How many consumer levels can survive in a chemotactic food chain?

Abstract

We investigate the effect and the impact of predator-prey interactions, diffusivity and chemotaxis on the ability of survival of multiple consumer levels in a predator-prey microbial food chain. We aim at answering the question of how many consumer levels can survive from a dynamical system point of view. To solve this standing issue on food-chain length, first we construct a chemotactic food chain model. A priori bounds of the steady state populations are obtained. Then under certain sufficient conditions combining the effect of conversion efficiency, diffusivity and chemotaxis parameters, we derive the co-survival of all consumer levels, thus obtaining the food chain length of our model. Numerical simulations not only confirm our theoretical results, but also demonstrate the impact of conversion efficiency, diffusivity and chemotaxis behavior on the survival and stability of various consumer levels.

Keywords

Food chain length / chemotaxis / stability / priori estimates / fixedpoint index theory

References

[1] Amann H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review, 1976, 18: 620-709

[2] Bally M, Dung L, Jones D A, Smith H L. Effects of random motility on microbial growth and competition in a flow reactor. SIAM J Appl Math, 1998, 59: 573-596

[3] Boer M P, Kooi B W, Kooijman S. Multiple attractors and boundary crises in a tri-trophic food chain. Math Biosci, 2001, 169: 109-128

[4] Butler G J, Hsu S B, Waltman P. A mathematical model of the chemostat with periodic washout rate. SIAM J Appl Math, 1985, 45(3): 435-449

[5] Cantrell R S, Cosner C. Models for predator-prey systems at multiple scales. SIAM Rev, 1996, 38(2): 256-286

[6] Chiu C, Hsu S. Extinction of top-predator in a three-level food-chain model. J Math Biol, 1998, 37(4): 372-380

[7] Du Y, Shi J. Allee effect and bistability in a spatially heterogeneous predator-prey model. Trans Amer Math Soc, 2007, 359(9): 4557-4593

[8] Dunbar S R. Travelling wave solutions of diffusive Lotka-Volterra equations. J Math Biol, 1983, 17(1): 11-32

[9] Dung L. Global attractors and steady state solutions for a class of reaction-diffusion systems. J Differential Equations, 1998, 147: 1-29

[10] Dung L. Coexistence with chemotaxis. SIAM J Math Anal, 2000, 32: 504-521

[11] Dung L, Smith H L. A parabolic system modeling microbial competition in an unmixed bio-reactor. J Differential Equations, 1996, 130(1): 59-91

[12] Dung L, Smith H L. Steady states of models of microbial growth and competition with chemotaxis. J Math Anal Appl, 1999, 229: 295-318

[13] Freedman H I, Ruan S. Hopf bifurcation in three-species food chain models with group defense. Math Biosci, 1992, 111(1): 73-87

[14] Fretwell S D. Food chain dynamics: the central theory of ecology? OIKOS, 1987, 50: 291-301

[15] Gourley S A, Kuang Y. Two-species competition with high dispersal: the winning strategy. Math Biosci Eng, 2005, 2(2): 345-362

[16] Herrero M, Velazquez J. Chemotactic collapse for the Keller-Segel model. J Math Biol, 1996, 35: 177-194

[17] Hofer T, Sherratt J, Maini P K. Cellular pattern formation during Dictyostelium aggregation. Physica D, 1995, 85: 425-444

[18] Hsu S B, Waltman P. Analysis of a model of two competitors in a chemostat with an external inhibitor. SIAM J Appl Math, 1992, 52(2): 528-540

[19] Hsu S B, Waltman P. On a system of reaction-diffusion equations arising from competition in an unstirred chemostat. SIAM J Appl Math, 1993, 53(4): 1026-1044

[20] Huang J, Lu G, Ruan S. Existence of traveling wave solutions in a diffusive predatorprey model. J Math Biol, 2003, 46(2): 132-152

[21] Keller E F, Segel L A. Initiation of slime mold aggregation viewed as an instability. J Math Biol, 1970, 26: 399-415

[22] Lauffenburger D, Calcagno P. Competition between two microbial populations in a non-mixed environment: effect of cell random motility. Biotech Bioengrg, 1983, 25: 2103-2125

[23] Lemesle V, Gouze J L. A simple unforced oscillatory growth model in the chemostat. Bull Math Biol, 2008, 70(2): 344-357

