Stability and Hopf bifurcations for a delayed diffusion system in population dynamics

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January 2012, 17(1): 367-399. doi: 10.3934/dcdsb.2012.17.367

Stability and Hopf bifurcations for a delayed diffusion system in population dynamics

Xiang-Ping Yan ^1, and Wan-Tong Li ^2,

1.	Department of Mathematics, Lanzhou Jiaotong University, Lanzhou, Gansu 730070, China
2.	School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received December 2010 Revised February 2011 Published October 2011

Abstract
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A generalized two-species Lotka-Volterra reaction-diffusion system with a discrete delay and subject to homogeneous Dirichlet boundary conditions is considered. By regarding the delay as the bifurcation parameter and analyzing in detail the spectrum of the associated linear operator, the stability of the positive steady state bifurcating from the zero solution is studied. In particular, it is shown that the system can undergo a forward Hopf bifurcation at the positive steady state solution when the delay take a sequence of critical values via the implicit function theorem. To verify the obtained theoretical results, some numerical simulations are also included.

Keywords: Hopf bifurcation., Diffusion effect, Discrete delay, Steady state, Generalized Lotka-Volterra system.

Mathematics Subject Classification: Primary: 35K57, 35B32, 35B10, 92D25, 92D4.

Citation: Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367

References:

[1]	S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.
[2]	S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982.
[3]	G. E. Hutchinson, Circular causal systems in ecology, Ann. New York Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.
[4]	L. C. Evans, "Partial Differential Equations," 2^nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
[5]	T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
[6]	J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.
[7]	J. Hale, "Theory of Functional Differential Equations," 2^nd edition, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977.
[8]	W. Huang, Global Dynamics for a Reaction-Diffusion Equation with Time Delay, J. Differential Equations, 143 (1998), 293-326.
[9]	H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.
[10]	Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Differential Integral Equations, 4 (1991), 117-128.
[11]	Y. Kuang and H. L. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-494. doi: 10.1017/S0334270000009036.
[12]	W.-T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with application to diffusion-competition system, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.
[13]	W.-T. Li, X.-P. Yan and C.-H. Zhang, Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos, Solitons and Fractals, 38 (2008), 227-237. doi: 10.1016/j.chaos.2006.11.015.
[14]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[15]	K. Ryu and I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc., 38 (2001), 643-655.
[16]	S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.
[17]	Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184.
[18]	C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.
[19]	J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.
[20]	X.-P. Yan and W.-T. Li, Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 441-453. doi: 10.1142/S0218127408020434.
[21]	X.-P. Yan and W.-T. Li, Stability of bifurcated periodic solutions in a delayed competition system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 857-871. doi: 10.1142/S0218127409023342.
[22]	X.-P. Yan and C.-H. Zhang, Direction of Hopf bifurcation in a delayed Lotka-Volterra competition diffusion system, Nonlinear Anal. Real World Appl., 10 (2009), 2758-2773. doi: 10.1016/j.nonrwa.2008.08.004.
[23]	X.-P. Yan and C.-H. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Appl. Math. Model., 34 (2010), 184-199. doi: 10.1016/j.apm.2009.03.040.
[24]	Y. Yang and J. W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation, in "Dynamical Systems and Differential Equations," Vol. II (eds. W. Chen and S. Hu) (Springfield, MO, 1996), Discrete Contin. Dynam. Systems, 1998, Added Volume II, 333-352.
[25]	Q. Ye and Z. Li, "An Introduction to Reaction-Diffusion Equations," (in Chinese), Science Press, Beijing, 1990.
[26]	K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[27]	L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals, 14 (2002), 1201-1225. doi: 10.1016/S0960-0779(02)00068-1.

