Special Issue "Partial Differential Equations in Ecology: Recent Advances and New Challenges"

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

Deadline for manuscript submissions: closed (31 July 2021).

Special Issue Editor

Prof. Dr. Sergei Petrovskii
E-Mail Website
Guest Editor
School of Mathematics and Actuarial Science, University of Leicester, UK
Interests: mathematical ecology; climate change; mass extinctions; nonlinear dynamics
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Special Issue Information

Dear Colleagues,

The application of partial differential equations (PDEs) in ecology has an 80-year long history, dating back to seminal works by Fisher (1937) and Kolmogorov et al. (1937) where nonlinear population waves were discovered and studied. A few decades later, Alan Turing’s (1953) discovery of diffusive instability was applied to ecological pattern formation, which led to an upsurge of research on all aspects of the population dynamics in space and time using PDEs of diffusion-reaction type. Today, at appropriate spatial and temporal scales, PDEs remain a powerful modeling tool widely used to shed new light on some old problems and provide insights into new ones. This Special Issue will highlight recent advances in the application of PDE-based models in ecology and population dynamics. We welcome papers where traditional diffusion-reaction models are applied to problems of ecological significance. We especially welcome papers where the PDE framework is extended beyond the standard diffusion-reaction paradigm, e.g., to include cross-diffusion, nonlocal effects, time-delay, etc. Both analytical studies and simulation-based studies will be considered.

Prof. Dr. Sergei Petrovskii
Guest Editor

Manuscript Submission Information

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Keywords

  • Population dynamics
  • Pattern formation
  • Biological invasions
  • Population waves
  • Extinctions

Published Papers (1 paper)

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Research

Article
A Multispecies Cross-Diffusion Model for Territorial Development
Mathematics 2021, 9(12), 1428; https://doi.org/10.3390/math9121428 - 19 Jun 2021
Viewed by 343
Abstract
We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model [...] Read more.
We develop an agent-based model on a lattice to investigate territorial development motivated by markings such as graffiti, generalizing a previously-published model to account for K groups instead of two groups. We then analyze this model and present two novel variations. Our model assumes that agents’ movement is a biased random walk away from rival groups’ markings. All interactions between agents are indirect, mediated through the markings. We numerically demonstrate that in a system of three groups, the groups segregate in certain parameter regimes. Starting from the discrete model, we formally derive the continuum system of 2K convection–diffusion equations for our model. These equations exhibit cross-diffusion due to the avoidance of the rival groups’ markings. Both through numerical simulations and through a linear stability analysis of the continuum system, we find that many of the same properties hold for the K-group model as for the two-group model. We then introduce two novel variations of the agent-based model, one corresponding to some groups being more timid than others, and the other corresponding to some groups being more threatening than others. These variations present different territorial patterns than those found in the original model. We derive corresponding systems of convection–diffusion equations for each of these variations, finding both numerically and through linear stability analysis that each variation exhibits a phase transition. Full article
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