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2 pages, 137 KiB

Open AccessEditorial

Progress in Mathematical Ecology

by Sergei Petrovskii

Mathematics 2018, 6(9), 167; https://doi.org/10.3390/math6090167 - 13 Sep 2018

Cited by 1 | Viewed by 2934

Abstract

n/a Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

Research

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21 pages, 709 KiB

Open AccessFeature PaperArticle

Analysis of the Incidence of Poxvirus on the Dynamics between Red and Grey Squirrels

by Fadi Barbara, Valentina La Morgia, Valerio Parodi, Giuseppe Toscano and Ezio Venturino

Mathematics 2018, 6(7), 113; https://doi.org/10.3390/math6070113 - 01 Jul 2018

Cited by 10 | Viewed by 4244

Abstract

A model for the interactions of the invasive grey squirrel species as asymptomatic carriers of the poxvirus with the native red squirrel is presented and analyzed. Equilibria of the dynamical system are assessed, and their sensitivity in terms of the ecosystem parameters is [...] Read more.

A model for the interactions of the invasive grey squirrel species as asymptomatic carriers of the poxvirus with the native red squirrel is presented and analyzed. Equilibria of the dynamical system are assessed, and their sensitivity in terms of the ecosystem parameters is investigated through numerical simulations. The findings are in line with both field and theoretical research. The results indicate that mainly the reproduction rate of the alien population should be drastically reduced to repel the invasion, and to achieve disease eradication, actions must be performed to reduce the intraspecific transmission rate; also, the native species mortality plays a role: if grey squirrels are controlled, increasing it may help in the red squirrel preservation, while the invaders vanish; on the contrary, decreasing it in favorable situations, the coexistence of the two species may occur. Preservation or restoration of the native red squirrel requires removal of the grey squirrels or keeping them at low values. Wildlife managers should exert a constant effort to achieve a harsh reduction of the grey squirrel growth rate and to protect the remnant red squirrel population. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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27 pages, 7669 KiB

Open AccessArticle

Linearization of the Kingman Coalescent

by Paul F. Slade

Mathematics 2018, 6(5), 82; https://doi.org/10.3390/math6050082 - 14 May 2018

Cited by 3 | Viewed by 4167

Abstract

Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term

(\begin{matrix} n \\ 2 \end{matrix})

where n denotes the number [...] Read more.

Kingman’s coalescent process is a mathematical model of genealogy in which only pairwise common ancestry may occur. Inter-arrival times between successive coalescence events have a negative exponential distribution whose rate equals the combinatorial term

(\begin{matrix} n \\ 2 \end{matrix})

where n denotes the number of lineages present in the genealogy. These two standard constraints of Kingman’s coalescent, obtained in the limit of a large population size, approximate the exact ancestral process of Wright-Fisher or Moran models under appropriate parameterization. Calculation of coalescence event probabilities with higher accuracy quantifies the dependence of sample and population sizes that adhere to Kingman’s coalescent process. The convention that probabilities of leading order

N^{- 2}

are negligible provided

n ≪ N

is examined at key stages of the mathematical derivation. Empirically, expected genealogical parity of the single-pair restricted Wright-Fisher haploid model exceeds 99% where

n \leq \frac{1}{2} \sqrt[3]{N}

; similarly, per expected interval where

n \leq \frac{1}{2} \sqrt{N / 6}

. The fractional cubic root criterion is practicable, since although it corresponds to perfect parity and to an extent confounds identifiability it also accords with manageable conditional probabilities of multi-coalescence. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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27 pages, 2768 KiB

Open AccessFeature PaperArticle

The “Lévy or Diffusion” Controversy: How Important Is the Movement Pattern in the Context of Trapping?

by Danish A. Ahmed, Sergei V. Petrovskii and Paulo F. C. Tilles

Mathematics 2018, 6(5), 77; https://doi.org/10.3390/math6050077 - 09 May 2018

Cited by 7 | Viewed by 5435

Abstract

Many empirical and theoretical studies indicate that Brownian motion and diffusion models as its mean field counterpart provide appropriate modeling techniques for individual insect movement. However, this traditional approach has been challenged, and conflicting evidence suggests that an alternative movement pattern such as [...] Read more.

