Simulation of fluctuations of aggressive alien species in continuous models with independent regulation

Traditional models in biology do not describe modern extraordinary situations when the whole species composition is mixed. The article deals with oscillatory equations and non-dissipative population dynamics for specific environmental situations that are associated with alien species in ecosystems. During the invasion of new species, the resistance of the biotic environment may be completely absent for a considerable time. Under such conditions, with a high specific fecundity, nonstationary regimes of change in numbers arise. A pest outbreak is realized with an explosive growth phase. All outbreaks of fish and insects are some brief extreme episodes that end with a new state of the environment and the aggressive new species. Completion options are varied even in the example of one malicious species of comb jelly Mnemiopsis leidyi in Azov and Caspian Sea. Transition to relaxation oscillations after a comb jelly outbreak is possible. A new species may become a small group or even disappear in case $\min N_*(t;\tau r)=0$. The paper proposes a model based on the lagging regulation for actual scenarios of population behaviour in a new environment. In computational experiments we have shown the conditions for stabilization after an outbreak in an extremely small group of individuals or complete disappearance after collapse. A separate scenario describes the complete depletion of environmental resources during fluctuations with a significant amplitude. The most relevant is the model scenario of stabilization at minimum values after a rapid change in the phases of the outbreak and a transition to the depression of the pest number in the modification of the differential equation of Bazykin and Hutchinson model.