Zhang, Jianbao; Fang, Hui Multiple periodic solutions for a discrete time model of plankton allelopathy. (English) Zbl 1134.39008 Adv. Difference Equ. 2006, Article ID 90479, 14 p. (2006). Summary: We study a discrete time model of the growth of two species of plankton with competitive and allelopathic effects on each other \[ \begin{aligned} N_1(k+1)&= N_1(k)\exp\{r_1(k)- a_{11}(k)N_1(k)- a_{12}(k)N_2(k)- b_1(k)N_1(k)N_2(k)\},\\ N_2(k+1)&= N_2(k)\exp\{r_2(k)- a_{21}(k)N_1(k)- a_{22}(k) N_2(k)- b_2(k)N_1(k)N_2(k)\}. \end{aligned} \]A set of sufficient conditions is obtained for the existence of multiple positive periodic solutions for this model. The approach is based on Mawhin’s continuation theorem of coincidence degree theory as well as some a priori estimates. Some new results are obtained. Cited in 57 Documents MSC: 39A11 Stability of difference equations (MSC2000) 39A12 Discrete version of topics in analysis 92D25 Population dynamics (general) Keywords:discrete time model; multiple positive periodic solutions; Mawhin’s continuation theorem; coincidence degree theory PDFBibTeX XMLCite \textit{J. Zhang} and \textit{H. Fang}, Adv. Difference Equ. 2006, Article ID 90479, 14 p. (2006; Zbl 1134.39008) Full Text: DOI EuDML References: [1] Arditi R, Ginzburg LR, Akcakaya HR: Variation in plankton densities among lakes: a case for ratio-dependent predation models.The American Naturalist 1991, 138: 1287-1296. 10.1086/285286 [2] Chattopadhyay J: Effect of toxic substances on a two-species competitive system.Ecological Modelling 1996,84(1-3):287-289. [3] Chen Y: Multiple periodic solutions of delayed predator-prey systems with type IV functional responses.Nonlinear Analysis: Real World Applications 2004,5(1):45-53. 10.1016/S1468-1218(03)00014-2 · Zbl 1066.92050 [4] Fan M, Wang K: Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system.Mathematical and Computer Modelling 2002,35(9-10):951-961. 10.1016/S0895-7177(02)00062-6 · Zbl 1050.39022 [5] Freedman HI, Wu J: Periodic solutions of single-species models with periodic delay.SIAM Journal on Mathematical Analysis 1992,23(3):689-701. 10.1137/0523035 · Zbl 0764.92016 [6] Gaines RE, Mawhin JL: Coincidence Degree and Nonlinear Differential Equations, Lecture Notes in Mathematics. Volume 568. Springer, Berlin; 1977:i+262. [7] Hellebust JA: Extracellular Products in Algal Physiology and Biochemistry, edited by W. D. P. Stewart. University of California Press, California; 1974. [8] Kuang Y: Delay Differential Equations with Applications in Population Dynamics, Mathematics in Science and Engineering. Volume 191. Academic Press, Massachusetts; 1993:xii+398. · Zbl 0777.34002 [9] Maynard-Smith J: Models in Ecology. Cambridge University Press, Cambridge, UK; 1974. · Zbl 0312.92001 [10] Mukhopadhyay A, Chattopadhyay J, Tapaswi PK: A delay differential equations model of plankton allelopathy.Mathematical Biosciences 1998,149(2):167-189. 10.1016/S0025-5564(98)00005-4 · Zbl 0946.92031 [11] Rice EL: Allelopathy. Academic Press, New York; 1984. [12] Zhang RY, Wang ZC, Chen Y, Wu J: Periodic solutions of a single species discrete population model with periodic harvest/stock.Computers & Mathematics with Applications 2000,39(1-2):77-90. 10.1016/S0898-1221(99)00315-6 · Zbl 0970.92019 [13] Zhen J, Ma Z: Periodic solutions for delay differential equations model of plankton allelopathy.Computers & Mathematics with Applications 2002,44(3-4):491-500. 10.1016/S0898-1221(02)00163-3 · Zbl 1094.34542 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.