A two-species predator-prey model in an environment enriched by a biotic resource

Abstract

Classical population growth models assume that the environmental carrying capacity is a fixed parameter, which is not often realistic. We propose a modified predator-prey model where the carrying capacity of the environment is dependent on the availability of a biotic resource. In this model both populations are able to consume the resource, thus altering the environment. Stability, bifurcation and numerical analyses are presented to illustrate the system's dynamical behaviour. Bistability occurs in certain parameter regions. This could describe the transition from a beneficial environment to a detrimental one. We examine special cases of the system and show that both permanence and extinction are possible. References J. Vandermeer. Seasonal isochronic forcing of Lotka Volterra equations. Prog. Theor. Phys. , 96:13–28, 1996. doi:10.1143/PTP.96.13 S. Ikeda and T. Yokoi. Fish population dynamics under nutrient enrichment–-A case of the East Seto Inland Sea. Ecol. Model. , 10:141–165, 1980. doi:10.1016/0304-3800(80)90057-5 S. P. Rogovchenko and Y. V. Rogovchenko. Effect of periodic environmental fluctuations on the Pearl–Verhulst model. Chaos, Solitons, Fractals , 39:1169–1181, 2009. doi:10.1016/j.chaos.2007.11.002 H. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. A simple model for the total microbial biomass under occlusion of healthy human skin. In Chan, F., Marinova, D. and Anderssen, R.S. (eds) MODSIM2011, 19th International Congress on Modelling and Simulation. Modelling and Simulation Society of Australia and New Zealand., 733–739, 2011. http://www.mssanz.org.au/modsim2011/AA/safuan.pdf P. Meyer and J. H. Ausubel. Carrying capacity: A model with logistically varying limits. Technol. Forecast. Soc. , 61:209–214, 1999. doi:10.1016/S0040-1625(99)00022-0 R. Huzimura and T. Matsuyama. A mathematical model with a modified logistic approach for singly peaked population processes. Theor. Popul. Biol. , 56:301–306, 1999. doi:10.1006/tpbi.1999.1426 J. H. M. Thornley and J. France. An open-ended logistic-based growth function. Ecol. Model. , 184:257–261, 2005. doi:10.1016/j.ecolmodel.2004.10.007 J. H. M. Thornley, J. J. Shepherd and J. France. An open-ended logistic-based growth function: Analytical solutions and the power-law logistic model. Ecol. Model. , 204:531–534, 2007. doi:10.1016/j.ecolmodel.2006.12.026 H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. Coupled logistic carrying capacity model. ANZIAM J , 53:C172–C184, 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/4972 P. H. Leslie and J. C. Gower. The properties of a stochastic model for the predator-prey type of interaction between two species. Biometrika , 47:219–234, 1960. doi:10.1093/biomet/47.3-4.219 B. Basener and D. S. Ross. Booming and crashing populations and Easter Island. SIAM J. Appl. Math. , 65:684–701, 2005. doi:10.1137/S0036139903426952 D. Lacitignola and C. Tebaldi. Symmetry breaking effects on equilibria and time dependent regimes in adaptive Lotka–Volterra systems. Int. J. Bifurcat. Chaos , 13:375–392, 2003. doi:10.1142/S0218127403006595 F. Wang and G. Pang. Chaos and Hopf bifurcation of a hybrid ratio-dependent three species food chain. Chaos, Solitons, Fractals , 36:1366–1376, 2008. doi:10.1016/j.chaos.2006.09.005 R. Ball. Understanding critical behaviour through visualization: A walk around the pitchfork. Comput. Phys. Commun. , 142:71–75, 2001. doi:10.1016/S0010-4655(01)00322-8 B. Ermentrout. XPP-Aut v. 6.00, 2011. http://www.math.pitt.edu/ bard/xpp/xpp.html R. Malka, E. Shochat and V. R. Kedar. Bistability and bacterial infections. PLOS ONE , 5:1–10, 2010. doi:10.1371/journal.pone.0010010 J. Elf, K. Nilsson, T. Tenson and M. Ehrenberg. Bistable bacterial growth rate in response to antibiotics with low membrane permeability. Phys. Rev. Lett. , 97:258104, 2006. doi:10.1103/PhysRevLett.97.258104 D. Dubnau and R. Losick. Bistability in bacteria. Mol. Microbiol. , 61:564–572, 2006. doi:10.1111/j.1365-2958.2006.05249.x M. Santillan. Bistable behavior in a model of the lac Operon in Escherichia coli with variable growth rate. Biophys. J. , 94:2065–2081, 2008. doi:10.1529/biophysj.107.118026 M. Rosenzweig. Paradox of enrichment: destabilization of exploitation ecosystem in ecological time. Science , 171:385–387, 1971. doi:10.1126/science.171.3969.385