Memory and mutualism in species sustainability: a time-fractional Lotka-Volterra model with harvesting

We first present a predator-prey model for two species and then extend the model to three species where the two predator species engage in mutualistic predation. Constant effort harvesting and the impact of by-catch issue are also incorporated. Necessary sufficient conditions for the existence and stability of positive equilibrium points are examined. It is shown that harvesting is sustainable, and the memory concept of the fractional derivative damps out oscillations in the population numbers so that the system as a whole settles on an equilibrium quicker than it would with integer time derivatives. Finally, some possible physical explanations are given for the obtained results. It is shown that the stability requires the memory concept in the model

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

Application of rational Chebyshev polynomials to optical problems

We present the use of the rational Chebyshev polynomials for discretising the transverse dimension(s) of beam propagation problems within the field of nonlinear optics. How a beam propagates in an optical medium, whether linear or nonlinear, is a common problem and important in both theoretical studies and optical design. The infinite domain and convergence properties of these polynomials allows one to handle the boundary conditions with greater correctness than methods that impose periodic boundary conditions such as Fourier methods. The beam is propagated forward by exponential integration for fast and accurate numerical simulations. The techniques employed to solve the beam propagation problems are easily applied to problems in other fields with mathematically similar models. References L. N. Trefethen. Spectral Methods in Matlab. Siam, Philadelphia, PA, 2000. http://www.comlab.ox.ac.uk/nick.trefethen/spectral.html M. Frigo and S. G. Johnson. The Design and Implementation of FFTW3. Proceedings of the IEEE, 93, 2005, 216--231. doi:10.1109/JPROC.2004.840301; Fastest Fourier Transform in the West. http://www.fftw.org/ J. A. C. Weideman and B. M. Herbst. Split-step methods for the solution of the nonlinear Schrodinger equation. SIAM J. Numer. Anal., 23, 1986, 485--507. http://www.jstor.org/stable/2157521 A. Taflove and S. C. Hageness. Computational Electrodynamics: The Finite-Difference Time-Domain Method. Norwood, MA: Artech, 2000. J. P. Boyd. Chebyshev and Fourier Spectral Methods. Dover Publications, 2nd edition, 2000. J. P. Boyd. Spectral methods using rational basis functions on an infinite interval. J. Comput. Phys., 69, 1987, 112--142. doi:10.1016/0021-9991(87)90158-6 J. A. C. Weideman and S. C. Reddy. A Matlab differentiation matrix suite. ACM Transactions on Mathematical Software, 26, 2000, 465--519. doi:10.1145/365723.365727 B. Minchev and W. Wright. A review of exponential integrators for first order semi-linear problems. Preprint Numerics No. 2/2005. Norwegian University of Science and Technology. http://www.math.ntnu.no/preprint/numerics/2005/N5-2005.pdf S. Krogstad. Generalized integrating factor methods for stiff pdes. J. Comput. Phys., 203, 2002, 72--88. doi:10.1016/j.jcp.2004.08.006 A. Kassam and L. N. Trefethen. Fourth-order time-stepping for stiff pdes. SIAM J. Sci. Comput., 6, 2005, 1214--1244. doi:10.1137/S1064827502410633 S. M. Cox and P. C. Matthews. Exponential time differencing for stiff systems. J. Comput. Phys., 176, 2002, 430--455. doi:10.1006/jcph.2002.6995 H. Berland, B. Skaflestad and W. M. Wright. Expint --- A Matlab package for exponential integrators. ACM Trans. Math. Softw., 33, 2007, 4. doi:10.1145/1206040.1206044 N. N. Akhmediev and A. Ankiewicz. Solitons---Nonlinear pulses and beams. Chapman and Hall, 1997. R. A. Fisher. The wave of advance of advantageous genes. Ann. Eugenics, 7, 1937, 353--369. http://digital.library.adelaide.edu.au/dspace/handle/2440/1512

Reducing the gap between the school and university mathematics: university lecturers' perspective

This paper deals with a very practical issue. In many countries there is a gap between school and university mathematics. The transition period from school to university can be hard for many students. Even students with good marks in school mathematics experience psychological difficulties at university and sometimes fail the first year university mathematics courses. Often the pass rate in the first year university mathematics is around 50%. Different parties – school teachers, university lecturers, first year university students, administrators, researchers – might have different views on the reasons for the gap and the ways to narrow or fill it. The purpose of this paper is to present responses of university lecturers worldwide to a short survey concerning the transition period between the school and university mathematics

Stochastic processes in a discrete model of ground combat

Discrete models of combat are a rare part of the combat modelling literature. Our work introduces a stochastic version of a discrete ground combat based on Epstein theory featuring two adversarial sides, namely an attacker and a defender. Noticeably, the Epstein model of ground combat features an evolving battle front through a withdrawal mechanism to capture the connection between attrition and movement of the front historically prevalent in ground war. Our extension from the deterministic setting of the Epstein model to the stochastic setting is achieved by taking the exchange ratio of attackers lost to defenders to be a mean-reverting stochastic process. The extension of the exchange ratio to a stochastic process is interrupted to be the result of changing strategies and engagements by either side as well of the generally uncertainty of warfare known as the Fog of War upon the outcome of combat. In the deterministic setting of our model, our toy numerical example results in an attacker victory. In the extension of the exchange ratio to a stochastic process, the attackers are no longer assured victory. However, the variations in the exchange ratio can be of benefit to the attackers in that they may achieve victory in a shorter combat duration and as a consequence suffer less attrition. Thus we interpret the stochastic process as introducing a risk vs reward scenario for the attackers where the risk is quantified through the volatility of the process. Our numerical simulations explore the shift in the outcome of combat for the attackers as they take on additional risk and more uncertainty is introduced into combat. We observe the probability that the attacker is victorious, the time till victory when the attacker is victorious, and the remaining ground force strength of the attacking forces for varying volatility. Our results show that for increasing values of the volatility of the exchange ratio process, the probability of an attacker victory increases but the combat duration decreases and the remaining combat power of the attacker forces increases

Shock Waves and Other Solutions to the Benjamin-Bona-Mahoney-Burgers Equation with Dual Power-Law Nonlinearity

We study the hybrid Benjamin-Bona-Mahoney-Burgers equation with dual power-law nonlinearity. Three different techniques - the ansatz method, Lie-symmetry analysis and the (G'/G)-expansion method - are used to find shock wave solutions. Several constraint conditions naturally emerge that guarantee the existence of shock waves. We discuss the nature of the solutions generated by the different methods