Competitive Lotka–Volterra equations

The competitive Lotka–Volterra equations are a simple model of the population dynamics of species competing for some common resource.

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.

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

A complete classification of this dynamics, even for all sign patterns of above coefficients, is available,[1][2] which is based upon equivalence to the 3-type replicator equation.

One can think of the populations and growth rates as vectors, α's as a matrix.

The definition of a competitive Lotka–Volterra system assumes that all values in the interaction matrix are positive or 0 (αij ≥ 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 (ri > 0 for all i), then some definite statements can be made about the behavior of the system.

A simple 4-dimensional example of a competitive Lotka–Volterra system has been characterized by Vano et al.[9] Here the growth rates and interaction matrix have been set to

This value is not a whole number, indicative of the fractal structure inherent in a strange attractor.

The coexisting equilibrium point, the point at which all derivatives are equal to zero but that is not the origin, can be found by inverting the interaction matrix and multiplying by the unit column vector, and is equal to

This point is unstable due to the positive value of the real part of the complex 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.

A detailed study of the parameter dependence of the dynamics was performed by Roques and Chekroun in.

[10] The authors observed that interaction and growth parameters leading respectively to extinction of three species, or coexistence of two, three or four species, are for the most part arranged in large regions with clear boundaries.

As predicted by the theory, chaos was also found; taking place however over much smaller islands of the parameter space which causes difficulties in the identification of their location by a random search algorithm.

This implies a high sensitivity of biodiversity with respect to parameter variations in the chaotic regions.

Additionally, in regions where extinction occurs which are adjacent to chaotic regions, the computation of local Lyapunov exponents [11] revealed that a possible cause of extinction is the overly strong fluctuations in species abundances induced by local chaos.

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

Therefore, if the competitive Lotka–Volterra equations are to be used for modeling such a system, they must incorporate this spatial structure.

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.

A simple, but non-realistic, example of this type of system has been characterized by Sprott et al.[12] The coexisting equilibrium point for these systems has a very simple form given by the inverse of the sum of the row

If the derivative of the function is equal to zero for some orbit not including the equilibrium point, then that orbit is a stable attractor, but it must be either a limit cycle or n-torus - but not a strange attractor (this is because the largest Lyapunov exponent of a limit cycle and n-torus are zero while that of a strange attractor is positive).

When searching a dynamical system for non-fixed point attractors, the existence of a Lyapunov function can help eliminate regions of parameter space where these dynamics are impossible.

The spatial system introduced above has a Lyapunov function that has been explored by Wildenberg et al.[13] If all species are identical in their spatial interactions, then the interaction matrix is circulant.

The Lyapunov function exists if the real part of the eigenvalues are positive (Re(λk) > 0 for k = 0, …, N/2).

Now, instead of having to integrate the system over thousands of time steps to see if any dynamics other than a fixed point attractor exist, one need only determine if the Lyapunov function exists (note: the absence of the Lyapunov function doesn't guarantee a limit cycle, torus, or chaos).

If α1 = 0.852 then the real part of one of the complex eigenvalue pair becomes positive and there is a strange attractor.

The eigenvalues of the circle system plotted in the complex plane form a trefoil shape.

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