Abstract:
A nonlinear system of three differential equations with distributed delay is studied. The first two equations describe the number of two populations of interacting particles, the third equation describes the number of complexes formed as a result of contact interactions of particles of different populations. The duration of the complexes' existence is determined by a distribution function focused on a finite period of time. The system of equations is complemented by non-negative initial data. The resulting Cauchy problem has a unique solution on the semimajor axis, and the components of the solution are non-negative. The system of differential equations has one equilibrium position with non-negative components. N.N. Krasovsky's theorem was used to study the asymptotic stability of the equilibrium position. It is shown that the roots of the characteristic equation for the linear approximation system have negative real parts separated from zero. The discarded nonlinear terms satisfy the necessary smallness condition. Special cases of the system of equations under study are considered ‒ a system with a concentrated (constant) delay and a system without delay. It is shown that in both special cases the equilibrium position is locally asymptotically stable. The task is to estimate the area of attraction of the equilibrium position and find the parameters reflecting the exponential convergence of the solution to the equilibrium position.
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