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Partial Differential Equations in Ecology: 80 Years and Counting

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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: closed (31 January 2020) | Viewed by 42341

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Prof. Dr. Sergei Petrovskii

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School of Mathematics and Actuarial Science, University of Leicester, Leicester, UK
Interests: mathematical ecology; climate change; mass extinctions; nonlinear dynamics
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Dear Colleagues,

The application of partial differential equations (PDEs) in ecology has a long history dating back to seminal works by Fisher (1937) and Kolmogorov et al. (1937), in which the travelling wave solutions of a scalar diffusion-reaction equation were discovered and studied. These papers laid the foundation of the mathematical theory of population travelling waves that was later applied, with great success, to biological invasions and other phenomena involving population spreading. Fifteen years later, Alan Turing’s work on chemical morphogenesis (1953) appeared to demonstrate that a system of two coupler PDEs gives rise to pattern formation due to diffusive instability. This discovery, especially after Segel & Jackson (1972) revealed its ecological context, led to an outbreak of research on all aspects of the population dynamics in space and time using PDEs of the diffusion-reaction type. The ecological significance of this was eventually examined and justified in an influential review by Holmes et al. (1994).

New times brought new challenges and new ideas. Over the last 25 years, certain limitations of the PDE-based framework were revealed and understood, and a number of alternative mathematical techniques were developed for ecological applications. However, on appropriate spatial and temporal scales, PDEs remain as fully relevant and powerful modelling tools, which are nowadays widely used both to bring new light to some old problems and to gain insight into new ones. This Special Issue aims to summarize and highlight the current role of PDE-based models in ecology and population dynamics. We welcome papers in which the traditional diffusion-reaction models are applied to clearly defined problems of ecological significance. We especially welcome papers in which the PDE framework is extended beyond the diffusion-reaction paradigm (e.g., the reaction–telegraph equation or Cahn–Hilliard equation). Of particular interest are papers in which emerging ecological problems and trends are addressed, such as the effect of the climate change. Both analytical studies and simulation-based studies will be considered.

Prof. Dr. Sergei Petrovskii
Guest Editor

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Keywords

Pattern formation
Spatiotemporal complexity and chaos
Population waves
Biological invasion
Biological control
Competition
Prey–predator interaction
Host–parasite interaction
Climate change
Landscape geometry
Habitat fragmentation
Agroecology

Published Papers (12 papers)

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Research

16 pages, 336 KiB

Open AccessArticle

Persistence for a Two-Stage Reaction-Diffusion System

by Robert Stephen Cantrell, Chris Cosner and Salomé Martínez

Mathematics 2020, 8(3), 396; https://doi.org/10.3390/math8030396 - 11 Mar 2020

Cited by 8 | Viewed by 1975

Abstract

In this article, we study how the rates of diffusion in a reaction-diffusion model for a stage structured population in a heterogeneous environment affect the model’s predictions of persistence or extinction for the population. In the case of a population without stage structure, faster diffusion is typically detrimental. In contrast to that, we find that, in a stage structured population, it can be either detrimental or helpful. If the regions where adults can reproduce are the same as those where juveniles can mature, typically slower diffusion will be favored, but if those regions are separated, then faster diffusion may be favored. Our analysis consists primarily of estimates of principal eigenvalues of the linearized system around

(0, 0)

and results on their asymptotic behavior for large or small diffusion rates. The model we study is not in general a cooperative system, but if adults only compete with other adults and juveniles with other juveniles, then it is. In that case, the general theory of cooperative systems implies that, when the model predicts persistence, it has a unique positive equilibrium. We derive some results on the asymptotic behavior of the positive equilibrium for small diffusion and for large adult reproductive rates in that case. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

