Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

Azzh Saad Alshehry; Safyan Mukhtar; Ali M. Mahnashi; Azzh Saad Alshehry; Safyan Mukhtar; Ali M. Mahnashi

doi:10.3934/math.2024773

AIMS Mathematics

2024, Volume 9, Issue 6: 15966-15987. doi: 10.3934/math.2024773

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Research article Special Issues

Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

1.
Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah Bint Abdulrahman University, P.O.Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Basic Sciences, Preparatory Year, King Faisal University, Al-Ahsa 31982, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia

We examine a biological population model of fractional order (FBPM) in this paper using the Riccati-Bernoulli sub-ODE approach. Many scenarios in computational biology make use of this fundamental fractional model. Of particular note is that our study's FBPM uses fractional derivatives to track changes in the density populations. The study is concerned with the construction of new solitary wave solutions for the FBPM, a system of two nonlinear fractional ordinary differential equations. In this investigation, we use the conformable derivative as the fractional derivative. The Backlund transformation is the foundation of the solution process. We create a variety of families of soliton wave solutions and explain different physical behaviours that are inherent in the problems we explore. In particular, we apply the suggested methods to investigate rational, periodic, and hyperbolic solutions. The solutions found in various classes provide insightful information about the underlying physical mechanisms. To sum up, our current methods are superior instruments for analyzing different families of solutions in fractional-order issues.

Keywords:

Citation: Azzh Saad Alshehry, Safyan Mukhtar, Ali M. Mahnashi. Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions[J]. AIMS Mathematics, 2024, 9(6): 15966-15987. doi: 10.3934/math.2024773

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Abstract

1. Introduction

Fractional calculus (FC) has been a key tool in many scientific fields in recent years, and it has been used to formulate many novel and state of the art models. Fractional derivative operators are important because they can improve the accuracy of models even when important real parameters are not precisely known. Fractional differential operators of many kinds, including Caputo-Fabrizio, Caputo, Riemann-Liouville, and Atangana-Baleanu, are covered in the scientific literature ^[1,2,3]. The latter operators have proven to be effective in a variety of real-world models, especially in the field of biology. As an example, Ghanbari and Cattani ^[4] have successfully used the Atangana-Baleanu derivative to simulate two Lotka-Volterra models with mutualistic predation. An ideal control plan for a novel variable order fractional tumor model under immune suppression was put forth by Sweilam et al. ^[5]. Using fractional differential mathematical models that included memory effects, Danane et al. ^[6] investigated the dynamics of the hepatitis B virus infection. A new fractional model of the human liver was presented by Baleanu et al. ^[7] using the Caputo-Fabrizio fractional derivative, which is based on the exponential kernel. A novel model of fractional SIRS-SI malaria transmission was introduced by Kumar et al. ^[8]. Using the Atangana-Baleanu operator, Singh et al. ^[9] examined the dynamic behaviour of a fractional fish farming model. Readers with an interest in practical models utilising fractional derivatives are referred to references ^{[10,11,12,13]} for additional investigation. It is crucial to remember that understanding interactions in these kinds of models depends on how they are solved. This means that there are two main ways to solve mathematical models, analytically or numerically to robustly apply approximate results. Analytical approaches assert themselves as powerful mathematical tools when numerical methods fail. The application of fractional derivatives to scientific investigations has been conducted both analytically and numerically in numerous studies ^{[14,15,16,17,18,19]}. The purpose of these studies were to explore fractional derivative properties and applications in many different contexts using mathematical expressions and computational simulations ^{[20,21,22,23,24]}.

The investigation of analytical solutions for fractional partial differential equations (FPDEs) poses a significant difficulty, leading to the emergence of many mathematical approaches aimed at tackling this complex issue ^[25,26,27]. Researchers are particularly drawn to analytical solutions due to their ability to offer a comprehensive understanding of the fundamental physical processes, and to reveal the exact behavior of the modeled system, surpassing the capabilities of numerical methods ^[28,29,30]. Hence, the pursuit of analytical solutions within the domain of fractional partial differential equations (FPDEs) is regarded as a pivotal and dynamic field of study ^[31]. Within the realm of scientific literature, a multitude of mathematical techniques have been utilized to effectively address the analytical resolution of fractional partial differential equations (FPDEs). This exemplifies the wide range of methodologies employed by researchers in this field. Several techniques ^{[32,33,34,35,36,37,38,39,40]} have been notably included among the methods being considered. Every method has distinct advantages, expanding the range of techniques that may be used to solve fractional partial differential equations (FPDEs) and enhancing our understanding of these intricate mathematical models ^{[41,42,43,44,45]}.

