Competitive Lotka-Volterra equations

The competitive Lotka–Volterra equations are a simple model of the population dynamics

Population dynamics

Population dynamics is the branch of life sciences that studies short-term and long-term changes in the size and age composition of populations, and the biological and environmental processes influencing those changes...

of species competing

Competition

Competition is a contest between individuals, groups, animals, etc. for territory, a niche, or a location of resources. It arises whenever two and only two strive for a goal which cannot be shared. Competition occurs naturally between living organisms which co-exist in the same environment. For...

for some common resource

Natural resource

Natural resources occur naturally within environments that exist relatively undisturbed by mankind, in a natural form. A natural resource is often characterized by amounts of biodiversity and geodiversity existent in various ecosystems....

. They can be further generalised

Generalized Lotka–Volterra equation

The generalized Lotka–Volterra equations are a set of equations which are more general than either the competitive or predator-prey examples of Lotka–Volterra types. They can be used to model direct competition and trophic relationships between an arbitrary number of species. Their...

to include trophic interactions.

Overview

The form is similar to the Lotka–Volterra equations for predation in that the equation for each species has one term for self-interaction and one term for the interaction with other species. In the equations for predation, the base population model is exponential

Exponential function

In mathematics, the exponential function is the function ex, where e is the number such that the function ex is its own derivative. The exponential function is used to model a relationship in which a constant change in the independent variable gives the same proportional change In mathematics,...

. For the competition equations, the logistic equation is the basis.

The logistic population model, when used by ecologists

Ecology

Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...

often takes the following form:

Here x is the size of the population at a given time, r is inherent per-capita growth rate, and K is the carrying capacity

Carrying capacity

The carrying capacity of a biological species in an environment is the maximum population size of the species that the environment can sustain indefinitely, given the food, habitat, water and other necessities available in the environment...

Two species

Given two populations, x₁ and x₂, with logistic dynamics, the Lotka–Volterra formulation adds an additional term to account for the species' interactions

Biological interaction

Biological interactions are the effects organisms in a community have on one another. In the natural world no organism exists in absolute isolation, and thus every organism must interact with the environment and other organisms...

. Thus the competitive Lotka–Volterra equations are:

Here, α₁₂ represents the effect species 2 has on the population of species 1 and α₂₁ represents the effect species 1 has on the population of species 2. These values do not have to be equal. Because this is the competitive version of the model, all interactions must be harmful (competition

Competition

) and therefore all α-values are positive. Also, note that each species can have its own growth rate and carrying capacity

Carrying capacity

N species

This model can be generalized to any number of species competing against each other. One can think of the populations and growth rates as vectors and the interaction αs as a matrix

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

. Then the equation for any species i becomes

or, if the carrying capacity

Carrying capacity

is pulled into the interaction matrix (this doesn't actually change the equations, only how the interaction matrix is defined),

where N is the total number of interacting species. For simplicity all self-interacting terms α_ii are often set to 1.

Possible dynamics

The definition of a competitive Lotka-Volterra system assumes that all values in the interaction matrix are positive or 0 (α_ij ≥ 0 for all i,j). If it is also assumed that the population of any species will increase in the absence of competition unless the population is already at the carrying capacity

Carrying capacity

(r_i > 0 for all i), then some definite statements can be made about the behavior of the system.

The populations of all species will be bounded between 0 and 1 at all times (0 ≤ x_i ≤ 1, for all i) as long as the populations started out positive.
Smale showed that Lotka-Volterra systems that meet the above conditions and have five or more species (N ≥ 5) can exhibit any asymptotic
Asymptote
In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...

behavior, including a fixed point
Fixed point (mathematics)
In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

, a limit-cycle
Limit-cycle
In mathematics, in the area of dynamical systems, a limit-cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such...

, an n-torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

, or attractors.
Hirsch proved that all of the dynamics of the attractor occur on a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

of dimension N-1. This basically says that the attractor cannot have dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

greater than N-1. Why is this important? A limit-cycle
Limit-cycle
In mathematics, in the area of dynamical systems, a limit-cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such...

cannot exist in fewer than two dimensions. A n-torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

cannot exist in less than n dimensions, and finally, chaos cannot occur in less than three dimensions. So, Hirsch proved that competitive Lotka–Volterra systems cannot exhibit a limit-cycle
Limit-cycle
In mathematics, in the area of dynamical systems, a limit-cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such...

for N < 3, or any torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

or chaos for N < 4. This is still in agreement with Smale that any dynamics can occur for N ≥ 5.
- More specifically, Hirsch showed there is an invariant
  Invariant (mathematics)
  In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
  
  set C that is homeomorphic
  Homeomorphism
  In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
  
  to the (N-1)-dimensional simplex
  Simplex
  In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
  
  and is a global attractor of every point excluding the origin. This carrying simplex contains all of the asymptotic
  Asymptote
  In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as they tend to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors...
  
  dynamics of the system.
To create a stable ecosystem the α_ij matrix must have all positive eigenvalues. For large N systems Lotka-Volterra models are either unstable or have low connectivity. Kondoh and Ackland and Gallagher have independently shown that large, stable Lotka-Volterra systems arise if the elements of α_ij (i.e. the features of the species) can evolve in accordance with natural selection.

