Continuous generation population model with discontinuous life cycle characteristics

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.