Furthermore, a specific group of fractional partial differential equations (PDEs) presents difficulties when attempting to solve them using traditional approaches. The proposed approach ^[46,47,48], known for its ability to do complex algebraic calculations, is utilized for the derivation of solitary wave solutions, peaked wave solutions, and precise wave solutions. By employing a traveling wave transformation and the Riccati-Bernoulli equation, the specific partial differential equations (PDEs) undergo a smooth conversion into a set of algebraic equations. This method simultaneously functions as a powerful mathematical tool for solving certain issues in mathematical biology, demonstrating strong and effective performance. Further, the employed methodology produces finite solutions, resulting in reliable and efficient outcomes for the equation under investigation. Significantly, the notable characteristic of this phenomenon resides in the incorporation of an unbounded assortment of solutions for the partial differential equations. The Backlund transformation is also utilized to reveal a limitless sequence of solutions. Moreover, the present study aims to employ this methodology in order to elucidate the fractional biological population model. The results obtained make a substantial contribution to the understanding of practical physical difficulties and complex scenarios, demonstrating the wide range of scientific fields in which the suggested methodology may be effectively used.

Biological population models are essential in many areas of life and greatly advance our knowledge of the dynamics of the environment, public health, and ecology. These models are essential to ecology because they help anticipate and control species interactions, population increase, and the effects of changing environmental conditions on ecosystems. They also act as essential instruments in conservation biology, assisting researchers in determining a species' susceptibility and creating successful conservation plans. Biological population models play a key role in public health research by helping to understand how infectious diseases spread, assess programs, and forecast the start of epidemics. Numerous scientific studies have been conducted by scholars from different fields to investigate a range of epidemic models with the goal of clarifying the complexities of disease transmission dynamics and offering crucial information for the creation of successful public health interventions ^{[49,50,51,52]}. In this study, we aim at solving the fractional biological population model with the help of the presented methodology based on the Backlund transformation and based on the previous work in fractional modeling. Traditional methods are often inadequate at capturing the nuanced dynamics of biological populations due to the intricacies of ecological modeling. The purpose of this study is to build upon the foundation laid by previous research and provide new insight into the complex interplay of factors influencing population dynamics. Moreover, this paper aims to investigate a wide range of soliton wave solutions, which could reveal hitherto undiscovered phenomena and improve our comprehension of ecological systems and capacity to tackle urgent environmental problems. The mathematical model of the general biological model can be represented as ^[53]:

$\begin{equation} \left\{ \begin{aligned} & D_t^\alpha(F) = D_x^\beta(D_x^\beta(F)) + D_y^\gamma(D_y^\gamma(F)) + \chi(F), & \quad & x, y \in \Omega, \quad t \geq 0. \\ & F = F(x, y, t), & \quad & 0 < \alpha, \beta, \gamma \leq 1. \end{aligned} \right. \end{equation}$

(1.1)

Consider the following: In mathematical biology, population density is presented as the variable ( $F$ ), and the population contribution from births and deaths is presented as the variable ( $\chi$ ). For a region ( $\Omega$ ), ( $F$ ) and ( $\chi$ ) are both time ( $t$ ) and position ( $x, y$ ) dependent functions that describe the distribution dynamics of the diffusion of a biological species over ( $\Omega$ ). For a specific time ( $t$ ) and position ( $x, y$ ), the $F (x, y, t)$ function measures the number of people in a unit volume. In each given subregion, the integral of $F (x, y, t)$ offers a thorough representation of the entire population at any given ( $t$ ). Conversely, the function $\chi(F)$ explicates the average rate of individual introduction per unit volume at ( $x, y$ ) as a result of births and deaths. Three relevant cases are presented, illustrating the intricate relationship between population dynamics and the underlying processes of birth and death, and emulating the constitutive equations for $\chi(F)$ for $\alpha\rightarrow1$ .

$\begin{align} (i) & \quad \chi(F) = k, \quad \text{where } k \text{ is an arbitrary constant. This leads to the Malthusian Law.} \\ (ii) & \quad \chi(F) = F(k_1 - k_2 F), \quad \text{where } k_1 \text{ and } k_2 \text{ are positive constants. This leads to the Verhulst Law.} \\ (iii) & \quad \chi(F) = -kF^\delta, \quad \text{where } k \geq 0 \text{ and } 0 < \delta < 1. \text{ This leads to porous media.} \end{align}$

In the current investigation, we consider several forms of Eq (1.1), defining the particular forms as follows:

$\begin{equation} D_t^\alpha(F) = D_x^\beta(D_x^\beta(F)) + D_y^\gamma(D_y^\gamma(F)) + K \left(F^2 - \delta \right). \end{equation}$

(1.2)

$\begin{equation} D_t^\alpha(F) = D_x^\beta(D_x^\beta(F)) + D_y^\gamma(D_y^\gamma(F)) + F \left(1 - \delta F \right). \end{equation}$

(1.3)

The operator that represents the derivatives of order $\alpha$ adheres to the definition provided in reference ^[54].

