The method of modeling sharp changes in biological processes
Institute for Informatics and Automation (SPIIRAS), RAS
14 Line, 39, St. Petersburg, 199178 Russia
*e-mail: temp_elf@mail.ru
In biological systems, there are processes that occur with abrupt state changes, and this is what we call threshold effects. Such nonlinear phenomena are becoming increasingly important and frequent due to the current instability of the ecosystems functioning. Migration of alien species and exploitation of biological resources provoke rapid changes that cannot be modeled by classical dynamic systems with completely continuous or only discrete time. For such systems cyclic trajectories can be obtained, and, at the limit of the cycle complication in asymptotic behavior, chaotic oscillations. We propose a method based on redefinable computational structures for describing explosive and transient processes in population dynamics. Our models operate with an extension of the stock–recruitment curve method for situations of threshold phenomena. A special addition into the continuous component of a dynamic system describes varying efficiency of population reproduction, which depends on the number of mature individuals. The method has been used for the modeling scenario of two extreme situations of different genesis. We examined situations of rapid collapse of a large fish stock and spontaneously ending outbreaks of insect pests. Those situations are considered using the collapse of Atlantic cod near Labrador in 1992 and psyllids Cardiaspina albitextura in the eucalyptus evergreen forest in Australia as an example. The mathematical effects that we use in the scenarios are: fractal boundaries of the regions of attraction, inverse tangential bifurcation, and the boundary crisis of the chaotic interval attractor. Nonlinear dynamic effects can be predicted and combined.