13 pages, 12041 KiB

Open AccessArticle

Quantitatively Inferring Three Mechanisms from the Spatiotemporal Patterns

by Kang Zhang, Wen-Si Hu and Quan-Xing Liu

Mathematics 2020, 8(1), 112; https://doi.org/10.3390/math8010112 - 10 Jan 2020

Cited by 2 | Viewed by 2575

Abstract

Although the diversity of spatial patterns has gained extensive attention on ecosystems, it is still a challenge to discern the underlying ecological processes and mechanisms. Dynamical system models, such partial differential equations (PDEs), are some of the most widely used frameworks to unravel the spatial pattern formation, and to explore the potential ecological processes and mechanisms. Here, comparing the similarity of patterned dynamics among Allen–Cahn (AC) model, Cahn–Hilliard (CH) model, and Cahn–Hilliard with population demographics (CHPD) model, we show that integrated spatiotemporal behaviors of the structure factors, the density-fluctuation scaling, the Lifshitz–Slyozov (LS) scaling, and the saturation status are useful indicators to infer the underlying ecological processes, even though they display the indistinguishable spatial patterns. First, there is a remarkable peak of structure factors of the CH model and CHPD model, but absent in AC model. Second, both CH and CHPD models reveal a hyperuniform behavior with scaling of −2.90 and −2.60, respectively, but AC model displays a random distribution with scaling of −1.91. Third, both AC and CH display uniform LS behaviors with slightly different scaling of 0.37 and 0.32, respectively, but CHPD model has scaling of 0.19 at short-time scales and saturation at long-time scales. In sum, we provide insights into the dynamical indicators/behaviors of spatial patterns, obtained from pure spatial data and spatiotemporal related data, and a potential application to infer ecological processes. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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28 pages, 2644 KiB

Open AccessArticle

Dynamics of a Diffusive Two-Prey-One-Predator Model with Nonlocal Intra-Specific Competition for Both the Prey Species

by Kalyan Manna, Vitaly Volpert and Malay Banerjee

Mathematics 2020, 8(1), 101; https://doi.org/10.3390/math8010101 - 07 Jan 2020

Cited by 19 | Viewed by 5336

Abstract

Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey–predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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19 pages, 1149 KiB

Open AccessArticle

Modelling Population Dynamics of Social Protests in Time and Space: The Reaction-Diffusion Approach

by Sergei Petrovskii, Weam Alharbi, Abdulqader Alhomairi and Andrew Morozov

Mathematics 2020, 8(1), 78; https://doi.org/10.3390/math8010078 - 03 Jan 2020

Cited by 5 | Viewed by 3841

Abstract

Understanding of the dynamics of riots, protests, and social unrest more generally is important in order to ensure a stable, sustainable development of various social groups, as well as the society as a whole. Mathematical models of social dynamics have been increasingly recognized as a powerful research tool to facilitate the progress in this field. However, the question as to what should be an adequate mathematical framework to describe the corresponding social processes is largely open. In particular, a great majority of the previous studies dealt with non-spatial or spatially implicit systems, but the literature dealing with spatial systems remains meagre. Meanwhile, in many cases, the dynamics of social protests has a clear spatial aspect. In this paper, we attempt to close this gap partially by considering a spatial extension of a few recently developed models of social protests. We show that even a straightforward spatial extension immediately bring new dynamical behaviours, in particular predicting a new scenario of the protests’ termination. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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12 pages, 913 KiB

Open AccessArticle

Carrying Capacity of a Population Diffusing in a Heterogeneous Environment

by D.L. DeAngelis, Bo Zhang, Wei-Ming Ni and Yuanshi Wang

Mathematics 2020, 8(1), 49; https://doi.org/10.3390/math8010049 - 01 Jan 2020

Cited by 17 | Viewed by 4904

Abstract

The carrying capacity of the environment for a population is one of the key concepts in ecology and it is incorporated in the growth term of reaction-diffusion equations describing populations in space. Analysis of reaction-diffusion models of populations in heterogeneous space have shown that, when the maximum growth rate and carrying capacity in a logistic growth function vary in space, conditions exist for which the total population size at equilibrium (i) exceeds the total population that which would occur in the absence of diffusion and (ii) exceeds that which would occur if the system were homogeneous and the total carrying capacity, computed as the integral over the local carrying capacities, was the same in the heterogeneous and homogeneous cases. We review here work over the past few years that has explained these apparently counter-intuitive results in terms of the way input of energy or another limiting resource (e.g., a nutrient) varies across the system. We report on both mathematical analysis and laboratory experiments confirming that total population size in a heterogeneous system with diffusion can exceed that in the system without diffusion. We further report, however, that when the resource of the population in question is explicitly modeled as a coupled variable, as in a reaction-diffusion chemostat model rather than a model with logistic growth, the total population in the heterogeneous system with diffusion cannot exceed the total population size in the corresponding homogeneous system in which the total carrying capacities are the same. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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18 pages, 2209 KiB

Open AccessArticle

Pattern Formation and Bistability in a Generalist Predator-Prey Model

by Vagner Weide Rodrigues, Diomar Cristina Mistro and Luiz Alberto Díaz Rodrigues

