Theoretical and Mathematical Ecology

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

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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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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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Published Papers (10 papers)

2024

Jump to: 2023, 2022

18 pages, 6510 KiB  
Article
Taxis-Driven Pattern Formation in Tri-Trophic Food Chain Model with Omnivory
by Evgeniya Giricheva
Mathematics 2024, 12(2), 290; https://doi.org/10.3390/math12020290 - 16 Jan 2024
Viewed by 594
Abstract
The spatiotemporal dynamics of a three-component model of a food web are considered. The model describes the interactions between populations of resources, prey, and predators that consume both species. It assumes that the predator responds to the spatial change in the resource and [...] Read more.
The spatiotemporal dynamics of a three-component model of a food web are considered. The model describes the interactions between populations of resources, prey, and predators that consume both species. It assumes that the predator responds to the spatial change in the resource and prey densities by occupying areas where species density is higher (prey-taxis) and that the prey population avoids areas with a high predator density (predator-taxis). This work studies the conditions for the taxis-driven instability leading to the emergence of stationary patterns resulting from Turing instability and autowaves caused by wave instability. The existence of nonconstant positive steady states for the system is assessed through a rigorous bifurcation analysis. Meanwhile, the conditions for the existence of both types of instabilities are obtained by linear stability analysis. It is shown that the presence of cross-diffusion in the system supports the formation of spatially heterogeneous patterns. For low values of the resource-tactic and predator-tactic coefficients, Turing and wave instabilities coexist. The system undergoes only Turing instability for high levels of these parameters. Full article
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13 pages, 1871 KiB  
Article
Predator–Prey Dynamics and Ideal Free Distribution in a Heterogeneous Environment
by Vyacheslav Tsybulin and Pavel Zelenchuk
Mathematics 2024, 12(2), 275; https://doi.org/10.3390/math12020275 - 15 Jan 2024
Viewed by 696
Abstract
The concept of an ideal free distribution (IFD) is extended to a predator–prey system in a heterogeneous environment. We consider reaction–diffusion–advection equations which describe the evolution of spatial distributions of predators and prey under directed migration. Modification of local interaction terms is introduced, [...] Read more.
The concept of an ideal free distribution (IFD) is extended to a predator–prey system in a heterogeneous environment. We consider reaction–diffusion–advection equations which describe the evolution of spatial distributions of predators and prey under directed migration. Modification of local interaction terms is introduced, if some coefficients depend on resource. Depending on coefficients of local interaction, the different scenarios of predator distribution are possible. We pick out three cases: proportionality to prey (and respectively to resource), indifferent distribution and inversely proportional to the prey. These scenarios apply in the case of nonzero diffusion and taxis under additional conditions on diffusion and migration rates. We examine migration functions for which there are explicit stationary solutions with nonzero densities of both species. To analyze solutions with violation of the IFD conditions, we apply asymptotic expansions and a numerical approach with staggered grids. The results for a two-dimensional domain with no-flux boundary conditions are presented. Full article
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2023

Jump to: 2024, 2022

12 pages, 308 KiB  
Article
Paradoxes of Competition in Periodic Environments: Delta Functions in Ecological Models
by Vitaly G. Il’ichev and Dmitry B. Rokhlin
Mathematics 2024, 12(1), 125; https://doi.org/10.3390/math12010125 - 29 Dec 2023
Viewed by 568
Abstract
We demonstrate a basic technique for simplifying time-periodic competition models, which is based on the utilization of periodic delta functions as population growth rates. We show that the Poincare mapping splits into a sequence of one-dimensional mappings. The study of the corresponding stable [...] Read more.
We demonstrate a basic technique for simplifying time-periodic competition models, which is based on the utilization of periodic delta functions as population growth rates. We show that the Poincare mapping splits into a sequence of one-dimensional mappings. The study of the corresponding stable equilibria allows us to make conclusions concerning the coexistence and selection of the family of competitors. In particular, in “all vs. all” systems, for one of the populations to dominate, it is enough to surpass the others with a certain margin, and the correspondent stock constant does not depend on the number of competitors. We present paradoxical examples, where (1) a low-productive population can displace a highly productive one, (2) the displacement is non-transitive, (3) the coexistence is non-transitive. We also show how the delta functions can be utilized for the analysis of a “predator–prey” system. Full article
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15 pages, 4657 KiB  
Article
Emergent Spatial–Temporal Patterns in a Ring of Locally Coupled Population Oscillators
by Alexey V. Rusakov, Dmitry A. Tikhonov, Nailya I. Nurieva and Alexander B. Medvinsky
Mathematics 2023, 11(24), 4970; https://doi.org/10.3390/math11244970 - 15 Dec 2023
Viewed by 614
Abstract
A closed chain of oscillators can be considered a model for ring-shaped ecosystems, such as atolls or the coastal zones of inland reservoirs. We use the logistic map, which is often referred to as an archetypical example of how complex dynamics can arise [...] Read more.
A closed chain of oscillators can be considered a model for ring-shaped ecosystems, such as atolls or the coastal zones of inland reservoirs. We use the logistic map, which is often referred to as an archetypical example of how complex dynamics can arise from very simple nonlinear equations, as a model for a separate oscillator in the chain. We present an original algorithm that allows us to find solutions to the spatiotemporal logistic equation quite efficiently or to state with certainty that there are no such solutions. Based on the Shannon formula, we propose formulas for estimating the spatial and temporal entropy, which allow us to classify our solutions as regular or irregular. We show that regular solutions can occur within the Malthus parameter region that corresponds to the irregular dynamics of a solitary logistic map. Full article
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13 pages, 484 KiB  
Article
Acoustic Wind in a Hyperbolic Predator—Prey System
by Andrey Morgulis
Mathematics 2023, 11(5), 1265; https://doi.org/10.3390/math11051265 - 6 Mar 2023
Viewed by 1016
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

Jump to: 2024, 2023

24 pages, 5213 KiB  
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
Cited by 1 | Viewed by 1680
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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14 pages, 878 KiB  
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
Cited by 2 | Viewed by 1549
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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23 pages, 6098 KiB  
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 10 | Viewed by 3645
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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16 pages, 2839 KiB  
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 3 | Viewed by 2866
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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12 pages, 528 KiB  
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 7 | Viewed by 2186
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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