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Global stability of a reaction-diffusion predator-prey model with a nonlocal delay. (English) Zbl 1185.35130
Summary: A reaction-diffusion predator-prey system with nonlocal delay due to the gestation of the predator and homogeneous Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each boundary steady state is established. The existence of Hopf bifurcations at the positive steady state is also discussed. Sufficient conditions are derived for the global stability of the positive steady state and the semi-trivial steady state of the proposed problem by using the method of upper-lower solutions and its associated monotone iteration scheme, respectively. Numerical simulations are carried out to illustrate the main results.
35K57Reaction-diffusion equations
92D25Population dynamics (general)
35B35Stability of solutions of PDE
[1]Wu, J.: Theory and applications of partial functional differential equations, (1996)
[2]Pao, C. V.: Dynamics of nonlinear parabolic systems with time delays, J. math. Anal. appl. 198, 751-779 (1996) · Zbl 0860.35138 · doi:10.1006/jmaa.1996.0111
[3]Pao, C. V.: Convergence of solutions of reaction-diffusion systems with time delays, Nonlinear anal. TMA 48, 349-362 (2002) · Zbl 0992.35105 · doi:10.1016/S0362-546X(00)00189-9
[4]Pao, C. V.: Global asymptotic stability of Lotka–Volterra 3-species reaction-diffusion systems with time delays, J. math. Anal. appl. 281, 186-204 (2003) · Zbl 1031.35071 · doi:10.1016/S0022-247X(03)00033-7
[5]Pao, C. V.: Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays, Nonlinear anal. RWA 5, 91-104 (2004) · Zbl 1066.92054 · doi:10.1016/S1468-1218(03)00018-X
[6]Britton, N. F.: Spatial structures and periodic traveling waves in an integrodifferential reaction-diffusion population model, SIAM J. Appl. math. 50, 1663-1688 (1990) · Zbl 0723.92019 · doi:10.1137/0150099
[7]Boshaba, K.; Ruan, S.: Instability in diffusive ecological models with nonlocal delay effects, J. math. Anal. appl. 258, 269-286 (2001) · Zbl 0982.35117 · doi:10.1006/jmaa.2000.7381
[8]Gourley, S. A.: Instability in a predator–prey system with delay and spatial averaging, IMA J. Appl. math. 56, 121-132 (1996) · Zbl 0848.92014 · doi:10.1093/imamat/56.2.121
[9]Gourley, S. A.; Britton, N. F.: Instability of traveling wave solutions of a population model with nonlocal effects, IMA J. Appl. math. 51, 299-310 (1993) · Zbl 0832.35068 · doi:10.1093/imamat/51.3.299
[10]Gourley, S. A.; Britton, N. F.: A predator–prey reaction-diffusion system with nonlocal effects, J. math. Biol. 34, 297-333 (1996) · Zbl 0840.92018
[11]Gourley, S. A.; Ruan, S.: Convergence and travelling fronts in functional differential equations with nonlocal terms: A competition model, SIAM J. Math. anal. 35, 806-822 (2003) · Zbl 1040.92045 · doi:10.1137/S003614100139991
[12]Gourley, S. A.; Ruan, S.: Spatio-temporal delays in a nutrient-plankton model on a finite domain: linear stability and bifurcations, Appl. math. Comput. 145, 391-412 (2003) · Zbl 1026.92050 · doi:10.1016/S0096-3003(02)00494-0
[13]Gourley, S. A.; So, J. W. H.: Dynamics of a food-limited population model incorporating nonlocal delays on a finite domain, J. math. Biol. 44, 49-78 (2002) · Zbl 0993.92027 · doi:10.1007/s002850100109
[14]Yamada, Y.: On a certain class of semilinear Volterra diffusion equations, J. math. Anal. appl. 88, 433-451 (1982) · Zbl 0515.45012 · doi:10.1016/0022-247X(82)90205-0
[15]Yamada, Y.: Asymptotic stability for some semilinear Volterra diffusion equations, J. differential equations 52, 295-326 (1984) · Zbl 0543.35053 · doi:10.1016/0022-0396(84)90165-7
[16]Redlinger, R.: Existence theorem for semilinear parabolic systems with functionals, Nonlinear anal. 8, 667-682 (1984) · Zbl 0543.35052 · doi:10.1016/0362-546X(84)90011-7
[17]Henry, D.: Geometric theory of semilinear parabolic equations, Lecture notes in mathematics 840 (1993)
[18]Wang, M.: Stationary patterns for a prey-predator model with prey-dependent and ratio-dependent functional responses and diffusion, Physica D 196, 172-192 (2004) · Zbl 1081.35025 · doi:10.1016/j.physd.2004.05.007
[19]Hirsch, M. W.: The dynamical systems approach to differential equations, Bull. amer. Math. soc. 11, 1-64 (1984) · Zbl 0541.34026 · doi:10.1090/S0273-0979-1984-15236-4