Application of homotopy perturbation and variational iteration methods for nonlinear imprecise prey–predator model with stability analysis

Abstract

A coupled nonlinear prey–predator system is presented. The system formulation is based on nonlinear ordinary differential equations with imprecise parameter values. In this paper, we find the equilibrium point and conduct a stability analysis of the prey–predator model using a Lyapunov functional. A comparison of our approximate analytical expressions with numerical simulation using MATLAB software is also presented. Furthermore, the proposed mathematical model is solved analytically by using the VIM and HPM for all possible parameter values in their specified ranges. Excellent agreement is noted on comparisons between the analytical and numerical results.

Appendices

Appendix A

Basic principle of the HPM

Let us consider the following equation:

$$C(u) - g(t) = 0,{\text{ t}} \in \Psi ,$$

(A.1)

with the boundary condition

$$D(\partial x/\partial n) = 0,{\text{ s}} \in \Lambda {\text{,}}$$

(A.2)

C is divided into two parts: L - a linear part and N - a nonlinear part.

Eq. (A1) can be rewritten as

$$L(x) + N(x) - g(t) = 0$$

(A.3)

We construct a homotopy of Eq. (A1) $x(s,p):\psi \times [0,1] \to \Re$, which satisfies

$$H(x,p) = (1 - p)[L(x) - L(x_{0} )] + p[C(x) - g(t)] = 0,{\text{ p}} \in {\text{[0,1], s}} \in \Psi$$

(A.4)

$$H(x,p) = L(x) - L(x_{0} ) + pL(x_{0} ) + p[N(x) - g(t)] = 0,$$

(A.5)

where $\rho \in [0,{\text{ }}1]$—an embedding parameter. It follows from (A.4) and (A.5) that

$$H(s,0) = L(x) - L(x_{0} ) = 0,{\text{ H(x,1)}} = {\text{C(x) - g(t)}} = {\text{0}}$$

(A.6)

Thus, the changing process of p from zero to unity is just that of x(s, p) to x(s). In topology, L(x)-L(x₀) is a deformation, and C(x)-g(t) are homotopic.

$$x = x_{0} + px_{1} + p^{2} x_{2} + ...........$$

(A.7)

The approximate solution of Eq. (A1) is

$$x = \mathop {\lim }\limits_{{p \to 1}} x = x_{0} + x_{1} + x_{2} + ........$$

(A.8)

Appendix B

Basic principle of the VIM

We consider the following differential equation:

$$L\left( x \right) + N\left( x \right) = g\left( t \right)$$

(B.1)

L (x)- linear term, N(x)- nonlinear term and g(t) - inhomogeneous term.

According to the VIM,

$$x_{{n + 1}} (t) = x_{n} (t) + \int\limits_{0}^{t} {\eta (Lx_{n} (\xi ) + N} \mathop x\limits^{\sim } (\xi ) - g(\xi ))d\xi$$

(B.2)

$\eta$—general Lagrangian multiplier.

The subscript n indicates the n^th approximation, and $\mathop {x_{n} }\limits^{\sim }$ is considered a restricted variation $\mathop {\delta x_{n} }\limits^{\sim } = 0$.

Appendix C

Implementation of the model without harvesting

Estimated analytical solutions of the system of Eqs. (1–3) using the HPM [21,22,23]

A homotopy is constructed as follows:

$$(1 - p)\left[ {\frac{{dx}}{{dt}} - rx} \right] + p\left[ {\frac{{dx}}{{dt}} - rx + \frac{{rx^{2} }}{k} + \alpha xy} \right] = 0$$

(C.1)

$$(1 - p)\left[ {\frac{{dy}}{{dt}} + dy} \right] + p\left[ {\frac{{dy}}{{dt}} + dy - m\alpha xy + \delta y^{2} } \right] = 0$$

(C.2)

$$x_{0} \left( 0 \right) = m_{1} ;y_{0} \left( 0 \right) = m_{2} ;$$

(C.3)

$$x = x_{0} + px_{1} + p^{2} x_{2} + ........$$

(C.4)

$$y = y_{0} + py_{1} + p^{2} y_{2} + ........$$

(C.5)

