Spiral chaos in Lotka-Volterra like models

Bakhanova, Yulia V.; Iitmm, Unn; Kazakov, Alexey O.; Korotkov, Alexander G.

doi:10.15507/2079-6900.19.201701.013-024

Cited by 4 publications

(6 citation statements)

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“…Remark 1. Note that for parameter values which do not lead to a strong contraction in the (y 1 + y 2 )-direction, system (3) cannot be described by a onedimensional map and presumably may exhibit such phenomena as Shil'nikov's spiral chaos and coexistence of at least three attractors (see [3,21,27]). As discussed below, this cannot occur in the one-dimensional model (4).…”

Section: Two-predators-one-prey Modelmentioning

confidence: 99%

Bistability in a one-dimensional model of a two-predators-one-prey population dynamics system

Kryzhevich¹,

Аврутин²,

Söderbacka³

2021

Preprint

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In this paper, we study the classical two-predators-one-prey model. The classical model described by a system of 3 ordinary differential equations can be reduced to a one-dimensional bimodal map. We prove that this map has at most two stable periodic orbits. Besides, we describe the structure of bifurcations of the map. Finally, we describe a mechanism that yields bistable regimes. We find several areas of bistability numerically.

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Section: Two-predators-one-prey Modelmentioning

confidence: 99%

Bistability in a one-dimensional model of a two-predators-one-prey population dynamics system

Kryzhevich¹,

Аврутин²,

Söderbacka³

2021

Preprint

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“…Интересно, что в этой модели не обнаружены траектории, покрывающие такие множества как «чайная кружка», которыми, в частности, описывают структуру спиральных аттракторов в трехмерной модификации модели Розенцвейга -Макартура, описывающей динамику систем типа жертва-хищник-суперхищник. [9]. По всей видимости, в системе (2) такого рода траектории, в принципе, могут сформироваться на основе единственного быстрого цикла S 1 и канарда C или C 1 , однако обнаружить этого,…”

Section: сценарий «удвоения» пачекunclassified

“…С другой стороны, вызывают особый интерес сложные нелинейные явления, наблюдаемые в математических моделях популяций, сообществ живых организмов, биохимических системах, моделях клеточной активности и др. В частности исследуются условия синхронного или асинхронного поведения динамики [8], механизмы возникновения странных, в том числе, гомоклинических аттракторов [9,10], закономерности формирования сложных пространственно-временных структур [4,11] и др. При казалось бы теоретическом характере подобных работ, нельзя не отметить, что глубокое понимание математических закономерностях и динамических эффектов, постоянно обнаруживаемых в таких системах, может оказаться решающим в понимании процессов происходящих на молекулярном и клеточном уровне, в сообществах живых организмов, популяциях и т.д.…”

Section: Introductionunclassified

Synchronization and Bursting Activity in the Model for Two Predator-Prey Systems Coupled By Predator Migration

Кулаков¹,

Kurilova²,

Frisman³

2019

Math.Biol.Bioinf.

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The papers is devoted to a model for two non-identical predator-prey communities coupled by migration and characterized by logistic growth of prey and Holling type II functional response. The coupling is a predator migration at constant weak rate. The non-identity is the difference in the prey growth rates or predator mortalities in each patch. We studied the equilibrium states of model and scenarios of loss of their stability and emerge of complex periodic solutions. It was shown that in some domains of the parameter space there is a bursting activity which are that the dynamics of two communities contain segments of slowly resting dynamic (as part of a fast-slow cycle or canard) and regular bursts of spikes. In the resting part, the dynamics of the second community, as a rule, follow the slow changes in the first community, i.e. the dynamics in different patches are synchronous. But in the fast part there is only phase synchronization between the fast-slow cycle in first patch and bursts in second. We classified the scenarios of transition between different types of bursting activity by location spiking manifold and canard. These types differ not so much in size, shape or numbers of spikes as in the order of bursts emerge relative a slow-fast cycle. In a typical case the start of burst (divergent fast oscillations) coincides with the minimum numbers or quasi-extinction of prey in the first patch. After a rapid increase in the prey number in the first patch, diverging fluctuations give way to damped in the second patch. Such dynamics correspond to the rhombus-wave shape of spikes cluster. Another case is interesting, when the burst of spikes is formed after the full recovery of prey and with a certain predator number in the first patch. In this case, the spikes cluster takes the shape of a triangle-wave or a truncated rhombus-wave. It was shown that transitions between these types of bursts are accompanied by a change in the oscillation period and the degree of synchronization. Triangular-wave bursters correspond to complete synchronization of the predator dynamics in the resting part and rhomboid-wave correspond to antiphase synchronization. In the fast part with many spikes, communities are completely asynchronous to each other.

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“…На протяжении долгого времени актуальными остаются вопросы изучения сложного поведения нелинейных динамических систем, способных демонстрировать разные, в том числе и весьма сложные режимы динамики. Одним из на-правлений, рассматриваемых в данных исследованиях, являются вопросы изучения возникновения колебаний в динамике численности взаимодействующих популяций [2,25].…”

Section: Introductionunclassified

Modeling the Dynamics of Interacting Predator-Prey Populations With Constant Migration of Individuals From Adjacent Territories

Kurilova,

Frisman

2024

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The article deals with the research of the dynamics of a local predator-prey community with constant migration of individuals from neighboring territories. We studied several models with a constant inflow of individuals into both predator and prey populations. It is shown that changes in the overall dynamics are significantly influenced by the number of predator migrants: their large influx leads to the rapid and almost complete extinction of preys. The model considering only a constant prey inflow is successfully applied to modeling of various processes based on the principles of predator-prey population interactions, for example, when studying the consumption of difficult-to-renewable or nonrenewable natural resources. The study of the model provides allows getting estimations of the resources use efficiency and and the degree of modernization of the consumer sector. It is shown that two-dimensional models - modifications of the basic Volterra and Bazykin models, lead to structurally stable fluctuation regimes corresponding to the focus and limit cycle. We found that these models also contain fast-slow periodic dynamics, corresponding to a maximum limit cycle with strong spikes and smoother declines in the population size. The emergence of such regimes corresponds to the alternation of periods of active extinctions of prey, accompanied by an increase in the number of predators, and periods of long-term restoration of the prey, during which there is no harvesting of prey. We used methods of dynamic systems analysis to study the models. The construction of two-dimensional parametric portraits shows that, to get stable dynamics of the basic models of two interacting biological species, no complication is required, for example, by adding nonlinear terms. Stable regimes are also observed in simpler models, such as the original Lotka-Volterra or Bazikin models, where the recovery rate of prey population is a constant value.

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Spiral chaos in Lotka-Volterra like models

Cited by 4 publications

References 0 publications

Bistability in a one-dimensional model of a two-predators-one-prey population dynamics system

Bistability in a one-dimensional model of a two-predators-one-prey population dynamics system

Synchronization and Bursting Activity in the Model for Two Predator-Prey Systems Coupled By Predator Migration

Modeling the Dynamics of Interacting Predator-Prey Populations With Constant Migration of Individuals From Adjacent Territories

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