$\begin{equation} \begin{split} D_\varphi^\beta{\rho(\varphi)} = \lim\limits_{t \to 0} \frac{\rho(t(\varphi)^{1-\beta}-\rho{(\varphi)})}{t}, 0 < \beta\leq1. \end{split} \end{equation}$

(1.4)

This investigation leverages the subsequent properties of this derivative:

$\begin{equation} \left\{ \begin{aligned} &D_\varphi^\beta \varphi^j = j \varphi^{j-\beta}.\\ &D_\varphi^\beta\left(j_1 \eta(\varphi) \pm j_2 t(\varphi)\right) = j_1 D_\varphi^\beta(\eta(\varphi)) \pm j_2 D_\varphi^\beta(t(\varphi)).\\ &D_\varphi^\beta \chi\left[\xi^\tau(\varphi)\right] = \chi_{\xi}^{\varphi}(\xi(\varphi)) D_\varphi^\beta \xi(\varphi). \end{aligned} \right. \end{equation}$

(1.5)

2. Methodology

Suppose the following fractional partial differential equation were to arise:

$\begin{equation} P_1\left(f, D^\alpha_t (f), D^\beta_{\zeta_1} (f), D^{\gamma}_{\zeta_2} (f), fD^{\beta}_{\zeta_1} (f), \ldots \right) = 0, 0 < \alpha, \beta, \gamma\leq 1. \end{equation}$

(2.1)

The above polynomial consists of both the fractional order derivatives and the nonlinear terms. The primary steps of this method will be discussed intensively, and we will consider the more sophisticated wave transformation as a possible means to observe the feasible solutions for Eq (1.1).

$\begin{equation} \begin{aligned} & F(x, y, t) = e^{i\psi} f(\psi). \end{aligned} \end{equation}$

(2.2)

where,

$\begin{equation} \begin{aligned} &\psi(x, y, t) = p \left ({\frac {{t}^{\alpha}}{\alpha}}\right )+ q \left ( {\frac {{x}^{\beta}}{\beta}} \right )+ r\left ( {\frac {{y}^{\gamma}}{\gamma}}\right ), \quad p, q, r \in R. \end{aligned} \end{equation}$

(2.3)

Whereas $\psi$ represents the phase function. These functions are related to the complex variables $x$ and $y$ , as well as the temporal variable $t$ , hence demonstrating their reliance on the precise coordinates and time. Equation (2.1) is subject to a modification resulting in the emergence of a nonlinear ordinary differential equation (NODE), assuming the modified mathematical expression.

$\begin{equation} P_2\left(f, \frac {df}{d\psi} , \frac {d^2f}{d\psi^2}, f\frac {df}{d\psi}, \ldots \right) = 0. \end{equation}$

(2.4)

Consider the formal solution for Eq (2.4)

$\begin{equation} {f}(\psi) = \sum\limits_{i = -n}^{n} b_i{\vartheta(\psi)}^i. \end{equation}$

(2.5)

The $b_i$ constants must be found under the constraint that $b_n \neq 0$ , $b_{-n} \neq 0$ at the same time. Meanwhile, the Backlund transformation that follows produces the function.

$\begin{equation} \vartheta(\psi) = \frac{-\tau Y + X \phi(\psi)}{X + Y \phi(\psi)}. \end{equation}$

(2.6)

Let ( $\tau$ ), ( $X$ ), and ( $Y$ ) be constants, with the condition that $Y\neq0$ . Additionally, let $\phi(\psi)$ be a function that may be defined as:

$\begin{equation} \frac{d{\phi}}{d\psi} = \tau +{\phi(\psi)^2}. \end{equation}$

(2.7)

The solutions of Eq (2.7) are widely recognized ^[55] to be as follows:

$\begin{align} (i) &\quad\text{If}\quad \tau < 0, \quad\text{then}\quad \phi(\psi) = -\sqrt{-\tau}\tanh(\sqrt{-\tau}\psi), \quad \text{or}\quad \phi(\psi) = -\sqrt{-\tau}coth(\sqrt{-\tau}\psi). \end{align}$

(2.8)

$\begin{align} (ii) &\quad\text{If}\quad \tau > 0, \quad\text{then}\quad \phi(\psi) = \sqrt{\tau}\tan(\sqrt{\tau}\psi), \quad\text{or}\quad \phi(\psi) = -\sqrt{\tau}cot(\sqrt{\tau}\psi). \end{align}$

(2.9)

$\begin{align} (iii) &\quad\text{If}\quad \tau = 0, \quad\text{then}\quad \phi(\psi) = \frac{-1}{\psi}. \end{align}$

(2.10)

It should be noted that, out of the theoretical framework for Eq (2.4), it is possible to describe the positive integer $(N)$ by utilizing the homogeneous balancing principles. These principles equate the nonlinear variables of Eq (2.5) in terms of the highest order derivatives. Indeed, for some degree $D[{f}(\psi)] = N$ , the degrees of the associated expressions are

$\begin{equation} D\left[\frac{d^p f}{d\psi^p}\right] = N + p, \quad D\left[f^J \frac{d^p f}{d\psi^p}\right]^s = NJ + s(p + N). \end{equation}$

(2.11)

In other words, combining Eq (2.5) and Eq (2.7) into Eq (2.4) and then redistributing the terms with respect to the power of ${f(\psi)}$ and setting them equal to zero, algebraic equations are created that may then be solved. When Maple is used, this system is algorithmically solved to yield numerical values for the parameters. As a result, this method allows one to solve accurately Eq (2.1) using the solutions, which have the characteristics of solitons.

