The Lotka–Volterra equations, also called the Lotka–Volterra predator–prey model, are two mathematical equations used to describe how populations of two species change over time. One species is a predator, and the other is prey. The equations show how the sizes of these populations change:

$$
frac{dx}{dt} = alpha x – beta xy, quad frac{dy}{dt} = -gamma y + delta xy
$$

Here, $ x $ represents the prey population, and $ y $ represents the predator population. The letters $ alpha, beta, gamma, delta $ are constants that describe how the populations grow or decline based on their interactions.

The solutions to these equations are predictable and smooth, meaning the predator and prey populations are always overlapping in their life cycles.

The Lotka–Volterra equations are an example of a Kolmogorov population model, a broader system that can describe changes in ecological systems, including predator-prey relationships, competition, disease, and mutualism. This should not be confused with the Kolmogorov equations, which are a different type of mathematical model.

Biological interpretation and model assumptions

The prey are assumed to have an endless food supply and to grow very quickly, unless they are hunted by predators. This fast growth is shown in the equation by the term αx. The rate at which predators hunt prey is thought to depend on how often predators and prey meet, which is shown in the equation by βxy. If there are no prey (x = 0) or no predators (y = 0), then no hunting can occur. These two terms explain that the change in the prey population depends on its own growth rate minus the rate at which it is hunted.

The term δxy shows how the predator population grows. (This is similar to the hunting rate, but a different number is used because predator growth does not always match the rate at which they eat prey.) The term γy shows how predator numbers decrease due to natural deaths or leaving the area, which causes their population to shrink quickly if there are no prey. This means the predator population's change depends on how much prey they eat minus how many predators die naturally.

The Lotka–Volterra predator-prey model includes several assumptions about the environment and the animals involved:

• The prey always have enough food available.
• The predators’ food supply depends only on how many prey there are.
• The rate at which populations change is proportional to their current size.
• The environment stays the same for both species, and genetic changes do not affect the outcome.
• Predators can eat as much prey as they want.
• Both populations can be described using one number each, meaning their size does not depend on where they live or their age.

Biological relevance of the model

The assumptions discussed earlier are unlikely to apply to natural populations. However, the Lotka–Volterra model highlights two important characteristics of predator and prey populations. These characteristics often remain true even in more complex versions of the model where the assumptions are adjusted.

First, predator and prey populations tend to change in cycles, increasing and decreasing over time. This pattern has been observed in nature, such as in the changing numbers of lynx and snowshoe hares recorded by the Hudson's Bay Company, and in the fluctuating populations of moose and wolves on Isle Royale National Park.

Second, the model shows that the balance between predator and prey populations depends on specific factors. The prey’s stable population level (calculated as x = γ / δ) is influenced by predator-related factors, while the predator’s stable population level (calculated as y = α / β) is influenced by prey-related factors. This means that improving conditions for prey, such as increasing the prey’s growth rate (α), can lead to higher predator numbers but not necessarily higher prey numbers. For example, during World War I (1914–18), reduced fishing activity increased prey growth rates, which in turn led to more predatory fish being caught.

Another example involves experiments where iron was added to the ocean to promote phytoplankton growth. Iron is a nutrient that limits phytoplankton growth, so scientists expected it to increase phytoplankton and reduce carbon dioxide in the atmosphere. However, the phytoplankton growth was short-lived and quickly consumed by other organisms, such as small fish or zooplankton. This resulted in more predators but limited carbon sequestration. This outcome matches predictions from the Lotka–Volterra model and is also seen in more complex models that relax the model’s original assumptions.

Applications to economics and marketing

The Lotka–Volterra model is also used in fields like economics and marketing. It helps explain how businesses compete in a market, how products or platforms that work together affect each other, and how services in a sharing economy operate. In some cases, one business might force others to leave the market, while in other cases, the market reaches a balance where each company keeps its share. The model can also describe situations where industries change in repeating patterns or where changes happen often and are hard to predict.

In economics, the Phillips curve shows how unemployment and wage increases are connected. The Goodwin model links this idea to the Lotka–Volterra model by comparing how predator and prey interact in nature to how different groups in society compete for resources. This model draws comparisons to conflicts between social classes as described by Marx. The Kolmogorov version of the predator-prey model, along with later improvements to the Goodwin model, has helped expand these ideas further.

History

The Lotka–Volterra predator–prey model was first introduced by Alfred J. Lotka in 1910 as part of his study on chemical reactions that increase over time. This idea was based on the logistic equation, which was first developed by Pierre François Verhulst. In 1920, Lotka expanded the model with help from Andrey Kolmogorov to explain living systems, using an example of a plant species and a herbivore species. In 1925, he used these equations to study predator–prey relationships in his book on biomathematics. The same equations were also published in 1926 by Vito Volterra, a mathematician and physicist who became interested in biology. Volterra’s work was inspired by Umberto D’Ancona, a marine biologist who noticed that the number of predatory fish caught in the Adriatic Sea increased during World War I (1914–18), even though fishing activity had decreased. This observation puzzled D’Ancona because fewer predators usually mean more prey. Volterra created his model to explain this, working independently of Lotka. He acknowledged Lotka’s earlier work in his publication, and the model became known as the "Lotka–Volterra model."

Later, the model was improved to include how prey populations grow based on their density and how predators respond to prey availability, as described by C. S. Holling. This version is called the Rosenzweig–MacArthur model. Both the Lotka–Volterra and Rosenzweig–MacArthur models have been used to study how predator and prey populations change over time in nature.

In the late 1980s, a different model called the ratio-dependent or Arditi–Ginzburg model was introduced as an alternative to the Lotka–Volterra model and its variations. Scientists have debated whether models based on prey numbers or predator–prey ratios are more accurate.

