The effect of landscape fragmentation on Turing-pattern formation

Nazanin Zaker; Christina A. Cobbold; Frithjof Lutscher; Nazanin Zaker; Christina A. Cobbold; Frithjof Lutscher

doi:10.3934/mbe.2022116

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 3: 2506-2537. doi: 10.3934/mbe.2022116

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The effect of landscape fragmentation on Turing-pattern formation

1.
Department of Mathematics and Statistics, University of Ottawa, Ottawa, Canada
2.
School of Mathematics and Statistics, University of Glasgow, Glasgow, UK
3.
Department of Mathematics and Statistics and Department of Biology, University of Ottawa, Ottawa, Canada

Academic Editor:Yang Kuang

Received: 11 October 2021 Revised: 13 December 2021 Accepted: 20 December 2021 Published: 07 January 2022

Diffusion-driven instability and Turing pattern formation are a well-known mechanism by which the local interaction of species, combined with random spatial movement, can generate stable patterns of population densities in the absence of spatial heterogeneity of the underlying medium. Some examples of such patterns exist in ecological interactions between predator and prey, but the conditions required for these patterns are not easily satisfied in ecological systems. At the same time, most ecological systems exist in heterogeneous landscapes, and landscape heterogeneity can affect species interactions and individual movement behavior. In this work, we explore whether and how landscape heterogeneity might facilitate Turing pattern formation in predator–prey interactions. We formulate reaction-diffusion equations for two interacting species on an infinite patchy landscape, consisting of two types of periodically alternating patches. Population dynamics and movement behavior differ between patch types, and individuals may have a preference for one of the two habitat types. We apply homogenization theory to derive an appropriately averaged model, to which we apply stability analysis for Turing patterns. We then study three scenarios in detail and find mechanisms by which diffusion-driven instabilities may arise even if the local interaction and movement rates do not indicate it.
- reaction-diffusion system,
- Turing pattern formation,
- diffusion-driven instability,
- interface conditions,
- population dynamics
Citation: Nazanin Zaker, Christina A. Cobbold, Frithjof Lutscher. The effect of landscape fragmentation on Turing-pattern formation[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2506-2537. doi: 10.3934/mbe.2022116

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Abstract

Diffusion-driven instability and Turing pattern formation are a well-known mechanism by which the local interaction of species, combined with random spatial movement, can generate stable patterns of population densities in the absence of spatial heterogeneity of the underlying medium. Some examples of such patterns exist in ecological interactions between predator and prey, but the conditions required for these patterns are not easily satisfied in ecological systems. At the same time, most ecological systems exist in heterogeneous landscapes, and landscape heterogeneity can affect species interactions and individual movement behavior. In this work, we explore whether and how landscape heterogeneity might facilitate Turing pattern formation in predator–prey interactions. We formulate reaction-diffusion equations for two interacting species on an infinite patchy landscape, consisting of two types of periodically alternating patches. Population dynamics and movement behavior differ between patch types, and individuals may have a preference for one of the two habitat types. We apply homogenization theory to derive an appropriately averaged model, to which we apply stability analysis for Turing patterns. We then study three scenarios in detail and find mechanisms by which diffusion-driven instabilities may arise even if the local interaction and movement rates do not indicate it.

