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Special Issue "Advances in the Mathematics of Ecological Modelling"

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

Deadline for manuscript submissions: 30 June 2023 | Viewed by 14416

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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

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Guest Editor

Institute of Mathematical Problems of Biology of the Russian Academy of Sciences, a branch of the M. V. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 1, Prof. Vitkevich Street, Pushchino, 142290 Moscow, Russia
Interests: statistical modelling in terrestrial ecosystems; functional groups of vegetation; modelling of vegetation dynamics; assessment and forecast of forest biodiversity; ecological informatics

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

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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

Published Papers (11 papers)

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Research

Open AccessArticle

Pattern-Multiplicative Average of Nonnegative Matrices: When a Constrained Minimization Problem Requires Versatile Optimization Tools

Vladimir Yu. Protasov

Tatyana I. Zaitseva

and

Dmitrii O. Logofet

Mathematics 2022, 10(23), 4417; https://doi.org/10.3390/math10234417 - 23 Nov 2022

Viewed by 403

Abstract

Given several nonnegative matrices with a single pattern of allocation among their zero/nonzero elements, the average matrix should have the same pattern as well. This is the first tenet of the pattern-multiplicative average (PMA) concept, while the second one suggests the multiplicative nature of averaging. The concept of PMA was motivated in a number of application fields, of which we consider the matrix population models and illustrate solving the PMA problem with several sets of model matrices calibrated in particular botanic case studies. The patterns of those matrices are typically nontrivial (they contain both zero and nonzero elements), the PMA problem thus having no exact solution for a fundamental reason (an overdetermined system of algebraic equations). Therefore, searching for the approximate solution reduces to a constrained minimization problem for the approximation error, the loss function in optimization terms. We consider two alternative types of the loss function and present a general algorithm of searching the optimal solution: basin-hopping global search, then local descents by the method of conjugate gradients or that of penalty functions. Theoretical disadvantages and practical limitations of both loss functions are discussed and illustrated with a number of practical examples. Full article

(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)

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Open AccessArticle

Complex Dynamics of a Predator–Prey Interaction with Fear Effect in Deterministic and Fluctuating Environments

Nirapada Santra

Sudeshna Mondal

and

Guruprasad Samanta

Mathematics 2022, 10(20), 3795; https://doi.org/10.3390/math10203795 - 14 Oct 2022

Cited by 2 | Viewed by 908

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 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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Open AccessArticle

Mathematical Model of Pancreatic Cancer Cell Dynamics Considering the Set of Sequential Mutations and Interaction with the Immune System

and

Mathematics 2022, 10(19), 3557; https://doi.org/10.3390/math10193557 - 29 Sep 2022

Viewed by 732

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 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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Open AccessArticle

Monitoring in a Discrete-Time Nonlinear Age-Structured Population Model with Changing Environment

and

Mathematics 2022, 10(15), 2707; https://doi.org/10.3390/math10152707 - 31 Jul 2022

Cited by 1 | Viewed by 581

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, 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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Open AccessArticle

Mathematical Model of Pest Control Using Different Release Rates of Sterile Insects and Natural Enemies

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Mathematics 2022, 10(6), 883; https://doi.org/10.3390/math10060883 - 10 Mar 2022

Cited by 1 | Viewed by 2131

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 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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Open AccessArticle

Asymptotic Properties of Solutions to Delay Differential Equations Describing Plankton—Fish Interaction

Maria A. Skvortsova

Mathematics 2021, 9(23), 3064; https://doi.org/10.3390/math9233064 - 28 Nov 2021

Cited by 1 | Viewed by 896

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 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)

Open AccessArticle

“Realistic Choice of Annual Matrices Contracts the Range of λ_S Estimates” under Reproductive Uncertainty Too

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

Viewed by 875

Abstract

Our study is devoted to a subject popular in the field of matrix population models, namely, estimating the stochastic growth rate, λ_S, 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 λ_S estimation following the defining of λ_S 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 λ_S 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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Open AccessArticle

Potential-Growth Indicators Revisited: Higher Generality and Wider Merit of Indication

Dmitrii O. Logofet

and

Valerii N. Razzhevaikin

Mathematics 2021, 9(14), 1649; https://doi.org/10.3390/math9141649 - 13 Jul 2021

Cited by 1 | Viewed by 1228

Abstract

The notion of a potential-growth indicator came to being in the field of matrix population models long ago, almost simultaneously with the pioneering Leslie model for age-structured population dynamics, although the term has been given and the theory developed only in recent years. The indicator represents an explicit function, R(L), of matrix L elements and indicates the position of the spectral radius of L relative to 1 on the real axis, thus signifying the population growth, or decline, or stabilization. Some indicators turned out to be useful in theoretical layouts and practical applications prior to calculating the spectral radius itself. The most senior (1994) and popular indicator, R₀(L), is known as the net reproductive rate, and we consider two others, R₁(L) and R_RT(A), developed later on. All the three are different in terms of their simplicity and the level of generality, and we illustrate them with a case study of Calamagrostis epigeios, a long-rhizome perennial weed actively colonizing open spaces in the temperate zone. While the R₀(L) and R₁(L) fail, respectively, because of complexity and insufficient generality, the R_RT(L) does succeed, justifying the merit of indication. Full article

(This article belongs to the Special Issue Advances in the Mathematics of Ecological Modelling)

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Open AccessArticle

The Dubovitskii and Milyutin Methodology Applied to an Optimal Control Problem Originating in an Ecological System

and

Mathematics 2021, 9(5), 479; https://doi.org/10.3390/math9050479 - 26 Feb 2021

Cited by 2 | Viewed by 988

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 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)

Open AccessArticle

Realistic Choice of Annual Matrices Contracts the Range of λ_S Estimates

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 4 | Viewed by 1428

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 λ_S, 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 λ_S 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 λ_S. 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 λ_S 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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Open AccessArticle

Species Distribution Models and Niche Partitioning among Unisexual Darevskia dahli and Its Parental Bisexual (D. portschinskii, D. mixta) Rock Lizards in the Caucasus

Alexander Varshavskiy

Felix Danielyan

and

Marine Arakelyan

Mathematics 2020, 8(8), 1329; https://doi.org/10.3390/math8081329 - 10 Aug 2020

Cited by 14 | Viewed by 2528

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 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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