The Lotka–Volterra equations have also been used in economic theories. Their first use in this area is often credited to Richard Goodwin in 1965 or 1967.

Solutions to the equations

The equations have solutions that repeat over time. These solutions cannot be written easily using common trigonometric functions, although they can still be studied and understood.

If none of the non-negative parameters α, β, γ, δ is zero, three of these parameters can be adjusted to simplify the equations, leaving only one parameter that influences the behavior of the solutions. This is because the first equation depends only on x, and the second depends only on y. Adjustments to β/α and δ/γ can be made to simplify the equations for y and x, and γ can be adjusted to simplify the equation for time. Only α/γ remains as a free parameter that affects the solutions.

If the equations are simplified, the solutions resemble those of simple harmonic motion, where the predator population lags behind the prey population by 90 degrees in each cycle.

Suppose there are two species: a rabbit (prey) and a fox (predator). If the starting numbers are 10 rabbits and 10 foxes per square kilometer, the changes in their populations over time can be shown in a graph. The growth rate of rabbits is 1.1, and their death rate is 0.4. For foxes, the growth rate is 0.1, and their death rate is 0.4. The choice of time units is not fixed.

Solutions can also be shown as paths in phase space, where one axis represents the number of prey and the other represents the number of predators at all times. This involves removing time from the equations to create a single equation that connects prey (x) and predator (y) populations. The solutions to this equation form closed loops. This equation can be solved by separating variables, leading to an implicit relationship:

V = constant, where V depends on the initial conditions and remains unchanged along each loop.

A note: These graphs highlight a possible problem in using this model for real-world biology. For certain parameter choices, the prey population drops to very low levels during each cycle, but recovers (while the predator population remains relatively high). In reality, small random changes in population numbers could cause prey to go extinct, which would also lead to predator extinction. This issue is called the "atto-fox problem," where an "atto-fox" represents an extremely small number of foxes. A density of 10 foxes per square kilometer corresponds to about 5×10 foxes globally, which is effectively zero in practical terms.

Since V(x, y) remains constant over time, it acts as a Hamiltonian function for the system. To explain this, the Poisson bracket can be defined as:
{f(x, y), g(x, y)} = −xy(∂f/∂x ∂g/∂y − ∂f/∂y ∂g/∂x).
Using this, Hamilton's equations become:
ẋ = {x, V} = αx − βxy,
ẏ = {y, V} = δxy − γy.
The variables x and y are not canonical because {x, y} = −xy ≠ 1. However, by using transformations p = ln(x) and q = ln(y), the equations can be rewritten in a canonical form with Hamiltonian H(q, p) = δe^p − γp + βe^q − αq. This leads to:
q̇ = ∂H/∂p = δe^p − γ,
ṗ = −∂H/∂q = α − βe^q.
The Poisson bracket for the canonical variables (q, p) now takes the standard form:
{F(q, p), G(q, p)} = (∂F/∂q ∂G/∂p − ∂F/∂p ∂G/∂q).

Another example uses:
α = 2/3, β = 4/3, γ = 1, δ = 1.
Assume x and y represent thousands of individuals. Circles show starting points for prey and predator populations, where x and y range from 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, predators grow in number when prey is abundant. However, as predators consume more prey, their food supply decreases, leading to a decline in predator numbers. When predator numbers are low, prey populations increase again. This pattern of growth and decline continues in a repeating cycle.

Population equilibrium happens when both predator and prey numbers remain unchanged. This occurs when the rates of change for both populations are zero:
x(α − βy) = 0,
−y(γ − δx) = 0.

Solving these equations gives two possible outcomes:
1. Both populations are zero (no predators or prey).
2. Predators and prey reach specific numbers: y = α/β and x = γ/δ.

The first solution means both species are extinct. If both populations are zero, they will stay that way forever. The second solution represents a balance where predator and prey numbers remain stable and non-zero indefinitely. The exact numbers depend on the values of the parameters α, β, γ, and δ.

To determine the stability of the balance point at zero (no life), scientists use a mathematical tool called the Jacobian matrix. For the point (0, 0), the Jacobian becomes:
J(0, 0) = [α, 0; 0, −γ].

The eigenvalues of this matrix are λ₁ = α and λ₂ = −γ. Since α and γ are always positive, the eigenvalues have opposite signs. This means the balance point at zero is unstable, like a saddle point. If this point were stable, populations might collapse to zero. However, because it is unstable, extinction is unlikely unless prey is completely removed (causing predators to starve) or predators are removed (allowing prey to grow endlessly).

At the second balance point (γ/δ, α/β), the Jacobian matrix becomes:
J(γ/δ, α/β) = [0, −βγ/δ; αδ/β, 0].

The eigenvalues here are λ₁ = i√(αγ) and λ₂ = −i√(αγ). These values are purely imaginary, meaning the balance point is either a center (with closed, repeating cycles) or a spiral. In this model, the balance point is a center, and populations oscillate in a repeating pattern around it. These cycles have a frequency of ω = √(αγ) and a period of T = 2π/√(αγ).

The conserved quantity V = δx − γ ln(x) + βy − α ln(y) helps describe these cycles. Closed orbits around the balance point show that predator and prey numbers rise and fall in a predictable, repeating pattern.

The constant K = y^α e^−βy x^γ e^−δx represents the value of V for these cycles. Increasing K moves the cycle closer to the balance point. The maximum value of K occurs at the balance point (γ/δ, α/β) and is calculated as K* = (α/(βe))²α (γ/(δe))²γ, where e is Euler’s number.