Global stability of prey-taxis systems

Abstract In this paper, we prove the global boundedness and stability of the predator–prey system with prey-taxis in a two-dimensional bounded domain with Neumann boundary conditions. By deriving an entropy-like equality and a boundedness criterion, we show that the intrinsic interaction between predators and preys is sufficient to prevent the population overcrowding even the prey-taxis is included and strong. Furthermore, by constructing appropriate Lyapunov functionals, we show that prey-only steady state is globally asymptotically stable if the predation is weak, and the co-existence steady state is globally asymptotically stable under some conditions (like the prey-taxis is weak or the prey diffuses fast) if the predation is strong. The convergence rates of solutions to the steady states are derived in the paper.

[1]  R. Arditi,et al.  The Role of Prey Taxis in Biological Control: A Spatial Theoretical Model , 2003, The American Naturalist.

[2]  Vincenzo Vespri,et al.  Holder Estimates for Local Solutions of Some Doubly Nonlinear Degenerate Parabolic Equations , 1993 .

[3]  Junjie Wei,et al.  Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey , 2011 .

[4]  Herbert Amann,et al.  Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems , 1990, Differential and Integral Equations.

[5]  Michael Winkler,et al.  Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model , 2010 .

[6]  Youshan Tao,et al.  Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity , 2011, 1106.5345.

[7]  T. Hillen,et al.  Pattern formation in prey-taxis systems , 2009, Journal of biological dynamics.

[8]  M A Lewis,et al.  Continuous Traveling Waves for Prey-Taxis , 2008, Bulletin of mathematical biology.

[9]  S. Sastry.  Nonlinear Systems: Analysis, Stability, and Control , 1999 .

[10]  M. Saunders,et al.  Plant-Provided Food for Carnivorous Insects: a Protective Mutualism and Its Applications , 2009 .

[11]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[12]  J. P. Lasalle.  Some Extensions of Liapunov's Second Method , 1960 .

[13]  Thomas Hillen,et al.  Classical solutions and pattern formation for a volume filling chemotaxis model. , 2007, Chaos.

[14]  Junjie Wei,et al.  Global bifurcation analysis and pattern formation in homogeneous diffusive predator–prey systems , 2016 .

[15]  Nicola Bellomo,et al.  Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues , 2015 .

[16]  Wendi Wang,et al.  Global bifurcation of solutions for a predator–prey model with prey‐taxis , 2015 .

[17]  Yuanfu Shao,et al.  Steady states of a predator–prey model with prey-taxis , 2014 .

[18]  K. Painter,et al.  Volume-filling and quorum-sensing in models for chemosensitive movement , 2002 .

[19]  H. Amann.  Dynamic theory of quasilinear parabolic systems , 1989 .

[20]  Sining Zheng,et al.  Global boundedness of solutions in a reaction-diffusion system of predator-prey model with prey-taxis , 2015, Appl. Math. Lett..

[21]  Michael Winkler,et al.  Large Time Behavior in a Multidimensional Chemotaxis-Haptotaxis Model with Slow Signal Diffusion , 2015, SIAM J. Math. Anal..

[22]  Youshan Tao,et al.  Global existence of classical solutions to a predator–prey model with nonlinear prey-taxis , 2010 .

[23]  R. Macarthur,et al.  Graphical Representation and Stability Conditions of Predator-Prey Interactions , 1963, The American Naturalist.

[24]  K. Painter,et al.  A User's Guide to Pde Models for Chemotaxis , 2022 .

[25]  G. Odell,et al.  Swarms of Predators Exhibit "Preytaxis" if Individual Predators Use Area-Restricted Search , 1987, The American Naturalist.

[26]  N. Mizoguchi,et al.  Nondegeneracy of blow-up points for the parabolic Keller–Segel system , 2014 .

[27]  William W. Murdoch,et al.  Consumer-resource dynamics , 2003 .

[28]  Louis Nirenberg,et al.  An extended interpolation inequality , 1966 .

[29]  T. Iwasaki,et al.  Formal analysis of resonance entrainment by central pattern generator , 2008, Journal of mathematical biology.

[30]  Herbert Amann,et al.  Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems , 1993 .

[31]  Michael Winkler,et al.  Global Large-Data Solutions in a Chemotaxis-(Navier–)Stokes System Modeling Cellular Swimming in Fluid Drops , 2012 .

[32]  C. Cosner.  Reaction-diffusion-advection models for the effects and evolution of dispersal , 2013 .

[33]  D Grünbaum,et al.  Using Spatially Explicit Models to Characterize Foraging Performance in Heterogeneous Landscapes , 1998, The American Naturalist.

[34]  Yihong Du,et al.  A diffusive predator–prey model with a protection zone☆ , 2006 .

[35]  Junjie Wei,et al.  Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system ✩ , 2009 .

[36]  Dirk Horstmann,et al.  F ¨ Ur Mathematik in Den Naturwissenschaften Leipzig from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences from 1970 until Present: the Keller-segel Model in Chemotaxis and Its Consequences , 2022 .

[37]  Michael Winkler,et al.  A critical exponent in a degenerate parabolic equation , 2002 .

[38]  Pavol Quittner,et al.  Superlinear Parabolic Problems , 2007, Birkhäuser Advanced Texts Basler Lehrbücher.

[39]  Youshan Tao,et al.  Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant , 2012 .

[40]  Manmohan Singh,et al.  Predator-prey model with prey-taxis and diffusion , 2007, Math. Comput. Model..

[41]  Boying Wu,et al.  Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis ✩ , 2016 .

[42]  Mostafa Bendahmane,et al.  A reaction–diffusion system modeling predator–prey with prey-taxis , 2008 .

[43]  Michael Winkler,et al.  Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system , 2011, 1112.4156.

[44]  Peter Chesson,et al.  Biological Control in Theory and Practice , 1985, The American Naturalist.

[45]  M. Winkler,et al.  Stabilization in a chemotaxis model for tumor invasion , 2015 .

[46]  Zuzanna Szymańska,et al.  On the global existence of solutions to an aggregation model , 2008 .

[47]  R. Hambrock,et al.  The Evolution of Conditional Dispersal Strategies in Spatially Heterogeneous Habitats , 2009, Bulletin of mathematical biology.

[48]  Yuan Lou,et al.  On the effects of migration and spatial heterogeneity on single and multiple species , 2006 .