Lotka–Volterra equation

The Lotka -Volterra equations ( also known as the predator-prey equations) are a system of two nonlinear, coupled differential equations of first order and describe the interaction of predator and prey populations. Under predator and prey, two classes of beings are meant, the one feeds on the other. [V 1] have been Established in 1925 by the equations Alfred J. Lotka and regardless of in 1926 by Vito Volterra. they read

Labeled [V 2]

The Lotka -Volterra equations are an important foundation of theoretical biology, and in particular population dynamics. The predators and prey, it must only be animals or individual species are not necessarily; In principle, the model is applicable to guilds - see, eg, Volterra fisheries data. The applicability of the Lotka -Volterra equations depends on the extent the reasoning of the mathematical model is correct in each case.

  • 3.1 periodicity
  • 3.2 Conservation of the mean values
  • 3.3 disorder of the average values
  • 4.1 Intraspecific competition Terme
  • 4.2 More than two classes of living beings
  • 5.1 Fisheries Data
  • 5.2 Medical Epidemiology
  • 5.3 Economics

Justification of the mathematical model

Volterra explains his system of equations as follows: [V 3 ]

  • The population numbers of the prey and the predator are designated respectively.
  • The uninterrupted growth rates per unit of time and are, with the sign are not yet fixed.
  • The ( average) number of encounters between prey and predator per unit of time is a positive real number, which is assumed to be constant within a biotope, but in general the habitat depends.
  • A sufficient number of matches have an effect on the number of population in the medium. When the prey creatures that is readily apparent: an encounter with a predator leads to a certain probability that the prey is eaten. In contrast, the impact of an encounter on the number of predators is only indirectly, in any case positive; for modeling an immediate impact is assumed on the population count at the robbers.

Taken together, the results in the equations

Division by leads to the equations


And one makes the border crossing, we obtain the Lotka -Volterra equations of the form mentioned above.

Mathematical treatment

Of course, Volterra was clear that the time-dependent population numbers and only integer values ​​can accept and therefore as functions of either constant or are not differentiable. But for large numbers of population made ​​by the transition to the continuous model relative error is small. The advantage of the two-dimensional Lotka -Volterra equation, however, is that some statements are mathematically provable that have an interesting relation to real data, as described below.

For the mathematical treatment of Lotka -Volterra systems are used today mostly the slightly simpler notation [M 1]

Where are positive constants and the number of prey and number of predators ( predators ), respectively.

Constant solutions

The constant solutions (also called equilibrium points or critical points) is obtained by setting the right hand sides of the Lotka -Volterra equations equal to zero:

So there are exactly two constant solutions, namely the trivial equilibrium point and the inner equilibrium point

A first integral

A method for detecting non-constant solutions is to find a first integral, ie an invariant of the motion. Volterra find such in the following way: [V 4] Multiplying the first equation and the second reason with, and then adds the two equations vanish, the terms with the product, and we obtain

By multiplying the first equation with reason and with the second and subsequent addition does one

Subtraction of these two equations gives

By integrating this last equation, we finally reached the relationship

Conversely, one can calculate the total derivative of the function thus defined by:

Then one arrives at the statement that is on the solutions of the basic equations constant ( invariant ); a solution of the Lotka -Volterra equation can therefore not leave their level lines.

Another way to locate an invariant of the motion is to reshape the Lotka -Volterra equations with the aid of a Euler's multiplier into an exact differential equation and integrating this then. [W 1]


Since the first integral is also a Lyapunov function, and since the inner equilibrium point has a strict local minimum, it follows from the first criterion of Lyapunov that this equilibrium point is stable.

The Lotka -Volterra laws

Using the first integral proves Volterra [V 5] three mathematical properties of the solutions ( "Acts" ) of the Lotka -Volterra equations have been used as Lotka -Volterra rules spread their biological interpretations.

From the edge of the functional behavior may be inferred that there is no trajectory that a point in the first quadrant

Has, this leaves: the first quadrant is invariant. The Lotka -Volterra laws generally apply to maximum solutions of the Lotka -Volterra equations in this quadrant; dies one of the two classes of animals, so this quadrant is exited and the Lotka -Volterra laws invalid.


