Abstract
A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered. The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation. The direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are derived by the center manifold theory and normal form method. In particular, the global continuation of the Hopf bifurcation branches are investigated with a careful estimate of the bounds and periods of the periodic orbits, and the existence of multiple periodic orbits are shown.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Azevedo K.A.G., Ladeira L.A.C.: Hopf bifurcation for a class of partial differential equation with delay. Funkcialaj Ekvacioj 47, 395–422 (2004)
Busenberg S., Huang W.: Stability and Hopf Bifurcation for a Population Delay Model with Diffusion Effects. J. Differ. Equ. 124, 80–107 (1996)
Chen, S., Shi, J.: Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect. Submitted (2011)
Chow S.N., Hale J.K.: Methods of Bifurcation Theory. Springer, New York (1982)
Cooke, K.L., Huang, W.: A theorem of George Seifert and an equation with state-dependent delay. In: Fink, A.M., Miller, R.K., Kliemann, W. (eds.) Delay and Differential Equations, pp. 65–77. World Scientific, Singapore (1992)
Cushing J.M.: Integrodifferential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomathematics, vol. 20. Springer, Berlin (1977)
Dos Santos J.S., Bená M.A.: The delay efect on reaction-diffusion equations. Appl. Anal. 83, 807–824 (2004)
Faria T.: Normal form for semilinear functional differential equations in Banach spaces and applications. Part II. Disc. Cont. Dyn. Syst. 7(1), 155–176 (2001)
Faria, T., Huang, W.: Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay. In: Differential Equations and Dynamical Systems (Lisbon, 2000), vol. 31, pp. 125–141. Fields Institute Communications, American Mathematical Society, Providence, RI (2002)
Faria T., Huang W., Wu J.: Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces. SIAM J. Math. Anal. 34((1), 173–203 (2002)
Friedman A.: Remarks on the maximum principle for parabolic equations and its applications. Pac. J. Math. 8, 201–211 (1958)
Friesecke G.: Convergence to equilibrium for delay-diffusion equations with small delay. J. Dyn. Differ. Equ. 5, 89–103 (1993)
Gopalsamy K.: Stability and oscillations in delay differential equations of population. In: Mathematics and its Applications, vol. 74. Kluwer, Dordrecht (1992)
Green, D., Stech, H.: Diffusion and hereditary effects in a class of population models. In: Busenberg, S., Cooke, C. (eds.) Differential Equation and Applications in Ecology, Epidemics and Population Problems, pp. 19–28, Academic Press, New York (1981)
Gurney M.S., Blythe S.P., Nisbet R.M.: Nicholson’s bowflies revisited. Nature 287, 17–21 (1980)
Hutchinson G.E.: Circular Causal Systems in Ecology. Ann. N. Y. Acad. Sci. 50, 221–246 (1948)
Henry D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Huang W.: Global dynamics for a reaction-diffusion equation with time delay. J. Differ. Equ. 143, 293–326 (1998)
Krawcewicz W., Wu J.: Theory of degrees with applications to bifurcations and differential equations. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)
Kuang Y., Smith H.L.: Global stability in diffusive delay Lotka-Volterra systems. Differ. Integr. Equ. 4, 117–128 (1991)
Kuang Y., Smith H.L.: Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks. J. Aust. Math. Soc. B 34, 471–493 (1993)
Lenhart S.M., Travis C.C.: Global stability of a biological model with time delay. Proc. Am. Math. Soc. 96, 75–78 (1986)
Li W.T., Yan X.P., Zhang C.H.: Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions. Chaos Solitons Fract. 38, 227–237 (2008)
May R.M.: Time-delay versus stability in population models with two and three trophic levels. Ecology 54, 315–325 (1973)
May R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973)
Maynard-Smith J.: Models in Ecology. Cambridge University Press, Cambridge (1978)
Memory M.C.: Bifurcation and asymptotic behaviour of solutions of a delay-differential equation with diffusion. SIAM J. Math. Anal. 20, 533–546 (1989)
Miller R.: On Volterra’s population equation. SIAM J. Appl. Math. 14, 446–452 (1996)
Morita Y.: Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions. Jpn. J. Appl. Math. 1, 39–65 (1984)
Nirenberg L.: A strong maximum principle for parabolic equations. Commun. Pure Appl. Math. 6, 167–177 (1953)
Pao C.V.: Dynamics of nonlinear parabolic systems with time delays. J. Math. Anal. Appl. 198, 751–779 (1996)
Parrot M.E.: Linearized stability and irreducibility for a functional differential equation. SIAM J. Math. Anal. 23, 649–661 (1993)
Pazy A.: Semigroups of Linear Operators and Application to Partial Differential Equations. Springer, Berlin (1983)
Ricklefs R.E., Miller G.: Ecology. W.H. Freeman, (1999)
Ruan, S.: Delay differential equations in single species dynamics. In: Delay Differential Equations and Applications (Marrakech, 2002), pp. 477–517. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 205. Springer, New York (2006)
Seifert G.: On a delay differential equation for single specie population variations. Nonlinear Anal. 11, 1051–1059 (1987)
Shi J., Shivaji R.: Persistence in reaction diffusion models with weak allee effect. J. Math. Biol. 52(6), 807–829 (2006)
Smith H.: An introduction to delay differential equations with applications to the life sciences. In: Texts in Applied Mathematics, vol. 57. Springer, New York (2011)
Smoller J.: Shock Waves and Reaction-Diffusion Equations Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer, New York (1983)
So J.W.-H., Yang Y.: Dirichlet problem for the diffusive Nicholson’s blowflies equation. J. Differ. Equ. 150, 317–348 (1998)
Su Y., Wei J., Shi J.: Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation. Nonlinear Anal. Real World Appl. 11, 1692–1703 (2010)
Su Y., Wei J., Shi J.: Hopf bifurcation in a reaction-diffusion population model with delay effect. J. Differ. Equ. 247, 1156–1184 (2009)
Travis C., Webb G.: Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395–418 (1974)
Wu J.: Theory and Applications of Partial Functional-Differential Equations. Springer, New York (1996)
Wu J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)
Yan X., Li W.: Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model. Nonlinearity 23, 1413–1431 (2010)
Yoshida K.: The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology. Hiroshima Math. J. 12, 321–348 (1982)
Zhou L., Tang Y., Hussein S.: Stability and Hopf bifurcation for a delay competition diffusion system. Chaos Solitons Fract. 14, 1201–1225 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Su, Y., Wei, J. & Shi, J. Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence. J Dyn Diff Equat 24, 897–925 (2012). https://doi.org/10.1007/s10884-012-9268-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-012-9268-z
Keywords
- Reaction-diffusion equation
- Logistic equation
- Delayed and instantaneous density dependence
- Stability
- Local Hopf bifurcation
- Global Hopf bifurcation