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Theoretical and Mathematical Ecology
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Topical Collection "Theoretical and Mathematical Ecology"

A topical collection in Mathematics (ISSN 2227-7390). This collection belongs to the section "Mathematical Biology".

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Editor

Prof. Dr. Yuri V. Tyutyunov

E-Mail Website
Collection 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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Topical Collection Information

Dear Colleagues,

Mathematics provides the most compact way of formulating theoretical knowledge and hypotheses, and ecological theory is not an exception. General laws of ecosystems’ functioning are formulated as theoretical models. Modern “Theoretical and Mathematical Ecology” began with seminal works of A. J. Lotka, V. Volterra, R. A. Fisher, V. A. Kostitzin, G. F. Gause and of other authors who have developed classical models that became a starting point for further studying of natural ecosystems. Confronting models with empirical data is the only way to test theoretical hypotheses to understand whether the theory should be improved, modified or rejected. Highlighting contradictions between natural observations and properties of mathematical models greatly stimulated the development of predation theory, theory of harvesting, theory of biological control of pests and weeds, theory of spatiotemporal population dynamics, theory of animal movements, theory of collective animal behavior, theory of natural selection and evolution. However, many interesting and urgent fundamental problems of “Theoretical and Mathematical Ecology” still remain unsolved, awaiting novel mathematical models and original approaches to the modelling investigation.

The purpose of this Topical Collection is to select and publish original research articles, review papers, and perspective papers, presenting achievements in the theory and applications of mathematical models in various fields of Theoretical and Mathematical Ecology.

Prof. Dr. Yuri V. Tyutyunov
Collection Editor

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2100 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (6 papers)

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2023

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Article
Acoustic Wind in a Hyperbolic Predator—Prey System
by
Andrey Morgulis
Mathematics 2023, 11(5), 1265; https://doi.org/10.3390/math11051265 - 06 Mar 2023
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Abstract
We address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is that the non-linear [...] Read more.
We address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is that the non-linear interplay of the short waves generically excites the slowly growing amplitude modulation for wide ranges of the model parameters. We have observed such a feature in the numerical experiments and support our conclusions with a short-wave asymptotic solution in the limit of vanishing prey diffusivity. Our reasoning relies on the so-called homogenized system that governs slow evolutions of the amplitudes of the short-wave parcels. It includes a term (called wind) which is absent in the original model and only comes from averaging over the short waves. It is the wind that (unlike any of the other terms!) is capable of exciting the instability and pumping the growth of solutions. There is quite a definite relationship between the predators’ transport coefficients to be held for getting rid of the wind. Interestingly, this relationship had been introduced in prior studies of small-scale mosaics in the spatial distributions of some real-life populations. Full article
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2022