3. Evaluation of problems

Here, we utilize the process outlined in Section 2 to determine exact single wave solutions for the fractional biological population model (1). We examine two particular scenarios within the biological population model, simplifying each by using the wave transformation described in Eq (2.2) to make the associated equations easier to understand.

3.1. Problem 1

Examine the first case of the biological population model provided by Eq (1.2). Here, we present the model following the transformation, which yields a nonlinear ordinary differential equation (NODE) from the initial fractional partial differential equation (FPDE).

$\begin{equation} \begin{aligned} p F' - \left ( q^2 + r^2 \right ) F'' - K \left ( F^2 - \delta \right) = 0. \end{aligned} \end{equation}$

(3.1)

In the study we incorporate the replacement given in Eq (2.5) into Eq (3.1) as well as Eqs (2.7) and (2.4). Carefully collecting coefficients associated with $\phi^{i}(\psi)$ , an algebraic system of equations is built and then set equal to zero. With the aid of the computational tool Maple, we are able to solve the system of algebraic equations given above and obtain the following results:

Case 1

$\begin{equation} \begin{split} &b_{{0}} = 1/2\, \sqrt {-{\tau}^{-1}}b_{{-1}}, b_{{1}} = 0, b_{{-1}} = b_{{-1}}, b_ {{-2}} = 1/2\, \tau\sqrt {-{\tau}^{-1}}b_{{-1}}, b_{{2}} = 0, p = 10\, \left( {q}^{2} +{r}^{2} \right) {\frac {1}{\sqrt {-{\tau}^{-1}}}}, q = q, \\ &r = r, \delta = -{ \frac {{b_{{-1}}}^{2}}{\tau}}, K = -12\, \tau \left( {q}^{2}+{r}^{2} \right) { \frac {1}{\sqrt {-{\tau}^{-1}}}}{b_{{-1}}}^{-1}. \end{split} \end{equation}$

(3.2)

Case 2

$\begin{equation} \begin{split} &b_{{0}} = 2\, b_{{2}}\tau, b_{{1}} = 4\, \sqrt {-\tau}b_{{2}}, b_{{-1}} = -4\, \sqrt { }b_{{2}}\tau, b_{{-2}} = b_{{2}}{\tau}^{2}, b_{{2}} = b_{{2}}, p = -20\, \sqrt {-\tau} \left( {q}^{2}+{r}^{2} \right) , q = q, \\ &r = r, \delta = 64\, {b_{{2}}}^{2}{\tau}^{ 2}, K = -6\, {\frac {{q}^{2}+{r}^{2}}{b_{{2}}}}. \end{split} \end{equation}$

(3.3)

Assuming Case 1, we get the following families of solutions for,

$\begin{equation} \begin{aligned} \psi = {\frac {q{x}^{\beta}}{\beta}}+{\frac {r{y}^{\gamma}}{\gamma}}+10\, \left( {r}^{2}+{q}^{2} \right) {t}^{\alpha}{\frac {1}{\sqrt {-{\tau}^{-1 }}}}{\alpha}^{-1}.\\ \end{aligned} \end{equation}$

(3.4)

Family 1: When $\tau < 0$ , Eq (1.2) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} F_1(x, y, t) = &e^{i\psi} \left[ \frac{1}{2}\tau\sqrt{-\tau^{-1}}b_{-1} \left( X-Y\sqrt{-\tau}\tanh \left( \sqrt{-\tau}\psi \right) \right)^{2} \right.\left( -{\tau}Y-X\sqrt{-\tau}\tanh \left( \sqrt{-\tau}\psi \right) \right)^{-2} \\ &+ \frac{b_{-1} \left( X-Y\sqrt{-\tau}\tanh \left( \sqrt{-\tau}\psi \right) \right)}{-{\tau}Y-X\sqrt{-\tau}\tanh \left( \sqrt{-\tau}\psi \right)}+ \left. \frac{1}{2}\sqrt{-\tau^{-1}}b_{-1} \right]. \end{split} \end{equation}$

(3.5)

$\begin{equation} \begin{split} F_2(x, y, t) = &e^{i\psi} \left[ \frac{1}{2}\tau\sqrt{-\tau^{-1}}b_{-1} \left( X-Y\sqrt{-\tau}\coth \left( \sqrt{-\tau}\psi \right) \right)^{2} \right.\left( -{\tau}Y-X\sqrt{-\tau}\coth \left( \sqrt{-\tau}\psi \right) \right)^{-2} \\ &+ \frac{b_{-1} \left( X-Y\sqrt{-\tau}\coth \left( \sqrt{-\tau}\psi \right) \right)}{-{\tau}Y-X\sqrt{-\tau}\coth \left( \sqrt{-{\tau}}\psi \right)}+ \left. \frac{1}{2}\sqrt{-{\tau}^{-1}}b_{-1} \right]. \end{split} \end{equation}$

(3.6)

Family 2: When ${\tau} > 0$ , Eq (1.2) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} F_3(x, y, t) = & e^{i\psi} \left[ \frac{1}{2}{\tau}\sqrt{-{\tau}^{-1}}b_{-1} \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^{2} \right. \left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^{-2} \\ &+ \frac{b_{-1} \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)}{-{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right)}+ \left. \frac{1}{2}\sqrt{-{\tau}^{-1}}b_{-1} \right]. \end{split} \end{equation}$

(3.7)

$\begin{equation} \begin{split} F_4(x, y, t) = & e^{i\psi} \left[ \frac{1}{2}{\tau}\sqrt{-{\tau}^{-1}}b_{-1} \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^{2} \right.\left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^{-2} \\ &+ \frac{b_{-1} \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right)}+ \left. \frac{1}{2}\sqrt{-{\tau}^{-1}}b_{-1} \right]. \end{split} \end{equation}$

(3.8)

Family 3: When ${\tau} = 0$ , Eq (1.2) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} &F_5(x, y, t) = e^{i\psi}\left[1/2\, {\tau}\sqrt {-{{\tau}}^{-1}}b_{{-1}} \left( X-{\frac {Y}{\psi}} \right) ^{2 } \left( -{\tau}Y-{\frac {X}{\psi}} \right) ^{-2}+b_{{-1}} \left( X-{\frac {Y}{\psi}} \right) \left( -{\tau}Y-{\frac {X}{\psi}} \right) ^{-1}+1/2\, \sqrt {-{{\tau}}^{-1}}b_{{-1}}\right]. \end{split} \end{equation}$

(3.9)

Assuming Case 2, we get the following families of solutions for,

$\begin{equation} \begin{aligned} \psi = {\frac {q{x}^{\beta}}{\beta}}+{\frac {r{y}^{\gamma}}{\gamma}}-20\, { \frac {\sqrt {-{\tau}} \left( {r}^{2}+{q}^{2} \right) {t}^{\alpha}}{\alpha}}.\\ \end{aligned} \end{equation}$

(3.10)

Family 1: When ${\tau} < 0$ , Eq (1.2) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} F_6(x, y, t) = & e^{i\psi}\left[ \frac{b_2{\tau}^2 \left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right.- 4 \frac{\sqrt{-{\tau}}b_2{\tau} \left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right)} \\ & + 2b_2{\tau} + 4 \frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)}{X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right)}+ \left. \frac{b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.11)

$\begin{equation} \begin{split} F_7(x, y, t) = & e^{i\psi}\left[ \frac{b_2{\tau}^2 \left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right.- 4 \frac{\sqrt{-{\tau}}b_2{\tau} \left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right)} \\ & + 2b_2{\tau} + 4 \frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)}{X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right)}+ \left. \frac{b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.12)

Family 2: When ${\tau} > 0$ , Eq (1.2) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} F_8(x, y, t) = & e^{i\psi}\left[ \frac{b_2{\tau}^2 \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2} \right.- 4\frac{\sqrt{-{\tau}}b_2{\tau} \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)}{-{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right)} + 2b_2{\tau} \\ & + \frac{4\sqrt{-{\tau}}b_2 \left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)}{X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right)}.+ \left. \frac{b_2 \left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.13)

$\begin{equation} \begin{split} F_9(x, y, t) = & e^{i\psi}\left[ \frac{b_2{\tau}^2 \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2} \right.- 4\frac{\sqrt{-{\tau}}b_2{\tau} \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right)}+ 2b_2{\tau} \\ & + \left. 4\frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)}{X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right)} \right.+ \left. \frac{b_2 \left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.14)

Family 3: When ${\tau} = 0$ , Eq (1.2) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} F_{10}(x, y, t) = & e^{i\psi} \left[ b_2{\tau}^2 \left( X-{\frac {Y}{\psi}} \right)^2 \left( -{\tau}Y-{\frac {X}{\psi}} \right)^{-2} \right.- 4\sqrt{-{\tau}}b_2{\tau} \left( X-{\frac {Y}{\psi}} \right) \left( -{\tau}Y-{\frac {X}{\psi}} \right)^{-1} + 2b_2{\tau} \\ & + \left. 4\sqrt{-{\tau}}b_2 \left( -{\tau}Y-{\frac {X}{\psi}} \right) \left( X-{\frac {Y}{\psi}} \right)^{-1} \right.+ \left. b_2 \left( -{\tau}Y-{\frac {X}{\psi}} \right)^2 \left( X-{\frac {Y}{\psi}} \right)^{-2} \right]. \end{split} \end{equation}$

(3.15)

3.2. Problem 2

Examine the second case of the biological population model provided by Eq (1.3). Here, we present the model following the transformation, which yields a nonlinear ordinary differential equation (NODE) from the initial fractional partial differential equation (FPDE).