References

[1]	A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. R. Soc., B, 237 (1952), 37–72. https://doi.org/10.1098/rstb.1952.0012. doi: 10.1098/rstb.1952.0012
[2]	J. D. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, Springer, New York, 2001.
[3]	L. E. Keshet, Mathematical Models in Biology. SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 2005. https://doi.org/10.1137/1.9780898719147.
[4]	A. Okubo, S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, 2001. https://doi.org/10.1007/978-1-4757-4978-6.
[5]	M. Rietkerk, S. C. Dekker, P. C. D. Ruiter, J. V. D. Koppel, Self-organized patchiness and catastrophic shifts in ecosystems, Science, 305 (2004), 1926–1929. https://doi.org/10.1126/science.1101867. doi: 10.1126/science.1101867
[6]	M. Rietkerk, J. V. D. Koppel, Regular pattern formation in real ecosystems, Trends Ecol. Evol., 23 (2008), 169–175. https://doi.org/10.1016/j.tree.2007.10.013. doi: 10.1016/j.tree.2007.10.013
[7]	L. Segel, J. Jackson, Dissipative structure: an explanation and an ecological example, J. Theor. Biol., 37 (1972), 545–559. https://doi.org/10.1016/0022-5193(72)90090-2. doi: 10.1016/0022-5193(72)90090-2
[8]	S. Levin, L. Segel, Hypothesis for origin of planktonic patchiness, Nature, 259 (1976). https://doi.org/10.1038/259659a0. doi: 10.1038/259659a0
[9]	J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Anim. Ecol., 44 (1975), 331–340. https://doi.org/10.2307/3866. doi: 10.2307/3866
[10]	R. Arditi, L. R. Ginzburg, Coupling in predator–prey dynamics: ratio dependence, J. Theor, Biol., 139 (1989), 311–326. https://doi.org/10.1016/S0022-5193(89)80211-5. doi: 10.1016/S0022-5193(89)80211-5
[11]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton University Press, 1974.
[12]	D. Alonso, F. Bartumeus, J. Catalan, Mutual interference between predators can give rise to Turing spatial patterns, Ecology, 83 (2002), 28–34. https://doi.org/10.1890/0012-9658(2002)083[0028:MIBPCG]2.0.CO;2. doi: 10.1890/0012-9658(2002)083[0028:MIBPCG]2.0.CO;2
[13]	B. Mukhopadhyay, R. Bhattacharyya, Modeling the role of diffusion coefficients on Turing instability in a reaction-diffusion predator–prey system, Bull. Math. Biol., 68 (2006), 293–313. https://doi.org/10.1007/s11538-005-9007-2. doi: 10.1007/s11538-005-9007-2
[14]	W. Wang, L. Zhang, Y. Xue, Z. Jin, Spatiotemporal pattern formation of Beddington–Deangelis-type predator–prey model, preprint, arXiv: 0801.0797v1.
[15]	S. Fasani, S. Rinaldi, Factors promoting or inhibiting Turing instability in spatially extended prey–predator systems, Ecol. Modell., 222 (2011), 3449–3452. https://doi.org/10.1016/j.ecolmodel.2011.07.002. doi: 10.1016/j.ecolmodel.2011.07.002
[16]	C. S. Holling, The components of predation as revealed by a study of small-mammal predation of the European pine sawfly, Can. Entomol., 91 (1959), 293–320. https://doi.org/10.4039/Ent91293-5. doi: 10.4039/Ent91293-5
[17]	C. S. Holling, Some characteristics of simple types of predation and parasitism, Can. Entomol., 91 (1959), 385–398. https://doi.org/10.4039/Ent91385-7. doi: 10.4039/Ent91385-7
[18]	C. A. Cobbold, F. Lutscher, J. A. Sherratt, Diffusion-driven instabilities and emerging spatial patterns in patchy landscapes, Ecol. Complexity, 24 (2015), 69–81. https://doi.org/10.1016/j.ecocom.2015.10.001. doi: 10.1016/j.ecocom.2015.10.001
[19]	N. Zaker, Population dynamics in patchy landscapes: steady states and pattern formation, PhD thesis, University of Ottawa, 2021.
[20]	D. L. Benson, P. K. Maini, J. A. Sherratt, Diffusion driven instability in an inhomogeneous domain, Bull. Math. Biol., 55 (1993), 365–384. https://doi.org/10.1016/S0092-8240(05)80270-8. doi: 10.1016/S0092-8240(05)80270-8
[21]	D. L. Benson, P. K. Maini, J. A. Sherratt, Analysis of pattern formation in reaction-diffusion models with spatially inhomogeneous coefficients, Math. Comput. Modell., 17 (1993), 29–34. https://doi.org/10.1016/0895-7177(93)90025-T. doi: 10.1016/0895-7177(93)90025-T
