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Bifurcation structure of coexistence states for a prey–predator model with large population flux by attractive transition

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  23 August 2021

Kousuke Kuto Open the ORCID record for Kousuke Kuto [Opens in a new window]  and
Kazuhiro Oeda
Kousuke Kuto
Affiliation:
Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan ([email protected])
Kazuhiro Oeda
Affiliation:
Center for Fundamental Education, Kyushu Sangyo University, 2-3-1 Matsukadai, Higashi-ku, Fukuoka 813-8503, Japan ([email protected])
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Abstract

This paper is concerned with a prey–predator model with population flux by attractive transition. Our previous paper (Oeda and Kuto, 2018, Nonlinear Anal. RWA, 44, 589–615) obtained a bifurcation branch (connected set) of coexistence steady states which connects two semitrivial solutions. In Oeda and Kuto (2018, Nonlinear Anal. RWA, 44, 589–615), we also showed that any positive steady-state approaches a positive solution of either of two limiting systems, and moreover, one of the limiting systems is an equal diffusive competition model. This paper obtains the bifurcation structure of positive solutions to the other limiting system. Moreover, this paper implies that the global bifurcation branch of coexistence states consists of two parts, one of which is a simple curve running in a tubular domain near the set of positive solutions to the equal diffusive competition model, the other of which is a connected set characterized by positive solutions to the other limiting system.

Keywords

Asymptotic behaviourattractive transitional fluxbifurcation analysiscoexistence steady statesLyapunov–Schmidt reductionprey–predator model

MSC classification

Primary: 35J57: Boundary value problems for second-order elliptic systems
Secondary: 35J61: Semilinear elliptic equations 35B32: Bifurcation 92D25: Population dynamics (general) 35B20: Perturbations 35B09: Positive solutions
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 4 , August 2022 , pp. 965 - 988
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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References