4-dimensional example

A simple 4-Dimensional example of a competitive Lotka–Volterra system has been characterized by Vano et al. Here the growth rates and interaction matrix have been set to

This system is chaotic and has a largest Lyapunov exponent

Lyapunov exponent

In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...

of 0.0203. From the theorems by Hirsch, it is one of the lowest dimensional chaotic competitive Lotka–Volterra systems. The Kaplan–Yorke dimension, a measure of the dimension

Dimension

In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...

ality of the attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

, is 2.074. This value is not a whole number, indicative of the fractal

Fractal

A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...

structure inherent in a strange attractor. The coexisting equilibrium point, the point at which all derivative

Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...

s are equal to zero but that is not the origin

Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In a Cartesian coordinate system, the origin is the point where the axes of the system intersect...

, can be found by inverting the interaction matrix and multiplying

Matrix multiplication

In mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an n-by-m matrix and B is an m-by-p matrix, the result AB of their multiplication is an n-by-p matrix defined only if the number of columns m of the left matrix A is the...

by the unit column vector, and is equal to

Note that there are always 2^N equilibrium points, but all others have at least one species' population equal to zero.

The eigenvalues

Eigenvalue, eigenvector and eigenspace

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

of the system at this point are 0.0414±0.1903i, -0.3342, and -1.0319. This point is unstable due to the positive value of the real part of the complex

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

eigenvalue pair. If the real part were negative, this point would be stable and the orbit would attract asymptotically. The transition between these two states, where the real part of the complex eigenvalue pair is equal to zero, is called a Hopf bifurcation

Hopf bifurcation

In the mathematical theory of bifurcations, a Hopf or Poincaré–Andronov–Hopf bifurcation, named after Henri Poincaré, Eberhard Hopf, and Aleksandr Andronov, is a local bifurcation in which a fixed point of a dynamical system loses stability as a pair of complex conjugate eigenvalues of...

Spatial arrangements

Background

There are many situations where the strength of species' interactions depends on the physical distance of separation. Imagine bee

Bee

Bees are flying insects closely related to wasps and ants, and are known for their role in pollination and for producing honey and beeswax. Bees are a monophyletic lineage within the superfamily Apoidea, presently classified by the unranked taxon name Anthophila...

colonies in a field. They will compete for food strongly with the colonies located near to them, weakly with further colonies, and not at all with colonies that are far away. This doesn't mean, however, that those far colonies can be ignored. There is a transitive

Transitive relation

In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....

effect that permeates through the system. If colony A interacts with colony B, and B with C, then C affects A through B. Therefore, if the competitive Lotka–Volterra equations are to be used for modeling such a system, they must incorporate this spatial structure.

Matrix organization

One possible way to incorporate this spatial structure is to modify the nature of the Lotka–Volterra equations to something like a
reaction-diffusion system. It is much easier, however, to keep the format of the equations the same and instead modify the interaction matrix. For simplicity, consider a five species example where all of the species are aligned on a circle

Circle

A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....

, and each interacts only with the two neighbors on either side with strength α_-1 and α₁ respectively. Thus, species 3 interacts only with species 2 and 4, species 1 interacts only with species 2 and 6, etc. The interaction matrix will now be

If each species is identical in its interactions with neighboring species, then each row of the matrix is just a permutation

Permutation

In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order...

of the first row. A simple, but non-realistic, example of this type of system has been characterized by Sprott et al. The coexisting equilibrium point for these systems has a very simple form given by the inverse

Inverse element

In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...

of the sum of the row

Lyapunov functions

A Lyapunov function

Lyapunov function

In the theory of ordinary differential equations , Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions are important to stability theory and control...

is a function

Function (mathematics)

In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...

of the system f = f(x) whose existence in a system demonstrates stability

Lyapunov stability

Various types of stability may be discussed for the solutions of differential equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Lyapunov...

