Mathematical Modelling in Studying Spatial Aspects of Population Dynamics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematical Biology".

Deadline for manuscript submissions: closed (31 May 2021) | Viewed by 10267

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Guest Editor
Southern Scientific Centre of the Russian Academy of Sciences (SSC RAS), Chekhov Street, 41, 344006 Rostov-on-Don, Russia
Interests: mathematical and theoretical ecology; spatiotemporal models of population dynamics; predator–prey interactions; models of animal movements; biological control; demogenetic models
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Special Issue Information

Dear Colleagues,

Classical theory of mathematical ecology began with population dynamics models that were built and validated on data collected from experiments performed with small laboratory microcosms. However, the understanding of the crucial role played by various spatial aspects in the functioning of population systems came quite soon, partly owing to the results obtained with the help of mathematical modelling. In turn, this understanding stimulated the development of new modelling approaches to the mathematical description of biophysical phenomena related to spatial factors.

The spatial extension of habitats is a key feature distinguishing natural ecosystems from small-scale laboratory microcosms that allow a modeler to admit the hypotheses of the well-mixed environment, random encounters, and unpurposive movements of animals. Taking into account the patchy distribution and directed movements of animals qualitatively alters the dynamics of population models and thus their response to external impacts. In other words, the inclusion of spatial factors into a population model can change the model’s prediction. Thus, the solution of applied or theoretical problems of biological population management should be based on adequate modeling tools correctly describing the interrelated processes of spatiotemporal population dynamics and spatial behavior of species. That is how modelling helped reveal the relationships between characteristics of spatiotemporal chaos and predictability of the population dynamics and spatial activity of phytophagous insects; its successful application for the biological control of weeds; the acclimatization of adventive invaders and their capability of spatial expansion by forming solitary population waves; the swarming behavior of fish and their vulnerability to predator attacks; the collective hunting of predators; and the emergence of the predator interference at the population level.

The purpose of this Special Issue is to gather a collection of articles presenting recent research dealing with spatial phenomena in biological populations and communities, obtained with models based on various mathematical techniques: ODE and PDE systems, integro-differential equations, difference equations (mapping), individual-based models, box models, matrix models, Markov models, and others.

Prof. Dr. Yuri V. Tyutyunov
Guest Editor

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Keywords

  • population dispersal
  • animal movements
  • predator–prey
  • trophic community
  • spatiotemporal heterogeneity
  • pattern formation
  • spatial behavior
  • swarming
  • emergent properties
  • taxis–diffusion–reaction
  • bifurcation
  • spatiotemporal chaos

Published Papers (5 papers)

