Bifurcation Analysis in a Diffusive Mussel-Algae Model with Delay

In this paper, we consider the dynamics of a delayed reaction-diffusion mussel-algae system subject to Neumann boundary conditions. When the delay is zero, we show the existence of positive solutions and the global stability of the boundary equilibrium. When the delay is not zero, we obtain the stability of the positive constant steady state and the existence of Hopf bifurcation by analyzing the distribution of characteristic values. By using the theory of normal form and center manifold reduction for partial functional differential equations, we derive an algorithm that determines the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, some numerical simulations are carried out to support our theoretical results.

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

[2]  Jianhong Wu,et al.  Centre manifolds for partial differential equations with delays , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[3]  Johan van de Koppel,et al.  Alternative mechanisms alter the emergent properties of self-organization in mussel beds , 2012, Proceedings of the Royal Society B: Biological Sciences.

[4]  Peter J. Wangersky,et al.  Time Lag in Prey‐Predator Population Models , 1957 .

[5]  Jianhong Wu,et al.  Periodic solutions of single-species models with periodic delay , 1992 .

[6]  Horst Malchow.  Nonlinear plankton dynamics and pattern formation in an ecohydrodynamic model system , 1996 .

[7]  Jonathan A Sherratt,et al.  How does tidal flow affect pattern formation in mussel beds? , 2016, Journal of theoretical biology.

[8]  Johan van de Koppel,et al.  Scale‐Dependent Feedback and Regular Spatial Patterns in Young Mussel Beds , 2005, The American Naturalist.

[9]  Marten Scheffer,et al.  Pattern formation at multiple spatial scales drives the resilience of mussel bed ecosystems , 2014, Nature Communications.

[10]  Inthira Chaiya,et al.  Nonlinear stability analyses of Turing patterns for a mussel-algae model , 2015, Journal of mathematical biology.

[11]  Junjie Wei,et al.  Hopf bifurcation analysis in a diffusive predator-prey system with delay and surplus killing effect. , 2018, Mathematical biosciences and engineering : MBE.

[12]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[13]  C. Klausmeier,et al.  Regular and irregular patterns in semiarid vegetation , 1999, Science.

[14]  Johan van de Koppel,et al.  Phase separation explains a new class of self-organized spatial patterns in ecological systems , 2013, Proceedings of the National Academy of Sciences.

[15]  Junjie Wei,et al.  Qualitative analysis of a neural network model with multiple time delays , 1999 .

[16]  M. Rietkerk,et al.  Experimental Evidence for Spatial Self-Organization and Its Emergent Effects in Mussel Bed Ecosystems , 2008, Science.

[17]  A. M. Turing,et al.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[18]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[19]  Yuan Yuan,et al.  Spatiotemporal Dynamics of the Diffusive Mussel-Algae Model Near Turing-Hopf Bifurcation , 2017, SIAM J. Appl. Dyn. Syst..

[20]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[21]  Xiao-Qiang Zhao,et al.  A nonlocal and periodic reaction-diffusion-advection model of a single phytoplankton species , 2016, Journal of mathematical biology.

[22]  Gregory Dunkel.  Single species model for population growth depending on past history , 1968 .

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

[24]  Junjie Wei,et al.  Dynamics in a diffusive plankton system with delay and toxic substances effect , 2015 .

[25]  Junjie Wei,et al.  Bifurcation analysis of a spruce budworm model with diffusion and physiological structures , 2017 .

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

[27]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[28]  K. Cooke,et al.  Discrete delay, distributed delay and stability switches , 1982 .

[29]  A. Ōkubo,et al.  Analysis of the self-shading effect on algal vertical distribution in natural waters , 1981 .

[30]  Junjie Wei,et al.  Local and Global Hopf bifurcation in a Delayed Hematopoiesis Model , 2004, Int. J. Bifurc. Chaos.

[31]  J. Sherratt.  History-dependent patterns of whole ecosystems , 2013 .

[32]  Junjie Wei,et al.  The Effect of Delay on a Diffusive Predator-Prey System with Holling Type-Ii Predator Functional Response , 2012 .

[33]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[34]  Zhen Jin,et al.  Nonlinear dynamic and pattern bifurcations in a model for spatial patterns in young mussel beds , 2008, Journal of The Royal Society Interface.

[35]  Teresa Faria,et al.  Normal forms and Hopf bifurcation for partial differential equations with delays , 2000 .