Mathematics 2020, 8(1), 20; https://doi.org/10.3390/math8010020 - 20 Dec 2019

Cited by 16 | Viewed by 3508

Abstract

Generalist predators have several food sources and do not depend on one prey species to survive. There has been considerable attention paid by modellers to generalist predator-prey interactions in recent years. Erbach and collaborators in 2013 found a complex dynamics with bistability, limit-cycles and bifurcations in a generalist predator-prey system. In this paper we explore the spatio-temporal dynamics of a reaction-diffusion PDE model for the generalist predator-prey dynamics analyzed by Erbach and colleagues. In particular, we study the Turing and Turing-Hopf pattern formation with special attention to the regime of bistability exhibited by the local model. We derive the conditions for Turing instability and find the region of parameters for which Turing and/or Turing-Hopf instability are possible. By means of numerical simulations, we present the main types of patterns observed for parameters in the Turing domain. In the Turing-Hopf range of the parameters, we observed either stable patterns or homogeneous periodic distributions. Our findings reveal that movement can break the effect of hysteresis observed in the local dynamics, what can have important implication in pest management and species conservation. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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17 pages, 2419 KiB

Open AccessFeature PaperArticle

Using G-Functions to Investigate the Evolutionary Stability of Bacterial Quorum Sensing

by Anne Mund, Christina Kuttler and Judith Pérez-Velázquez

Mathematics 2019, 7(11), 1112; https://doi.org/10.3390/math7111112 - 15 Nov 2019

Viewed by 1984

Abstract

In ecology, G-functions can be employed to define a growth function G for a population b, which can then be universally applied to all individuals or groups

b_{i}

within this population. We can further define a strategy

v_{i}

for [...] Read more.

In ecology, G-functions can be employed to define a growth function G for a population b, which can then be universally applied to all individuals or groups

b_{i}

within this population. We can further define a strategy

v_{i}

for every group

b_{i}

. Examples for strategies include diverse behaviour such as number of offspring, habitat choice, and time of nesting for birds. In this work, we employ G-functions to investigate the evolutionary stability of the bacterial cooperation process known as quorum sensing. We employ the G-function ansatz to model both the population dynamics and the resulting evolutionary pressure in order to find evolutionary stable states. This results in a semi-linear parabolic system of equations, where cost and benefit are taken into account separately. Depending on different biological assumptions, we analyse a variety of typical model functions. These translate into different long-term scenarios for different functional responses, ranging from single-strategy states to coexistence. As a special feature, we distinguish between the production of public goods, available for all subpopulations, and private goods, from which only the producers can benefit. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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22 pages, 11618 KiB

Open AccessArticle

Continuum Modeling of Discrete Plant Communities: Why Does It Work and Why Is It Advantageous?

by Ehud Meron, Jamie J. R. Bennett, Cristian Fernandez-Oto, Omer Tzuk, Yuval R. Zelnik and Gideon Grafi

Mathematics 2019, 7(10), 987; https://doi.org/10.3390/math7100987 - 17 Oct 2019

Cited by 8 | Viewed by 3606

Abstract

Understanding ecosystem response to drier climates calls for modeling the dynamics of dryland plant populations, which are crucial determinants of ecosystem function, as they constitute the basal level of whole food webs. Two modeling approaches are widely used in population dynamics, individual (agent)-based models and continuum partial-differential-equation (PDE) models. The latter are advantageous in lending themselves to powerful methodologies of mathematical analysis, but the question of whether they are suitable to describe small discrete plant populations, as is often found in dryland ecosystems, has remained largely unaddressed. In this paper, we first draw attention to two aspects of plants that distinguish them from most other organisms—high phenotypic plasticity and dispersal of stress-tolerant seeds—and argue in favor of PDE modeling, where the state variables that describe population sizes are not discrete number densities, but rather continuous biomass densities. We then discuss a few examples that demonstrate the utility of PDE models in providing deep insights into landscape-scale behaviors, such as the onset of pattern forming instabilities, multiplicity of stable ecosystem states, regular and irregular, and the possible roles of front instabilities in reversing desertification. We briefly mention a few additional examples, and conclude by outlining the nature of the information we should and should not expect to gain from PDE model studies. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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17 pages, 575 KiB