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Article
The Harvest Effect on Dynamics of Northern Fur Seal Population: Mathematical Modeling and Data Analysis Results
by
Oksana Zhdanova
,
Alexey Kuzin
and
Efim Frisman
Mathematics 2022, 10(17), 3067; https://doi.org/10.3390/math10173067 - 25 Aug 2022
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Abstract
We examine population trends in light of male harvest data considering the long-time series of population data on northern fur seals at Tyuleniy Island. To answer the question has the way males were harvested influenced the population trajectory, we analyzed the visual harem [...] Read more.
We examine population trends in light of male harvest data considering the long-time series of population data on northern fur seals at Tyuleniy Island. To answer the question has the way males were harvested influenced the population trajectory, we analyzed the visual harem size and birth rate dynamics of the population, as well as the strategy and intensity of the harvest. We analyzed the dynamics of the sex ratio in the early (1958–1988) period to estimate parameters in the late period (1989–2013) based on the observed number of bulls and pups, while utilizing the distribution of reproductive rates obtained from pelagic sealing. Using a matrix population model for the observed part of the population (i.e., the male population), we analyzed the population growth rate associated with changes in both birth and survival rates considering the stochastic effects. Observations allow us to reject the hypothesis of nonselective harvest. Among the variety of natural and anthropogenic factors that could contribute to the decrease in the birth rate in the population, the effect of selective harvesting seems to be the most realistic. Full article
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Article
Internal Prices and Optimal Exploitation of Natural Resources
by
Vitaly G. Il’ichev
and
Dmitry B. Rokhlin
Mathematics 2022, 10(11), 1860; https://doi.org/10.3390/math10111860 - 29 May 2022
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Abstract
Within the framework of traditional fishery management, we propose an interpretation of natural resource prices. It leads to an economic taxation mechanism based on internal prices and reduces a complex problem of optimal long-term exploitation to a sequence of one-year optimization problems. Internal [...] Read more.
Within the framework of traditional fishery management, we propose an interpretation of natural resource prices. It leads to an economic taxation mechanism based on internal prices and reduces a complex problem of optimal long-term exploitation to a sequence of one-year optimization problems. Internal prices obey natural, economic patterns: the increase in resource amount diminishes taxes, and the rise in the number of “fishers” raises taxes. These taxes stimulate cooperative agent behavior. We consider new problems of optimal fishing, taking into account an adaptive migration of the fish population in two regions. To analyze these problems, we use evolutionary ecology models. We propose a paradoxical method to increase the catch yield through the so-called fish “luring” procedure. In this case, a kind of “giveaway” game occurs, where the region with underfishing becomes more attractive for fish and for catches in the future. Full article
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Article
The Ricker Competition Model of Two Species: Dynamic Modes and Phase Multistability
by
Matvey Kulakov
,
Galina Neverova
and
Efim Frisman
Mathematics 2022, 10(7), 1076; https://doi.org/10.3390/math10071076 - 27 Mar 2022
Cited by 2 | Viewed by 1866
Abstract
The model of two species competing for a resource proposed by R. May and A.P. Shapiro has not yet been fully explored. We study its dynamic modes. The model reveals complex dynamics: multistable in-phase and out-of-phase cycles, and their bifurcations occur. The multistable [...] Read more.
The model of two species competing for a resource proposed by R. May and A.P. Shapiro has not yet been fully explored. We study its dynamic modes. The model reveals complex dynamics: multistable in-phase and out-of-phase cycles, and their bifurcations occur. The multistable out-of-phase dynamic modes can bifurcate via the Neimark–Sacker scenario. A value variation of interspecific competition coefficients changes the number of in-phase and out-of-phase modes. We have suggested an approach to identify the bifurcation (period-doubling, pitchfork, or saddle-node bifurcations) due to which in-phase and out-of-phase periodic points appear. With strong interspecific competition, the population’s survival depends on its growth rate. However, with a specific initial condition, a species with a lower birth rate can displace its competitor with a higher one. With weak interspecific competition and sufficiently high population growth rates, the species coexist. At the same time, the observed dynamic mode or the oscillation phase can change due to altering of the initial condition values. The influence of external factors can be considered as an initial condition modification, leading to dynamics shift due to the coexistence of several stable attractors. Full article
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Article
Optimal Harvest Problem for Fish Population—Structural Stabilization
by
Aleksandr Abakumov
and
Yuri Izrailsky
Mathematics 2022, 10(6), 986; https://doi.org/10.3390/math10060986 - 18 Mar 2022
Cited by 1 | Viewed by 1728
Abstract
The influence of environmental conditions and fishery on a typical pelagic or semi-pelagic fish population is studied. A mathematical model of population dynamics with a size structure is constructed. The problem of the optimal harvest of a population in unstable environment conditions is [...] Read more.
The influence of environmental conditions and fishery on a typical pelagic or semi-pelagic fish population is studied. A mathematical model of population dynamics with a size structure is constructed. The problem of the optimal harvest of a population in unstable environment conditions is investigated and an optimality system to the problem research is constructed. The solutions properties in various cases have also been investigated. Environmental conditions influence the fish population through recruitment. Modelling of recruitment rate is made by using a stochastic imitation of environmental conditions. In the case of stationary environment, a population model admits nontrivial equilibrium state. The parameters of fish population are obtained from this equilibrium condition. The variability of environment leads to large oscillations of generation size. The fluctuations of the fish population density follow the dynamics of recruitment rate fluctuations but have smaller gradients than recruitment. The dynamics of the optimal fishing effort is characterized by high variability. The population and the average size of individuals decrease under the influence of fishery. In general, the results of computer calculations indicate the stabilization of the population dynamics under influence of size structure. Optimal harvesting also contributes to stabilization. Full article
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Article
Effect of Slow–Fast Time Scale on Transient Dynamics in a Realistic Prey-Predator System
by
Pranali Roy Chowdhury
,
Sergei Petrovskii
and
Malay Banerjee
Mathematics 2022, 10(5), 699; https://doi.org/10.3390/math10050699 - 23 Feb 2022
Cited by 5 | Viewed by 1323
Abstract
Systems with multiple time scales, often referred to as `slow–fast systems’, have been a focus of research for about three decades. Such systems show a variety of interesting, sometimes counter-intuitive dynamical behaviors and are believed to, in many cases, provide a more realistic [...] Read more.
Systems with multiple time scales, often referred to as `slow–fast systems’, have been a focus of research for about three decades. Such systems show a variety of interesting, sometimes counter-intuitive dynamical behaviors and are believed to, in many cases, provide a more realistic description of ecological dynamics. In particular, the presence of slow–fast time scales is known to be one of the main mechanisms resulting in long transients—dynamical behavior that mimics a system’s asymptotic regime but only lasts for a finite (albeit very long) time. A prey–predator system where the prey growth rate is much larger than that of the predator is a paradigmatic example of slow–fast systems. In this paper, we provide detailed investigation of a more advanced variant of prey–predator system that has been overlooked in previous studies, that is, where the predator response is ratio-dependent and the predator mortality is nonlinear. We perform a comprehensive analytical study of this system to reveal a sequence of bifurcations that are responsible for the change in the system dynamics from a simple steady state and/or a limit cycle to canards and relaxation oscillations. We then consider how those changes in the system dynamics affect the properties of long transient dynamics. We conclude with a discussion of the ecological implications of our findings, in particular to argue that the changes in the system dynamics in response to an increase of the time scale ratio are counter-intuitive or even paradoxical. Full article
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