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This paper deals with solving numerically partial integrodifferential equations appearing in biological dynamics models when nonlocal interaction phenomenon is considered. An explicit finite difference scheme is proposed to get a numerical solution preserving qualitative properties of the solution. Gauss quadrature rules are used for the computation of the integral part of the equation taking advantage of its accuracy and low computational cost. Numerical analysis including consistency, stability, and positivity is included as well as numerical examples illustrating the efficiency of the proposed method.

Since the seminal papers by Fisher [

Important theoretical results about existence of solutions and qualitative properties of nonlocal biological dynamic problems have been treated in [

In this paper we consider the nonlocal interaction biological dynamic model described by the partial integro-differential reaction-diffusion problem (PIDE); see [

From the biological point of view, the first term of the right-hand side models the diffusion, and the second includes the pure logistic quadratic term and the consumption of resources in some area around the average location. Note that if the kernel

Numerical analysis of the problem is suitable because the best model may be wasted by a disregarded numerical treatment. Numerical methods dealing with problem (

In this paper we develop an explicit finite difference scheme for the numerical computation and analysis of qualitative properties preserving numerical solutions of problem (

Positivity of the numerical solutions is essential dealing with a population problem and needs to be guaranteed. It is also important to check that numerical solutions are bounded by the carrying capacity of the problem, in agreement with the behaviour of the theoretical solution [

This paper is organized as follows. Section

In this section, we perform the discretization of the continuous problem, with the goal to reach an explicit finite difference scheme. Hereafter, we will work in a suitable bounded numerical domain. Let us consider the numerical domain

The approximation

According to the expression for the Gauss-Hermite quadrature, we have

Given a node

With previous notation, the approximation

The Gauss-Legendre quadrature rule is appropriate in the case that the kernel function has a compact support. For instance, let us consider

By using the change of variables

Applying Gauss-Legendre quadrature together with bilinear interpolation in analogously way to (

Regarding the differential part of PIDE (

According to Lemma 3.1 and Theorem 3.2 of [

This section is devoted to the numerical analysis of the proposed scheme, guaranteeing the preservation of the qualitative properties of the theoretical solution. In the following we show that under appropriate step size conditions the numerical solution

Taking into account the equality (

Regarding the boundedness of the numerical solution, since the term

Taking partial derivatives with respect to

Then, under the assumption

In conclusion, assuming that

Note that stability and positivity step size condition is linked to the problem dimension. In particular, for the one dimensional case, the condition becomes

In this section we study the consistency of the numerical solution, given by the scheme (

Let us consider the (

In accordance with [

Regarding the local truncation error

It can be verified, see [

Moreover,

It can be verified from the expression of

This section illustrates the behaviour of the numerical solution of the problem (

Let us consider the nonlocal logistic diffusion model (

The spatial and temporal step sizes are chosen as

Numerical solution in the case of one space dimension and unbounded domain.

The convergence rate

Relative root mean squared error RRMSE and convergence rate

| | | |
---|---|---|---|

| 0.0014733979 | 0.0006292592 | 0.0002095394 |

| - | 1.22742 | 1.58643 |

Let us consider the same model as in the previous example, with identical parameters values, initial condition, kernel function

Numerical solution when the positivity and stability condition is broken.

In this example, we consider the case of two unbounded spatial dimensions. Here, the parameters are chosen to take the values

Note that, with this data, the problem presents radial symmetry. The spatial and temporal step sizes are chosen as

Numerical solution for two space dimension in an unbounded domain for

Taking the same data as in Example

Numerical solution for two space dimension in an unbounded domain when the positivity and stability condition is broken, for

In this example, using the same data and step sizes as in Example

The numerical solution

Numerical solution for two space dimension in an unbounded domain, taking a Gaussian kernel, for

No data were used to support this study.

The authors of this work declare that there are no conflicts of interest regarding the publication of this paper.

This work has been partially supported by the Ministerio de Economía y Competitividad Spanish grant MTM2017-89664-P.