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Linear and Weakly Nonlinear Stability Analyses of Turing Patterns for Diffusive Predator–Prey Systems in Freshwater Marsh Landscapes

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Abstract

We study a diffusive predator–prey model describing the interactions of small fishes and their resource base (small invertebrates) in the fluctuating freshwater marsh landscapes of the Florida Everglades. The spatial model is described by a reaction–diffusion system with Beddington–DeAngelis functional response. Uniform bound, local and global asymptotic stability of the steady state of the PDE model under the no-flux boundary conditions are discussed in details. Sufficient conditions on the Turing (diffusion-driven) instability which induces spatial patterns in the model are derived via linear analysis. Existence of one-dimensional and two-dimensional spatial Turing patterns, including rhombic and hexagonal patterns, are established by weakly nonlinear analyses. These results provide theoretical explanations and numerical simulations of spatial dynamical behaviors of the wetland ecosystems of the Florida Everglades.

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References

  • Alonson D, Bartumeus F, Catalan J (2002) Mutual interference between predators can give rise to Turing spatial patterns. Ecology 83:28–34

    Article  Google Scholar 

  • Andow DA, Kareiva PM, Levin SA, Okubo A (1990) Spread of invading organisms. Landsc Ecol 4:177–188

    Article  Google Scholar 

  • Armsworth PR, Roughgarden JE (2005) The impact of directed versus random movement on population dynamics and biodiversity patterns. Am Nat 165:449–465

    Article  Google Scholar 

  • Aronson DG, Weinberger HF (1975) Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Goldstein JA (ed) Partial differential equations and related topics. Lecture notes in mathematics, vol 446. Springer, Berlin, pp 5–49

  • Beddington JR (1975) Mutual interference between parasites or predators and its effect on searching efficiency. J Anim Ecol 44:331–340

    Article  Google Scholar 

  • Blaine TW, DeAngelis DL (1997) The interaction of spatial scale and predator-prey functional response. Ecol Model 95:319–328

    Article  Google Scholar 

  • Brown KJ, Dunne PC, Gardner RA (1981) A semilinear parabolic system arising in the theory of superconductivity. J Differ Equ 40:232–252

    Article  MathSciNet  MATH  Google Scholar 

  • Cantrell RS, Cosner C (2001) On the dynamics of predator-prey models with the Beddington–DeAngelis functional response. J Math Anal Appl 257:206–222

    Article  MathSciNet  MATH  Google Scholar 

  • Cosner C (2005) A dynamic model for the ideal-free distribution as a partial differential equation. Theor Popul Biol 67:101–108

    Article  MATH  Google Scholar 

  • DeAngelis DL, Goldstein RA, O’Neill RV (1975) A model for trophic interaction. Ecology 56:881–892

    Article  Google Scholar 

  • DeAngelis DL, Trexler JC, Loftus WF (2005) Life history tradeoffs and community dynamics of small fishes in a seasonally pulsed wetland. Can J Fish Aquat Sci 62:781–790

    Article  Google Scholar 

  • DeAngelis DL, Trexler JC, Cosner C, Obazac A, Jopp F (2010) Fish population dynamics in a seasonally varying wetland. Ecol Model 221:1131–1137

    Article  Google Scholar 

  • Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugen 7:355–369

    Article  MATH  Google Scholar 

  • Gaff H, DeAngelis DL, Gross LJ, Salinas R, Shorrosh M (2000) A dynamic landscape model for fish in the Everglades and its application to restoration. Ecol Model 127:33–52

    Article  Google Scholar 

  • Gaff H, Chick J, Trexler J, DeAngelis D, Gross L, Salinas R (2004) Evaluation of and insights from ALFISH: a spatially explicit, landscape-level simulation of fish populations in the Everglades. Hydrobiologia 520:73–87

    Article  Google Scholar 

  • Gawlik DE (2002) The effects of prey availability on the numerical response of wading birds. Ecol Monogr 72:329–346

    Article  Google Scholar 

  • Golubitsky M, Swift JK, Knoblich Z (1984) Symmetries and pattern selection in Kayleigh–Benard convection. Phys D 10:249–276

    Article  MathSciNet  MATH  Google Scholar 

  • Hwang TW (2003) Global analysis of the predator–prey system with Beddington–DeAngelis functional response. J Math Anal Appl 281:395–401

    Article  MathSciNet  MATH  Google Scholar 

  • Hwang TW (2004) Uniqueness of limit cycles of the predator–prey system with Beddington–DeAngelis functional response. J Math Anal Appl 290:113–122

    Article  MathSciNet  MATH  Google Scholar 

  • Jopp F, DeAngelis DL, Trexler JC (2010) Modeling seasonal dynamics of small fish cohorts in fluctuating freshwater marsh landscapes. Landsc Ecol 25:1041–1054

    Article  Google Scholar 

  • Kawata M, Hayashi M, Hara T (2001) Interactions between neighboring algae and snail grazing in structuring microdistribution patterns of periphyton. Oikos 92:404–416

    Article  Google Scholar 

  • Koch AJ, Meinhardt H (1994) Biological pattern formation: from basic mechanisms to complex structures. Rev Mod Phys 66:1481–1501

