The Lotka–Volterra equations, also known as the predator–prey equations, are a pair of first-order, non-linear, 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 growth rates of the two populations over time;
- t represents time; and
- α, β, γ, δ are positive real parameters describing the interaction of the two species.
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.
- 1 History
- 2 Physical meanings of the equations
- 3 Solutions to the equations
- 4 Dynamics of the system
- 5 See also
- 6 Notes
- 7 References
- 8 External links
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, which was originally derived by Pierre François Verhulst. In 1920 Lotka extended, via Kolmogorov (see above), the model to "organic systems" using a plant species and a herbivorous animal species as an example  and in 1925 he utilised the equations to analyse predator-prey interactions in his book on biomathematics  arriving at the equations that we know today. Vito Volterra, who made a statistical analysis of fish catches in the Adriatic  independently investigated the equations in 1926.
C.S. Holling extended this model yet again, in two 1959 papers, in which he proposed the idea of functional response. Both the Lotka–Volterra model and Holling's extensions have been used to model the moose and wolf populations in Isle Royale National Park, which with over 50 published papers is one of the best studied predator-prey relationships.
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 two are the extremes of the spectrum of predator interference models. According to the authors of the alternative view, the data show that true interactions in nature are so far from the Lotka–Volterra extreme on the interference spectrum that the model can simply be discounted as wrong. They are much closer to the ratio dependent extreme, so if a simple model is needed one can use the Arditi–Ginzburg model as the first approximation.
The Lotka–Volterra equations have a long history of use in economic theory; their initial application is commonly credited to Richard Goodwin in 1965 or 1967. In economics, links are between many if not all industries; a proposed way to model the dynamics of various industries has been by introducing trophic functions between various sectors, and ignoring smaller sectors by considering the interactions of only two industrial sectors.
Physical meanings of the equations
The Lotka–Volterra model makes a number of assumptions about the environment and evolution of the predator and prey populations:
- 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.
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: the change in the prey's numbers is given by its own growth minus the rate at which it is preyed upon.
The predator equation becomes:
In this equation, 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). 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 the change in the predator population as growth fueled by the food supply, minus natural death.
Solutions to the equations
The equations have periodic solutions and do not have a simple expression in terms of the usual trigonometric functions, although they are quite tractable. If all the parameters α,β,γ,δ do not vanish, three can be absorbed into the normalization of variables to leave but one behind. 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.
An example problem
Suppose there are two species of animals, a baboon (prey) and a cheetah (predator). If the initial conditions are 80 baboons and 40 cheetahs, one can plot the progression of the two species over time. The choice of time interval is arbitrary.
One may also plot solutions parametrically in phase-space, without representing time, but with one axis representing the number of prey and the other axis representing the number of predators. This is to say, eliminating time from the two differential equations above results in one such,
whose solutions are closed curves; integrating yields an evident constant quantity V depending on the initial conditions, which is conserved on each curve,
These graphs clearly 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, chance fluctuations of the discrete numbers of individuals, as well as the family structure and life-cycle of baboons, would 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, in the context of rabies modelling in the UK.
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 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.
When solved for x and y the above system of equations yields
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.
Stability of the fixed points
The Jacobian matrix of the predator-prey 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 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 importance. 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, we find that the extinction of both species is difficult in the model. (In fact, this can only occur if the prey are artificially completely eradicated, causing the predators to die of starvation. If the predators are eradicated, the prey population grows without bound in this simple model).
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, this fixed point is not hyperbolic, so no conclusions can be drawn from the linear analysis. However, as illustrated above, the system admits a constant of motion V, or, equivalently, exp(V),
and the level curves, for each constant K, are closed trajectories surrounding the fixed point: the levels of the predator and prey populations cycle, and oscillate around this fixed point. Increasing K moves the closed trajectory closer to the fixed point.
Finally, the largest value of the constant K is obtained by solving the optimization problem
The maximal value of K is attained at the stationary point and amounts to
where e is Euler's Number.
- Competitive Lotka–Volterra equations
- Generalized Lotka–Volterra equation
- Mutualism and the Lotka–Volterra equation
- Community matrix
- Population dynamics
- Population dynamics of fisheries
- Nicholson–Bailey model
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- Interactive Lotka–Volterra Predator–Prey Model (Based on historical Isle Royale Data)
- Lotka–Volterra Predator–Prey Model by Elmer G. Wiens
- Lotka–Volterra Predator–Prey Model as a multi-agents system.
- Lotka–Volterra Model
- NANIA Lotka–Volterra applet Archived from the Original on 2012-07-10.
- From the Wolfram Demonstrations Project — requires CDF player (free):