Coexistence in a one-predator, two-prey system with indirect effects. (English) Zbl 1266.92063

Colucci, Renato

J. Appl. Math. 2013, Article ID 625391, 13 p. (2013).

Summary: We study the dynamics of a one-predator, two-prey system in which the predator has an indirect effect on the prey. We show that, in presence of the indirect effect term, the system admits coexistence of the three populations while, if we disregard it, at least one of the populations goes to extinction.

MSC:

92D40

Ecology

References:

[1]	B. Bolker, M. Holyoak, V. K\vrivan, L. Rowe, and O. Schmitz, “Connecting theoretical and empirical studies of trait-mediated interactions,” Ecology, vol. 84, no. 5, pp. 1101-1114, 2003.
[2]	D. Cariveau, R. E. Irwin, A. K. Brody, L. S. Garcia-Mayeya, and A. Von der Ohe, “Direct and indirect effects of pollinators and seed predators to selection on plant and floral traits,” Oikos, vol. 104, no. 1, pp. 15-26, 2004.
[3]	B. A. Menge, “Indirect effects in marine rocky intertidal interaction webs: patterns and importance,” Ecological Monographs, vol. 65, no. 1, pp. 21-74, 1995.
[4]	W. E. Snyder and A. R. Ives, “Generalist predators disrupt biological control by a specialist parasitoid,” Ecology, vol. 82, no. 3, pp. 705-716, 2001.
[5]	J. T. Wootton, “Indirect effects, prey susceptibility, and habitat selection: impacts of birds on limpets and algae,” Ecology, vol. 73, no. 3, pp. 981-991, 1992.
[6]	J. Estes, K. Crooks, and R. Holt, Ecological Role of Predators. Enciclopedia of Biodiversity, vol. 4, Academic Press, New York, NY, USA, 2001.
[7]	V. Lundgren and E. Granéli, “Grazer-induced defense in Phaeocystis globosa (prymnesiophyceae): Influence of different nutrient conditions,” Limnology and Oceanography, vol. 55, no. 5, pp. 1965-1976, 2010.
[8]	M. F. Caruselaa, F. R. Momoa, and L. Romanellia, “Competition, predation and coexistence in a three trophic system,” Ecological Modelling, vol. 220, no. 10, pp. 2349-2352, 2009.
[9]	“Indirect effects affects ecosystem dynamics,” 2011, http://www.ictp-saifr.org/.
[10]	R. Margalef, “Life forms of Phytoplanktos as survival alternative in an unstable environment,” Oceanologica Acta, vol. 134, no. 1, pp. 493-509, 1978.
[11]	O. Sarnelle, “Daphnia as keystone predators: effects on phytoplankton diversity and grazing resistance,” Journal of Plankton Research, vol. 27, no. 12, pp. 1229-1238, 2005.
[12]	D. O. Hessen, T. Andersen, P. Brettum, and B. A. Faafeng, “Phytoplankton contribution to sestonic mass and elemental ratios in lakes: implications for zooplankton nutrition,” Limnology and Oceanography, vol. 48, no. 3, pp. 1289-1296, 2003.
[13]	L. R. Devaney, M. W. Hirsh, and S. Smale, Differential Equations, Dynamical Systems, and an Introduction to Chaos, Elvesier, New York, NY, USA, 2004. · Zbl 1135.37002
[14]	S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer, New York, NY, USA, 2nd edition, 2010. · Zbl 1027.37002
[15]	A. Klebanoff and A. Hastings, “Chaos in one-predator, two-prey models: general results from bifurcation theory,” Mathematical Biosciences, vol. 122, no. 2, pp. 221-233, 1994. · Zbl 0802.92017
[16]	S. Gakkhar and R. K. Naji, “Existence of chaos in two-prey, one-predator system,” Chaos, Solitons and Fractals, vol. 17, no. 4, pp. 639-649, 2003. · Zbl 1034.92033
[17]	J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, UK, 1998. · Zbl 0914.90287
[18]	P. van den Driessche and M. L. Zeeman, “Three-dimensional competitive Lotka-Volterra systems with no periodic orbits,” SIAM Journal on Applied Mathematics, vol. 58, no. 1, pp. 227-234, 1998. · Zbl 0910.34050
[19]	W. Liu, D. Xiao, and Y. Yi, “Relaxation oscillations in a class of predator-prey systems,” Journal of Differential Equations, vol. 188, no. 1, pp. 306-331, 2003. · Zbl 1094.34025
[20]	H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, American Mathematical Society, Providence, RI, USA, 2011. · Zbl 1214.37002
[21]	G. Butler, H. I. Freedman, and P. Waltman, “Uniformly persistent systems,” Proceedings of the American Mathematical Society, vol. 96, no. 3, pp. 425-430, 1986. · Zbl 0603.34043
[22]	J. Huisman and F. J. Weissing, “Biodiversity of plankton by species oscillations and chaos,” Nature, vol. 402, no. 6760, pp. 407-410, 1999.
[23]	D. Li, J. Li, and Z. Zheng, “Measuring nonequilibrium stability and resilience in an n-competitor system,” Nonlinear Analysis. Real World Applications, vol. 11, no. 3, pp. 2016-2022, 2010. · Zbl 1188.93100
[24]	H. Nie and J. Wu, “Coexistence of an unstirred chemostat model with Beddington-DeAngelis functional response and inhibitor,” Nonlinear Analysis. Real World Applications, vol. 11, no. 5, pp. 3639-3652, 2010. · Zbl 1203.35128
[25]	R. A. de Assis, S. Bonelli, M. Witek et al., “A model for the evolution of parasite-host interactions based on the Maculinea-Myrmica system: numerical simulations and multiple host behavior,” Nonlinear Analysis. Real World Applications, vol. 13, no. 4, pp. 1507-1524, 2012. · Zbl 1401.92172
[26]	M. R. Walsh and D. N. Reznick, “Interactions between the direct and indirect effects of predators determine life history evolution in a killifish,” Pnas, vol. 105, no. 2, pp. 594-599, 2008.
[27]	X. Liu and L. Huang, “Periodic solutions for impulsive semi-ratio-dependent predator-prey systems,” Nonlinear Analysis. Real World Applications, vol. 10, no. 5, pp. 3266-3274, 2009. · Zbl 1187.34091
[28]	R. L. Devaney, An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Reading, Mass, USA, 2nd edition, 1989. · Zbl 0695.58002
[29]	J.-M. Ginoux, B. Rossetto, and J.-L. Jamet, “Chaos in a three-dimensional Volterra-Gause model of predator-prey type,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 5, pp. 1689-1708, 2005. · Zbl 1092.37541
[30]	Z. He and X. Lai, “Bifurcation and chaotic behavior of a discrete-time predator-prey system,” Nonlinear Analysis. Real World Applications, vol. 12, no. 1, pp. 403-417, 2011. · Zbl 1202.93038