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

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.171.3969.38

Student interpretations of the terms in first-order ordinary differential equations in modelling contexts

A study of first-year undergraduate students′ interpretational difficulties with first-order ordinary differential equations (ODEs) in modelling contexts was conducted using a diagnostic quiz, exam questions and follow-up interviews. These investigations indicate that when thinking about such ODEs, many students muddle thinking about the function that gives the quantity to be determined and the equation for the quantity's rate of change, and at least some seem unaware of the need for unit consistency in the terms of an ODE. It appears that shifting from amount-type thinking to rates-of-change-type thinking is difficult for many students. Suggestions for pedagogical change based on our results are made

Piecewise linear approximation of nonlinear ordinary differential equations

The study of linear ordinary differential equations (ODEs) is an important component of the undergraduate engineering curriculum. However, most of the interesting behaviour of nature is described by nonlinear ODEs whose solutions are analytically intractable. We present a simple method based on the idea that the curve of the nonlinear terms of the dependent variable can be replaced by an approximate curve consisting of a set of line segments tangent to the original curve. This enables us to replace a nonlinear ODE with a finite set of linear inhomogeneous ODEs for which analytic solutions are possible. We apply this method to the cooling of a body under the combined effects of convection and radiation and demonstrate very accurate solutions with a relatively few number of line segments. Furthermore, we discuss how a number of key and usually disparate concepts of calculus are needed to apply this method, including continuity and differentiability, Taylor polynomials and optimisation. References S. Theodorakis and E. Svoukis. Piecewise linear emulation of propagating fronts as a method for determining their speeds. Physical Review E 68, 2003, 027201. doi:10.1103/PhysRevE.68.027201 D. D. Ganji, M. J. Hosseini and J. Shayegh. Some nonlinear heat transfer equations solved by three approximate methods. International Communications in Heat and Mass Transfer 34, 2007, 1003--1016. doi:10.1016/j.icheatmasstransfer.2007.05.010 H. D. Young. University Physics. 8th edition, Addison-Wesley Publishing Company, Massachusetts, 1992. F. P. Incropera, D. P. Dewitt, T. L. Bergman and A. S. Lavine. Fundamentals of Heat and Mass Transfer. 6th edition, John Wiley and Sons, New Jersey, 2007

Piecewise linear approximation to Fisher's equation

A simple method is presented which allows the replacement of a nonlinear differential equation with a piecewise linear differential equation. The method is based on the idea that a curve of the nonlinear terms of the dependent variable in a differential equation can be replaced by an approximate curve consisting of a set of line segments tangent to the original curve. We apply this method to the ubiquitous Fisher's equation and demonstrate that accurate solutions are obtained with a relatively small number of line segments. References R. A. Fisher. The wave of advance of advantageous genes. Ann. of Eugenics., 7, 355--369, 1937. V. M. Kenkre and M. N. Kuperman. Applicability of the Fisher equation to bacterial population dynamics. Phys. Rev. E., 67, 2003, 051921. doi:10.1103/PhysRevE.67.051921 D. A. Kessler, Z. Ner and L. M. Sander. Front propagation: precursors, cutoffs, and structural stability. Phys. Rev. E., 58, 1998, 107--114. P. K. Maini, D. L. S. McElwain and D. I. Leavesley. Traveling wave model to interpret a wound-healing cell migration assay for human peritoneal mesothelial cells. Tissue Eng., 10, 2004, 475--482. M. J. Ablowitz and A. Zeppetella. Explicit solutions of Fisher's equation for a special wave speed. Bull. Math. Biol., 41, 1979, 835--840. X. Y. Wang. Exact and explicit solitary wave solutions for the generalised Fisher equation. Phys. Lett. A, 131, 1988, 277--279. J. Rinzel and J. B. Keller. Traveling wave solutions of a nerve conduction equation. Biophys. J., 13, 1973, 1313--1337. Z. Jovanoski and S. Soo. Piecewise linear approximation of nonlinear ordinary differential equations. ANZIAM J. (E), 51, 2010, C570--C585. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/2622 T. Hagstrom and H. B. Keller. The numerical calculation of traveling wave solutions of nonlinear parabolic equations. SIAM J. Sci. Stat. Comput., 7, 1986, 978--988

The Use of Microsoft Excel to Illustrate Wave Motion and Fraunhofer Diffraction in First Year Physics Courses

