### Logistic equation is a simple stochastic carrying capacity

#### Abstract

The logistic model has long been used in ecological modelling for its simplicity and effectiveness. Variations on the logistic model are prolific but, to date, there are a limited number of models that incorporate the stochastic nature of the carrying capacity. This study proposes a modification to the logistic model to incorporate a second differential equation which describes the changes in the carrying capacity, thus treating the carrying capacity as a state variable. The carrying capacity is modelled via a stochastic differential equation that accounts for stochastic ('noisy') variations in the finite resources that the population relies on. The extinction probability distribution, expected solution paths, variance of the solution paths, and distribution of the population are computed using the Monte Carlo method.

**References**- R. B. Banks.
*Growth and Diffusion Phenomena*. Springer, 1994. doi:10.1007/978-3-662-03052-3 - F. Brauer and C. Castillo-Chavez.
*Mathematical Models in Population Biology and Epidemiology*. Springer, 2001. doi:10.1007/978-1-4614-1686-9 - A. Tsoularis and J. Wallace. Analysis of logistic growth models.
*Math. Biosci.*, 179:21–55, 2002. doi:10.1016/S0025-5564(02)00096-2 - S. Oppel, G. Hilton, N. Ratcliffe, C. Fenton, J. Daley, G. Gray, J. Vickery and D. Gibbons. Assessing population viability while accounting for demographic and environmental uncertainty.
*Ecology*, 95:1809–1818, 2014. doi:10.1890/13-0733.1 - J. E. Cohen. Population growth and earth's human carrying capacity.
*Science*, 269:341–346 1995. doi:10.1126/science.7618100 - J. M. Cushing. Periodic time-dependent predator-prey systems.
*SIAM J. Appl. Math.*, 32:82–95 1977. doi:10.1137/0132006 - B. D. Coleman. Nonautonomous logistic equation as models of the adjustment of populations to environmental change.
*Math. Biosci.*, 45:159–173, 1979. doi:10.1016/0025-5564(79)90057-9 - J. Vandermeer. Seasonal ioschronic forcing of Lotka Volterra equations.
*Prog. Theor. Phys.*, 96:13–28, 1996. doi:10.1143/PTP.96.13 - H. M. Safuan, Z. Jovanoski, I. N. Towers and H. S. Sidhu. Exact solution of a non-autonomous logistic population model.
*Ecol. Model.*, 251:99–102, 2013. doi:10.1016/j.ecolmodel.2012.12.016 - P. Del Monte-Luna, B. W. Brook, M. J. Zetina-Rejon and V. H. Cruz-Escalona. The carrying capacity of ecosystems.
*Global Ecol. Biogeogr.*, 13:485–495, 2004. doi:10.1111/j.1466-822X.2004.00131.x - 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.
*19th International Congress on Modelling and Simulation*, Perth, Australia, December 2011, 733–739. http://mssanz.org.au/modsim2011/AA/safuan.pdf - H. M. Safuan, I. N. Towers, Z. Jovanoski and H. S. Sidhu. Coupled logistic carrying capacity model.
*EMAC2011, ANZIAM J.*, 53:C172–C184 2012. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/4972 - H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers. Impacts of biotic resource enrichment on predator-prey population.
*B. Math. Biol.*, 75:1798–1812, 2013. doi:10.1007/s11538-013-9869-7 - H. M. Safuan, H. S. Sidhu, Z. Jovanoski and I. N. Towers. A two-species predator-prey model in an environment enriched by a biotic resource.
*CTAC2012, ANZIAM J.*, 54:C768–C787, 2014 . http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6376 - S. M. Henson and J. M. Cushing. The effect of periodic habitat fluctuations on a nonlinear insect population model.
*J. Math. Biol.*, 36:201–226, 1997. doi:10.1007/s002850050098 - V. Mendez, I. Llopis, D. Campos and W. Horsthemke. Extinction conditions for isolated populations affected by environmental stochasticity.
*Theor. Popul. Biol.*, 77:250–256, 2010. doi:10.1016/j.tpb.2010.02.006 - P. Foley. Predicting extinction times from environmental stochasticity and carrying capacity.
*Conserv. Biol.*, 8:124–137, 1994. http://www.jstor.org/stable/2386727 - D. J. Higham. An algorithmic introduction to numerical simulation of stochastic differential equations.
*SIAM Rev.*, 43:525–546, 2001. doi:10.1137/S0036144500378302 - C. W. Gardiner.
*Handbook of Stochastic Methods*, 2nd ed. Springer, 1985. http://trove.nla.gov.au/version/46588286

#### Keywords

carrying capacity, stochastic, logistic equation, environmental change

DOI: http://dx.doi.org/10.21914/anziamj.v56i0.9386

**Remember,**for most actions you have to record/upload into this online system

**and then**inform the editor/author via clicking on an email icon or Completion button.

ANZIAM Journal, ISSN 1446-8735, copyright Australian Mathematical Society.