[24] Li B T, Kuang Y. Simple food chain in a Chemostat with distinct removal rates. J Math Anal Appl, 2000, 242: 75-92

[25] Lin C S, Ni W M, Takagi I. Large amplitude stationary solutions to a chemotaxis system. J Differential Equations, 1988, 72: 1-27

[26] Liu J, Zheng S N. A Reaction-diffusion system arising from food chain in an unstirred Chemostat. J Biomath, 2002, 3: 263-272

[27] Martiel J L, Goldbeter A. A model based on receptor desensitization for cyclic AMP signaling in Dictyostelium cells. Biophys J, 1987, 52: 807-828

[28] Monk P B, Othmer H G. Cyclic AMP oscillations in suspensions of Dictyostelium discoideum. Proc Roy Soc Lond B, 1989, 240: 555-589

[29] Muratori S, Rinaldi S. Low- and high-frequency oscillations in three-dimensional food chain systems. SIAM J Appl Math, 1992, 52(6): 1688-1706

[30] Murray J D. Mathematical Biology, I and II. New York: Springer-Verlag, 2002

[31] Odum E P. Trophic structure and productivity of Silver Springs. Florida, -Ecol Monogr, 1957, 27: 55-112

[32] Ou C, Wu J. Spatial spread of rabies revisited: influence of age-dependent diffusion on nonlinear dynamics. SIAM J Appl Math, 2006, 67(1): 138-163

[33] Ou C, Wu J. Persistence of wavefronts in delayed nonlocal reaction-diffusion equations. J Differential Equations, 2007, 235(1): 219-261

[34] Ou C, Wu J. Traveling wavefronts in a delayed food-limited population model. SIAM J Math Anal, 2007, 39(1): 103-125

[35] Owen M R, Lewis M A. How predation can slow, stop or reverse a prey invasion. Bull Math Bio, 2001, 63: 655-684

[36] Painter K J, Hillen T. Volume-filling and quorum-fensing in models for chemosensitive and movement. Canadian Applied Mathematics Quarterly, 2002, 10: 501-542

[37] Pang P Y H,Wang M X. Strategy and stationary pattern in a three-species predatorprey model. Journal of Differential Equations, 2004, 200(2): 245-273

[38] Peng R, Shi J, Wang M. Stationary pattern of a ratio-dependent food chain model with diffusion. SIAM J Appl Math, 2007, 67(5): 1479-1503

[39] Peng R, Wang M X. On multiplicity and stability of positive solutions of a diffusive prey-predator model. Journal of Mathematical Analysis and Applications, 2006, 316(1): 256-268

[40] Pimm S L, Kitching R L. The determinants of food chain lengths. OIKOS, 1987, 50: 302-307

[41] Post D M. The long and short of food chain length. Trends in Ecology and Evolution, 2002, 17: 269-277

[42] Potapov A B, Hillen T. Metastability in chemotaxis models. J Dynam Differential Equations, 2005, 17(2): 293-330

[43] Ruan S, He X. Global stability in chemostat-type competition models with nutrient recycling. SIAM J Appl Math, 1998, 58(1): 170-192

[44] Smith H L. Competitive coexistence in an oscillating chemostat. SIAM J Appl Math, 1981, 40(3): 498-522

[45] Wang X. Qualitative behavior of solutions of chemotactic diffusion systems: effects of motility and chemotaxis and dynamics. SIAM J Math Anal, 2000, 31(3): 535-560

[46] Wang Y F, Yin J X. Predator-prey in an unstirred chemostat with periodical input and washout. Nonlinear Analysis: Real World Applications, 2002, 3: 597-610

[47] Wu J, Nie H, Wolkowicz G S K. A mathematical model of competition for two essential resources in the unstirred chemostat. SIAM J Appl Math, 2004, 65(1): 209-229

[48] Wu J, Nie H, Wolkowicz G S K. The effect of inhibitor on the plasmid-bearing and plasmid-free model in the unstirred chemostat. SIAM J Math Anal, 2007, 38(6): 1860-1885

[49] Wu J, Wolkowicz G S K. A system of resource-based growth models with two resources in the unstirred chemostat. J Differential Equations, 2001, 172(2): 300-332

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg