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Advances in the Mathematics of Ecological Modelling
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Advances in the Mathematics of Ecological Modelling

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

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 28120

Special Issue Editors

Prof. Dr. Dmitrii O. Logofet

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Guest Editor
A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences, 3, Pyzhevskii Pereulok, 119017 Moscow, Russia
Interests: stability problems in models of population, community, and ecosystem dynamics; the hierarchy of stability subsets in matrices; "stability versus complexity" issues; mathematical models of plant successions; ecological risk analysis; matrix models of structured population dynamics
Dr. Larisa Khanina

E-Mail Website
Guest Editor
Institute of Mathematical Problems of Biology of RAS, Branch of the Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, 1 Prof. Vitkevich Str., Pushchino 142290, Russia
Interests: forest ecology; plant and soil ecology; biodiversity assessment; element cycles
Special Issues, Collections and Topics in MDPI journals
Dr. Pavel Grabarnik

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Guest Editor
1. Institute of Physico-Chemical and Biological Problems of Soil Science of the Russian Academy of Sciences, Institutskaya, 2, Pushchino, 142290 Moscow, Russia
2. Pushchino Scientific Center for Biological Research of the Russian Academy of Sciences, Prospect Nauki, 3, Pushchino, 142290 Moscow, Russia
Interests: space-time structure of forest ecosystems modelling; spatial patterns analysis; spatial statistics; Monte Carlo inference for spatial processes; multiple hypotheses testing

Special Issue Information

Dear Colleagues,

Current practices of ecological modelling often motivate new mathematical problems to be posed, which may sometimes be solvable or lead to further mathematical research. Though classical works by Alfred Lotka and Vito Volterra initiated the discipline of Mathematical Ecology with ordinary and integro-differential equations almost a hundred years ago, it is now difficult to imagine any formalism of applied mathematics that has never been applied in this area. Many excellent mathematical topics have roots in ecological problems.

We initiate this Special Issue aiming to sample the current state-of-the-art in the mathematics of ecological modelling. Manuscripts are welcome that are devoted to the following topics (but not limited to these):
Population and community dynamics;
Biodiversity assessment;
Inference from ecological data;
Interacting agents system;
Matrix population models;
Migrations and stability in metapopulations;
Stability and bifurcations in multispecies systems;
Stability vs complexity in randomly structured communities;
Vegetation successions;
Environmental stochasticity;
Biogeochemical cycles in ecosystems;
Species distribution models;
Pattern formation;
Spatial patterns and processes;
Uncertainties in model calibration;
Rational-/integer-valued formalisms.

We will prioritize contributions that develop a new kind of mathematical formalism or open a new field of ecological applications.

Prof. Dr. Dmitrii O. Logofet
Dr. Larisa Khanina
Dr. Pavel Grabarnik
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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 2600 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.

Keywords

  • Population dynamics
  • Community dynamics
  • Ecosystem dynamics
  • Migrations and stability in metapopulations
  • Stability and bifurcations in multispecies systems
  • Matrix population models
  • Environmental stochasticity
  • Spatial patterns and processes
  • Uncertainties in model calibration

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Published Papers (12 papers)

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Research

12 pages, 869 KiB  
Article
Global Asymptotic Stability Analysis of Fixed Points for a Density-Dependent Single-Species Population Growth Model
by Meilin He, Mingjue Zhu, Xuyang Teng, Zhirui Hu, Wei Feng, Huina Song, Xiyuan Chen and Haiquan Wang
Mathematics 2023, 11(20), 4345; https://doi.org/10.3390/math11204345 - 19 Oct 2023
Viewed by 1243
Abstract
In a density-dependent single-species population growth model, a simple method is proposed to explicitly and directly derive the analytic expressions of reliable regions for local and global asymptotic stability. Specifically, first, a reliable region ΛLAS is explicitly represented by solving the fixed [...] Read more.
In a density-dependent single-species population growth model, a simple method is proposed to explicitly and directly derive the analytic expressions of reliable regions for local and global asymptotic stability. Specifically, first, a reliable region ΛLAS is explicitly represented by solving the fixed point and utilizing the asymptotic stability criterion, over which the fixed point is locally asymptotically stable. Then, two types of auxiliary Liapunov functions are constructed, where the variation of the Liapunov function is decomposed into the product of two functions and is always negative at the non-equilibrium state. Finally, based on the Liapunov stability theorem, a closed-form expression of reliable region ΛGAS is obtained, where the fixed point is globally asymptotically stable in the sense that all the solutions tend to fixed point. Numerical results show that our analytic expressions of reliable regions are accurate for both local and global asymptotic stability. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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15 pages, 321 KiB  
Article
Novel Roles of Standard Lagrangians in Population Dynamics Modeling and Their Ecological Implications
by Diana T. Pham and Zdzislaw E. Musielak
Mathematics 2023, 11(17), 3653; https://doi.org/10.3390/math11173653 - 24 Aug 2023
Viewed by 1101
Abstract
The Lagrangian formalism based on the standard Lagrangians, which are characterized by the presence of the kinetic and potential energy-like terms, is established for selected population dynamics models. A general method that allows for constructing such Lagrangians is developed, and its specific applications [...] Read more.
The Lagrangian formalism based on the standard Lagrangians, which are characterized by the presence of the kinetic and potential energy-like terms, is established for selected population dynamics models. A general method that allows for constructing such Lagrangians is developed, and its specific applications are presented and discussed. The obtained results are compared with the previously found Lagrangians, whose forms were different as they did not allow for identifying the energy-like terms. It is shown that the derived standard Lagrangians for the population dynamics models can be used to study the oscillatory behavior of the models and the period of their oscillations, which may have ecological and environmental implications. Moreover, other physical and biological insights that can be gained from the constructed standard Lagrangians are also discussed. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
8 pages, 2237 KiB  
Article
COVID-19: From Limit Cycle to Stable Focus
by Alexander Sokolov and Vladimir Voloshinov
Mathematics 2023, 11(14), 3226; https://doi.org/10.3390/math11143226 - 22 Jul 2023
Viewed by 1027
Abstract
The study aims at investigating a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time—qualitative change (bifurcation) in the behavior of the “virus vs. human” system with an increase in contagiousness. Numerical experiments with a model of the [...] Read more.
The study aims at investigating a new fundamental property of infectious diseases with natural adaptive immunity that weakens over time—qualitative change (bifurcation) in the behavior of the “virus vs. human” system with an increase in contagiousness. Numerical experiments with a model of the COVID-19 epidemic in Moscow have demonstrated that when the reproduction number R0 is about 4, a qualitative change (bifurcation) occurs in the behavior of the virus–human system. Below this value, the long-term forecast tends toward undamped oscillations; above it, the forecast shows damped oscillations: the amplitudes of epidemic waves decrease gradually, with a constant, very high background level of morbidity that keeps natural immunity near 100%. To confirm this result analytically, we use an original modification of the Euler–Lotka renewal equation, which describes the dynamics of infected patients distributed by disease duration (time since infection) and accounts for immunity. To construct a bifurcation diagram, which illustrates the dependence of the equilibrium stability on the parameter R0, we linearize the equation in the vicinity of the equilibrium point and examine its numerical approximation (discrete form). This approximation can be interpreted as a Leslie model, with the matrix elements dependent on the parameter R0. By examining the roots of the corresponding Lotka polynomial, we can assess the stability of the equilibrium point and verify the basic assumption about the change in the properties of the system with increasing R0—about the transition from undamped oscillations to damped ones. For the bifurcation diagram, we use the functions obtained from the simulation of the COVID-19 epidemic in Moscow. However, observations of the epidemic in other cities and countries support the primary finding of our study regarding the attenuation of epidemic waves. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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38 pages, 7515 KiB  
Article
Complex Dynamics of a Predator–Prey Interaction with Fear Effect in Deterministic and Fluctuating Environments
by Nirapada Santra, Sudeshna Mondal and Guruprasad Samanta
