Abstract
Abstract-This article focuses on the modeling of crisis and threshold development of the population process during the formation of a new population in a competitive environment. As a population spreads, a deep population crisis may arise as a result an abrupt triggering of biotic countermeasures before resources for a further increase in population size are exhausted. A bottleneck occurred in the history of many populations, including humans at the time of the Neolithic crash in Europe. Invaders with high reproductive potential often exert deleterious effects on biosystems. The emergence of efficient competition can not only cause classical cyclical fluctuations, but also lead to a complete extinction of the population after a series of high peaks in its abundance. Two alternative scenarios provide classical examples of induced population crises. One was observed in Gause's experiments where an introduction of a predatory ciliate drove another ciliate species to extinction. The other scenario was observed in a series of experiments where bacteriophages were introduced into colonies of actively dividing bacteria that had a dynamically adapting antiviral mechanism. In this work, modifications to the model were proposed to describe the actual scenarios of crisis effects in population dynamics. Equations with deviating arguments in the time variable allowed a threshold effect of conditions on reproduction of the invasive species and an aggregated nature of the lagging regulation with two time factors. The computational scenarios described both completion of the process after a repeated outbreak and successful elimination of the population crisis via rapid adaptation. Deep crisis phenomena are characteristic of local population dynamics when organisms interact with viruses that are new to them.