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References:

[1]	S. Busenberg and W. Huang, Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differential Equations, 124 (1996), 80-107.
[2]	S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982.
[3]	G. E. Hutchinson, Circular causal systems in ecology, Ann. New York Acad. Sci., 50 (1948), 221-246. doi: 10.1111/j.1749-6632.1948.tb39854.x.
[4]	L. C. Evans, "Partial Differential Equations," 2^nd edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010.
[5]	T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.
[6]	J. A. Goldstein, "Semigroups of Linear Operators and Applications," Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985.
[7]	J. Hale, "Theory of Functional Differential Equations," 2^nd edition, Applied Mathematical Sciences, 3, Springer-Verlag, New York-Heidelberg, 1977.
[8]	W. Huang, Global Dynamics for a Reaction-Diffusion Equation with Time Delay, J. Differential Equations, 143 (1998), 293-326.
[9]	H. Kielhöfer, "Bifurcation Theory. An Introduction with Applications to PDEs," Applied Mathematical Sciences, 156, Springer-Verlag, New York, 2004.
[10]	Y. Kuang and H. L. Smith, Global stability in diffusive delay Lotka-Volterra systems, Differential Integral Equations, 4 (1991), 117-128.
[11]	Y. Kuang and H. L. Smith, Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks, J. Austral. Math. Soc. Ser. B, 34 (1993), 471-494. doi: 10.1017/S0334270000009036.
[12]	W.-T. Li, G. Lin and S. Ruan, Existence of traveling wave solutions in delayed reaction-diffusion systems with application to diffusion-competition system, Nonlinearity, 19 (2006), 1253-1273. doi: 10.1088/0951-7715/19/6/003.
[13]	W.-T. Li, X.-P. Yan and C.-H. Zhang, Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions, Chaos, Solitons and Fractals, 38 (2008), 227-237. doi: 10.1016/j.chaos.2006.11.015.
[14]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[15]	K. Ryu and I. Ahn, Positive coexistence of steady states for competitive interacting system with self-diffusion pressures, Bull. Korean Math. Soc., 38 (2001), 643-655.
[16]	S. Ruan, Absolute stability, conditional stability and bifurcation in Kolmogorov-type predator-prey systems with discrete delays, Quart. Appl. Math., 59 (2001), 159-173.
[17]	Y. Su, J. Wei and J. Shi, Hopf bifurcations in a reaction-diffusion population model with delay effect, J. Differential Equations, 247 (2009), 1156-1184.
[18]	C. Travis and G. Webb, Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418. doi: 10.1090/S0002-9947-1974-0382808-3.
[19]	J. Wu, "Theory and Applications of Partial Functional-Differential Equations," Applied Mathematical Sciences, 119, Springer-Verlag, New York, 1996.
[20]	X.-P. Yan and W.-T. Li, Stability and Hopf bifurcation for a delayed cooperative system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 18 (2008), 441-453. doi: 10.1142/S0218127408020434.
[21]	X.-P. Yan and W.-T. Li, Stability of bifurcated periodic solutions in a delayed competition system with diffusion effects, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), 857-871. doi: 10.1142/S0218127409023342.
[22]	X.-P. Yan and C.-H. Zhang, Direction of Hopf bifurcation in a delayed Lotka-Volterra competition diffusion system, Nonlinear Anal. Real World Appl., 10 (2009), 2758-2773. doi: 10.1016/j.nonrwa.2008.08.004.
[23]	X.-P. Yan and C.-H. Zhang, Asymptotic stability of positive equilibrium solution for a delayed prey-predator diffusion system, Appl. Math. Model., 34 (2010), 184-199. doi: 10.1016/j.apm.2009.03.040.
[24]	Y. Yang and J. W.-H. So, Dynamics for the diffusive Nicholson's blowflies equation, in "Dynamical Systems and Differential Equations," Vol. II (eds. W. Chen and S. Hu) (Springfield, MO, 1996), Discrete Contin. Dynam. Systems, 1998, Added Volume II, 333-352.
[25]	Q. Ye and Z. Li, "An Introduction to Reaction-Diffusion Equations," (in Chinese), Science Press, Beijing, 1990.
[26]	K. Yosida, "Functional Analysis," Reprint of the sixth (1980) edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[27]	L. Zhou, Y. Tang and S. Hussein, Stability and Hopf bifurcation for a delay competition diffusion system, Chaos Solitons Fractals, 14 (2002), 1201-1225. doi: 10.1016/S0960-0779(02)00068-1.

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