$\begin{equation} \begin{aligned} p F' - \left ( q^2 + r^2 \right ) F'' - F \left ( 1 - \delta F \right) = 0. \end{aligned} \end{equation}$

(3.16)

In the study we incorporate the replacement given in Eq (2.5) into Eq (3.16) as well as Eqs (2.7) and (2.4). Carefully collecting coefficients associated with $\phi^{i}(\psi)$ , an algebraic system of equations is built and then set equal to zero. With the aid of the computational tool Maple, we are able to solve the system of algebraic equations given above and obtain the following results:

Case 1

$\begin{equation} \begin{split} &b_{{0}} = -{\frac {b_{{-2}}}{{\tau}}}, b_{{1}} = 0, b_{{-1}} = 2\, \sqrt {-{{\tau}}^{-1} }b_{{-2}}, b_{{-2}} = b_{{-2}}, b_{{2}} = 0, p = -{\frac {5}{12}}\, \sqrt {-{{\tau}}^ {-1}}, \\ &q = 1/12\, \sqrt {-{\frac {6+144\, {r}^{2}{\tau}}{{\tau}}}}, r = r, \delta = -1/4\, { \frac {{\tau}}{b_{{-2}}}}, K = K. \end{split} \end{equation}$

(3.17)

Case 2

$\begin{equation} \begin{split} &b_{{0}} = 10\, b_{{2}}{\tau}, b_{{1}} = 4\, \sqrt {-{\tau}}b_{{2}}, b_{{-1}} = -4\, \sqrt {-{\tau}}b_{{2}}{\tau}, b_{{-2}} = b_{{2}}{{\tau}}^{2}, b_{{2}} = b_{{2}}, p = -{\frac {5}{24} }\, {\frac {\sqrt {-{\tau}}}{{\tau}}}, \\ &q = 1/24\, \sqrt {-{\frac {-6+576\, {r}^{2}{\tau}}{{\tau} }}}, r = r, \delta = 1/16\, {\frac {1}{b_{{2}}{\tau}}}, K = K. \end{split} \end{equation}$

(3.18)

Assuming Case 1, we get the following families of solutions for,

$\begin{equation} \begin{aligned} &\psi = 1/12\, \sqrt {-{\frac {6+144\, {r}^{2}{\tau}}{{\tau}}}}{x}^{\beta}{\beta}^{-1}+{ \frac {r{y}^{\gamma}}{\gamma}}-{\frac {5}{12}}\, \sqrt {-{{\tau}}^{-1}}{t}^{ \alpha}{\alpha}^{-1}. \end{aligned} \end{equation}$

(3.19)

Family 1: When ${\tau} < 0$ , Eq (1.3) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} G_1(x, y, t) = & e^{i\psi} \left[ \frac{b_{-2} \left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\Psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right.+ 2\sqrt{-{\tau}^{-1}}b_{-2} \left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)\\ &\left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^{-1} - \left. \frac{b_{-2}}{{\tau}} \right]. \end{split} \end{equation}$

(3.20)

$\begin{equation} \begin{split} G_2(x, y, t) = & e^{i\psi} \left[ \frac{b_{-2} \left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\Psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right.+ 2\sqrt{-{\tau}^{-1}}b_{-2} \left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)\\ &\left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^{-1} - \left. \frac{b_{-2}}{{\tau}} \right]. \end{split} \end{equation}$

(3.21)

Family 2: When ${\tau} > 0$ , Eq (1.3) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} G_3(x, y, t) = & e^{i\psi} \left[ \frac{b_{-2} \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2} \right. + 2\sqrt{-{\tau}^{-1}}b_{-2} \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)\\ &\left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^{-1} - \left. \frac{b_{-2}}{{\tau}} \right]. \end{split} \end{equation}$

(3.22)

$\begin{equation} \begin{split} G_4(x, y, t) = & e^{i\psi} \left[ \frac{b_{-2} \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2} \right. + 2\sqrt{-{\tau}^{-1}}b_{-2} \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right) \\ &\left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^{-1}- \left. \frac{b_{-2}}{{\tau}} \right]. \end{split} \end{equation}$

(3.23)

