Generic model of stochastic population dynamics
Mother nature is noisy, therefore, it is time to deal with stochasticity. The evolution of deterministic systems is ﬁxed by the initial and boundary conditions, though a forecast is impossible for chaotic dynamics on large time scales. In stochastic systems, the noise leads to diﬀerent realizations for the same initial conditions. The statistical ensemble of inﬁnitely many realizations deﬁnes a stochastic process. For simplicity, only Markov processes will be considered here, where the present state determines the further evolution. The interplay of stochasticity and determinism can be modeled by stochastic (partial) diﬀerential equations [S(P)DE’s] on the level of state variables (Langevin equations) or by dynamical equations for diﬀerent probability densities (diﬀusion-reaction master and Fokker-Planck equations). Nice introductions to the theory of stochastic processes and its applications have been written by Gardiner (1985); Allen (2003); Anishenko et al. (2003), and with regard to spatial processes by Garc´ıa-Ojalvo and Sancho (1999).