Abstract
The evolution of states in a spatial population model is studied. The model describes an infinite system of point entities in mathbb {R}^d which reproduce themselves at distant points (disperse) and die with rate that includes a competition term. The system’s states are probability measures on the space of configurations, and their evolution is obtained from a hierarchical chain of differential equations for the corresponding correlation functions derived from the Fokker–Planck equation for the states. Under natural conditions imposed on the model parameters it is proved that the correlation functions evolve in a scale of Banach spaces in such a way that at each moment of time the correlation function corresponds to a unique sub-Poissonian state. Some further properties of the evolution of states constructed in this way are described.
Highlights
1.1 PosingThe development of a mathematical theory of complex living systems is a challenging task of modern mathematics [4]
If their increase is controlled by affine functions of n, the evolution is obtained with the help of a stochastic semigroup, see e.g. [3,22] and the papers quoted in these works
If λn and μn increase faster than n, one would not expect that the evolution takes place, for all t > 0, in one and the same Banach space and is described by a C0-semigroup of operators acting in this space
Summary
The development of a mathematical theory of complex living systems is a challenging task of modern mathematics [4]. The time evolution of the probability of having n particles in the system is obtained from the Kolmogorov equation in which the generator is a tridiagonal infinite matrix containing the birth and death rates λn and μn, respectively. If their increase is controlled by affine functions of n, the evolution is obtained with the help of a stochastic semigroup, see e.g. The main result of the present paper consists in constructing the evolution of states of an infinite birth-and-death system of particles placed in Rd with ‘rates’ that roughly speaking increase as n2, cf (3.9) below. They do satisfy if a− is strictly positive in some vicinity of the origin, and a+ has finite range
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