Modeling recovery effect from the invasive population depression by the threshold and delayed equation

The article discusses the modeling of extreme invasive processes using the formalization of the threshold resistance. To analyze the diversity of invasion scenarios into biosystems, it is interesting to model the crisis in the equation, where time factors in the regulation of reproduction and resistance compete. The aim of the work is to substantiate a model for describing the effect of the passage of a crisis during a rapid seizure of an area with an interpretation of the parameters of the time lag. To demonstrate the ways of development of the population process during invasion, we have developed equations with delay for actual ecological situations. The use of population models with a deviating argument 𝑥˙ = (𝑥𝑘(𝑡 τ)) Ψ(𝑥𝑚(𝑡 ν)) phenomenologically describes specificsituations in ecological resistance and in rapidly developing invasive processes without the occurrence of cycles. The action of delay in the model is divided by us into regulatory internal and adaptive, from the side of the biotic environment. The effect of a sharp but brief population depression is described in a computational experiment. A computational study of the influence of a given initial state of the invasive group and the activation time of the counteraction on overcoming the depression was carried out. The practical significance lies in modeling the introduction scenario against the invader of its natural enemy, for whose activity a high population is important. A scenario has been studied when a special release of parasites against an unwanted alien species will not be an effective means, which is shown by the example of an adaptive bacterial antivirus. It has been confirmed that it is inappropriate to identify the effect of delay in modeling invasions with the characteristics of biological species. The model is applicable to the analysis of the infectious process in a suppressed immune response.