Random collision model for interacting populations of two species and the fluctuation-dissipation theorem --- the law of large numbers and the central limit theorem (Q1344206)

This paper is concerned with the interaction of two species with population sizes \(X_ j^{(M)} (t)\), \(t \geq 0\), \(j = 1,2\), where \(X_ 1^{(M)} (t) + X_ 2^{(M)} (t) = M\), a fixed number, \(X_ 1^{(M)} (t)\) satisfies the equation \[ X_ 1^{(M)} (t) = X_ 1^{(M)} (0) + N \left( {\lambda \over M} \int^ t_ 0 X_ 1^{(M)} (s)X_ 2^{(M)} (s)ds \right), \] with \(N(\cdot)\) a Poisson process, and \(\lambda > 0\). The authors first prove that this equation has a unique solution, and derive its form. They then examine the stochastic structure of the model in some detail, showing that the random time is a stopping time, and the solution is a Markov process. They go on to prove that if \(u_ j(t)\), \(j=1,2\), are the unique solutions of the Lotka-Volterra equation \[ {du_ 1 \over dt} = \lambda u_ 1u_ 2, \quad u_ 1 + u_ 2 = 1, \] then as \(M \to \infty\), \(X_ j^{(M)} (t)/M \to u_ j (t)\). First a weak law of large numbers is derived, and then refined to a strong law of large numbers. The authors conclude by showing that if \[ Y_ 1^{(M)} (t) = \sqrt M \biggl\{ X_ 1^{(M)} (t)M^{-1} - u_ 1(t) \biggr\}, \] then \(Y_ 1^{(M)} (t)\) converges weakly to a one-dimensional Gaussian diffusion process.