Mathematics 2022, 10(20), 3795; https://doi.org/10.3390/math10203795 - 14 Oct 2022
Cited by 8 | Viewed by 2518
Abstract
Many ecological models have received much attention in the past few years. In particular, predator–prey interactions have been examined from many angles to capture and explain various environmental phenomena meaningfully. Although the consumption of prey directly by the predator is a well-known ecological [...] Read more.
Many ecological models have received much attention in the past few years. In particular, predator–prey interactions have been examined from many angles to capture and explain various environmental phenomena meaningfully. Although the consumption of prey directly by the predator is a well-known ecological phenomenon, theoretical biologists suggest that the impact of anti-predator behavior due to the fear of predators (felt by prey) can be even more crucial in shaping prey demography. In this article, we develop a predator–prey model that considers the effects of fear on prey reproduction and on environmental carrying capacity of prey species. We also include two delays: prey species birth delay influenced by fear of the predator and predator gestation delay. The global stability of each equilibrium point and its basic dynamical features have been investigated. Furthermore, the “paradox of enrichment” is shown to exist in our system. By analysing our system of nonlinear delay differential equations, we gain some insights into how fear and delays affect on population dynamics. To demonstrate our findings, we also perform some numerical computations and simulations. Finally, to evaluate the influence of a fluctuating environment, we compare our proposed system to a stochastic model with Gaussian white noise terms. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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12 pages, 407 KiB  
Article
Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System
by Alexander S. Bratus, Nicholas Leslie, Michail Chamo, Dmitry Grebennikov, Rostislav Savinkov, Gennady Bocharov and Daniil Yurchenko
Mathematics 2022, 10(19), 3557; https://doi.org/10.3390/math10193557 - 29 Sep 2022
Viewed by 1848
Abstract
Pancreatic cancer represents one of the difficult problems of contemporary medicine. The development of the illness evolves very slowly, happens in a specific place (stroma), and manifests clinically close to a final stage. Another feature of this pathology is a coexistence (symbiotic) effect [...] Read more.
Pancreatic cancer represents one of the difficult problems of contemporary medicine. The development of the illness evolves very slowly, happens in a specific place (stroma), and manifests clinically close to a final stage. Another feature of this pathology is a coexistence (symbiotic) effect between cancer cells and normal cells inside stroma. All these aspects make it difficult to understand the pathogenesis of pancreatic cancer and develop a proper therapy. The emergence of pancreatic pre-cancer and cancer cells represents a branching stochastic process engaging populations of 64 cells differing in the number of acquired mutations. In this study, we formulate and calibrate the mathematical model of pancreatic cancer using the quasispecies framework. The mathematical model incorporates the mutation matrix, fineness landscape matrix, and the death rates. Each element of the mutation matrix presents the probability of appearing as a specific mutation in the branching sequence of cells representing the accumulation of mutations. The model incorporates the cancer cell elimination by effect CD8 T cells (CTL). The down-regulation of the effector function of CTLs and exhaustion are parameterized. The symbiotic effect of coexistence of normal and cancer cells is considered. The computational predictions obtained with the model are consistent with empirical data. The modeling approach can be used to investigate other types of cancers and examine various treatment procedures. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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16 pages, 2035 KiB  
Article
Monitoring in a Discrete-Time Nonlinear Age-Structured Population Model with Changing Environment
by Inmaculada López, Zoltán Varga, Manuel Gámez and József Garay
Mathematics 2022, 10(15), 2707; https://doi.org/10.3390/math10152707 - 31 Jul 2022
Cited by 1 | Viewed by 1370
Abstract
This paper is a contribution to the modeling–methodological development of the application of mathematical systems theory in population biology. A discrete-time nonlinear Leslie-type model is considered, where both the reproduction and survival rates decrease as the total population size increases. In this context, [...] Read more.
This paper is a contribution to the modeling–methodological development of the application of mathematical systems theory in population biology. A discrete-time nonlinear Leslie-type model is considered, where both the reproduction and survival rates decrease as the total population size increases. In this context, the monitoring problem means that, from the observation of the size of certain age classes as a function of time, we want to recover (estimate) the whole state process (i.e., the time-dependent size of the rest of the classes). First, for the linearization approach, conditions for the existence and asymptotic stability of a positive equilibrium are obtained, then the discrete-time observer design method is applied to estimate an unknown state trajectory near the equilibrium, where we could observe a single age class. It is also shown how the observer design can be used to detect an unknown change in the environment that affects the population dynamics. The environmental change is supposed to be generated by additional dynamics (exosystem). Now, the Leslie-type model is extended with this exosystem, and the observer design is applied to this extended system. In this way, an estimation can be obtained for different (constant or periodic) environmental changes as well. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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18 pages, 1575 KiB  