[22]	K. Page, P. K. Maini, N. A. M. Monk, Pattern formation in spatially heterogeneous Turing reaction-diffusion models, Phys. D, 181 (2003), 80–101. https://doi.org/10.1016/S0167-2789(03)00068-X. doi: 10.1016/S0167-2789(03)00068-X
[23]	K. Page, P. K. Maini, N. A. M. Monk, Complex pattern formation in reaction-diffusion systems with spatially varying patterns, Phys. D, 202 (2005), 95–115. https://doi.org/10.1016/j.physd.2005.01.022. doi: 10.1016/j.physd.2005.01.022
[24]	M. Kozák, E. A. Gaffney, V. Klika, Pattern formation in reaction-diffusion systems with piecewise kinetic modulation: an example study of heterogeneous kinetics, Phys. Rev. E, 100 (2019), 042220. https://doi.org/10.1103/PhysRevE.100.042220. doi: 10.1103/PhysRevE.100.042220
[25]	E. Sheffer, J. V. Hardenberg, H. Yizhaq, M. Shachak, E. Meron, Emerged or imposed: a theory on the role of physical templates and self-organisation for vegetation patchiness, Ecol. Lett., 16 (2013), 127–139. https://doi.org/10.1111/ele.12027. doi: 10.1111/ele.12027
[26]	O. Ovaskainen, S. Cornell, Biased movement at a boundary and conditional occupancy times for diffusion processes, J. Appl. Probab., 40 (2003), 557–580. https://doi.org/10.1239/jap/1059060888. doi: 10.1239/jap/1059060888
[27]	G. A. Maciel, F. Lutscher, How individual movement response to habitat edge affects population persistence and spatial spread, Am. Nat., 182 (2013), 42–52. https://doi.org/10.1086/670661. doi: 10.1086/670661
[28]	B. Yurk, C. A. Cobbold, Homogenization techniques for population dynamics in strongly heterogeneous landscapes, J. Biol. Dyn., 12 (2018), 171–193. https://doi.org/10.1080/17513758.2017.1410238. doi: 10.1080/17513758.2017.1410238
[29]	C. A. Cobbold, F. Lutscher, B. Yurk, Bridging the scale gap: predicting large-scale population dynamics from small-scale variation in strongly heterogeneous landscapes, Methods Ecol. Evol., 2021. https://doi.org/10.1111/2041-210X.13799. doi: 10.1111/2041-210X.13799
[30]	G. Maciel, C. Cosner, R. S. Cantrell, F. Lutscher, Evolutionarily stable movement strategies in reaction–diffusion models with edge behavior, J. Math. Biol., 80 (2020), 61–92. https://doi.org/10.1007/s00285-019-01339-2. doi: 10.1007/s00285-019-01339-2
[31]	Y. Alqawasmeh, F. Lutscher, Persistence and spread of stage-structured populations in heterogeneous landscapes, J. Math. Biol., 78 (2019), 1485–1527. https://doi.org/10.1007/s00285-018-1317-8. doi: 10.1007/s00285-018-1317-8
[32]	P. Turchin, Quantitative Analysis of Movement: Measuring and Modeling Population Redistribution of Plants and Animals, Sinauer Associates, Sunderland, 1998.
[33]	P. Turchin, Complex Population Dynamics, Princeton University Press, 2001.
[34]	E. E. Crone, L. M. Brown, J. A. Hodgson, F. Lutscher, C. B. Schultz, Faster movement in nonhabitat matrix promotes range shifts in heterogeneous landscapes, Ecology, 100 (2019), e02701. https://doi.org/10.1002/ecy.2701. doi: 10.1002/ecy.2701
[35]	J. C. Strikwerda, Finite difference schemes and partial differential equations, SIAM: Society for Industrial and Applied Mathematics, Philadelphia, 2004. https://doi.org/10.1137/1.9780898717938.
[36]	A. L. Krause, V. Klika, T. E. Woolley, E. A. Gaffney, From one pattern into another: analysis of Turing patterns in heterogeneous domains via WKBJ, J. R. Soc. Interface, 17 (2020), 20190621. https://doi.org/10.1098/rsif.2019.0621. doi: 10.1098/rsif.2019.0621
[37]	M. J. Garlick, J. A. Powell, M. B. Hooten, L. R. McFarlane, Homogenization of large-scale movement models in ecology, Bull. Math. Biol., 73 (2011), 2088–2108. https://doi.org/10.1007/s11538-010-9612-6. doi: 10.1007/s11538-010-9612-6
[38]	B. Yurk, Homogenization of a directed dispersal model for animal movement in a heterogeneous environment, Bull. Math. Biol., 78 (2016), 2034–2056. https://doi.org/10.1007/s11538-016-0210-0. doi: 10.1007/s11538-016-0210-0

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