On the existence, uniqueness and computational analysis of a fractional order spatial model for the squirrel population dynamics under the Atangana-Baleanu-Caputo operator

: pp. 432–443
Received: April 25, 2021
Accepted: May 19, 2021

Mathematical Modeling and Computing, Vol. 8, No. 3, pp. 432–443 (2021)

Department of Mathematics, Faculty of Science, Ekiti State University
Department of Mathematics, Faculty of Science, Ekiti State University
Department of Mathematics, Faculty of Physical Science, University of Ilorin

In this paper, we examine the fractional order analysis of a diffusion competition spatial model describing the interactions between the externally introduced grey and local red squirrel under the Atangana-Baleanu-Caputo (ABC) sense.  Also, we establish the existence and uniqueness analysis of the fractional order spatial model of the squirrel population dynamics, while the numerical computation of the fractional order spatial model is carried out using the two dimensional Fractional Order Differential Transform Method (FODTM).  Simulations of the variables of the model reveal that as the system evolves, the grey squirrels increase in density with increase in time, while the red squirrels decrease in density with increase in time.  Also the simulations show that the FODTM is efficient and convergent with low computational cost.

  1. Hirsch M. W.  Systems of differential equations that are competitive or cooperative II: Convergence almost everywhere.  SIAM Journal on Mathematical Analysis. 16 (3), 423–439 (1985).
  2. Kermack W. O., Mckendrick A. G.  A contribution to the mathematical theory of epidemics.  Proceedings of the Royal Society A. 115 (772), 700–721 (1927).
  3. Anderson R. M., May R. M.  Population biology of infectious diseases: Part I. Nature. 280, 361–367 (1979).
  4. Bolt D. J.  Changes in the concentration of lutenizing hormone in plasma rams following the administration of oestradiol, progesterone or testosterone.  Journal of Reproductive Fertility. 24 (3), 435–438 (1971).
  5. Lloyd H. G.  The Distribution of Squirrels in England and Wales, 1959.  Journal of Animal Ecology. 31 (1), 157–165 (1962).
  6. Lloyd H. G.  Past and present distribution of red and grey squirrels.  Mammal Review. 13 (1), 69–80 (1983).
  7. Murray J. D., Stanley E. A., Brown D. L.  On the spatial spread of rabies among foxes.  Proceedings of the Royal Society B. 229 (1255), 111–150 (1986).
  8. Reynolds J. C.  The interaction of red and grey squirrels. Ph.D. thesis, University of East Anglia, Norwich, U.K. (1981).
  9. Shorten M.  A survey of the distribution of the American grey squirrel (Sciurus carolinensis) and the British red squirrel (S. Vulgaris leucourus) in England and Wales in 1944-5.  Journal of Animal Ecology. 15 (1), 82–92 (1946).
  10. Shorten M.  Notes on the distribution of the grey squirrel (Sciurus carolinensis) and the red squirrel (Sciurus vulgaris leucourus) in England and Wales from 1945 to 1952.  Journal of Animal Ecology. 22 (1), 134–140 (1953).
  11. Shorten M.  Squirrels in England, Wales and Scotland, 1955.  Journal of Animal Ecology. 26 (2), 287–294 (1957).
  12. Shorten M.  Squirrels in Britain. In Symposium on the gray squirrel (ed. V. Flyger), pp. 375–378.  Md Dept Res. Ed. Maryland, U.S.A.: publication no. 162 (1959).
  13. Shorten M., Courtier E. A.  A population study of the grey squirrel (Sciurus carolinensis) in May 1954.  Annals of Applied Biology. 43, 494-510 (1955).
  14. Murray J. D.  Spatial dispersal of species.  Trends in Ecology & Evolution. 3 (11), 307–309 (1988).
  15. Murray J. B.  Mathematical Biology, Biomathematics. 3, 105–115, Springer Verlag, Berlin, Germany  (1993).
  16. Diethelm K., Ford N. J.  Analysis of fractional differential equations.  Journal of Mathematical Analysis and Applications. 265 (2), 229–248 (2002).
  17. Diethelm K., Freed A.  The FracPECE Subroutine for the numerical solution of differential equation of fractional order.  Orschung und Wissenschaftliches Rechnen. 57–71 (1999).
  18. Ogunmiloro O. M.  Mathematical analysis and approximate solution of a fractional order Caputo fascioliasis disease model.  Chaos Solitons & Fractals. 146, 110851 (2021).
  19. Caputo M., Fabrizio M.  A new definition of fractional derivative without singular kernel.  Progress in Fractional Differentiation and Applications. 2 (1), 1–11  (2015).
  20. Caputo M., Fabrizio M.  On the notion of fractional derivative and applications to the hysteresis phenomena.  Meccanica. 52, 3043–3052  (2017).
  21. Atangana A., Baleanu D.  New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model.  Thermal Science. 20 (2), 763–769 (2016).
  22. Baleanu D., Jajarmi A., Hajipour M.  On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel.  Nonlinear Dynamics. 94, 397–414 (2018).
  23. Zhou J. K.  Differential transformation and its Application for Electrical Circuit.  Huazhong University Press, Wuhan, China (1986).
  24. Abazari R.,  Borhanifar A.  Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method.  Computers & Mathematics with Applications. 59 (8), 2711–2722 (2010).
  25. Arikoglu A., Ozkol I.  Solution of fractional differential equations by using differential transformation method.  Chaos, Solitons & Fractals. 34 (5), 1473–1481 (2007).
  26. Arikoglu A.  Application of differential transforms method to linear-nonlinear engineering problems.  MS thesis, Istanbul Technical University (2004).
  27. Arikoglu A., Ozkol I.  Solution of difference equations by using differential transformation method.  Applied Mathematics and  Computation. 174 (2), 1216–1228 (2006).
  28. Ayaz F.  Solutions of the systems of differential equations by differential transform method.  Applied Mathematics and Computation. 147 (2), 547–567 (2004).
  29. Ogunmiloro O. M., Abedo F. O., Kareem H. A.  Numerical and stability analysis of the transmission dynamics of SVIR epidemic model with standard incidence rate.  Malaysian Journal of Computing. 4 (2), 349–361 (2019).
  30. Borhanifar A., Abazari R.  Exact solutions for non-linear Schrödinger equations by differential transformation method.  Journal of Applied Mathematics and Computing. 35 (1), 37–51 (2011).
  31. Jang M. J., Chen C. L., Liu Y. C.  On solving the initial value problems using differential transformation method.  Applied Mathematics and Computations. 115 (2–3), 145–160 (2000).
  32. Soltanalizadeh B.,  Branch S.  Application of differential transformation method for solving a fourth-order parabolic partial differential equations.  International Journal of Pure and Applied Mathematics. 78 (3), 299–308 (2012).
  33. Okubo A., Maini P. K., Williamson M. H., Murray J. D.  On the spatial spread of the grey squirrel in Britain.  Proceedings of the Royal Society B. 238 (1291),  113–125 (1989).