zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Global asymptotic stability of positive equilibrium of three-species Lotka-Volterra mutualism models with diffusion and delay effects. (English) Zbl 1201.35030
Summary: In the mutualism system with three species if the effects of dispersion and time delays are both taken into consideration, then the densities of the cooperating species are governed by a coupled system of reaction-diffusion equations with time delays. The aim of this paper is to investigate the asymptotic behavior of the time-dependent solution in relation to a positive uniform solution of the corresponding steady-state problem in a bounded domain with Neumann boundary condition, including the existence and uniqueness of a positive steady-state solution. A simple and easily verifiable condition is given to ensure the global asymptotic stability of the positive steady-state solution. This result leads to the permanence of the mutualism system, the instability of the trivial and all forms of semitrivial solutions, and the nonexistence of nonuniform steady-state solution. The condition for the global asymptotic stability is independent of diffusion and time-delays as well as the net birth rate of species, and the conclusions for the reaction-diffusion system are directly applicable to the corresponding ordinary differential system and 2-species cooperating reaction-diffusion systems. Our approach to the problem is based on inequality skill and the method of upper and lower solutions for a more general reaction-diffusion system. Finally, the numerical simulation is given to illustrate our results.
35B25Singular perturbations (PDE)
92D25Population dynamics (general)
[1]Goh, B. S.: Stability in models of mutualism, Am. nat. 113, 261-275 (1979)
[2]Diekmann, O.; Nisbet, R.; Gurney, W.; Bosch, F.: Simple mathematical models for cannibalism: a critique and a new approach, Math. biosci. 78, 21-46 (1986) · Zbl 0587.92020 · doi:10.1016/0025-5564(86)90029-5
[3]Freedman, H. I.: Deterministic mathematical models in population ecology, (1980)
[4]Murray, J. D.: Mathematical biology, biomathematics, Mathematical biology, biomathematics 19 (1989) · Zbl 0682.92001
[5]Takeuchi, Y.: Global dynamical properties of Lotka – Volterra systems, (1996) · Zbl 0844.34006
[6]Aiello, W. G.; Freedman, H. I.: A time-delay model of single-species growth with stage structure, Math. biosci. 101, 139-153 (1990) · Zbl 0719.92017 · doi:10.1016/0025-5564(90)90019-U
[7]Aiello, W. G.; Freedman, H. I.; Wu, J.: Analysis of a model representing stage-structured population growth with state-dependent time delay, SIAM J. Appl. math. 52, 855-869 (1992) · Zbl 0760.92018 · doi:10.1137/0152048
[8]Al-Omari, J. F. M.; Gourley, S. A.: Stability and traveling fronts in Lotka – Volterra competition models with stage structure, SIAM J. Appl. math. 63, 2063-2086 (2003) · Zbl 1058.92037 · doi:10.1137/S0036139902416500
[9]Freedman, H. I.; Wu, J.: Persistence and global asymptotical stability of single species dispersal model with stage structure, Quart. appl. Math. 49, 351-371 (1991) · Zbl 0732.92021
[10]Gourley, S. A.; Kuang, Y.: A stage structured predator – prey model and its dependence on maturation delay and death rate, J. math. Biol. 49, 188-200 (2004) · Zbl 1055.92043 · doi:10.1007/s00285-004-0278-2
[11]Joseph, W. -H. So; Wu, J.; Zou, X.: A reaction – diffusion model for a single species with age structure. I. travelling fronts on unbounded domains, Proc. R. Soc. lond. A 457, 1-13 (2001) · Zbl 0999.92029 · doi:10.1098/rspa.2001.0789
[12]Kuang, Y.: Delay differential equations with applications in population dynamics, (1993) · Zbl 0777.34002
[13]Faria, T.: Stability and bifurcation for a delayed predator – prey model and the effect of diffusion, J. math. Anal. appl. 254, 433-463 (2001) · Zbl 0973.35034 · doi:10.1006/jmaa.2000.7182
