Scenario of the invasive process in the modification of Bazykins population equation with delayed regulation and high reproductive potential

The paper discusses modeling of the variant of the development of a rapid invasive process in competitive biosystems. The emergence of dangerous alien species leads to extreme phenomena in the dynamics of populations. Invasions generate a phase of active spread of the alien species, but outbreaks are often followed by a phase of sharp depression. Changes in the process are associated with active resistance, which has a delayed activation time interval and a threshold level of maximizing the impact $J$. For the mathematical formalization of the successively following stages of the outbreak/crisis, equations with a deviating argument are used. In a variant of the equation with a delayed tuning of the biotic reaction $\dot x=rf(x(t-\tau))-\mathfrak{F}(x^m(t-\nu);J)$ a variant of the passage of the crisis that occurs it is in the phase of rapid growth until a balance is reached with the resources of the environment. Due to the threshold feedback, the competitive pressure after a deep crisis is weakened and the invasive population goes into a mode of damped oscillations. The asymptotic level of equilibrium in the scenario with a crisis turns out to be much less than the theoretically permissible limiting level of abundance for an alien species in a given environment. The new Equation also has an interpretation to describe the weakening development of the immune response in a situation of chronicity of the infectious process.