    Article  Google Scholar 

  • Lamberti GA, Resh VH (1983) Stream periphyton and insect herbivores: an experimental study of grazing by a caddisfly population. Ecology 64:1124–1135

    Article  Google Scholar 

  • Levin SA (1974) Dispersion and population interactions. Am Nat 108:207–228

    Article  Google Scholar 

  • Levin SA, Segel LA (1985) Pattern generation in space and aspect. SIAM Rev 27:45–67

    Article  MathSciNet  MATH  Google Scholar 

  • Loftus WF, Kushlan JA (1987) Freshwater fishes of Southern Florida. Bull Fla State Mus Biol Sci 31:147–344

    Google Scholar 

  • Loftus WF, Johnson RA, Anderson GH (1992) Ecological impacts of the reduction of groundwater levels in short-hydroperiod marshes of the Everglades. In: Stanford JA, Simons JJ (eds) Proceedings of the first international conference on ground water ecology. American Water Resources Association, pp 199–208

  • McIntosh AR, Peckarsky BL, Taylor BW (2004) Predator-induced resource heterogeneity in a stream food web. Ecology 85:2279–2290

    Article  Google Scholar 

  • Murray JD (1993) Mathematical biology. Springer, Berlin

    Book  MATH  Google Scholar 

  • Ni W-M, Tang M (2005) Turing patterns in the Lengyel–Epstein system for the CIMA reaction. Trans Am Math Soc 357:3953–3969

    Article  MathSciNet  MATH  Google Scholar 

  • Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives. Springer, New York

    Book  MATH  Google Scholar 

  • Rehage JS, Trexler JC (2006) Assessing the net effect of anthropogenic disturbance on aquatic communities in wetlands: community structure relative to distance from canals. Hydrobiologia 569:359–373

    Article  Google Scholar 

  • Rietkerk M, van de Koppel J (2008) Regular pattern formation in real ecosystems. Trends Ecol Evol 23:169–175

    Article  Google Scholar 

  • Segel LA, Jackson JL (1972) Dissipative structure: an explanation and an ecological example. J Theor Biol 37:545–549

    Article  Google Scholar 

  • Segel LA, Levin SA (1976) Application of nonlinear stability theory to the steady of the effects of diffusion on predatorprey interactions. In: Piccirelli R (ed) Selected topics in statistical mechanics and biophysics. American Institute of Physics Symposium, vol 27, pp 123–152

  • Shi HB, Ruan S (2015) Spatial, temporal and spatiotemporal patterns of diffusive predator–prey models with mutual interference. IMA J Appl Math 80(5):1534–1568

    Article  MathSciNet  MATH  Google Scholar 

  • Skellam JG (1951) Random dispersal in theoretical populations. Biometrika 38:196–218

    Article  MathSciNet  MATH  Google Scholar 

  • Trexler JC, Loftus WF, Jordan CF, Chick J, Kandl KL, McElroy TC, Bass OL (2001) Ecological scale and its implications for freshwater fishes in the Florida Everglades. In: Porter JW, Porter KG (eds) The Everglades, Florida Bay, and Coral reefs of the Florida Keys: an ecosystem sourcebook. CRC, Boca Raton, pp 153–181

    Google Scholar 

  • Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc B237:37–72

    Article  MathSciNet  Google Scholar 

  • Wollkind DJ, Manoranjan VS, Zhang L (1994) Weakly nonlinear stability analyses of prototype reaction-diffusion model equations. SIAM Rev 36:176–214

    Article  MathSciNet  MATH  Google Scholar 

  • Wollkind DJ, Stephenson LE (2000) Chemical Turing pattern formation analysis: comparison of theory with experiment. SIAM J Appl Math 61:384–431

    Article  MATH  Google Scholar 

  • Yan X-P, Zhang C-H (2014) Stability and Turing instability in a diffusive predator–prey system with Beddington–DeAngelis functional response. Nonlinear Anal Real World Appl 20:1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Yurek S, DeAngelis DL, Trexler JC, Jopp F, Donalson DD (2013) Simulating mechanisms for dispersal, production and stranding of small forage fish in temporary wetland habitats. Ecol Model 250(2013):391–401

    Article  Google Scholar 

  • Zhang X-C, Sun G-Q, Jin Z (2012) Spatial dynamics in a predator–prey model with Beddington–DeAngelis functional response. Phys Rev E 85:021924

    Article  Google Scholar 

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Acknowledgements

We would like to thank Donald L. DeAngelis for his assistance and input on this project.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xidian University, Xi’an, 710071, Shaanxi, China

    Li Zhang

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33146, USA

    Fan Zhang & Shigui Ruan

Authors
  1. Li Zhang
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  2. Fan Zhang
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  3. Shigui Ruan
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Corresponding author

Correspondence to Shigui Ruan.

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Research was partially supported by National Science Foundation (DMS-1412454).

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Zhang, L., Zhang, F. & Ruan, S. Linear and Weakly Nonlinear Stability Analyses of Turing Patterns for Diffusive Predator–Prey Systems in Freshwater Marsh Landscapes. Bull Math Biol 79, 560–593 (2017). https://doi.org/10.1007/s11538-017-0245-x

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