Hoppensteadt, F. (2006). "Predator-prey model".*Scholarpedia*.**1**(10): 1563. Bibcode: 2006SchpJ...1.1563H. doi: 10.4249/scholarpedia.1563.**^**Lotka, A. J. (1910). "Contribution to the Theory of Periodic Reaction".*J. Phys. Chem.***14**(3): 271ā274. doi: 10.1021/j150111a004.- ^
^{a}^{b}Goel, N. S.; et al. (1971).*On the Volterra and Other Nonlinear Models of Interacting Populations*. Academic Press. ISBN 0-12-287450-1. **^**Berryman, A. A. (1992). "The Origins and Evolution of Predator-Prey Theory" (PDF).*Ecology*.**73**(5): 1530ā1535. doi: 10.2307/1940005. JSTOR 1940005. Archived from the original (PDF) on 2010-05-31.**^**Verhulst, P. H. (1838). "Notice sur la loi que la population poursuit dans son accroissement".*Corresp. MathĆ©matique et Physique*.**10**: 113ā121.**^**Lotka, A. J. (1920). "Analytical Note on Certain Rhythmic Relations in Organic Systems".*Proc. Natl. Acad. Sci. U.S.A.***6**(7): 410ā415. Bibcode: 1920PNAS....6..410L. doi: 10.1073/pnas.6.7.410. PMC 1084562. PMID 16576509.**^**Lotka, A. J. (1925).*Elements of Physical Biology*. Williams and Wilkins.**^**Volterra, V. (1926). "Variazioni e fluttuazioni del numero d'individui in specie animali conviventi".*Mem. Acad. Lincei Roma*.**2**: 31ā113.**^**Volterra, V. (1931). "Variations and fluctuations of the number of individuals in animal species living together". In Chapman, R. N. (ed.).*Animal Ecology*. McGrawāHill.**^**Kingsland, S. (1995).*Modeling Nature: Episodes in the History of Population Ecology*. University of Chicago Press. ISBN 978-0-226-43728-6.**^**Rosenzweig, M. L.; MacArthur, R.H. (1963). "Graphical representation and stability conditions of predator-prey interactions".*American Naturalist*.**97**(895): 209ā223. doi: 10.1086/282272. S2CID 84883526.**^**Gilpin, M. E. (1973). "Do hares eat lynx?".*American Naturalist*.**107**(957): 727ā730. doi: 10.1086/282870. S2CID 84794121.**^**Jost, C.; Devulder, G.; Vucetich, J.A.; Peterson, R.; Arditi, R. (2005). "The wolves of Isle Royale display scale-invariant satiation and density dependent predation on moose".*J. Anim. Ecol*.**74**(5): 809ā816. doi: 10.1111/j.1365-2656.2005.00977.x.**^**Arditi, R.; Ginzburg, L. R. (1989). "Coupling in predator-prey dynamics: ratio dependence" (PDF).*Journal of Theoretical Biology*.**139**(3): 311ā326. Bibcode: 1989JThBi.139..311A. doi: 10.1016/s0022-5193(89)80211-5.**^**Abrams, P. A.; Ginzburg, L. R. (2000). "The nature of predation: prey dependent, ratio dependent or neither?".*Trends in Ecology & Evolution*.**15**(8): 337ā341. doi: 10.1016/s0169-5347(00)01908-x. PMID 10884706.**^**Gandolfo, G. (2008). "Giuseppe Palomba and the LotkaāVolterra equations".*Rendiconti Lincei*.**19**(4): 347ā357. doi: 10.1007/s12210-008-0023-7. S2CID 140537163.**^**Goodwin, R. M. (1967). "A Growth Cycle". In Feinstein, C. H. (ed.).*Socialism, Capitalism and Economic Growth*. Cambridge University Press.**^**Desai, M.; Ormerod, P. (1998). "Richard Goodwin: A Short Appreciation" (PDF).*The Economic Journal*.**108**(450): 1431ā1435. CiteSeerX 10.1.1.423.1705. doi: 10.1111/1468-0297.00350. Archived from the original (PDF) on 2011-09-27. Retrieved 2010-03-22.**^**"PREDATOR-PREY DYNAMICS".*www.tiem.utk.edu*. Retrieved 2018-01-09.**^**Cooke, D.; Hiorns, R. W.; et al. (1981).*The Mathematical Theory of the Dynamics of Biological Populations*. Vol. II. Academic Press.**^**Steiner, Antonio; Gander, Martin Jakob (1999). "Parametrische LĆ¶sungen der RĆ¤uber-Beute-Gleichungen im Vergleich".*Il Volterriano*.**7**: 32ā44.**^**Evans, C. M.; Findley, G. L. (1999). "A new transformation for the Lotka-Volterra problem".*Journal of Mathematical Chemistry*.**25**: 105ā110. doi: 10.1023/A:1019172114300. S2CID 36980176.**^**Leconte, M.; Masson, P.; Qi, L. (2022). "Limit cycle oscillations,response time,and the time-dependent solution to the Lotka-Volterra predator-prey model".*Physics of Plasmas*.**29**(2): 022302. arXiv: 2110.11557. doi: 10.1063/5.0076085. S2CID 239616189.**^**Tong, H. (1983).*Threshold Models in Non-linear Time Series Analysis*. SpringerāVerlag.**^**Lobry, Claude; Sari, Tewfik (2015). "Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem" (PDF).*Arima*.**20**: 95ā125.**^**Mollison, D. (1991). "Dependence of epidemic and population velocities on basic parameters" (PDF).*Math. Biosci*.**107**(2): 255ā287. doi: 10.1016/0025-5564(91)90009-8. PMID 1806118.

## Further reading

- Hofbauer, Josef;
Sigmund, Karl (1998). "Dynamical Systems and LotkaāVolterra Equations".
*Evolutionary Games and Population Dynamics*. New York: Cambridge University Press. pp. 1ā54. ISBN 0-521-62570-X. - Kaplan, Daniel;
Glass, Leon (1995).
*Understanding Nonlinear Dynamics*. New York: Springer. ISBN 978-0-387-94440-1. - Leigh, E. R. (1968). "The ecological role of Volterra's equations".
*Some Mathematical Problems in Biology*. – a modern discussion using Hudson's Bay Company data on lynx and hares in Canada from 1847 to 1903. - Murray, J. D. (2003).
*Mathematical Biology I: An Introduction*. New York: Springer. ISBN 978-0-387-95223-9.

## External links

Wikimedia Commons has media related to LotkaāVolterra equations. |