Many empirical and theoretical studies indicate that Brownian motion and diffusion models as its mean field counterpart provide appropriate modeling techniques for individual insect movement. However, this traditional approach has been challenged, and conflicting evidence suggests that an alternative movement pattern such as Lévy walks can provide a better description. Lévy walks differ from Brownian motion since they allow for a higher frequency of large steps, resulting in a faster movement. Identification of the ‘correct’ movement model that would consistently provide the best fit for movement data is challenging and has become a highly controversial issue. In this paper, we show that this controversy may be superficial rather than real if the issue is considered in the context of trapping or, more generally, survival probabilities. In particular, we show that almost identical trap counts are reproduced for inherently different movement models (such as the Brownian motion and the Lévy walk) under certain conditions of equivalence. This apparently suggests that the whole ‘Levy or diffusion’ debate is rather senseless unless it is placed into a specific ecological context, e.g., pest monitoring programs. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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24 pages, 2278 KiB

Open AccessArticle

Effects of Viral and Cytokine Delays on Dynamics of Autoimmunity

by Farzad Fatehi, Yuliya N. Kyrychko and Konstantin B. Blyuss

Mathematics 2018, 6(5), 66; https://doi.org/10.3390/math6050066 - 28 Apr 2018

Cited by 7 | Viewed by 3493

Abstract

A major contribution to the onset and development of autoimmune disease is known to come from infections. An important practical problem is identifying the precise mechanism by which the breakdown of immune tolerance as a result of immune response to infection leads to [...] Read more.

A major contribution to the onset and development of autoimmune disease is known to come from infections. An important practical problem is identifying the precise mechanism by which the breakdown of immune tolerance as a result of immune response to infection leads to autoimmunity. In this paper, we develop a mathematical model of immune response to a viral infection, which includes T cells with different activation thresholds, regulatory T cells (Tregs), and a cytokine mediating immune dynamics. Particular emphasis is made on the role of time delays associated with the processes of infection and mounting the immune response. Stability analysis of various steady states of the model allows us to identify parameter regions associated with different types of immune behaviour, such as, normal clearance of infection, chronic infection, and autoimmune dynamics. Numerical simulations are used to illustrate different dynamical regimes, and to identify basins of attraction of different dynamical states. An important result of the analysis is that not only the parameters of the system, but also the initial level of infection and the initial state of the immune system determine the progress and outcome of the dynamics. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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15 pages, 1120 KiB

Open AccessArticle

Critical Domain Problem for the Reaction–Telegraph Equation Model of Population Dynamics

by Weam Alharbi and Sergei Petrovskii

Mathematics 2018, 6(4), 59; https://doi.org/10.3390/math6040059 - 17 Apr 2018

Cited by 16 | Viewed by 4355

Abstract

A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that [...] Read more.

A telegraph equation is believed to be an appropriate model of population dynamics as it accounts for the directional persistence of individual animal movement. Being motivated by the problem of habitat fragmentation, which is known to be a major threat to biodiversity that causes species extinction worldwide, we consider the reaction–telegraph equation (i.e., telegraph equation combined with the population growth) on a bounded domain with the goal to establish the conditions of species survival. We first show analytically that, in the case of linear growth, the expression for the domain’s critical size coincides with the critical size of the corresponding reaction–diffusion model. We then consider two biologically relevant cases of nonlinear growth, i.e., the logistic growth and the growth with a strong Allee effect. Using extensive numerical simulations, we show that in both cases the critical domain size of the reaction–telegraph equation is larger than the critical domain size of the reaction–diffusion equation. Finally, we discuss possible modifications of the model in order to enhance the positivity of its solutions. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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13 pages, 2076 KiB

Open AccessArticle

Prey-Predator Model with a Nonlocal Bistable Dynamics of Prey

by Malay Banerjee, Nayana Mukherjee and Vitaly Volpert

Mathematics 2018, 6(3), 41; https://doi.org/10.3390/math6030041 - 08 Mar 2018

Cited by 18 | Viewed by 5266

Abstract

Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra- and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with [...] Read more.

Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra- and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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16 pages, 3154 KiB

Open AccessArticle

Role of Bi-Directional Migration in Two Similar Types of Ecosystems

by Nikhil Pal, Sudip Samanta, Maia Martcheva and Joydev Chattopadhyay

Mathematics 2018, 6(3), 36; https://doi.org/10.3390/math6030036 - 02 Mar 2018

Cited by 7 | Viewed by 3574

Abstract

Migration is a key ecological process that enables connections between spatially separated populations. Previous studies have indicated that migration can stabilize chaotic ecosystems. However, the role of migration for two similar types of ecosystems, one chaotic and the other stable, has not yet [...] Read more.

