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

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]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[2]  R. May,et al.  Population biology of infectious diseases: Part I , 1979, Nature.

[3]  A. Ōkubo,et al.  On the spatial spread of the grey squirrel in Britain , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

[4]  J. Murray,et al.  On the spatial spread of rabies among foxes , 1986, Proceedings of the Royal Society of London. Series B. Biological Sciences.

[5]  Ibrahim Özkol,et al.  Solution of difference equations by using differential transform method , 2006, Appl. Math. Comput..

[6]  Ming-Jyi Jang,et al.  On solving the initial-value problems using the differential transformation method , 2000, Appl. Math. Comput..

[7]  Oluwatayo Michael Ogunmiloro,et al.  Mathematical analysis and approximate solution of a fractional order caputo fascioliasis disease model , 2021 .

[8]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[9]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[10]  D. Baleanu,et al.  On the nonlinear dynamical systems within the generalized fractional derivatives with Mittag–Leffler kernel , 2018 .

[11]  Monica Shorten.  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 , 1953 .

[12]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[13]  R. May,et al.  Population biology of infectious diseases: Part II , 1979, Nature.

[14]  H. Kareem,et al.  NUMERICAL AND STABILITY ANALYSIS OF THE TRANSMISSION DYNAMICS OF SVIR EPIDEMIC MODEL WITH STANDARD INCIDENCE RATE , 2019 .

[15]  A. Borhanifar,et al.  Numerical study of the solution of the Burgers and coupled Burgers equations by a differential transformation method , 2010, Comput. Math. Appl..

[16]  Michele Caputo,et al.  On the notion of fractional derivative and applications to the hysteresis phenomena , 2017 .

[17]  D. Bolt,et al.  Changes in the concentration of luteinizing hormone in plasma of rams following administration of oestradiol, progesterone or testosterone. , 1971, Journal of reproduction and fertility.

[18]  Fatma Ayaz,et al.  Solutions of the system of differential equations by differential transform method , 2004, Appl. Math. Comput..

[19]  H. G. Lloyd.  Past and present distribution of Red and Grey squirrels , 1983 .

[20]  Reza Abazari,et al.  Exact solutions for non-linear Schrödinger equations by differential transformation method , 2011 .

[21]  J. D. Murray,et al.  Spatial dispersal of species. , 1988, Trends in ecology & evolution.

[22]  M. Hirsch.  Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere , 1985 .

[23]  Aytac Arikoglu,et al.  Solution of fractional differential equations by using differential transform method , 2007 .