. It is often useful to imagine a Lyapunov function

Lyapunov function

as the energy of the system. If the derivative

Derivative

of the function is equal to zero for some orbit

Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is different orbits do not intersect in the...

not including the equilibrium point, then that orbit

Orbit (dynamics)

is a stable attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

, but it must be either a limit-cycle

Limit-cycle

In mathematics, in the area of dynamical systems, a limit-cycle on a plane or a two-dimensional manifold is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity. Such...

or n-torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

- but not a strange attractor (this is because the largest Lyapunov exponent

Lyapunov exponent

In mathematics the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories...

of a limit-cycle

Limit-cycle

and n-torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

are zero while that of a strange attractor is positive). If the derivative

Derivative

is less than zero everywhere except the equilibrium point, then the equilibrium point is a stable fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

. When searching a dynamical system

Dynamical system

A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...

for non-fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

s, the existence of a Lyapunov function

Lyapunov function

can help eliminate regions of parameter space where these dynamics are impossible.

The spatial system introduced above has a Lyapunov function

Lyapunov function

that has been explored by Wildenberg et al. If all species are identical in their spatial interactions, then the interaction matrix is circulant

Circulant matrix

In linear algebra, a circulant matrix is a special kind of Toeplitz matrix where each row vector is rotated one element to the right relative to the preceding row vector. In numerical analysis, circulant matrices are important because they are diagonalized by a discrete Fourier transform, and hence...

. The eigenvalues

Eigenvalue, eigenvector and eigenspace

of a circulant matrix

Circulant matrix

are given by

for k = 0_N − 1 and where

the Nth root of unity

Root of unity

In mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...

. Here c_j is the jth value in the first row of the circulant matrix

Circulant matrix

.

The Lyapunov function

Lyapunov function

exists if the real part of the eigenvalues

Eigenvalue, eigenvector and eigenspace

are positive (Re(λ_k > 0 for k = 0, …, N/2). Consider the system where α_-2 = a, α_-1 = b, α₁ = c, and α₂ = d. The Lyapunov function

Lyapunov function

exists if

for k = 0, … ,N − 1. Now, instead of having to integrate

Numerical integration

In numerical analysis, numerical integration constitutes a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of...

the system over thousands of time steps to see if any dynamics other than a fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

exist, one need only determine if the Lyapunov function

Lyapunov function

exists (note: the absence of the Lyapunov function

Lyapunov function

doesn't guarantee a limit-cycle

Limit-cycle

, torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...

, or chaos

Chaos theory

Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...

).

Example: Let α₋₂ = 0.451, α₋₁ = 0.5, and α₂ = 0.237. If α₁ = 0.5 then all eigenvalues are negative and the only attractor

Attractor

An attractor is a set towards which a dynamical system evolves over time. That is, points that get close enough to the attractor remain close even if slightly disturbed...

is a fixed point

Fixed point (mathematics)

In mathematics, a fixed point of a function is a point that is mapped to itself by the function. A set of fixed points is sometimes called a fixed set...

. If α₁ = 0.852 then the real part of one of the complex eigenvalue

Eigenvalue, eigenvector and eigenspace

pair becomes positive and there is a strange attractor. The disappearance of this Lyapunov function

Lyapunov function

coincides with a Hopf bifurcation

Hopf bifurcation

Line systems and eigenvalues

It is also possible to arrange the species into a line. The interaction matrix for this system is very similar to that of a circle except the interaction terms in the lower left and upper right of the matrix are deleted (those that describe the interactions between species 1 and N, etc.).

This change eliminates the Lyapunov function

Lyapunov function

described above for the system on a circle, but most likely there are other Lyapunov function

Lyapunov function

s that have not been discovered.

The eigenvalues

Eigenvalue, eigenvector and eigenspace

of the circle system plotted in the complex plane

Complex plane

In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...

form a trefoil

Trefoil

Trefoil is a graphic form composed of the outline of three overlapping rings used in architecture and Christian symbolism...

shape. The eigenvalues

Eigenvalue, eigenvector and eigenspace

from a short line form a sideways Y, but those of a long line begin to resemble the trefoil

Trefoil

Trefoil is a graphic form composed of the outline of three overlapping rings used in architecture and Christian symbolism...

shape of the circle. This could be due to the fact that a long line is indistinguishable from a circle to those species far from the ends.

Cellular automata

A rule-based automaton model is equivalent to a Lotka–Volterra system and introduces 2D space, with either periodic or fixed boundaries.

Each site has three states, fox, bare, rabbit.
Rules are as follows:

Pick a site (only stochastic updates allowed), and a neighbour.
If fox is adjacent to rabbit, rabbit gets eaten (becomes fox with probability r). Else fox dies with probability p.
If rabbit is adjacent to bare ground, reproduces with probability q.
If bare ground is adjacent to anything, the thing moves into bare ground.

These rules give a model like Lotka–Volterra, but the additional feature of a correlation length between regions oscillating differently. The correlation length is very long, and the model develops a wave structure.

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