# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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 stability of single-species diffusion Volterra models with continuous time delays. (English) Zbl 0627.92021
Consider an ecological system composed of multiple heterogeneous patches connected by discrete diffusion, and each patch is assumed to be occupied by a single species whose evolution equation for the $i\sp{th}$ patch is: $$(1)\quad \dot x\sb i=x\sb i(e\sb i-a\sb ix\sb i+\gamma\sb i\int\sp{t}\sb{-\infty}F\sb i(t-\tau)x\sb i(\tau)d\tau)+\sum\sp{n}\sb{\mu =1}D\sb{i\mu}(x\sb{\mu}-x\sb i),\quad i\in N,$$ where $N=\{1,...,n\}$, n is the number of patches and $x\sb i$ is the population density in the $i\sp{th}$ patch. (1) may be thought as a generalization of the Volterra integral-differential equation to the n-patch case in which $a\sb i, e\sb i\in {\bbfR}\sp+$; $\gamma\sb i\in {\bbfR}$ for all $i\in N$, where $e\sb i$, $i\in N$, are the intrinsic growth rates, and $a\sb i$, $i\in N$, represent the intraspecific relationships. By introducing the supplementary functions $x\sb i\sp{(j)}$, $j=1,...,k\sb i$, $i\in N$, (1) is transformed into the expanded system of O.D.E.: $$(2)\quad \dot x\sb i=x\sb i(e\sb i-a\sb ix\sb i+\gamma\sb i\sum\sp{k\sb i}\sb{j=1}C\sb i\sp{(j)}x\sb i\sp{(j)})+\sum\sp{n}\sb{\mu =1}D\sb{i\mu}(x\sb{\mu}-x\sb i),$$ $$\dot x\sb i\sp{(j)}=\alpha\sb ix\sb i\sp{(j-1)}-\alpha\sb ix\sb i\sp{(j)},\quad j=1,...,k\sb i,\quad x\sb i\sp{(0)}=x\sb i\text{ for all } i\in N.$$ The dynamical behavior of (2) implies the same kind of dynamical behavior of (1). By applying homotopy function techniques [see e.g. {\it C. B. Garcia} and {\it W. I. Zangwill}, Pathways to solutions, fixed points, and equilibria (1981; Zbl 0512.90070)] the authors give sufficient conditions for the existence of a positive equilibrium and for its global and local stability. The biological meanings of the results are considered and compared with some known results.
Reviewer: Li Bingxi

##### MSC:
 92D25 Population dynamics (general) 34D20 Stability of ODE 45J05 Integro-ordinary differential equations 92D40 Ecology
Full Text:
##### References:
 [1] Beretta, E. and F. Solimano. 1985. ”A Generalization of Integro-differential Volterra Models in Population Dynamics: Boundedness and Global Asymptotic Stability.” To apper inSIAM J. appl. Math. · Zbl 0659.92020 [2] Beretta, E. and Y. Takeuchi. 1986. ”Global Asymptotic Stability of Lotka-Volterra Diffusion Models with Continuous Time Delay.” To appear inSIAM J. appl. Math. · Zbl 0661.92018 [3] Berman, A. and R. J. Plemmons. 1979.Non-negative Matrices in the Mathematical Sciences. New York: Academic Press. · Zbl 0484.15016 [4] Cushing, J. M. 1977.Integro-differential Equations and Delay Models in Population Dynamics.Lect. Notes in Biomath. 20. Berlin: Springer. · Zbl 0363.92014 [5] D’Ancona, U. 1954.The Struggle for Existence. Leiden, Netherlands: E. J. Brill. [6] Freedman, H. I., B. Rai and P. Waltman. 1986. ”Mathematical Models of Population Interactions with Dispersal--II. Differential Survival in a Change of Habitat.”J. math. Anal. Applic. 115, 140--154. · Zbl 0588.92020 · doi:10.1016/0022-247X(86)90029-6 [7] Garcia, C. B. and W. I. Zangwill. 1981.Pathways to Solutions, Fixed Points and Equilibria. Englewood Cliffs, NJ: Prentice-Hall. · Zbl 0512.90070 [8] Hastings, A. 1978. ”Global Stability in Lotka-Volterra Systems with Diffusion.”J. math. Biol. 6, 163--168. · Zbl 0393.92013 · doi:10.1007/BF02450786 [9] --. 1982. ”Dynamics of a Single Species in a Spatially Varying Environment: The Stabilizing Role of Higher Dispersal Rates.”J. math. Biol. 16, 49--55. · Zbl 0496.92010 · doi:10.1007/BF00275160 [10] Holt, R. D. 1985. ”Population Dynamics in Two-patch Environments: Some Anomalous Consequences of an Optimal Habitat Distribution.”Theor. Pop. Biol. 28, 181--208. · Zbl 0584.92022 · doi:10.1016/0040-5809(85)90027-9 [11] Levin, S. A. 1974. ”Dispersion and Population Interactions.”Am. Nat. 108, 207--228. · doi:10.1086/282900 [12] --. 1976. ”Spatial Partitioning and the Structure of Ecological Communities.” InSome Mathematical Questions in Biology, Vol. VII. Providence, RI: American Mathematical Society. · Zbl 0338.92017 [13] -- and L. A. Segel. 1976. ”Hypothesis to Explain the Origin of Planktonic Patchness.”Nature, Lond. 259, 659. · doi:10.1038/259659a0 [14] MacDonald, N. 1978.Time Lags in Biological Models.Lect. Notes in Biomath. 27. Berlin: Springer. · Zbl 0403.92020 [15] Miller, R. K. 1966. ”On Volterra’s Population Equation.”SIAM J. appl. Math. 14, 446--452. · Zbl 0161.31901 · doi:10.1137/0114039 [16] Okubo, A. 1980.Diffusion and Ecological Problems: Mathematical Models. Berlin: Springer. · Zbl 0422.92025 [17] Pozio, M. A. 1984. ”Conditions for Global Asymptotic Stability of Equilibria in Some Models with Delay.”Atti del 40 Simposio di Dinamica di Popolazioni (Parma 22--24 Ottobre 1981), pp. 271--275. [18] Scudo, F. M. and J. R. Ziegler. 1978.The Golden Age of Theoretical Ecology: 1923--1940Lect. Notes in Biomath. 22. Berlin: Springer. · Zbl 0372.92002 [19] Skellam, J. G. 1951. ”Random Dispersal in Theoretical Populations.”Biometrika 38, 196--218. · Zbl 0043.14401 · doi:10.1093/biomet/38.1-2.196 [20] Solimano, F. and E. Beretta. 1983. ”Existence of a Globally Asymptotically Stable Equilibrium in Volterra Models with Continuous Time Delay.”J. math. Biol. 18, 93--102. · Zbl 0523.92013 · doi:10.1007/BF00280659 [21] Takeuchi, Y. 1986. ”Global Stability in Generalized Lotka-Volterra Diffusion Models.”J. math. Anal. Applic. 116, 209--221. · Zbl 0595.92013 · doi:10.1016/0022-247X(86)90053-3 [22] --. 1987. ”Diffusion Effect on Stability of Lotka-Volterra Models.”Bull. math. Biol. 48, 585--601. · Zbl 0613.92025 · doi:10.1007/BF02462325 [23] Vance, R. R. 1984. ”The Effect of Dispersal on Population Stability in One Species, Discrete-Space Population Growth Models.”Am. Nat. 123, 230--254. · doi:10.1086/284199