Since the function of the quadrant is strictly convex and assumes its minimum at the inner point of equilibrium, which form contour lines of a closed curve in the phase space. There must be any solution in a level of line included, it follows from a consideration of the clarity and direction of the local field, the periodicity of the solutions. [V 6 ] [W 2]

Maintaining the mean values

From the periodicity of the solutions follows invoice with a few lines of the

That is, the time averages satisfy the equations

Confusing at first glance is here that the mean of the prey population depends only on mortality and feeding rate of the predator population and not on the reproduction rate of prey. In contrast, the mean value of the predator population only of reproduction and mortality rate of the prey population and not of feeding and death rate of predators depends. The equilibrium number of prey is higher, the less favorable are the parameters for the predators. The equilibrium number of robbers, however, is the higher, the more favorable are the parameters for the prey.

Naturally, this property of the Lotka -Volterra model, if you look at the coming here for application modeling look: control over the population number of a class of animals shall be incumbent exclusively of the other class.

Disturbance of the mean values

The most interesting because of its biological interpretation of these laws is the

In fact, Volterra proves a quantitative version: Is the destruction rate of the prey organisms and the rate of destruction of the robbers, as are the mean values ​​for the solutions of the perturbed Lotka -Volterra equations

This means the average over a Lotka -Volterra period Number of prey creatures if and only increases when predators are decimated - quite independent of depletion of prey, as long as it is not eradicated. Conversely, decreases the average number of predators whenever the prey creatures are decimated, and this decrease does not depend on how much the robbers in addition decimated (as long as they are not eradicated ).


In theoretical ecology, the Lotka -Volterra equations are the starting point for the development of more complex models, some of which have already been described in Volterra book.

Intraspecific competition Terme

A first extension of the Lotka -Volterra equations generated by subtraction of terms proportional to or that model the intraspecific competition [W 3] There are several ways to establish the form and the newly added terms.:

  • With the empirical studies on the population development by Pierre -François Verhulst, see logistic equation.
  • By the assumption that the (unperturbed ) growth rate of a population is proportional to the difference between a capacity limit and the actual population number.
  • Through an analysis of the influence of intraspecific encounters on the population numbers similar to Volterra reasoning of the term to the modeling of interspecific competition: A intraspecific encounter is with a certain probability a competition for a resource when an individual draws the short straw.

The resulting Lotka -Volterra competition equations of theoretical biology are a classical approach to describe the dynamics of a simplified biological communities, consisting of a renewable resource and therefore at least 2 competing species:

Where a, b exponential growth rates and the m ( mortality rate ) represent death rates. The available amount of resources at a time is assumed to be:

This results in:

By multiplying out and replacing the coefficients

One arrives at equations of the form

Allow the turn two equilibrium positions: the trivial equilibrium point, and the inner equilibrium point, which is given by a linear system of equations:

By solving this system of equations we find the equilibrium point

Which is under the condition in the first quadrant.

There is in this expanded Lotka -Volterra system, a Lyapunov function:

With the conditions of the second criterion of Lyapunov are met for the equilibrium point. It follows that the equilibrium point is now asymptotically stable. [W 4]

More than two classes of organisms

Much of Volterra's book [V 7 ] refers to extensions of its system to more than two classes of organisms that interact in different ways with each other.


Fisheries data

In the introduction to Volterra's book [V 8] there is a table that contains each of the percentage of cartilaginous fish ( Sélaciens ), ie in particular the Sharks in the total fishing the fishing port to 1905 and from 1910 to 1923 and three fishing ports:

These statistics show in 1915 until 1920, when fishing in the Mediterranean was less intense because of the First World War, an increased proportion of predatory fish, which then goes back again to the intensification of the fishery after 1920. The third Lotka -Volterra law, the shift of the mean values ​​, this provides a plausible explanation.

Medical Epidemiology

In theoretical biology as well as in medical epidemiology will find models of the Lotka -Volterra model to describe the spread of many diseases using. Some examples can be found in the SI model, SIR model and the SIS model.


The Goodwin model for the explanation of economic fluctuations are Lotka -Volterra equations based on where the wage share the role of the predator and the employment rate plays the role of prey.

Gerold Blümle developed an economic model in which ( mathematically ) the investment ratio plays the role of predators, and the dispersion or variance of the profits of the role of prey. At Frank Schohl the variance of yield changes of businesses comes the role of predators, the variance of the offer changes to the company to the role of prey.