Article
Mathematical Model of Pest Control Using Different Release Rates of Sterile Insects and Natural Enemies
by Toni Bakhtiar, Ihza Rizkia Fitri, Farida Hanum and Ali Kusnanto
Mathematics 2022, 10(6), 883; https://doi.org/10.3390/math10060883 - 10 Mar 2022
Cited by 4 | Viewed by 4463
Abstract
In the framework of integrated pest management, biological control through the use of living organisms plays important roles in suppressing pest populations. In this paper, the complex interaction between plants and pest insects is examined under the intervention of natural enemies releases coupled [...] Read more.
In the framework of integrated pest management, biological control through the use of living organisms plays important roles in suppressing pest populations. In this paper, the complex interaction between plants and pest insects is examined under the intervention of natural enemies releases coupled with sterile insects technique. A set of nonlinear ordinary differential equations is developed in terms of optimal control model considering characteristics of populations involved. Optimal control measures are sought in such a way they minimize the pest density simultaneously with the control efforts. Three different strategies relating to the release rate of sterile insects and predators as natural enemies, namely, constant, proportional, and saturating proportional release rates, are examined for the attainability of control objective. The necessary optimality conditions of the control problem are derived by using Pontryagin maximum principle, and the forward–backward sweep method is then implemented to numerically calculate the optimal solution. It is shown that, in an environment consisting of rice plants and brown planthoppers as pests, the releases of sterile planthoppers and ladybeetles as natural enemies can deteriorate the pest density and thus increase the plant biomass. The release of sterile insects with proportional rate and the release of natural enemies with constant rate are found to be the most cost-effective strategy in controlling pest insects. This strategy successfully decreases the pest population about 35 percent, and thus increases the plant density by 13 percent during control implementation. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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11 pages, 259 KiB  
Article
Asymptotic Properties of Solutions to Delay Differential Equations Describing Plankton—Fish Interaction
by Maria A. Skvortsova
Mathematics 2021, 9(23), 3064; https://doi.org/10.3390/math9233064 - 28 Nov 2021
Cited by 1 | Viewed by 1381
Abstract
We consider a system of differential equations with two delays describing plankton–fish interaction. We analyze the case when the equilibrium point of this system corresponding to the presence of only phytoplankton and the absence of zooplankton and fish is asymptotically stable. In this [...] Read more.
We consider a system of differential equations with two delays describing plankton–fish interaction. We analyze the case when the equilibrium point of this system corresponding to the presence of only phytoplankton and the absence of zooplankton and fish is asymptotically stable. In this case, the asymptotic behavior of solutions to the system is studied. We establish estimates of solutions characterizing the stabilization rate at infinity to the considered equilibrium point. The results are obtained using Lyapunov–Krasovskii functionals. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
15 pages, 5749 KiB  
Article
“Realistic Choice of Annual Matrices Contracts the Range of λS Estimates” under Reproductive Uncertainty Too
by Dmitrii O. Logofet, Leonid L. Golubyatnikov, Elena S. Kazantseva and Nina G. Ulanova
Mathematics 2021, 9(23), 3007; https://doi.org/10.3390/math9233007 - 24 Nov 2021
Cited by 3 | Viewed by 1456
Abstract
Our study is devoted to a subject popular in the field of matrix population models, namely, estimating the stochastic growth rate, λS, a quantitative measure of long-term population viability, for a discrete-stage-structured population monitored during many years. “Reproductive uncertainty [...] Read more.
Our study is devoted to a subject popular in the field of matrix population models, namely, estimating the stochastic growth rate, λS, a quantitative measure of long-term population viability, for a discrete-stage-structured population monitored during many years. “Reproductive uncertainty” refers to a feature inherent in the data and life cycle graph (LCG) when the LCG has more than one reproductive stage, but when the progeny cannot be associated to a parent stage in a unique way. Reproductive uncertainty complicates the procedure of λS estimation following the defining of λS from the limit of a sequence consisting of population projection matrices (PPMs) chosen randomly from a given set of annual PPMs. To construct a Markov chain that governs the choice of PPMs for a local population of Eritrichium caucasicum, an short-lived perennial alpine plant species, we have found a local weather index that is correlated with the variations in the annual PPMs, and we considered its long time series as a realization of the Markov chain that was to be constructed. Reproductive uncertainty has required a proper modification of how to restore the transition matrix from a long realization of the chain, and the restored matrix has been governing random choice in several series of Monte Carlo simulations of long-enough sequences. The resulting ranges of λS estimates turn out to be more narrow than those obtained by the popular i.i.d. methods of random choice (independent and identically distributed matrices); hence, we receive a more accurate and reliable forecast of population viability. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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17 pages, 344 KiB  
