Modeling Abrupt Changes in Population Dynamics with Two Threshold States *
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MODELING ABRUPT CHANGES IN POPULATION DYNAMICS WITH TWO THRESHOLD STATES*
A. Yu. Perevaryukha
UDC 532.22
Abstract. The paper solves the problem of modeling an outbreak of the species, which begins with the achievement of a specific nontrivial state of a dynamic system. Population process is divided into a sequence of substantially different stages. The model simulates the effect of spontaneous transition from weak fluctuations through accelerated reproductive activity to exceeded ecological capacity of the environment followed by the phase of rapid spontaneous decay. To describe the transition between final stages of the outbreak, the method of controlled functional realization of metamorphoses of trajectory behavior are used. Differential equations are combined with predicative constructions in software environment. The original computational unit is based on discrete-continuous time and allows us to apply the scenario approach to the consideration of environmental situations. The significance of the results is justified by the analysis of the data about outbreaks of small insect pests that eat on a limited number of suitable plants and, under certain conditions, can get out of control of parasitic wasps. Keywords: population models, hybrid computing structures. INTRODUCTION An analysis of prompt and unexpected jumps in the number of some biological species remains an interdisciplinary problem. One of the variants of spasmodic development of the process is described by E. Odum in his well-known book [1] on the basis of observations of outbreaks of eucalyptus worms by Australian entomologists. The example is interesting because it eliminates, in the conditions of evergreen forest, the influence of random weather factors on survival rate of wintering generation. Reproduction of monophages (psyllids, eating plants of only one type) can be modeled in case of formalization of biotic governing mechanisms, which lead to a certain size of the population. However, such threshold states should not be easily obtained from any other state of model, otherwise outbreaks of one species would be observed everywhere and regularly, instead of being localized and unexpected. In Odum’s hypothesis, it is overcoming of the neighborhood of unstable equilibrium that initiates the phase of prompt growth in the specific increase in pest population. The proposed model of a dangerous ecological phenomenon is based on differentiated description of the force of influence of biotic factors on reproductive activity in different states of the population. To simulate stochastic fluctuations, which are typical of the populations of poorly migrating insects, it is proposed to use a specific mode of deterministic transitive chaos.
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The study was carried out within the framework of the Project of the Russian Foundation for Basic Research (RFFI) No. 15-04-01226. St. Petersburg Institute for Informatics and Automation, Russian Academy of Sciences, St. Petersburg, Russia, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July
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