Open AccessFeature PaperArticle

Optimal Control of a PDE Model of an Invasive Species in a River

by Rebecca Pettit and Suzanne Lenhart

Mathematics 2019, 7(10), 975; https://doi.org/10.3390/math7100975 - 15 Oct 2019

Viewed by 3750

Abstract

Managing invasive species in rivers can be assisted by appropriate adjustment of flow rates. Using a partial differential equation (PDE) model representing an invasive population in a river, we investigate controlling the water discharge rate as a management strategy. Our goal is to see how controlling the water discharge rate will affect the invasive population, and more specifically how water discharges may force the invasive population downstream. We complete the analysis of a flow control problem, which seeks to minimize the invasive population upstream while minimizing the cost of this management. Using an optimality system, consisting of our population PDE, an adjoint PDE, and corresponding optimal control characterization, we illustrate some numerical simulations in which parameters are varied to determine how far upstream the invasive population reaches. We also change the river’s cross-sectional area to investigate its impact on the optimal control. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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21 pages, 592 KiB

Open AccessArticle

Individual Variability in Dispersal and Invasion Speed

by Aled Morris, Luca Börger and Elaine Crooks

Mathematics 2019, 7(9), 795; https://doi.org/10.3390/math7090795 - 01 Sep 2019

Cited by 9 | Viewed by 3210

Abstract

We model the growth, dispersal and mutation of two phenotypes of a species using reaction–diffusion equations, focusing on the biologically realistic case of small mutation rates. Having verified that the addition of a small linear mutation term to a Lotka–Volterra system limits it to only two steady states in the case of weak competition, an unstable extinction state and a stable coexistence state, we exploit the fact that the spreading speed of the system is known to be linearly determinate to show that the spreading speed is a nonincreasing function of the mutation rate, so that greater mixing between phenotypes leads to slower propagation. We also find the ratio at which the phenotypes occur at the leading edge in the limit of vanishing mutation. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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11 pages, 501 KiB

Open AccessFeature PaperArticle

Directionally Correlated Movement Can Drive Qualitative Changes in Emergent Population Distribution Patterns

by Jonathan R. Potts

Mathematics 2019, 7(7), 640; https://doi.org/10.3390/math7070640 - 18 Jul 2019

Cited by 2 | Viewed by 2274

Abstract

A fundamental goal of ecology is to understand the spatial distribution of species. For moving animals, their location is crucially dependent on the movement mechanisms they employ to navigate the landscape. Animals across many taxa are known to exhibit directional correlation in their movement. This work explores the effect of such directional correlation on spatial pattern formation in a model of between-population taxis (i.e., movement of each population in response to the presence of the others). A telegrapher-taxis formalism is used, which generalises a previously studied diffusion-taxis system by incorporating a parameter T, measuring the characteristic time for directional persistence. The results give general criteria for determining when changes in T will drive qualitative changes in the predictions of linear pattern formation analysis for

N \geq 2

populations. As a specific example, the

N = 2

case is explored in detail, showing that directional correlation can cause one population to ‘chase’ the other across the landscape while maintaining a non-constant spatial distribution. Overall, this study demonstrates the importance of accounting for directional correlation in movement for understanding both quantitative and qualitative aspects of species distributions. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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25 pages, 3250 KiB

Open AccessArticle

Cross Diffusion Induced Turing Patterns in a Tritrophic Food Chain Model with Crowley-Martin Functional Response

by Nitu Kumari and Nishith Mohan

Mathematics 2019, 7(3), 229; https://doi.org/10.3390/math7030229 - 01 Mar 2019

Cited by 10 | Viewed by 3453

Abstract

Diffusion has long been known to induce pattern formation in predator prey systems. For certain prey-predator interaction systems, self diffusion conditions ceases to induce patterns, i.e., a non-constant positive solution does not exist, as seen from the literature. We investigate the effect of cross diffusion on the pattern formation in a tritrophic food chain model. In the formulated model, the prey interacts with the mid level predator in accordance with Holling Type II functional response and the mid and top level predator interact via Crowley-Martin functional response. We prove that the stationary uniform solution of the system is stable in the presence of diffusion when cross diffusion is absent. However, this solution is unstable in the presence of both self diffusion and cross diffusion. Using a priori analysis, we show the existence of a inhomogeneous steady state. We prove that no non-constant positive solution exists in the presence of diffusion under certain conditions, i.e., no pattern formation occurs. However, pattern formation is induced by cross diffusion because of the existence of non-constant positive solution, which is proven analytically as well as numerically. We performed extensive numerical simulations to understand Turing pattern formation for different values of self and cross diffusivity coefficients of the top level predator to validate our results. We obtained a wide range of Turing patterns induced by cross diffusion in the top population, including floral, labyrinth, hot spots, pentagonal and hexagonal Turing patterns. Full article

(This article belongs to the Special Issue Partial Differential Equations in Ecology: 80 Years and Counting)

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