As explained by Alexander Sazonov, Junior Researcher at IAMR KarRC RAS, classical metapopulation models are two-dimensional, meaning they describe the dynamics of two variables – the ‘predator’ and the ‘prey’.
– The processes in the nature, however, are more complex. We created a 3D model by adding a third variable to account for the food attractiveness of the patch for the predator. When it’s positive, the predator stays in the patch because there is sufficient food. However, if the predator needs a large amount of prey for sustenance, the food attractiveness may become negative, urging the predator to move away. An extremely interesting result from a mathematical perspective is that we managed to find a closed trajectory for such a complex, hybrid system. The existence of a closed trajectory means that, for certain initial data, the predator will definitely return to the patch after they had moved out, – shared Alexander Sazonov.
Alexander Sazonov, Junior Researcher, Laboratory of Computer Information Technologies, Institute of Applied Mathematical Research KarRC RAS (photo courtesy of the scientist)
The new model offers a practical opportunity to solve the ecologically important problem of estimating the time periods in the patch re-colonization process. This development can be applied, e.g., in combating agricultural pests. In this case, there is an isolated patch – a crop field, and one prey species – the pest. To avoid using chemicals, a predator that naturally preys on this pest species can be introduced into the field. Mathematical methods enable calculating the behavior of both species and, hence, the effectiveness of the action.
The model proposed by the Karelian mathematicians is a hybrid one, meaning it consists of simpler systems, but switching between them helps account for complex dynamics, in particular the predators’ outmigration and whether or not they will return.
The model consists of five subsystems with switchings between them. The first is a classical one, describing a periodic process – the rise and decline of one or another population. Another system is the refuge mode. It occurs when prey numbers become too low for predators to detect them. This situation triggers a decline of predators and a rapid growth of the prey population. The third system is the absence of predators after they had moved away. Here, intraspecific competition among prey and self-regulation of their population are taken into account. Finally, two auxiliary systems describe the departure of predators and their return.
In their research, the scientists at IPMI also examined the case where the predator needs a small number of prey. In this case, food attractiveness always remains positive, and predator outmigration does not occur.
The model makes use of the ‘memory method’, developed by Alexander Kirillov, Leading Researcher at IARM KarRC RAS. This method introduces a lag for switching between subsystems, something that does not happen in classical approaches to describing such switchings. The proposed method provides a better representation of the real dynamics of ecological systems.
– Effectively, we are expanding the mathematical toolkit of the dynamical systems theory, which is not too extensive so far, so such an analysis contributes to the future development of the more general methods, – noted Alexander Sazonov.
The results of the study were published in the journal Izvestiya VUZ. Applied Nonlinear Dynamics. In 2025, Alexander Sazonov won in the academic article contest among KarRC RAS young scientists in the STEM category.
The article was published in the journal Izvestiya VUZ. Applied Nonlinear Dynamics
The focus for the young researcher and his colleagues at the IAMR KarRC RAS is the theory of dynamical systems. This discipline studies models of the processes of change over time. Its essence is the qualitative analysis of differential equations without solving them since explicit solutions can only be obtained for rare, specific classes of differential equations.
– A differential equation describes the law governing the rate of change of a variable. Based on this, we try to draw conclusions regarding the dynamics of that variable, – explains the scientist.
This is how mathematics helps to describe and model processes of varying nature – mechanical, physical, biological, economic, and so on – anything that has an initial value and dynamics of change over time.
The doctoral dissertation that Alexander Sazonov defended successfully in 2021 dealt with macroeconomic processes. The researcher has developed a methodology and software for mathematically modeling the dynamics of capital distribution across performance levels, taking into account the limitations of economic growth and enabling relevant forecasts. These models are underpinned by the theory of creative destruction, developed by economists in the 20th century, where innovations replace obsolete technologies. For companies to remain on top and competitive, they have to employ dynamical strategies of abrupt change and creative destruction. Mathematical models allow predicting the behavior of economic systems. Notably, the 2025 Nobel Prize in Economic Sciences was awarded to scientists who constructed a mathematical model of creative destruction for explaining innovation-based economic growth. This confirms the relevance and importance of mathematicians' work in this field.
Alexander Sazonov plans to continue developing hybrid systems and promoting their application across various subject domains.
