An efficient numerical technique for a biological population model of fractional order

Abstract

In the present paper, a biological population model of fractional order (FBPM) with one carrying capacity has been examined with the help of reproducing kernel Hilbert space method (RKHSM). This important fractional model arises in many applications in computational biology. It is worth noting that, the considered FBPM is used to provide the changes that is made on the densities of the predator and prey populations by the fractional derivative. The technique employed to construct new numerical solutions for the FBPM which is considered of a system of two nonlinear fractional ordinary differential equations (FODEs). In the proposed investigation, the utilised fractional derivative is the Caputo derivative. The most valuable advantages of the RKHSM is that it is easily and fast implemented method. The solution methodology is based on the use of two important Hilbert spaces, as well as on the construction of a normal basis through the use of Gram-Schmidt orthogonalization process. We illustrate the high competency and capacity of the suggested approach through the convergence analysis. The computational results, which are compared with the homotopy perturbation Sumudu transform method (HPSTM), clearly show: On the one hand, the effect of the fractional derivative in the obtained outcomes, and on the other hand, the great agreement between the mentioned methods, also the superior performance of the RKHSM. The numerical computational are presented in illustrated graphically to show the variations of the predator and prey populations for various fractional order derivatives and with respect to time.

Introduction

The predator-prey interaction in any natural ecosystem is recognized as one of the important events which has gain more and more attention over the years. Predator-prey is a dynamic system in which the predator feeds on the prey. The pairs list of predator-prey creatures is endless. So, it is sufficient to quote as examples: Lion and goat, snake and frog, fox and rabbit, and so on. Here, the same perception (The predators feed on their prey) used in the case of creatures is also applied in the case of plants. The predator ceases to live because of the non-availability of the prey to it. from this point, we can say that the predator is always dependent on its prey. Recently, mathematical models describing all these interactions, taking into account the populations into it and ecosystem conditions, have become the increasingly subject of study in various disciplines, especially in biology. Freedman [1] used mathematical modeling in population ecology. He investigated the classical predator-prey model which is presented as a Gaussian system. Recently, researchers studied how the group-defense mechanism of prey effects in predator population [2].

In recent years, the fractional calculus (FC) has been used in different hot areas, with a remarkable growth concerning the development of new and trendy models. The importance of including the fractional derivative operator in a model is that is increases accuracy although the ignorance of significant real parameters. Within the literature, there are various kinds of fractional differential operators including, Caputo-Fabrizio, Caputo, Riemann-Liouville and Atangana-Baleanu etc [3], [4], [5]. The latter fractional differential operators are successfully used in almost all practical models, with a special mention to different directions of computational biology. From this light, we refer to some examples: The Atangana-Baleanu derivative has been utilized to simulate two Lotka-Volterra models with mutualistic predation by Ghanbari and Cattani [6]. Sweilam et al. [7] provide an optimal control for a new variable order fractional Tumor under immune suppression. Danane et al. [8] investigated the dynamics of hepatitis B viral infection by using fractional differential mathematical models (taking into account the effect of memory). A novel fractional model of human liver is proposed with using Caputo-Fabrizio fractional derivative based on the exponential kernel by Baleanu et al. [9]. Kumar et al. [10] have presented a novel fractional SIRS-SI malaria model transmission. The dynamical behaviour of fractional fish farming model with using Atangana-Baleanu operator has been analysed by Singh at al[11]., interested reader can refer to [12], [13], [14], [15] for more practical models involving fractional derivatives.

It is worth noting that only with solving such models, the interactions can be understood. In this light, the resolution of mathematical models can be achieved in only two main way: the first way is to solve them in an exact manner using analytical methods to get the exact solutions and the second way is to solve them in an approximate manner using numerical methods to get the approximate solutions. When the implementation with the analytical methods are being unsuccessful, numerical methods impose themselves as a very strong mathematical tools.

To the best of our knowledge, there is little literature that has examined the FBPM with carrying capacity and obtained its numerical solutions (see, for instance, [16]), also there is no research where this model was studied by RKHSM. Hence, in this research, we apply for the first time the RKHSM to investigate the FBPM with one carrying capacity. The considered model represented by a non-linear system of FODEs. This study help us to understand the nature of the presented predator-prey model.

Reproducing kernels (RK, for short) have attracted much attention for the solution of a large variety of differential equations with integer or fractional order derivatives. The RK theory goes back to Zaremba research [20] that it focused on BVPs about harmonic and biharmonic functions which have Dirichlet condition. The RKHSM is a representative of the RK theory and it is a successfully applied method. This method has many advantages, the most valuables are : firstly, it provides that the approximate solution and its derivatives converge uniformly to the exact solution and its derivatives, respectively. Secondly, it is an easily and fast implemented method because it is mesh-free. It means that this method doesn’t need discretization. RKHSM has been applied for computation of accurate solutions of various kinds of differential and integral equations such as, forced Duffing equations with nonlocal boundary condition [21], nonlocal fractional boundary value problems [22], nonlinear fractional Volterra integro-differential equations [23], fractional Bratu-type equations [24], Riccati differential equations [25], fractional Bagley-Torvik equation [26], fractional differential equations under ABC-fractional derivative [27], fractional population dynamics system [17].

The motivation of our study is : Firstly, we consider the FBPM with one carrying capacity. Secondly, RKHSM is constructed to obtain numerical solutions of the considered model by using RK theory in The Hilbert space.

The second section of this paper presents some basic concepts and definitions to ease the application of the method. The third section is where the mathematical formulation of FBPM with one carrying capacity is presented, the RKHSM is described and applied to the presented model. The fourth section provides numerical results which are presented graphically to show the method’s effectiveness and the accuracy of the solutions. Finally, the concluding paragraph is given.