Family 3: When ${\tau} = 0$ , Eq (1.3) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} &G_5(x, y, t) = e^{i\psi} \left[b_{{-2}} \left( X-{\frac {Y}{\psi}} \right) ^{2} \left( -{\tau}Y-{\frac {X} {\psi}} \right) ^{-2}+2\, \sqrt {-{{\tau}}^{-1}}b_{{-2}} \left( X-{\frac {Y} {\psi}} \right) \left( -{\tau}Y-{\frac {X}{\psi}} \right) ^{-1}-{\frac {b_ {{-2}}}{{\tau}}}\right]. \end{split} \end{equation}$

(3.24)

Assuming Case 2, we get the following families of solutions for,

$\begin{equation} \begin{aligned} &\psi = -{\frac {1}{96}}\, {\frac { \left( -1+96\, {r}^{2}{\tau} \right) {x}^{\beta}} {{\tau}\beta}}+{\frac {r{y}^{\gamma}}{\gamma}}-{\frac {5}{24}}\, {\frac { \sqrt {-{\tau}}{t}^{\alpha}}{{\tau}\alpha}}. \end{aligned} \end{equation}$

(3.25)

Family 4: When ${\tau} < 0$ , Eq (1.3) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} G_6(x, y, t) = & e^{i\psi} \left[ \frac{b_2{\tau}^2 \left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right. - 4\frac{\sqrt{-{\tau}}b_2{\tau} \left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right)} \\ & + 10b_2{\tau} + 4\frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)}{X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right)} + \left. \frac{b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( X-Y\sqrt{-{\tau}}\tanh \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.26)

$\begin{equation} \begin{split} G_7(x, y, t) = & e^{i\psi} \left[ \frac{b_2{\tau}^2 \left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right. - 4\frac{\sqrt{-{\tau}}b_2{\tau} \left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right)} \\ & + 10b_2{\tau} + 4\frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)}{X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right)} + \left. \frac{b_2 \left( -{\tau}Y-X\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2}{\left( X-Y\sqrt{-{\tau}}\coth \left( \sqrt{-{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.27)

Family 5: When ${\tau} > 0$ , Eq (1.3) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} G_8(x, y, t) = & e^{i\psi} \left[ \frac{b_2{\tau}^2 \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2} \right.- 4\frac{\sqrt{-{\tau}}b_2{\tau} \left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)}{-{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right)} \\ & + 10b_2{\tau} + 4\frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)}{X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right)}+ \left. \frac{b_2 \left( -{\tau}Y+X\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( X+Y\sqrt{{\tau}}\tan \left( \sqrt{{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.28)

$\begin{equation} \begin{split} G_9(x, y, t) = & e^{i\psi} \left[ \frac{b_2{\tau}^2 \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2} \right. - 4\frac{\sqrt{-{\tau}}b_2{\tau} \left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)}{-{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right)} \\ & + 10b_2{\tau} + 4\frac{\sqrt{-{\tau}}b_2 \left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)}{X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right)} + \left. \frac{b_2 \left( -{\tau}Y-X\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2}{\left( X-Y\sqrt{{\tau}}\cot \left( \sqrt{{\tau}}\psi \right) \right)^2} \right]. \end{split} \end{equation}$

(3.29)

Family 6: When ${\tau} = 0$ , Eq (1.3) brings about the resulting single-wave solutions:

$\begin{equation} \begin{split} G_{10}(x, y, t) = & e^{i\psi} \Bigg[ b_2{{\tau}}^2 \left( X-{\frac{Y}{\psi}} \right)^2 \left( -{\tau}Y-{\frac{X}{\psi}} \right)^{-2} - 4\sqrt{-{\tau}}b_2{\tau} \left( X-{\frac{Y}{\psi}} \right) \left( -{\tau}Y-{\frac{X}{\psi}} \right)^{-1} \\ & + 10b_2{\tau} + 4\sqrt{-{\tau}}b_2 \left( -{\tau}Y-{\frac{X}{\psi}} \right) \left( X-{\frac{Y}{\psi}} \right)^{-1} + b_2 \left( -{\tau}Y-{\frac{X}{\psi}} \right)^2 \left( X-{\frac{Y}{\psi}} \right)^{-2} \Bigg]. \end{split} \end{equation}$

(3.30)

4. Results and discussion

When it comes to biological population models, the Riccati-Bernoulli sub-ODE approach is a potent analytical technique that is especially useful in the field of advanced biological sciences. The efficiency of the approach resides in its capacity to provide a multitude of periodic and single traveling wave solutions, each with unique properties. Interestingly, this is accomplished without the need of discretization and linearization steps, which are frequently used in problem-solving techniques. Consequently, the results provided by this method are accurate solutions for various biological population models in modern biological science. Specifically, this method contributes to knowing the dynamics of the constituents of biological populations through revealing the complicated rules that regulate biological events. This method is context-aware for the subtleties of biological systems and is further complimented by its ability to produce multiple solutions with different parameters.