Migration is a key ecological process that enables connections between spatially separated populations. Previous studies have indicated that migration can stabilize chaotic ecosystems. However, the role of migration for two similar types of ecosystems, one chaotic and the other stable, has not yet been studied properly. In the present paper, we investigate the stability of ecological systems that are spatially separated but connected through migration. We consider two similar types of ecosystems that are coupled through migration, where one system shows chaotic dynamics, and other shows stable dynamics. We also note that the direction of the migration is bi-directional and is regulated by the population densities. We propose and analyze the coupled system. We also apply our proposed scheme to three different models. Our results suggest that bi-directional migration makes the coupled system more regular. We have performed numerical simulations to illustrate the dynamics of the coupled systems. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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12 pages, 790 KiB

Open AccessArticle

The Collapse of Ecosystem Engineer Populations

by José F. Fontanari

Mathematics 2018, 6(1), 9; https://doi.org/10.3390/math6010009 - 12 Jan 2018

Cited by 5 | Viewed by 5890

Abstract

Humans are the ultimate ecosystem engineers who have profoundly transformed the world’s landscapes in order to enhance their survival. Somewhat paradoxically, however, sometimes the unforeseen effect of this ecosystem engineering is the very collapse of the population it intended to protect. Here we [...] Read more.

Humans are the ultimate ecosystem engineers who have profoundly transformed the world’s landscapes in order to enhance their survival. Somewhat paradoxically, however, sometimes the unforeseen effect of this ecosystem engineering is the very collapse of the population it intended to protect. Here we use a spatial version of a standard population dynamics model of ecosystem engineers to study the colonization of unexplored virgin territories by a small settlement of engineers. We find that during the expansion phase the population density reaches values much higher than those the environment can support in the equilibrium situation. When the colonization front reaches the boundary of the available space, the population density plunges sharply and attains its equilibrium value. The collapse takes place without warning and happens just after the population reaches its peak number. We conclude that overpopulation and the consequent collapse of an expanding population of ecosystem engineers is a natural consequence of the nonlinear feedback between the population and environment variables. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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695 KiB

Open AccessArticle

Impact of Parameter Variability and Environmental Noise on the Klausmeier Model of Vegetation Pattern Formation

by Merlin C. Köhnke and Horst Malchow

Mathematics 2017, 5(4), 69; https://doi.org/10.3390/math5040069 - 23 Nov 2017

Cited by 4 | Viewed by 4075

Abstract

Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential [...] Read more.

Semi-arid ecosystems made up of patterned vegetation, for instance, are thought to be highly sensitive. This highlights the importance of understanding the dynamics of the formation of vegetation patterns. The most renowned mathematical model describing such pattern formation consists of two partial differential equations and is often referred to as the Klausmeier model. This paper provides analytical and numerical investigations regarding the influence of different parameters, including the so-far not contemplated evaporation, on the long-term model results. Another focus is set on the influence of different initial conditions and on environmental noise, which has been added to the model. It is shown that patterning is beneficial for semi-arid ecosystems, that is, vegetation is present for a broader parameter range. Both parameter variability and environmental noise have only minor impacts on the model results. Increasing mortality has a high, nonlinear impact underlining the importance of further studies in order to gain a sufficient understanding allowing for suitable management strategies of this natural phenomenon. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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Review

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28 pages, 614 KiB

Open AccessReview

Ecological Diversity: Measuring the Unmeasurable

by Aisling J. Daly, Jan M. Baetens and Bernard De Baets

Mathematics 2018, 6(7), 119; https://doi.org/10.3390/math6070119 - 10 Jul 2018

Cited by 163 | Viewed by 20047

Abstract

Diversity is a concept central to ecology, and its measurement is essential for any study of ecosystem health. But summarizing this complex and multidimensional concept in a single measure is problematic. Dozens of mathematical indices have been proposed for this purpose, but these [...] Read more.

Diversity is a concept central to ecology, and its measurement is essential for any study of ecosystem health. But summarizing this complex and multidimensional concept in a single measure is problematic. Dozens of mathematical indices have been proposed for this purpose, but these can provide contradictory results leading to misleading or incorrect conclusions about a community’s diversity. In this review, we summarize the key conceptual issues underlying the measurement of ecological diversity, survey the indices most commonly used in ecology, and discuss their relative suitability. We advocate for indices that: (i) satisfy key mathematical axioms; (ii) can be expressed as so-called effective numbers; (iii) can be extended to account for disparity between types; (iv) can be parameterized to obtain diversity profiles; and (v) for which an estimator (preferably unbiased) can be found so that the index is useful for practical applications. Full article

(This article belongs to the Special Issue Progress in Mathematical Ecology)

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