Volume 71, Nº 3 (2025): Proceedings of the Crimean Autumn Mathematical School-Symposium

Using a predator-prey model in a heterogeneous environment, we have created a mathematical model that describes the interaction between populations with different evolutionary strategies. The model is based on partial differential equations and allows for the consideration of multifactor taxis. We propose modified functions of local predator-prey interactions, which provide a variety of evolutionary strategies for the system. Key parameters responsible for the formation of ideal free distribution strategies have been investigated, and conditions for the parameters of diffusion and migration have been given under which ideal free distribution-like strategies can be implemented. Results from computational experiments demonstrating stationary and oscillating modes have been presented.

In this paper, an initial-boundary value problem is studied for one-dimensional equations of the dynamics of a compressible viscous mixture. A theorem is proved for the existence and uniqueness of a solution to the initial-boundary value problem without any restrictions on the structure of the viscosity matrix other than the standard physical requirements of symmetry and positive definiteness.

Traditionally, continuous models of mathematical biology are focused on the dynamics of interacting populations as stationary homogeneous entities. The state of the populations in the equations is governed by factors common to all individuals \( \forall t,N(t) \): reproductive efficiency, mortality, living space limitations, or resource limitations. Many species exist with nonoverlapping generation sequences, replacing each other under different seasonal conditions. The number of annual generations is an important characteristic of the ecology of a species when occupying a new range. The length of the life cycle and the index of reproductive activity r in adjacent generations of insects in a range vary due to the need for wintering. Fluctuations in these values affect rapid population outbreaks. It is shown that the use of discrete models \( x_{n+1}=\psi(x_n;r)\varphi(x_{n-i})-\Xi \) is unrealistic for fundamental reasons. The appearance of cycles \( p\neq2^i \) in the order of Sharkovsky's theorem is excessive for the analysis of populations and the forecast of mass reproductions of insects. The article proposes a method for organizing models of the conjugate development of a succession of generations in a system of discontinuous differential equations as a sequence of boundary-value problems. The model is event-based redefined to obtain a solution on time intervals corresponding to the conditions of the season. The model taking into account competition and delayed regulation is relevant for the analysis of a sequence of peaks in pest activity, which are characterized by individual extremely numerous generations.

We consider a self-adjoint operator acting in a Krein space and possessing an invariant subspace that is maximal nonnegative and decomposes into a direct sum of a uniformly positive (i.e., equivalent to a Hilbert space with respect to the inner pseudoscalar product) and a finite-dimensional neutral subspace. We prove the existence of a difference expression that transforms the moment sequence generated by this operator into a sequence representable as the difference of positive moment sequences. In the case of a cyclic operator, this result is applied to construct a function space in which the operator under study is modeled as the operator of multiplication by an independent variable.