Modeling of adaptive counteraction of the induced biotic environment during the invasive process

Purpose is to develop a mathematical model for the analysis of a variant in the development of a population process with a non-trivially regulated confrontation between an invading species and a biotic environment. Relevance. The situation we are studying arises in invasive processes, but is a previously unexplored special variant of their development. The task of modeling is to describe the transition to a deep $\nu$-shaped crisis after intensive growth. The model is based on examples of the adaptive dynamics of a bacterial colony and the suppression of mollusk populations, carriers of dangerous parasitic diseases, after targeted anti-epidemic introduction of their antagonists. Methods. In our work equations with a retarded argument in the range of parameter values that have a biological interpretation were studied. The model uses a logarithmic form of species regulation, taking into account the theoretically permissible capacity of the medium. In the equation we included the function of external influence with flexible threshold regulation relative to the current and previous population size. Results. It is shown that the proposed form of impact regulation leads to the formation of a stable adapted population after the crisis, which does not have a destructive impact on the habitat. With an increase in the reproductive potential of an invasive species, a deep crisis becomes critically dangerous. The form of the crisis passage depends on the reproductive potential, on the size of the initial group of individuals, and also on the time of activation of the adaptive counteraction from the environment. It is established that at a sufficient level of resistance, a non destructive equilibrium is established. Conclusion. The actual scenario of sudden depression of an actively spreading population with a large reproductive $r$-parameter, which is caused by the delayed activity of its natural antagonists, has been studied. The threshold form of biotic regulation is characteristic of insects, the abundance of which is regulated by competing species of parasitic hymenoptera. The variant of rapid phase change considered by us in the model is relevant as a description of one of the forms of developing the body's immune response to the development of an acute infection with a significant delay. If the immune response is prematurely inhibited by the body itself, then the chronic focus of the disease persists. Examples of the dynamics of two real biological processes in experiments with biological suppression methods are given, which correspond to the invasion scenario obtained in the new model.