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Global dynamics and spatio-temporal patterns of predator–prey systems with density-dependent motion

Part of: Parabolic equations and systems Equations of mathematical physics and other areas of application Partial differential equations

Published online by Cambridge University Press:  12 August 2020

HAI-YANG JIN  and
ZHI-AN WANG Open the ORCID record for ZHI-AN WANG [Opens in a new window]
HAI-YANG JIN
Affiliation:
Department of Mathematics, South China University of Technology, Guangzhou510640, China, email: [email protected]
ZHI-AN WANG
Affiliation:
Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, email: [email protected]
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Abstract

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator–prey systems with density-dependent prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions, where the coefficients of motility (diffusiq‘dfdon) and mobility (prey-taxis) of the predator are correlated through a prey density-dependent motility function. We establish the existence of classical solutions with uniform-in time bound and the global stability of the spatially homogeneous prey-only steady states and coexistence steady states under certain conditions on parameters by constructing Lyapunov functionals. With numerical simulations, we further demonstrate that spatially homogeneous time-periodic patterns, stationary spatially inhomogeneous patterns and chaotic spatio-temporal patterns are all possible for the parameters outside the stability regime. We also find from numerical simulations that the temporal dynamics between linearised system and nonlinear systems are quite different, and the prey density-dependent motility function can trigger the pattern formation.

Keywords

Predator–prey systemprey-taxisboundednessglobal stabilityLyapunov functional

MSC classification

Primary: 35K51: Initial-boundary value problems for second-order parabolic systems
Secondary: 35A01: Existence problems 35B40: Asymptotic behavior of solutions 35K57: Reaction-diffusion equations 35Q92: PDEs in connection with biology and other natural sciences
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 4 , August 2021 , pp. 652 - 682
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Alikakos, N. D. (1979) ${L^p}$ bounds of solutions of reaction-diffusion equations. Comm. Part. Differ. Equations 4, 827–868.CrossRefGoogle Scholar
Ainseba, B. E., Bendahmane, M. & Noussair, A. (2008) A reaction-diffusion system modeling predator-prey with prey-taxis. Nonlinear Anal. Real World Appl. 9(5), 2086–2105.CrossRefGoogle Scholar
Amann, H. (1989) Dynamic theory of quasilinear parabolic equations III. Global existence. Math. Z. 202, 219–250.CrossRefGoogle Scholar
Amann, H. (1990) Dynamic theory of quasilinear parabolic equations II. Reaction-diffusion systems. Differ. Integral Equ. 3(1), 13–75.Google Scholar
Amann, H. (1992) Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Function Spaces, Differential Operators and Nonlinear Analysis (Friedrichroda, 1992), Vol. 133. Teubner-Texte Math., Teubner, Stuttgart, pp. 9–126.Google Scholar
Chakraborty, A., Singh, M., Lucy, D. & Ridland, P. (2007) Predator-prey model with prey-taxis and diffusion. Math. Comp. Mod. 46, 482–498.CrossRefGoogle Scholar
Chesson, P. & Murdoch, W. (1986) Aggregation of risk: relationships among host-parasitoid models. Am. Nat. 127, 696–715.CrossRefGoogle Scholar
Cosner, C. (2014) Reaction-diffusion-advection models for the effects and evolution of dispersal. Discrere. Contin. Dyn. Syst. 34, 1701–1745.CrossRefGoogle Scholar
Fu, X., Tang, L.-H., Liu, C., Huang, J.-D., Hwa, T. & Lenz, P. (2012) Stripe formation in bacterial system with density-suppressed motility. Phys. Rev. Lett. 108, 198102.CrossRefGoogle Scholar
Grünbaum, D. (1998) Using spatially explicit models to characterize foraging performance in heterogeneous landscapes. Am. Nat. 151, 97–115.CrossRefGoogle ScholarPubMed
Holling, C. S. (1965) The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entom. Soc. Can. 45, 1–60.Google Scholar
Jin, H. Y. & Wang, Z. A. (2017) Global stability of prey-taxis systems. J. Differ. Equations 262, 1257–1290.CrossRefGoogle Scholar
Jin, H. Y., Kim, Y. J. & Wang, Z. A. (2018) Boundedness, stabilization and pattern formation driven by density-suppressed motility. SIAM J. Appl. Math. 78, 1632–1657.CrossRefGoogle Scholar
Jüngel, A. (2010) Diffusive and nondiffusive population models. In: Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Boston, MA: Birkhäuser Boston, Inc., pp. 397–425.CrossRefGoogle Scholar
Jüngel, A., Kuehn, C. & Trussardi, L. (2017) A meeting point of entropy and bifurcations in cross-diffusion herding. European J. Appl. Math. 28(2), 317–356.CrossRefGoogle Scholar
Kareiva, P. & Odell, G. T. (1987) Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search. Am. Nat. 130, 233–270.CrossRefGoogle Scholar
Keller, E. F. & Segel, L. A. (1971) Model for chemotaxis. J. Theor. Biol. 30(2), 225–234.CrossRefGoogle ScholarPubMed
Kelley, W. G. & Peterson, A. C. (2010) The Theory of Differential Equations - Classical and Qualitative, Springer.CrossRefGoogle Scholar
Kowalczyk, R. & Szymańska, Z. (2008) On the global existence of solutions to an aggregation model. J. Math. Anal. Appl. 343, 379–398.CrossRefGoogle Scholar
Ladyzhenskaya, O., Solonnikov, V. & Uralceva, N. (1968) Linear and Quasilinear Equations of Parabolic Type. Providence, RI: AMS.CrossRefGoogle Scholar
LaSalle, J. P. (1960) Some extensions of Lyapunov’s second method. IRE Trans. Circ. Theo. CT-7, 520–527 (1960).CrossRefGoogle Scholar
Lee, J. M., Hillen, T. & Lewis, M. A. (2008) Continuous traveling waves for prey-taxis. Bull. Math. Biol. 70, 654–676.CrossRefGoogle ScholarPubMed
Lee, J. M., Hillen, T. & Lewis, M. A. (2009) Pattern formation in prey-taxis systems. J. Biol. Dyn. 3(6), 551–573.CrossRefGoogle ScholarPubMed
Liu, C., Fu, X., Liu, L., Ren, X., Chau, C. K. L., Li, S., Zeng, H., Chen, G., Tang, L., Lenz, P., Cui, X., Huang, W., Hwa, T. & Huang, J. (2011) Sequential establishment of stripe patterns in an expanding cell population. Science 334, 238–241.CrossRefGoogle Scholar
Lotka, A. J. (1925) Elements of Physical Biology. Baltimore: Williams and Wilkins Co.Google Scholar
Ma, M., Peng, R. & Wang, Z. (2020) Stationary and non-stationary patterns of the density-suppressed motility model. Phys. D 402, 132259, 13 pages.CrossRefGoogle Scholar
Mimura, M. & Kawasaki, K. (1980) Spatial segregation in competitive interaction-diffusion equations. J. Math. Biol. 9(1), 49–64.CrossRefGoogle Scholar
Mizoguchi, N. & Souplet, P. (2014) Nondegeneracy of blow-up points for the parabolic Keller-Segel system. Ann. Inst. H. Poincaré Anal. Non Linéaire 31, 851–875.CrossRefGoogle Scholar
Murdoch, W. W., Briggs, C. J. & Nisbert, R. M. (2003) Consumer-Resource Dynamics (Monographs in Population Biology-36). Princeton University Press.Google Scholar
Murdoch, W., Chesson, J. & Chesson, P. (1985) Biological control in theory and practice. Am. Nat. 125, 344–366.CrossRefGoogle Scholar
Okubo, A. & Levin, S. A. (2001) Diffusion and Ecological Problems: Modern Perspective . Interdisciplinary Applied Mathematics, Vol. 14. 2nd ed. Berlin: Springer.Google Scholar
Porzio, M. M. & Vespri, V. (1993) Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differ. Equations 103(1), 146–178.CrossRefGoogle Scholar
Rosenzweig, M. L. & MacArthur, R. H. (1963) Graphical representation and stability conditions of predator-prey interactions. Am. Nat. 97, 209–223.CrossRefGoogle Scholar
Sapoukhina, N., Tyutyunov, Y. & Arditi, R. (2003) The role of prey taxis in biological control: a spatial theoretical model. Am. Nat. 162, 61–76.CrossRefGoogle ScholarPubMed
Smith-Roberge, J., Iron, D. & Kolokolnikov, T. (2019) Pattern formation in bacterial colonies with density-dependent diffusion. European J. Appl. Math. 30(1), 196–218.CrossRefGoogle Scholar
Souplet, P. & Quittner, P. (2007) Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States. Basel, Boston, Berlin: Birkhäuser Advanced Texts.Google Scholar
Tao, Y. S. (2010) Global existence of classical solutions to a predator-prey model with nonlinear prey-taxis. Nonlinear Anal. Real World Appl. 11(3), 2056–2064.CrossRefGoogle Scholar
Tao, Y. S. & Winkler, M. (2011) A chemotaxis-haptotaxis model: the roles of nonlinear diffusion and logistic source. SIAM J. Math. Anal. 43(2), 685–704.CrossRefGoogle Scholar
Tao, Y. S. & Winkler, M. (2012) Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity. J. Differ. Equations 252(1), 692–715.CrossRefGoogle Scholar
Turchin, P. (2003) Complex Population Dynamics: A Theoretical/Empirical Synthesis. Monographs in Population Biology-35. Princeton University Press.Google Scholar
Volterra, V. (1926) Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560.CrossRefGoogle Scholar
Wang, K., Wang, Q. & Yu, F. (2017) Stationary and time periodic patterns of two predator and one-prey systems with prey-taxis. Discrere. Contin. Dyn. Syst. 37(1), 505–543.CrossRefGoogle Scholar
Wang, M. X. (2018) Note on the Lyapunov functional method. Appl. Math. Lett. 75, 102–107.CrossRefGoogle Scholar
Wang, Q., Song, Y. & Shao, L. J. (2017) Nonconstant positive steady states and pattern formation of 1D prey-taxis systems. J. Nonlinear Sci. 27(1), 71–97.CrossRefGoogle Scholar
Wang, J. P. & Wang, M. X. (2018) Boundedness and global stability of the two-predator and one-prey models with nonlinear prey-taxis. Z. Angew. Math. Phys. 69(3), 63, 24 pages.CrossRefGoogle Scholar
Wu, S., Shi, J. P. & Wu, B. (2016) Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis. J. Differ. Equations 260(7), 5847–5874.CrossRefGoogle Scholar
Wu, S., Wang, J. & Shi, J. (2018) Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis. Math. Models Method Appl. Sci. 28(11), 2275–2312.CrossRefGoogle Scholar
Yi, F., Wei, J. & Shi, J. (2009) Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system. J. Differ. Equations 246, 1944–1977.CrossRefGoogle Scholar