Persistence and global asymptotic stability of single species dispersal models with stage structure
Authors:
H. I. Freedman and J. H. Wu
Journal:
Quart. Appl. Math. 49 (1991), 351-371
MSC:
Primary 92D25; Secondary 34K20
DOI:
https://doi.org/10.1090/qam/1106397
MathSciNet review:
MR1106397
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Abstract: A system of retarded functional differential equations is proposed as a model of single-species population growth with dispersal in a multi-patch environment where individual members of the population have a life history that takes them through two stages, immature and mature. The persistence of the system as well as the existence and global asymptotic stability of a positive equilibrium is proved by using the monotone dynamical systems theory due to Hirsch and Smith, and a convergence theorem established in this paper for nonautonomous retarded equations by using limiting equations theory.
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H. D. Landahl and B. D. Hanson, A three stage population model with cannibalism, Bull. Math. Biol. 37, 11–17 (1975)
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- Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966. MR 0208086
W. G. Aiello and H. I. Freedman, A time-delay model of single-species growth with stage structure, Math. Biosci. 101, 139–153 (1990)
F. S. Anderson, Competition in populations of one age group, Biometrica 16, 19–27 (1960)
Z. Artstein, Uniform asymptotic stability via the limiting equations, J. Differential Equations 27, 172–189 (1978)
H. J. Barclay and P. Van den Driessche, A model for a species with two life history stages and added mortality, Ecol. Model 11, 157–166 (1980)
E. Beretta, F. Solimano, and Y. Takeuchi, Global stability and periodic orbits for two-patch predator-prey diffusion-delay models, Math. Biosci. 85, 153–183 (1987)
E. Beretta and Y. Takeuchi, Global stability of single-species diffusion models with continuous time delays, Bull. Math. Biol. 49, 431–448 (1987)
G. J. Butler, H. I. Freedman, and P. Waltman, Uniformly persistent systems, Proc. Amer. Math. Soc. 96, 425–430 (1986)
G. J. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations 63, 255–263 (1986)
H. I. Freedman, Single species migration in two habitats: persistence and extinction, Math. Model 8, 778–780 (1987)
H. I. Freedman, Persistence and extinction in models of two-habitat migration, Math. Comput. Modelling 12 105–112 (1989)
H. I. Freedman, B. Rai, and P. Waltman, Mathematical models of population interactions with dispersal II: differential survival in a change of habitat, J. Math. Anal. Appl. 115, 140–154 (1986)
H. I. Freedman, J. B. Shukla, and Y. Takeuchi, Population diffusion in a two-patch environment, Math. Biosci. 95, 111–123 (1989)
H. I. Freedman and Y. Takeuchi, Global stability and predator dynamics in a model of prey dispersal in a patchy environment, Nonlinear Anal. 13, 993–1002 (1989)
H. I. Freedman and Y. Takeuchi, Predator survival versus extinction as a function of dispersal in a predator-prey model with patchy environment, Appl. Anal. 31, 247–266 (1989)
H. I. Freedman and P. Waltman, Mathematical models of population interaction with dispersal I: Stability of two habitats with and without a predator, SIAM J. Appl. Math. 32, 631–648 (1977)
H. I. Freedman and J. H. Wu, Steady state analysis in a model for population diffusion in a multi-patch environment, preprint
W. S. C. Gurney and R. M. Nisbet, Fluctuating periodicity, generation separation, and the expression of larval competition, Theor. Pop. Biol. 28, 150–180 (1985)
W. S. C. Gurney, R. M. Nisbet, and J. H. Lawton, The systematic formulation of tractable single species population models incorporating age structure, J. Animal Ecol. 52, 479–495 (1983)
J. R. Haddock, T. Krisztin, and J. H. Wu, Asymptotic equivalence of neutral equations and retarded equations with infinite delay, Nonlinear Anal. 14, 369–377 (1990)
J. R. Haddock and J. Terjeki, Liapunov-Razumikhin functions and invariance principle for functional differential equations, J. Differential Equations 48, 95–122 (1983)
J. K. Hale, Theory of functional Differential Equations, Springer-Verlag, New York, 1979
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, Vol. 25, Amer. Math. Soc., Providence, 1988
H. Hastings, Dynamics of a single species in a spatially varying environment: the stabilizing role of high dispersal rates, J. Math. Biol. 16, 49–55 (1982)
M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math. Soc. 11, 1–64 (1984)
R. D. Holt, Population dynamics in two patch environment: some anomalous consequences of optional habitat selection, Theor. Pop. Biol. 28, 181–208 (1985)
Yu. S. Koslesov, Properties of solutions of a class of equations with lag which describe the dynamics of change in the population of a species with the age structure taken into account, Math. USSR Sb. 45, 91–100 (1983)
V. Lakshmikantham and S. Leela, Differential and Integral Inequalities, Vol. 2, Academic Press, New York, 1969
H. D. Landahl and B. D. Hanson, A three stage population model with cannibalism, Bull. Math. Biol. 37, 11–17 (1975)
S. A. Levin, Dispersion and population interactions, Am. Nat. 108, 207–228 (1974)
R. H. Martin and H. L. Smith, Reaction-diffusion systems with time delays: monotonicity invariance, comparison and convergence, preprint
S. W. Pacala and J. Roughgarden, Spatial heterogeneity and interspecific competition, Theor. Pop. Biol. 21, 92–113 (1982)
G. R. Sell, Nonautonomous differential equations and topological dynamics, I. The basic theory, Trans. Amer. Math. Soc. 127, 241–262 (1967)
G. R. Sell, Nonautonomous differential equations and topological dynamics, II. Limiting equations, Trans. Amer. Math. Soc. 127, 263–283 (1967)
N. Shigesada and J. Roughgarden, The role of rapid dispersal in the population dynamics of competition, Theor. Pop. Biol. 21, 353–372 (1982)
J. G. Skellam, Random dispersal in theoretical populations, Biometrika 38, 196–218 (1951)
H. L. Smith, Monotone semiflows generated by functional differential equations, J. Differential Equations 66, 420–442 (1987)
Y. Takeuchi, Global stability in generalized Lotka-Volterra diffusion systems, J. Math. Anal. Appl. 116, 209–221 (1986)
Y. Takeuchi, Diffusion effect on stability of Lotka-Volterra models, Bull. Math. Biol. 48, 585–601 (1986)
R. R. Vance, The effect of dispersal on population stability in one-species, discrete-space population growth models, Am. Nat. 123, 230–254 (1984)
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985
S. N. Wood, S. P. Blythe, S. C. Gurney, and R. M. Nisbet, Instability in mortality estimation schemes related to stage-structure population models, IMA J. Math. Appl. Med. Biol. 6, 47–68 (1989)
T. Yoshizawa, Stability Theory by Liapunov’s Second Method, Publications of the Mathematical Society of Japan, No. 9, The Mathematical Society of Japan, Tokyo, 1966
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