[14]Amann, H.: Dynamic theory of quasi-linear parabolic equations. II: reaction – diffusion systems, Diff. int. Eqn. 3, 13-75 (1990) · Zbl 0729.35062
[15]Pao, C. V.: Systems of parabolic equations with continuous and discrete delays, J. math. Anal. appl. 205, 157-185 (1997) · Zbl 0880.35126 · doi:10.1006/jmaa.1996.5177
[16]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
[17]Ruan, S.; Wu, J.: Reaction – diffusion equations with infinite delay, Can. appl. Math. quart. 2, 485-550 (1994) · Zbl 0836.35158
[18]Wu, J.; Zou, X.: Traveling wave fronts of reaction – diffusion systems with delay, J. dynam. Diff. eqn. 13, 651-687 (2001) · Zbl 0996.34053 · doi:10.1023/A:1016690424892
[19]Li, Z. Y.: Asymptotic behavior of solution of a cooperative – competitive reaction — diffusion equation, J. acta math. Appl. sin. 7, 436-450 (1984) · Zbl 0556.35076
[20]Feng, W.; Lu, X.: Some coexistence and extinction results in a three species ecological model, Diff. int. Eqn. 8, 617-626 (1995) · Zbl 0832.35070
[21]Zheng, S.: A reaction — diffusion system of a competitor – competitor – mutualist model, J. math. Ana1. appl. 124, 254-280 (1993) · Zbl 0658.35053 · doi:10.1016/0022-247X(87)90038-2
[22]Fu, S. M.; Cui, S. B.: Persistence in a periodic competitor – competitor – mutualist diffusion system, J. math. Ana1. appl. 263, 234-245 (2001) · Zbl 0995.35008 · doi:10.1006/jmaa.2001.7612
[23]Pao, C. V.: Global asymptotic stability of Lotka – Volterra 3-species reaction – diffusion systems with time delays, J. math. Anal. appl. 281, 86-204 (2003) · Zbl 1031.35071 · doi:10.1016/S0022-247X(03)00033-7
[24]Leung, A.: A study of 3-species prey – predator reaction – diffusions by monotone schemes, J. math. Anal. appl. 100, 583-604 (1984) · Zbl 0568.92016 · doi:10.1016/0022-247X(84)90103-3
[25]Kim, K. I.; Lin, Z.: Blowup in a three-species cooperating model, Appl. math. Lett. 17, 89-94 (2004) · Zbl 1047.35055 · doi:10.1016/S0893-9659(04)90017-1
[26]Kim, K. I.; Lin, Z.: Blowup estimates for a parabolic system in a three-species cooperating model, J. math. Anal. appl. 293, 663-676 (2004) · Zbl 1052.35081 · doi:10.1016/j.jmaa.2004.01.026
[27]Pao, C. V.: Nonlinear parabolic and elliptic equations, (1992)
[28]Pao, C. V.: Convergence of solutions of reaction – diffusion systems with time delays, Nonlinear anal. 48, 349-362 (2002) · Zbl 0992.35105 · doi:10.1016/S0362-546X(00)00189-9
[29]Leung, A.: Systems of nonlinear partial differential equations, (1989)
[30]Takeuchi, Y.: Global dynamical properties of Lotka – Volterra systems, (1996) · Zbl 0844.34006
[31]Wu, J.: Theory and applications of partial functional differential equations, (1996)
[32]Xia, Y.; Lin, M.: Existence of positive periodic solution of mutualism system with infinite delays, Ann. diff. Eqn. 21, 448-453 (2005) · Zbl 1090.34590
[33]Yang, F.; Ying, D.: Existence of positive solution of multidelays facultative mutualism system, J. eng. Math. 3, 64-68 (2002)
[34]Gopalsamy, K.; He, X. Z.: Persistence, attractivity, and delay in facultative mutualism, J. math. Anal. appl. 215, 154-173 (1997)
[35]Wang, C. Y.: Existence and stability of periodic solutions for parabolic systems with time delays, J. math. Anal. appl. 339, 1354-1361 (2008)
[36]Wang, C. Y.; Wang, S.: Oscillation of partial population model with diffusion and delay, Appl. math. Lett. 22, 1793-1797 (2009) · Zbl 1182.35219 · doi:10.1016/j.aml.2009.06.021
[37]Wang, C. Y.; Wang, S.; Yan, X. P.; Li, L. R.: Oscillation of a class of partial functional population model, J. math. Ana1. appl. 368, 32-42 (2010) · Zbl 1189.35344 · doi:10.1016/j.jmaa.2010.03.005
[38]Wang, C. Y.; Gong, F.; Wang, S.: Singular perturbation problems of three-species food-chain reaction diffusion models, J. pure appl. Math.: adv. Appl. 3, 137-146 (2010) · Zbl 1198.35022