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August  2015, 20(6): 1805-1819. doi: 10.3934/dcdsb.2015.20.1805

Positive steady state solutions of a plant-pollinator model with diffusion

 1 Institute of Mathematics and Information Science, Baoji University of Arts and Sciences, Baoji, Shaanxi 721013, China, China, China

Received  September 2014 Revised  December 2014 Published  June 2015

In this paper, a plant-pollinator population system with diffusion is investigated, which is described by a cooperative model with B-D functional response. Using the Leray-Schauder degree theory, we discuss the existence of positive steady state solutions of the model. The result shows when the growth rate of plants is large and the death rate of pollinators is small, the plants and pollinators can coexist. By the regular perturbation theorem and monotone dynamical system theory, the uniqueness and stability of positive solutions have been studied. Especially, we show that the unique positive solution is a global attractor under some conditions. Furthermore, we present some numerical simulations, which is not only to check our theoretical results but also to supply some conjectures out of theoretical analysis.
Citation: Lijuan Wang, Hongling Jiang, Ying Li. Positive steady state solutions of a plant-pollinator model with diffusion. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1805-1819. doi: 10.3934/dcdsb.2015.20.1805
References:
 [1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975), 331-340. doi: 10.2307/3866. [2] J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM Journal on Mathematical Analysis, 17 (1986), 1339-1353. doi: 10.1137/0517094. [3] R. S. Cantrell and C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math., 13 (1987), 337-352. [4] C. S. Cassanova, Existece and strueture of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X. [5] P. R. Crane, E. M. Friis and K. R. Pedersen, Lower cretaceous angiosperm flowers: Fossil evidence on early radiation of dicotyledons, Science, 232 (1986), 852-854. doi: 10.1126/science.232.4752.852. [6] P. R. Crane, E. M. Friis and K. R. Pedersen, The origin and early diversification of angiosperms, Nature, 374 (1994), 27-33. doi: 10.1038/374027a0. [7] E. N. Daneer, On the indices of fixed poins of mappings in cones and applications, Journal of Mathematical Analysis and Applications, 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7. [8] C. Darwin, The Origin of Species, Penguin Books, London, UK. 1859. [9] C. Darwin, The Effects of Cross and Self-Fertilisation in the Vegetable Kingdom, Appelton, New York, 1876. doi: 10.1017/CBO9780511694202. [10] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298. [11] Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4. [12] M. A. Fishman and L. Hadany, Plant-pollinator population dynamics, Theoretical Population Biology, 78 (2010), 270-277. doi: 10.1016/j.tpb.2010.08.002. [13] S. R. Jang, Dynamics of herbivore-plant-pollinator models, Journal of Mathematical Biology, 44 (2002), 129-149. doi: 10.1007/s002850100117. [14] L. G. Li, Coexistence theorems of steady states for predator-prey interacting systems, Transactions of the American Mathematical Society, 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1. [15] L. Lou and S. Martínez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model, Journal of Differential Equations, 230 (2006), 720-742. doi: 10.1016/j.jde.2006.04.005. [16] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [17] J. M. Soberon and C. M. Del Rio, The dynamics of a plant-pollinator interaction, Journal of Theoretical Biology, 91(1981), 363-378. doi: 10.1016/0022-5193(81)90238-1. [18] Y. Wang, H. Wu and S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81 (2012), 243-250. doi: 10.1016/j.tpb.2012.01.004. [19] L. J. Wang and H. L. Jiang, Properties and numerical simulations of positive solutions for a variable-territory model, Applied Mathematics and Computation, 236 (2014), 647-662. doi: 10.1016/j.amc.2014.03.080.

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References:
 [1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, Journal of Animal Ecology, 44 (1975), 331-340. doi: 10.2307/3866. [2] J. Blat and K. J. Brown, Global bifurcation of positive solutions in some systems of elliptic equations, SIAM Journal on Mathematical Analysis, 17 (1986), 1339-1353. doi: 10.1137/0517094. [3] R. S. Cantrell and C. Cosner, On the steady-state problem for the Volterra-Lotka competition model with diffusion, Houston J. Math., 13 (1987), 337-352. [4] C. S. Cassanova, Existece and strueture of the set of positive solutions of a general class of sublinear elliptic non-classical mixed boundary value problems, Nonlinear Analysis: Theory, Methods & Applications, 49 (2002), 361-430. doi: 10.1016/S0362-546X(01)00116-X. [5] P. R. Crane, E. M. Friis and K. R. Pedersen, Lower cretaceous angiosperm flowers: Fossil evidence on early radiation of dicotyledons, Science, 232 (1986), 852-854. doi: 10.1126/science.232.4752.852. [6] P. R. Crane, E. M. Friis and K. R. Pedersen, The origin and early diversification of angiosperms, Nature, 374 (1994), 27-33. doi: 10.1038/374027a0. [7] E. N. Daneer, On the indices of fixed poins of mappings in cones and applications, Journal of Mathematical Analysis and Applications, 91 (1983), 131-151. doi: 10.1016/0022-247X(83)90098-7. [8] C. Darwin, The Origin of Species, Penguin Books, London, UK. 1859. [9] C. Darwin, The Effects of Cross and Self-Fertilisation in the Vegetable Kingdom, Appelton, New York, 1876. doi: 10.1017/CBO9780511694202. [10] D. L. DeAngelis, R. A. Goldstein and R. V. O'Neill, A model for trophic interaction, Ecology, 56 (1975), 881-892. doi: 10.2307/1936298. [11] Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc., 349 (1997), 2443-2475. doi: 10.1090/S0002-9947-97-01842-4. [12] M. A. Fishman and L. Hadany, Plant-pollinator population dynamics, Theoretical Population Biology, 78 (2010), 270-277. doi: 10.1016/j.tpb.2010.08.002. [13] S. R. Jang, Dynamics of herbivore-plant-pollinator models, Journal of Mathematical Biology, 44 (2002), 129-149. doi: 10.1007/s002850100117. [14] L. G. Li, Coexistence theorems of steady states for predator-prey interacting systems, Transactions of the American Mathematical Society, 305 (1988), 143-166. doi: 10.1090/S0002-9947-1988-0920151-1. [15] L. Lou and S. Martínez and P. Poláčik, Loops and branches of coexistence states in a Lotka-Volterra competition model, Journal of Differential Equations, 230 (2006), 720-742. doi: 10.1016/j.jde.2006.04.005. [16] J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer, New York, 1994. doi: 10.1007/978-1-4612-0873-0. [17] J. M. Soberon and C. M. Del Rio, The dynamics of a plant-pollinator interaction, Journal of Theoretical Biology, 91(1981), 363-378. doi: 10.1016/0022-5193(81)90238-1. [18] Y. Wang, H. Wu and S. Sun, Persistence of pollination mutualisms in plant-pollinator-robber systems, Theoretical Population Biology, 81 (2012), 243-250. doi: 10.1016/j.tpb.2012.01.004. [19] L. J. Wang and H. L. Jiang, Properties and numerical simulations of positive solutions for a variable-territory model, Applied Mathematics and Computation, 236 (2014), 647-662. doi: 10.1016/j.amc.2014.03.080.

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