We have given an extension to the study of Kierstead, Slobodkin, and Skellam (KiSS) model. We present the theoretical results based on the survival and permanence of the species. To guarantee the long-term existence and permanence, the patch size denoted as

The study of reaction-diffusion problems has gained much attention over the years; such models are largely encountered in various fields and become increasingly useful tools for field engineers, mathematicians, conservation biologists, and population ecologists. In this paper, we describe the current state of the linear Kierstead-Slobodkin [

Understanding the conditions that could guarantee the extinction and persistence of species populations in large but finite domains is another focus area of this paper. Further, determination of the critical patch size ensuring the sustenance of population is another vital area to consider, and the critical patch size depends on some factor, namely, the species population in the patch, geometrical patch, boundaries type, and the reproduction kinetics of species population. In what follows, we will present the extended nonlinear KiSS model and determine its critical patch size and reproduction processes with hostile boundaries.

To start with, we let the critical patch size be

For linear dynamics, it is permitted to seek for solutions of the form

By taking the limit as

In other physical contexts, (

Let

The proof of Theorem

Considerations have been given to the use of spectral methods as alternative to conventional finite differences [

By adopting the notation of order

Next, we formulate the fourth-order exponential time-differencing Runge-Kutta (ETDRK4) method by following closely most notations used by Cox and Matthews [

In this section, we intend to present the numerical results in one and two dimensions. We expect our results to reflect the mathematical results. We simulate the KiSS model in one and two dimensions with the ETDRK4 method presented in Section

In one dimension, consider the nonlinear KiSS model

Permanence and existence of one-dimensional model (

In the second experiment, we utilize a random initial condition which we computed as

In Figure

The snapshots (a, c) and surface plots (b, d) of one-dimensional KiSS model (

Bear in mind that it is in higher dimensions that the mathematical ideas reported in this paper become of serious value. Hence, we give an extension to the numerical experiments in two-dimensional space. Also, we experiment with two different initial conditions to study the behavior of the problem.

In two dimensions, we consider the reaction-diffusion problem

Persistence and existence of two-dimensional KiSS model (

For the second case, we consider the initial condition

In Figure

Persistence and existence of two-dimensional KiSS model (

In this research paper, we have given an extension to the study of nonlinear reaction-diffusion problem. The case study is that of Kierstead, Slobodkin, and Skellam (KiSS) model, for population growing on a patch of finite size

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research contained in this paper is supported by the South African National Research Foundation.