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Research

13 pages, 1210 KiB  
Article
Emergence of Self-Organized Dynamical Domains in a Ring of Coupled Population Oscillators
by Alexey V. Rusakov, Dmitry A. Tikhonov, Nailya I. Nurieva and Alexander B. Medvinsky
Mathematics 2021, 9(6), 601; https://doi.org/10.3390/math9060601 - 11 Mar 2021
Cited by 1 | Viewed by 1246
Abstract
We show that interactions of inherently chaotic oscillators can lead to coexistence of regular oscillatory regimes and chaotic oscillations in the rings of coupled oscillators provided that the level of interaction between the oscillators exceeds a threshold value. The transformation of the initially [...] Read more.
We show that interactions of inherently chaotic oscillators can lead to coexistence of regular oscillatory regimes and chaotic oscillations in the rings of coupled oscillators provided that the level of interaction between the oscillators exceeds a threshold value. The transformation of the initially chaotic dynamics into the regular dynamics in a number of the coupled oscillators is shown to result from suppression of chaos by separation of certain oscillation periods from the continuous spectra, which are characteristic of chaotic oscillations. Full article
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15 pages, 881 KiB  
Article
A Moving-Mesh Finite-Difference Method for Segregated Two-Phase Competition-Diffusion
by Michael John Baines and Katerina Christou
Mathematics 2021, 9(4), 386; https://doi.org/10.3390/math9040386 - 15 Feb 2021
Cited by 1 | Viewed by 2046
Abstract
A moving-mesh finite-difference solution of a Lotka-Volterra competition-diffusion model of theoretical ecology is described in which the competition is sufficiently strong to spatially segregate the two populations, leading to a two-phase problem with a coupling condition at the moving interface. A moving mesh [...] Read more.
A moving-mesh finite-difference solution of a Lotka-Volterra competition-diffusion model of theoretical ecology is described in which the competition is sufficiently strong to spatially segregate the two populations, leading to a two-phase problem with a coupling condition at the moving interface. A moving mesh approach preserves the identities of the two species in space and time, so that the parameters always refer to the correct population. The model is implemented numerically with a variety of parameter combinations, illustrating how the populations may evolve in time. Full article
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18 pages, 2160 KiB  
Article
Nonlocal Reaction–Diffusion Models of Heterogeneous Wealth Distribution
by Malay Banerjee, Sergei V. Petrovskii and Vitaly Volpert
Mathematics 2021, 9(4), 351; https://doi.org/10.3390/math9040351 - 10 Feb 2021
Cited by 7 | Viewed by 1957
Abstract
Dynamics of human populations can be affected by various socio-economic factors through their influence on the natality and mortality rates, and on the migration intensity and directions. In this work we study an economic–demographic model which takes into account the dependence of the [...] Read more.
Dynamics of human populations can be affected by various socio-economic factors through their influence on the natality and mortality rates, and on the migration intensity and directions. In this work we study an economic–demographic model which takes into account the dependence of the wealth production rate on the available resources. In the case of nonlocal consumption of resources, the homogeneous-in-space wealth–population distribution is replaced by a periodic-in-space distribution for which the total wealth increases. For the global consumption of resources, if the wealth redistribution is small enough, then the homogeneous distribution is replaced by a heterogeneous one with a single wealth accumulation center. Thus, economic and demographic characteristics of nonlocal and global economies can be quite different in comparison with the local economy. Full article
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22 pages, 1444 KiB  
Article
Indirect Taxis on a Fluctuating Environment
by Andrey Morgulis and Konstantin Ilin
Mathematics 2020, 8(11), 2052; https://doi.org/10.3390/math8112052 - 17 Nov 2020
Cited by 3 | Viewed by 1636
Abstract
In this article, we study a Patlak–Keller–Siegel (PKS) model of a community of two species placed in the inhomogeneous environment. We employ PKS law for modeling tactic movement due to interspecific taxis and in response to the environmental fluctuations. These fluctuations can arise [...] Read more.
In this article, we study a Patlak–Keller–Siegel (PKS) model of a community of two species placed in the inhomogeneous environment. We employ PKS law for modeling tactic movement due to interspecific taxis and in response to the environmental fluctuations. These fluctuations can arise for natural reasons, e.g., the terrain relief, the sea currents and the food resource distribution, and there are artificial ones. The main result in the article elucidates the effect of the small-scale environmental fluctuations on the large-scale pattern formation in PKS systems. This issue remains uncharted, although numerous studies have addressed the pattern formation while assuming an homogeneous environment. Meanwhile, exploring the role of the fluctuating environment is substantial in many respects, for instance, for predicting the side effects of human activity or for designing the control of biological systems. As well, it is necessary for understanding the roles played in the dynamics of trophic communities by the natural environmental inhomogeneities—those mentioned above, for example. We examined the small-scale environmental inhomogeneities in the spirit of Kapitza’s theory of the upside-down pendulum, but we used the homogenization instead of classical averaging. This approach is novel for the dynamics of PKS systems (though used commonly for other areas). Employing it has unveiled a novel mechanism of exerting the effect from the fluctuating environment on the pattern formation by the drift of species arising upon the homogenization of the fluctuations. Full article
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15 pages, 5332 KiB  
Article
Spatiotemporal Pattern Formation in a Prey-Predator System: The Case Study of Short-Term Interactions Between Diatom Microalgae and Microcrustaceans
by Yuri V. Tyutyunov, Anna D. Zagrebneva and Andrey I. Azovsky
Mathematics 2020, 8(7), 1065; https://doi.org/10.3390/math8071065 - 1 Jul 2020
Cited by 15 | Viewed by 2100
Abstract
A simple mathematical model capable of reproducing formation of small-scale spatial structures in prey–predator system is presented. The migration activity of predators is assumed to be determined by the degree of their satiation. The hungrier individual predators migrate more frequently, randomly changing their [...] Read more.
A simple mathematical model capable of reproducing formation of small-scale spatial structures in prey–predator system is presented. The migration activity of predators is assumed to be determined by the degree of their satiation. The hungrier individual predators migrate more frequently, randomly changing their spatial position. It has previously been demonstrated that such an individual response to local feeding conditions leads to prey–taxis and emergence of complex spatiotemporal dynamics at population level, including periodic, quasi-periodic and chaotic regimes. The proposed taxis–diffusion–reaction model is applied to describe the trophic interactions in system consisting of benthic diatom microalgae and harpacticoid copepods. The analytical condition for the oscillatory instability of the homogeneous stationary state of species coexistence is given. The model parameters are identified on the basis of field observation data and knowledge on the species ecology in order to explain micro-scale spatial patterns of these organisms, which still remain obscure, and to reproduce in numerical simulations characteristic size and the expected lifetime of density patches. Full article
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