Additionally, the preferred families of analytical solutions produced by the method provide appropriate references for solving biological modeling analytically. From an analytical perspective, it enables accurate prediction of analytical results and also supports stability investigations, which helps make computer models of biological populations accurate. Soliton generation due to the solitons, ability to travel with minimum energy loss is another feature of the method. When describing wave motion in biological population models, soliton dynamics follows the same rules. Solitons represent population dynamics and phenomena that can spread a lot and maintain their shape even after interacting with other population waves in a unique biological population model. This fact that phenomena resulting in solitons emerge as a consequence of this unique emphasis on linear and nonlinear effects grants us a fresh perspective on how populations act in higher biological science. An effective and computationally practical way for addressing complex algebraic computations is the Riccati-Bernoulli sub-ODE method. It is commonly known that creating a general analytical technique that can be used for any type of nonlinear partial differential equation (NLPDE) is extremely difficult, and the Riccati-Bernoulli sub-ODE method is no different. This approach, like other analytical techniques, aims to get exact solutions for Eq 1.1. It is crucial to remember that efforts to improve the effectiveness of the Backlund and sub-ODE transformations are still underway. Subsequent efforts will concentrate on developing these approaches to unleash even more potent potential, which will enable the identification of precise solutions for a wider range of NLPDEs. For the problems considered, the proposed approach produces three different families of solitary wave solutions. The solitary wave solutions that fall into these categories are rational when ( $\tau = 0$ ), hyperbolic when ( $\tau > 0$ ), and periodic when ( $\tau < 0$ ). Figures 1–6 show the wave behaviors of these solutions graphically, giving a thorough knowledge of their dynamic properties.

Figure 1. Within these visualizations, differing degrees of granularity are displayed for the complex factors of

$F_{5}(x, y, t)$ .

DownLoad: Full-Size Img PowerPoint

Figure 2. Within these visualizations, differing degrees of granularity are displayed for the complex factors of

$F_{8}(x, y, t)$ .

DownLoad: Full-Size Img PowerPoint

Figure 3. Within these visualizations, differing degrees of granularity are displayed for the complex factors of

$F_{10}(x, y, t)$ . in this plots.

DownLoad: Full-Size Img PowerPoint

Figure 4. Within these visualizations, differing degrees of granularity are displayed for the complex factors of

$G_{1}(x, y, t)$ .

DownLoad: Full-Size Img PowerPoint

Figure 5. Within these visualizations, differing degrees of granularity are displayed for the complex factors of

$G_{5}(x, y, t)$ .

DownLoad: Full-Size Img PowerPoint

Figure 6. Within these visualizations, differing degrees of granularity are displayed for the complex factors of

$G_{9}(x, y, t)$ .

DownLoad: Full-Size Img PowerPoint

5. Conclusions

In this current work, Riccati-Bernoulli sub-ODE techniques are systematically applied to fractional-order biological population models. The graphical illustrations of the acquired solutions validate the effectiveness of the suggested strategies. The analysis indicates a change in the geometric features of the issues under various circumstances. Notably, the new approaches provide a more insightful and efficient comprehension of the dynamics inherent in the researched physical phenomena when compared to alternative analytical methods. The investigation includes three important traveling wave solutions: solutions for rational, hyperbolic, and trigonometric functions. These answers add to a thorough comprehension of the various behaviors displayed by various physical events. In addition, the current method's adaptability is investigated in order to obtain new families of solutions for the given problem. The improved method makes it easier to design solutions that are closely matched with the physical challenges that arise in applied research by adding a free parameter to the solution.

Authors contributions

All authors equally contributed this manuscript and approved the final version.

Use of AI tools declaration

The authors declare they have not used AI tools in the creation of this article.

Acknowledgments

Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R183), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. This work was supported by the Deanship of Scientific Research, Vice Presidency for Graduate Studies and Scientific Research, King Faisal University, Saudi Arabia (GrantA134).

Data availability

The data related to this study is available from the corresponding author upon reasonable request.

Funding

Conflict of interest

The authors declare no conflicts of interest.

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AIMS Mathematics

Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

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3.2. Problem 2

4. Results and discussion

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2. Methodology

3. Evaluation of problems

3.1. Problem 1

3.2. Problem 2

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5. Conclusions

Authors contributions

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AIMS Mathematics

Analytical methods in fractional biological population modeling: Unveiling solitary wave solutions

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Abstract

1. Introduction

2. Methodology

3. Evaluation of problems

3.1. Problem 1

3.2. Problem 2

4. Results and discussion

5. Conclusions

Authors contributions

Use of AI tools declaration

Acknowledgments

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Abstract

1. Introduction

2. Methodology

3. Evaluation of problems

3.1. Problem 1

3.2. Problem 2

4. Results and discussion

5. Conclusions

Authors contributions

Use of AI tools declaration

Acknowledgments

Data availability

Funding

Conflict of interest

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