The
Lotka–Volterra equations, also known as the
predator–prey equations, are a pair of first-order, non-linear,
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s frequently used to describe the dynamics of
biological systemsSystems biology is a biology-based inter-disciplinary study field that focuses on the systematic study of complex interactions in biological systems, thus using a new perspective to study them. Particularly from year 2000 onwards, the term is used widely in the biosciences, and in a variety of...
in which two species interact, one a predator and one its prey. They were proposed independently by
Alfred J. LotkaAlfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist best known for his proposal of the predator-prey model, developed simultaneously but independently of Vita Volterra...
in 1925 and
Vito VolterraVito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....
in 1926.
where
- y is the number of some predator (for example, wolves);
- x is the number of its prey (for example, rabbit
Rabbits are small mammals in the family Leporidae of the order Lagomorpha, found in several parts of the world. There are seven different genera in the family classified as rabbits, including the European rabbit , Cottontail rabbit , and the Amami rabbit...
s);
- dy/dt and dx/dt represents the growth of the two populations against time;
- t represents the time; and
- α, β, γ and δ are parameter
In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that serves to relate functions and variables using a common variable when such a relationship would be difficult to explicate with an equation...
s representing the interaction of the two speciesIn biology, a species is:* a taxonomic rank or* a unit at that rank ....
.
When multiplied out, the equations take a form useful for physical interpretation.
The
Lotka–Volterra equations, also known as the
predator–prey equations, are a pair of first-order, non-linear,
differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
s frequently used to describe the dynamics of
biological systemsSystems biology is a biology-based inter-disciplinary study field that focuses on the systematic study of complex interactions in biological systems, thus using a new perspective to study them. Particularly from year 2000 onwards, the term is used widely in the biosciences, and in a variety of...
in which two species interact, one a predator and one its prey. They were proposed independently by
Alfred J. LotkaAlfred James Lotka was a US mathematician, physical chemist, and statistician, famous for his work in population dynamics and energetics. An American biophysicist best known for his proposal of the predator-prey model, developed simultaneously but independently of Vita Volterra...
in 1925 and
Vito VolterraVito Volterra was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations....
in 1926.
where
- y is the number of some predator (for example, wolves);
- x is the number of its prey (for example, rabbit
Rabbits are small mammals in the family Leporidae of the order Lagomorpha, found in several parts of the world. There are seven different genera in the family classified as rabbits, including the European rabbit , Cottontail rabbit , and the Amami rabbit...
s);
- dy/dt and dx/dt represents the growth of the two populations against time;
- t represents the time; and
- α, β, γ and δ are parameter
In mathematics, statistics, and the mathematical sciences, a parameter is a quantity that serves to relate functions and variables using a common variable when such a relationship would be difficult to explicate with an equation...
s representing the interaction of the two speciesIn biology, a species is:* a taxonomic rank or* a unit at that rank ....
.
Physical meanings of the equations
When multiplied out, the equations take a form useful for physical interpretation. Their origin should be considered from a more general framework,
where both functions represent per capita growth rates of the prey and predator, respectively. These functions are too general, so a
Taylor seriesIn mathematics, the Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a single point. It may be regarded as the limit of the Taylor polynomials. Taylor series are named after the English mathematician Brook Taylor...
approximation is performed to obtain linearized per capita rates,
The signs of the coefficients arise from assumptions of population regulation, and by choosing nonzero coefficients appropriately, an ecologist can obtain predator-prey, competition, disease, and mutualism models that provide general insight into ecological systems.
ASSUMPTIONS
1) the prey population finds ample food at all times.
2) the food supply of the predator population depends entirely on the prey populations.
3) the rate of change of population is proportional to its size.
4) During the process, the environment does not change in favor of one species and the genetic adaptation is sufficiently slow.
Prey
The prey equation becomes:
The prey are assumed to have an unlimited food supply, and to reproduce exponentially unless subject to predation; this
exponential growthExponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
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.
Predators
The predator equation becomes:
In this equation,
δxy 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).
γy represents the natural death of the predators; 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
periodicIn mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π...
solutions which do not have a simple expression in terms of the usual
trigonometric functionIn mathematics, the trigonometric functions are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle...
s. However, an approximate
linearisedIn mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
solution yields a
simple harmonic motionIn physics, simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. A body in simple harmonic motion experiences a single force which is given by Hooke's law; that is, the force is directly proportional to the displacement x and points in...
with the population of predators following that of prey by 90°.
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. Time is dimensionless.
One can also plot a solution which corresponds to the oscillatory nature of the population of the two species. At any given time, the solution is somewhere on the inside of these elliptical solutions.
These graphs clearly illustrate a serious problem with this as a biological model: 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). Given chance fluctuations, discrete numbers of individuals, and the family structure and lifecycle of baboons, the baboons actually go extinct and by consequence the cheetahs as well. This modelling problem has been called the "atto-fox problem", an atto-fox being an imaginary 10
−18 of a fox, in relation to 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
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
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 parameters, α, β, γ, and δ.
Stability of the fixed points
The stability of the fixed point at the origin can be determined by performing a
linearizationIn mathematics and its applications, linearization refers to finding the linear approximation to a function at a given point. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or...
using
partial derivativeIn mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant...
s, while the other fixed point requires a slightly more sophisticated method.
The Jacobian matrix of the predator-prey model is
First fixed point
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 pointIn mathematics, a saddle point is a point in the domain of a function of two variables which is a stationary point but not a local extremum. At such a point, in general, the surface resembles a saddle that curves up in one direction, and curves down in a different direction...
.
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
Evaluating
J at the second fixed point we get
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, the system admits a
constant of motionIn mechanics, a constant of motion is a quantity that is conserved throughout the motion, imposing in effect a constraint on the motion. However, it is a mathematical constraint, the natural consequence of the equations of motion, rather than a physical constraint...
and the level curves
K=constant are closed trajectories surrounding the fixed point.
Consequently, the levels of the predator and prey populations cycle, and oscillate around this fixed point.
See also
- Lotka–Volterra inter-specific competition equations
- Population dynamics
Population dynamics is the branch of life sciences that studies short- and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...
- Nicholson–Bailey model
External links