This report introduces the ideas behind particle filters, looking at the Kalman filter and the SIS and SIR filters to learn about the latent state of state space models. It then introduces particle MCMC as a way of learning about the parameters behind these models. Finally, the SIR filter and particle MCMC algorithms are applied to reaction networks, in particular the Lotka Volterra model.

The equation for the orbit of the classical Lotka-Volterra oscillator is solved for one of two variables in terms of the other by using two inverse functions of x exp(x). Moreover, the period of the orbit is expressed as an integral, which is approximated numerically by Gauss-Tschebyscheff integration rule of the first kind.

For autonomous Lotka-Volterra systems modelling the dynamics of N competing species, a new condition has been found to prevent a particular species from dying out. Based on this condition, criteria have been established for all or some of the N species to stabilise at a steady state whilst the others, if any, die out.

The diffusive Lotka–Volterra predator–prey system with two delays is reconsidered here. The stability of the coexistence equilibrium and associated Hopf bifurcation are investigated by analyzing the characteristic equations, and our results complement earlier ones. We also obtain that in a special case, a Hopf bifurcation of spatial inhomogeneous periodic solutions occurs in the system. © 2011 ...

Abstract. In this paper, we study a non-autonomous n-species “pure-delay-type” competitive system, which can be seen as the generalization of the traditional “puredelay-type” Lotka-Volterra competition system. By constructing a suitable Lyapunov functional, some sufficient conditions which guarantee the global attractivity of the positive solutions of the system are obtained. An example shows t...

The Lotka – Volterra equation for three species are integrated in an original way obtaining the existence of cycle. No studies of the cycle stability are held. These would be very interesting kind as in two species. However our results are of value.

The research is supported by National Natural Science Foundation of China (No. 10771215) Abstract In this paper, a class of Lotka-Volterra system with multiple time delays is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantees the existence of at least a positive periodic solution.

In this paper, we study m-species time-delayed competitive Lotka–Volterra type dispersal system with periodic coefficients in n-patches. By utilizing analytic method, sufficient and realistic conditions on permanence and the existence of periodic solution are established. The theoretical results are confirmed by a special example and numerical simulations. 2006 Elsevier Inc. All rights reserved.

A complete classification for the global dynamics of a Lotka-Volterra two species competition model with seasonal succession is obtained via the stability analysis of equilibria and the theory of monotone dynamical systems. The effects of two death rates in the bad season and the proportion of the good season on the competition outcomes are also discussed.