Blat, J. and Brown, K. J.. Bifurcation of steady state solutions in predator-prey and competition systems. Proc. Roy. Soc. Edin. 97 A (1984), 21–34.CrossRefGoogle Scholar
Blat, J. and Brown, K. J.. Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal. 17 (1986), 1339–1353.CrossRefGoogle Scholar
Crandall, M. G. and Rabinowitz, P. H.. Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (1971), 321–340.CrossRefGoogle Scholar
Crandall, M. G. and Rabinowitz, P. H.. Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rational Mech. Anal. 52 (1972), 161–180.CrossRefGoogle Scholar
Dancer, E. N.. On positive solutions of some pairs of differential equations, II. J. Diff. Equ. 60 (1985), 236–258.CrossRefGoogle Scholar
Dancer, E. N., López-Gómez, J. and Ortega, R.. On the spectrum of some linear noncooperative weakly coupled elliptic systems. Diff. Int. Equ. 8 (1995), 515–523.Google Scholar
Gilbarg, D. and Trudinger, N. S.. Elliptic partial differential equations of second order, Second edition (Berlin: Springer-Verlag, 1983).Google Scholar
Gui, C. and Lou, Y.. Uniqueness and nonuniqueness of coexistence states in the Lotka–Volterra competition model. Commun. Pure Appl. Math. 47 (1994), 1571–1594.CrossRefGoogle Scholar
Kan-on, Y.. Stability of singularly perturbed solutions to nonlinear diffusion systems arising in population dynamics. Hiroshima Math. J. 28 (1993), 509–536.Google Scholar
Kan-on, Y.. On the structure of positive solutions for the Shigesada–Kawasaki–Teramoto model with large interspecific competition rate. Int. J. Bifur. Chaos Appl. Sci. Eng. 30, 2050001 (2020) (9p.).CrossRefGoogle Scholar
Kan-on, Y.. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete Contin. Dyn. Syst. A 40 (2020), 3561–3570.CrossRefGoogle Scholar
Kuto, K.. Stability of steady-state solutions to a prey–predator system with cross-diffusion. J. Diff. Equ. 197 (2004), 297–314.Google Scholar
Kuto, K.. Bifurcation branch of stationary solutions for a Lotka–Volterra cross-diffusion system. Nonlinear Anal. RWA 10 (2009), 943–965.CrossRefGoogle Scholar
Kuto, K.. Stability and Hopf bifurcation of coexistence steady-states to an SKT model in spatially heterogeneous environment. Discrete Contin. Dyn. Syst. A 24 (2009), 489–509.CrossRefGoogle Scholar
Kuto, K.. Limiting structure of shrinking solutions to the stationary Shigesada–Kawasaki–Teramoto model with large cross-diffusion. SIAM J. Math. Anal. 47 (2015), 3993–4024.CrossRefGoogle Scholar
Kuto, K. and Tsujikawa, T.. Limiting structure of steady-states to the Lotka–Volterra competition model with large diffusion and advection. J. Diff. Equ. 258 (2015) 1801–1858.CrossRefGoogle Scholar
Kuto, K. and Yamada, Y.. Multiple coexistence states for a prey–predator system with cross-diffusion. J. Diff. Equ. 197 (2004), 315–348.CrossRefGoogle Scholar
Kuto, K. and Yamada, Y.. Positive solutions for Lotka–Volterra competition systems with large cross-diffusion. Appl. Anal. 89 (2010), 1037–1066.CrossRefGoogle Scholar
Li, L.. On the uniqueness and ordering of steady-states of predator-prey systems. Proc. Roy. Soc. Edin. A 110 (1989), 295–303.CrossRefGoogle Scholar
Li, Q. and Wu, Y.. Stability analysis on a type of steady state for the SKT competition model with large cross diffusion. J. Math. Anal. Appl. 462 (2018), 1048–1078.CrossRefGoogle Scholar
Li, Q. and Wu, Y.. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete Contin. Dyn. Syst. A 40 (2020), 3657–3682.CrossRefGoogle Scholar
López-Gómez, J.. Spectral theory and nonlinear functional analysis, Research Notes in Mathematics Series, vol. 426 (Boca Raton, FL: CRC Press, 2001).CrossRefGoogle Scholar
López-Gómez, J. and Pardo, R.. Coexistence regions in Lotka–Volterra models with diffusion. Nonlinear Anal. TMA 19 (1992), 11–28.CrossRefGoogle Scholar
López-Gómez, J. and Pardo, R.. Existence and uniqueness of coexistence states for the predator-prey model with diffusion: the scalar case. Diff. Integral Equ. 6 (1993), 1025–1031.Google Scholar
Lou, Y. and Ni, W.-M.. Diffusion, self-diffusion and cross-diffusion. J. Diff. Equ. 131 (1996), 79–131.CrossRefGoogle Scholar
Lou, Y. and Ni, W.-M.. Diffusion vs cross-diffusion: an elliptic approach. J. Diff. Equ. 154 (1999), 157–190.CrossRefGoogle Scholar
Lou, Y., Ni, W.-M and Yotsutani, S.. On a limiting system in the Lotka–Volterra competition with cross-diffusion. Discrete Contin. Dyn. Syst. A 10 (2004), 435–458.CrossRefGoogle Scholar
Lou, Y., Ni, W.-M and Yotsutani, S.. Pattern formation in a cross-diffusion system. Discrete Contin. Dyn. Syst. A 35 (2015), 1589–1607.CrossRefGoogle Scholar
Mimura, M.. Stationary pattern of some density-dependent diffusion system with competitive dynamics. J. Hiroshima Math 11 (1981), 621–635.CrossRefGoogle Scholar
Mimura, M. and Kawasaki, K.. Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9 (1980), 49–64.CrossRefGoogle Scholar
Mimura, M., Nishiura, Y., Tesei, A. and Tsujikawa, T.. Coexistence problem for two competing species models with density-dependent diffusion. J. Hiroshima Math. 14 (1984), 425–449.CrossRefGoogle Scholar
Oeda, K. and Kuto, K.. Positive steady states for a prey–predator model with population flux by attractive transition. Nonlinear Anal. RWA 44 (2018), 589–615.CrossRefGoogle Scholar
Okubo, A. and Levin, S. A.. Diffusion and ecological problems: modern perspective, Second edition. Interdisciplinary Applied Mathematics, vol. 14 (New York: Springer-Verlag, 2001).CrossRefGoogle Scholar
Ryu, K. and Ahn, I.. Positive steady-states for two interacting species models with linear self-cross diffusions. Discrete Contin. Dyn. Syst. A 9 (2003), 1049–1061.CrossRefGoogle Scholar
Ryu, K. and Ahn, I.. Coexistence theorem of steady states for nonlinear self-cross diffusion systems with competitive dynamics. J. Math. Anal. Appl. 283 (2003), 46–65.CrossRefGoogle Scholar
Shi, J. and Wang, X.. On global bifurcation for quasilinear elliptic systems on bounded domains. J. Diff. Equ. 246 (2009), 2788–2812.CrossRefGoogle Scholar
Shigesada, N., Kawasaki, K. and Teramoto, E.. Spatial segregation of interacting species. J. Theor. Biol. 79 (1979), 83–99.CrossRefGoogle ScholarPubMed
Wang, Q., Gai, C. and Yan, J.. Qualitative analysis of a Lotka–Volterra competition system with advection. Discrete Contin. Dyn. Syst. A. 35 (2015), 1239–1284.CrossRefGoogle Scholar
Wang, J., Wu, S. and Shi, J.. Pattern formation in diffusive predator-prey systems with predator-taxis and prey-taxis. Discrete Contin. Dyn. Syst. B. 26 (2021), 1273–1289.Google Scholar