Article
The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System
by Aníbal Coronel, Fernando Huancas, Esperanza Lozada and Marko Rojas-Medar
Mathematics 2021, 9(5), 479; https://doi.org/10.3390/math9050479 - 26 Feb 2021
Cited by 3 | Viewed by 1909
Abstract
We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control [...] Read more.
We research a control problem for an ecological model given by a reaction–diffusion system. The ecological model is given by a nonlinear parabolic PDE system of three equations modelling the interaction of three species by considering the standard Lotka-Volterra assumptions. The optimal control problem consists of the determination of a coefficient such that the population density of predator decreases. We reformulate the control problem as an optimal control problem by introducing an appropriate cost function. Then, we introduce and prove three types of results. A first contribution of the paper is the well-posedness framework of the mathematical model by considering that the interaction of the species is given by a general functional responses. Second, we study the differentiability properties of a cost function. The third result is the existence of optimal solutions, the existence of an adjoint state, and a characterization of the control function. The first result is proved by the application of semigroup theory and the second and third result are proved by the application of Dubovitskii and Milyutin formalism. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
15 pages, 1438 KiB  
Article
Realistic Choice of Annual Matrices Contracts the Range of λS Estimates
by Dmitrii O. Logofet, Leonid L. Golubyatnikov and Nina G. Ulanova
Mathematics 2020, 8(12), 2252; https://doi.org/10.3390/math8122252 - 20 Dec 2020
Cited by 7 | Viewed by 2142
Abstract
In matrix population modeling the multi-year monitoring of a population structure results in a set of annual population projection matrices (PPMs), which gives rise to the stochastic growth rate λS, a quantitative measure of long-term population viability. This measure is usually [...] Read more.
In matrix population modeling the multi-year monitoring of a population structure results in a set of annual population projection matrices (PPMs), which gives rise to the stochastic growth rate λS, a quantitative measure of long-term population viability. This measure is usually found in the paradigm of population growth in a variable environment. The environment is represented by the set of PPMs, and λS ensues from a long sequence of PPMs chosen at random from the given set. because the known rules of random choice, such as the iid (independent and identically distributed) matrices, are generally artificial, the challenge is to find a more realistic rule. We achieve this with the a following a Markov chain that models, in a certain sense, the real variations in the environment. We develop a novel method to construct the ruling Markov chain from long-term weather data and to simulate, in a Monte Carlo mode, the long sequences of PPMs resulting in the estimates of λS. The stochastic nature of sequences causes the estimates to vary within some range, and we compare the range obtained by the “realistic choice” from 10 PPMs for a local population of a Red-Book species to those using the iid choice. As noted in the title of this paper, this realistic choice contracts the range of λS estimates, thus improving the estimation and confirming the Red-Book status of the species. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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21 pages, 4448 KiB  
Article
Species Distribution Models and Niche Partitioning among Unisexual Darevskia dahli and Its Parental Bisexual (D. portschinskii, D. mixta) Rock Lizards in the Caucasus
by Varos Petrosyan, Fedor Osipov, Vladimir Bobrov, Natalia Dergunova, Andrey Omelchenko, Alexander Varshavskiy, Felix Danielyan and Marine Arakelyan
Mathematics 2020, 8(8), 1329; https://doi.org/10.3390/math8081329 - 10 Aug 2020
Cited by 25 | Viewed by 3842
Abstract
Among vertebrates, true parthenogenesis is known only in reptiles. Parthenogenetic lizards of the genus Darevskia emerged as a result of the hybridization of bisexual parental species. However, uncertainty remains about the mechanisms of the co-existence of these forms. The geographical parthenogenesis hypothesis suggests [...] Read more.
Among vertebrates, true parthenogenesis is known only in reptiles. Parthenogenetic lizards of the genus Darevskia emerged as a result of the hybridization of bisexual parental species. However, uncertainty remains about the mechanisms of the co-existence of these forms. The geographical parthenogenesis hypothesis suggests that unisexual forms can co-exist with their parental species in the “marginal” habitats. Our goal is to investigate the influence of environmental factors on the formation of ecological niches and the distribution of lizards. For this reason, we created models of species distribution and ecological niches to predict the potential geographical distribution of the parthenogenetic and its parental species. We also estimated the realized niches breadth, their overlap, similarities, and shifts in the entire space of predictor variables. We found that the centroids of the niches of the three studied lizards were located in the mountain forests. The “maternal” species D. mixta prefers forest habitats located at high elevations, “paternal” species D. portschinskii commonly occurs in arid and shrub habitats of the lower belt of mountain forests, and D. dahli occupies substantially an intermediate or “marginal” position along environmental gradients relative to that of its parental species. Our results evidence that geographical parthenogenesis partially explains the co-existence of the lizards. Full article
(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)
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