Spatio-temporal pattern formation in predator-prey systems with fitness taxis

Irene T. Heilmann, Uffe Høgsbro Thygesen, Mads Peter Sørensen*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

Abstract

We pose a spatial predator–prey model in which the movement of animals is not purely diffusive, but also contains a drift term in the direction of higher specific growth rates. We refer to this as fitness taxis. We conduct a linear stability analysis of the resulting coupled reaction–advection–diffusion equations and derive conditions under which spatial patterns form. We find that for some parameters the problem is ill posed and short waves grow with unbounded speeds. To eliminate this, we introduce spatial kernels in the model, yielding coupled integro-differential equations, and conduct a similar stability analysis for this system. Through numerical simulation, we find that a variety of patterns can emerge, including stationary spatial patterns, standing and travelling waves, and seemingly chaotic spatio-temporal patterns. We argue that fitness taxis represents a simple and generic extension of diffusive motion, is ecologically plausible, and provides an alternative mechanism for formation of patterns in spatially explicit ecosystem models, with emphasis on non-stationary spatio-temporal dynamics.

Original languageEnglish
JournalEcological Complexity
Volume34
Pages (from-to)44-57
ISSN1476-945X
DOIs
Publication statusPublished - 1 May 2018

Keywords

  • Cross-diffusion
  • Fitness taxis
  • Pattern formation
  • Predator–prey systems

Cite this

@article{7bfa3d09998f4447b690229a58dba630,
title = "Spatio-temporal pattern formation in predator-prey systems with fitness taxis",
abstract = "We pose a spatial predator–prey model in which the movement of animals is not purely diffusive, but also contains a drift term in the direction of higher specific growth rates. We refer to this as fitness taxis. We conduct a linear stability analysis of the resulting coupled reaction–advection–diffusion equations and derive conditions under which spatial patterns form. We find that for some parameters the problem is ill posed and short waves grow with unbounded speeds. To eliminate this, we introduce spatial kernels in the model, yielding coupled integro-differential equations, and conduct a similar stability analysis for this system. Through numerical simulation, we find that a variety of patterns can emerge, including stationary spatial patterns, standing and travelling waves, and seemingly chaotic spatio-temporal patterns. We argue that fitness taxis represents a simple and generic extension of diffusive motion, is ecologically plausible, and provides an alternative mechanism for formation of patterns in spatially explicit ecosystem models, with emphasis on non-stationary spatio-temporal dynamics.",
keywords = "Cross-diffusion, Fitness taxis, Pattern formation, Predator–prey systems",
author = "Heilmann, {Irene T.} and Thygesen, {Uffe H{\o}gsbro} and S{\o}rensen, {Mads Peter}",
year = "2018",
month = "5",
day = "1",
doi = "10.1016/j.ecocom.2018.04.003",
language = "English",
volume = "34",
pages = "44--57",
journal = "Ecological Complexity: An International Journal on Biocomplexity in the Environment and Theoretical Ecology",
issn = "1476-945X",
publisher = "Elsevier",

}

Spatio-temporal pattern formation in predator-prey systems with fitness taxis. / Heilmann, Irene T.; Thygesen, Uffe Høgsbro; Sørensen, Mads Peter.

In: Ecological Complexity, Vol. 34, 01.05.2018, p. 44-57.

Research output: Contribution to journalJournal articleResearchpeer-review

TY - JOUR

T1 - Spatio-temporal pattern formation in predator-prey systems with fitness taxis

AU - Heilmann, Irene T.

AU - Thygesen, Uffe Høgsbro

AU - Sørensen, Mads Peter

PY - 2018/5/1

Y1 - 2018/5/1

N2 - We pose a spatial predator–prey model in which the movement of animals is not purely diffusive, but also contains a drift term in the direction of higher specific growth rates. We refer to this as fitness taxis. We conduct a linear stability analysis of the resulting coupled reaction–advection–diffusion equations and derive conditions under which spatial patterns form. We find that for some parameters the problem is ill posed and short waves grow with unbounded speeds. To eliminate this, we introduce spatial kernels in the model, yielding coupled integro-differential equations, and conduct a similar stability analysis for this system. Through numerical simulation, we find that a variety of patterns can emerge, including stationary spatial patterns, standing and travelling waves, and seemingly chaotic spatio-temporal patterns. We argue that fitness taxis represents a simple and generic extension of diffusive motion, is ecologically plausible, and provides an alternative mechanism for formation of patterns in spatially explicit ecosystem models, with emphasis on non-stationary spatio-temporal dynamics.

AB - We pose a spatial predator–prey model in which the movement of animals is not purely diffusive, but also contains a drift term in the direction of higher specific growth rates. We refer to this as fitness taxis. We conduct a linear stability analysis of the resulting coupled reaction–advection–diffusion equations and derive conditions under which spatial patterns form. We find that for some parameters the problem is ill posed and short waves grow with unbounded speeds. To eliminate this, we introduce spatial kernels in the model, yielding coupled integro-differential equations, and conduct a similar stability analysis for this system. Through numerical simulation, we find that a variety of patterns can emerge, including stationary spatial patterns, standing and travelling waves, and seemingly chaotic spatio-temporal patterns. We argue that fitness taxis represents a simple and generic extension of diffusive motion, is ecologically plausible, and provides an alternative mechanism for formation of patterns in spatially explicit ecosystem models, with emphasis on non-stationary spatio-temporal dynamics.

KW - Cross-diffusion

KW - Fitness taxis

KW - Pattern formation

KW - Predator–prey systems

U2 - 10.1016/j.ecocom.2018.04.003

DO - 10.1016/j.ecocom.2018.04.003

M3 - Journal article

VL - 34

SP - 44

EP - 57

JO - Ecological Complexity: An International Journal on Biocomplexity in the Environment and Theoretical Ecology

JF - Ecological Complexity: An International Journal on Biocomplexity in the Environment and Theoretical Ecology

SN - 1476-945X

ER -