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Erschienen in:
Short Introduction to Stability Theory of Deterministic Functional Differential Equations Kapitel lesen Erstes Kapitel lesen

2013 | OriginalPaper | Buchkapitel

10. Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations

verfasst von : Leonid Shaikhet

Erschienen in: Lyapunov Functionals and Stability of Stochastic Functional Differential Equations

Verlag: Springer International Publishing

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Abstract

In Chap. 10 a system of two nonlinear differential equations is considered that is destined to unify different known mathematical models, in particular, very often investigated models of predator–prey type. In particular, a ratio-dependent predator–prey model is considered. The system under consideration is exposed to stochastic perturbations and is linearized in a neighborhood of the positive point of equilibrium. Two different ways of construction of asymptotic mean-square stability conditions are considered. The obtained asymptotic mean-square stability conditions for the trivial solution of the constructed linear system are at the same time sufficient conditions for the stability in probability of the positive equilibrium point of the initial nonlinear system under stochastic perturbations. The obtained stability regions are illustrated by six figures.

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Vorheriges Kapitel Stability of Equilibrium Points of Nicholson’s Blowflies Equation with Stochastic Perturbations
Nächstes Kapitel Stability of SIR Epidemic Model Equilibrium Points
Literatur
15.
Arino O, Elabdllaoui A, Mikaram J, Chattopadhyay J (2004) Infection on prey population may act as a biological control in a ratio-dependent predator–prey model. Nonlinearity 17:1101–1116 MathSciNetMATHCrossRef
19.
Bandyopadhyay M, Chattopadhyay J (2005) Ratio dependent predator–prey model: effect of environmental fluctuation and stability. Nonlinearity 18:913–936 MathSciNetMATHCrossRef
23.
Beretta E, Kuang Y (1998) Global analysis in some delayed ratio-dependent predator–prey systems. Nonlinear Anal 32(4):381–408 MathSciNetMATHCrossRef
47.
Cai L, Li X, Song X, Yu J (2007) Permanence and stability of an age-structured prey–predator system with delays. Discrete Dyn Nat Soc 2007:54861. 15 pages MathSciNet
50.
Chen F (2005) Periodicity in a ratio-dependent predator–prey system with stage structure for predator. J Appl Math 2:153–169 CrossRef
53.
Colliugs JB (1997) The effects of the functional response on the bifurcation behavior of a mite predator–prey interaction model. J Math Biol 36:149–168 MathSciNetCrossRef
60.
65.
69.
Fan M, Wang Q, Zhou X (2003) Dynamics of a nonautonomous ratio-dependent predator–prey system. Proc R Soc Edinb A 133:97–118 MATHCrossRef
70.
Fan YH, Li WT, Wang LL (2004) Periodic solutions of delayed ratio-dependent predator–prey model with monotonic and no-monotonic functional response. Nonlinear Anal, Real World Appl 5(2):247–263 MathSciNetMATHCrossRef
72.
82.
Garvie M (2007) Finite-difference schemes for reaction–diffusion equations modelling predator–prey interactions in MATLAB. Bull Math Biol 69(3):931–956 MathSciNetMATHCrossRef
83.
94.
Gourley SA, Kuang Y (2004) A stage structured predator–prey model and its dependence on maturation delay and death rate. J Math Biol 4:188–200 MathSciNet
107.
108.
112.
113.
Huo HF, Li WT (2004) Periodic solution of a delayed predator–prey system with Michaelis–Menten type functional response. J Comput Appl Math 166:453–463 MathSciNetMATHCrossRef
116.
Ji C, Jiang D, Li X (2011) Qualitative analysis of a stochastic ratio-dependent predator–prey system. J Comput Appl Math 235:1326–1341 MathSciNetMATHCrossRef
127.
Ko W, Ahn I (2013) A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: I. Long time behavior and stability of equilibria. J Math Anal Appl 397(1):9–28 MathSciNetMATHCrossRef
128.
Ko W, Ahn I (2013) A diffusive one-prey and two-competing-predator system with a ratio-dependent functional response: II. Stationary pattern formation. J Math Anal Appl 397(1):29–45 MathSciNetMATHCrossRef
153.
Korobeinikov A (2009) Stability of ecosystem: global properties of a general predator–prey model. IMA J Math Appl Med Biol 26(4):309–321 MathSciNetMATHCrossRef
180.
235.
Peschel M, Mende W (1986) The predator–prey model: do we live in a Volterra world. Akademie-Verlag, Berlin MATH
249.
Ruan S, Xiao D (2001) Global analysis in a predator–prey system with nonmonotonic functional response. SIAM J Appl Math 61:1445–1472 MathSciNetMATHCrossRef
267.
Shaikhet L (1998) Stability of predator–prey model with aftereffect by stochastic perturbations. Stab Control: Theory Appl 1(1):3–13 MathSciNet
278.
Shaikhet L (2011) Lyapunov functionals and stability of stochastic difference equations. Springer, London MATHCrossRef
283.
Shi X, Zhou X, Song X (2010) Dynamical properties of a delay prey-predator model with disease in the prey species only. Discrete Dyn Nat Soc. Article ID 196204, 16 pages
288.
Song XY, Chen LS (2002) Optimal harvesting and stability for a predator–prey system with stage structure. Acta Math Appl Sin 18(3):423–430 (English series) MathSciNetMATHCrossRef
304.
Wang WD, Chen LS (1997) A predator–prey system with stage structure for predator. Comput Math Appl 33(8):83–91 MathSciNetCrossRef
305.
Wang LL, Li WT (2003) Existence and global stability of positive periodic solutions of a predator–prey system with delays. Appl Math Comput 146(1):167–185 MathSciNetMATHCrossRef
306.
Wang LL, Li WT (2004) Periodic solutions and stability for a delayed discrete ratio-dependent predator–prey system with Holling-type functional response. Discrete Dyn Nat Soc 2004(2):325–343 MATHCrossRef
309.
Wang Q, Fan M, Wang K (2003) Dynamics of a class of nonautonomous semi-ratio-dependent predator–prey system with functional responses. J Math Anal Appl 278:443–471 MathSciNetMATHCrossRef
311.
Wangersky PJ, Cunningham WJ (1957) Time lag in prey–predator population models. Ecology 38(1):136–139 CrossRef
314.
Xiao D, Ruan S (2001) Multiple bifurcations in a delayed predator–prey system with nonmonotonic functional response. J Differ Equ 176:494–510 MathSciNetMATHCrossRef
317.
Zeng X (2007) Non-constant positive steady states of a prey–predator system with cross-diffusions. J Math Anal Appl 332(2):989–1009 MathSciNetMATHCrossRef
321.
Zhang X, Chen L, Neumann UA (2000) The stage-structured predator–prey model and optimal harvesting policy. Math Biosci 168:201–210 MathSciNetMATHCrossRef
325.
Zuo W, Wei J (2011) Stability and Hopf bifurcation in a diffusive predator–prey system with delay effect. Nonlinear Anal, Real World Appl 12(4):1998–2011 MathSciNetMATHCrossRef
Metadaten
Titel
Stability of Positive Equilibrium Point of Nonlinear System of Type of Predator–Prey with Aftereffect and Stochastic Perturbations
verfasst von
Leonid Shaikhet
Copyright-Jahr
2013
Verlag
Springer International Publishing
Buch
Lyapunov Functionals and Stability of Stochastic Functional Differential Equations

Print ISBN: 978-3-319-00100-5

Electronic ISBN: 978-3-319-00101-2

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-319-00101-2_10

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