In this paper we present an Excel package that can be used to demonstrate physical phenomena in which variables may be automatically adjusted in real-time. This is accomplished by interrogating the system clock through the use of an appropriate macro, and using the clock reading to update the relevant variable. The package has been used for a number of years in first year physics courses to illustrate two phenomena: i) waves, including travelling waves, standing waves, the addition of waves and the interference of waves in general, and also Lissajous figures, and ii) Fraunhofer diffraction and the effects of varying such quantities as the wavelength of the incoming light, the number of slits, the slit width and the slit separation. A number of illustrative examples, generated by the package and taken from a fist year physics course, are presented graphically. The package, which is available for downloading from the web, may be used interactively by the student and is easily modified by them. The use of Excel has the advantage that it is accessible to a much wider audience than if it were written in, say, Matlab. We envisage that it may be useful for first year university courses in wave motion and optics, and may also be useful in physics courses in the last year of secondary school. The package has been tested under Excel 2003, 2007 and 2010, and runs satisfactorily in all three versions

Shepherded solitons

We investigate the onset of vector solitons in a three level, cascade, atomic system. We present an existence curve in the model parameter space for bright vector solitons. Approximate analytical solutions are given and the stability of the solutions discussed. Numerical simulations confirm the analytical predictions. References M. Segev. Optical spatial solitons. Optical and Quantum Electronics, 30, 1998, 503--533. J. Li, J. Liu and X. Yang. Superluminal optical soliton via resonant tunneling in coupled quantum dots. Physica E, 40, 2008, 2916--2920. doi:10.1016/j.physe.2008.02.004 G. Huang, L. Deng and M. G. Payne. Dynamics of ultraslow optical solitons in a cold three-state atomic system. Phys. Rev. E, 72, 2005, 016617. doi:10.1103/PhysRevE.72.016617 N. A. Ansari, Z. Jovanoski, H. S. Sidhu and I. N. Towers. Non-linear interaction of two intense fields with a three-level atomic system. J. Nonlin. Opt. Phys. Mat., 15, 2006, 401--413. doi:10.1142/S0218863506003402 N. A. Ansari, I. N. Towers, Z. Jovanoski and H. S. Sidhu. A semi-classical approach to two-frequency solitons in a three-level cascade atomic system. Opt. Commun., 274, 2007, 66--73. doi:10.1016/j.optcom.2007.02.019 C. R. Menyuk, IEEE J. Quantum Electron., {QE-23}, 174 (1987). Y. S. Kivshar and G. P. Agrawal. Optical Solitons: From fibers to photonic crystals, Chapter 9, San Diego: Academic Press, 2003. N. N. Akhmediev and A. Ankiewicz. Solitons: Nonlinear pulses and beams, p.27, London : Chapman and Hall, 1997. S. Flugge. Practical quantum mechanics, Vol. 1, pg. 94--100, Berlin; New York : Springer-Verlag, 1971. A. W. Snyder and J. D. Love. Optical Waveguide Theory, pp. 264--8, London; New York : Chapman and Hall, 1983. F. W. J. Olver. Introduction to Asymptotics and Special Functions, p.169, New York; London: Academic Press, 1974. L. F. Shampine, P. H. Muir and H. Xu. A User-Friendly BVP solver. JNAIAM, 1, 2006, 201--17. http://cs.stmarys.ca/ muir/JNAIAM_Shampine_Muir_Xu2006.pdf O. V. Sinkin, R. Holzlohner, J. Zweck and C. R. Menyuk. Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems. J. Lightw. Technol., 21, 2003, 61--8. doi:10.1109/JLT.2003.80862

Variational analysis of solitary waves in a homogeneous cubic-quintic nonlinear medium

Fundamental solitary waves in a cubic-quintic nonlinear medium are investigated both numerically and variationally. An equivalent particle model is used to understand the behaviour of the numerical solutions, and while a super-Gaussian trial function is found to have only a limited range of applicability in the variational analysis, a super-sech trial function is found to match the numerical solutions well for all input powers. Various limits are also investigated

Domain walls and their stability

Using Jacobi elliptic functions, we find a new class of exact periodic and the associated soliton solutions for two waves co-propagating in a defocussing nonlinear medium. We show, using both linear stability analysis and numerical simulation, that these solutions are stable

Solitary Waves, Shock Waves and Singular Solitons of the Generalized Ostrovsky-Benjamin-Bona-Mahoney Equation

This paper obtains the solitary wave, shock wave as well as singular soliton solutions to the generalized Ostrovsky- Benjamin-Bona-Mahoney (gO-BBM) equation. The ansatz method is applied to obtain the solutions. Several constraint conditions for the parameters are derived that establish the existence of the soliton solutions