By substituting Eqs. (C.4–C.5) into Eqs. (C.1) and (C.2) and comparing the coefficients of the like powers of P, we obtain

$$p^{0} :\frac{{dx_{0} }}{{dt}} - rx_{0} = 0$$

(C.6)

$$p^{1} :\frac{{dx_{1} }}{{dt}} - rx_{1} + \frac{{rx_{0} ^{2} }}{k} + \alpha x_{0} y_{0} = 0$$

(C.7)

$$p^{2} :\frac{{dx_{2} }}{{dt}} - rx_{2} + \frac{{2rx_{0} x_{1} }}{k} + \alpha x_{0} y_{1} + \alpha x_{1} y_{0} = 0$$

(C.8)

$$p^{0} :\frac{{dy_{0} }}{{dy}} + dy_{0} = 0$$

(C.9)

$$p^{1} :\frac{{dy_{1} }}{{dt}} + dy_{1} - m\alpha x_{0} y_{0} + \delta y_{0} ^{2} = 0$$

(C.10)

$$p^{2} :\frac{{dy_{2} }}{{dt}} + dy_{2} - m\alpha x_{0} y_{1} - m\alpha x_{1} y_{0} + 2\delta y_{0} y_{1} = 0$$

(C.11)

By solving Eqs. (C.6–C.11) with the conditions (C.3), we obtain the following solutions:

$$x_{0} (t) = m_{1} e^{{rt}}$$

(C.12)

$$y_{0} (t) = m_{2} e^{{ - dt}}$$

(C.13)

$$x_{1} (t) = \left( {\frac{{m_{1} ^{2} }}{k} - \frac{{\alpha m_{1} m_{2} }}{d}} \right)e^{{rt}} - \frac{{m_{1} ^{2} }}{k}e^{{2rt}} + \frac{{\alpha m_{1} m_{2} }}{d}e^{{(r - d)t}}$$

(C.14)

$$y_{1} (t) = \left( {\frac{{ - \alpha mm_{{_{1} }} m_{2} }}{r} - \frac{{\delta m_{2} ^{2} }}{d}} \right)e^{{ - {\text{d}}t}} + \frac{{\alpha mm_{{_{1} }} m_{2} }}{r}e^{{(r - d)t}} + \frac{{\delta m_{2} ^{2} }}{d}e^{{ - 2{\text{d}}t}}$$

(C.15)

$$\begin{gathered} x_{2} (t) = \left( \begin{gathered} \frac{{2m_{1} ^{3} }}{{k^{2} }} - \frac{{2\alpha m_{1} ^{2} m_{2} }}{{dk}} - \frac{{m_{1} ^{3} }}{{k^{2} }} + \frac{{2r\alpha m_{1} ^{2} m_{2} }}{{dk(r - d)}} + \frac{{\alpha ^{2} m_{1} ^{2} mm_{2} }}{{rd}} + \frac{{\delta \alpha m_{1} m_{2} ^{2} }}{{d^{2} }} + \hfill \\ \frac{{\alpha ^{2} mm_{1} ^{2} m_{2} }}{{r(r - d)}} - \frac{{\delta \alpha m_{1} m_{2} ^{2} }}{{2d^{2} }} - \frac{{\alpha m_{1} ^{2} m_{2} }}{{dk}} + \frac{{\alpha ^{2} m_{1} m_{2} ^{2} }}{{d^{2} }} - \frac{{\alpha m_{1} ^{2} m_{2} }}{{(r - d)k}} - \frac{{\alpha m_{1} m_{2} ^{2} }}{{2d^{2} }} \hfill \\ \end{gathered} \right)e^{{rt}} - \frac{{2m_{1} ^{3} }}{{k^{2} }}e^{{2rt}} \hfill \\ {\text{ }} + \frac{{2\alpha m_{1} ^{2} m_{2} }}{{dk}}e^{{2rt}} {\text{ }} + \frac{{m_{1} ^{3} }}{{k^{2} }}e^{{3rt}} - \frac{{2r\alpha m_{1} ^{2} m_{2} }}{{dk(r - d)}}e^{{(2r - d)t}} - \frac{{\alpha ^{2} m_{1} ^{2} mm_{2} }}{{rd}}e^{{(r - d)t}} - \frac{{\delta \alpha m_{1} m_{2} ^{2} }}{{d^{2} }}e^{{(r - d)t}} \hfill \\ {\text{ }} - \frac{{\alpha ^{2} mm_{1} ^{2} m_{2} }}{{r(r - d)}}e^{{(2r - d)t}} + \frac{{\delta \alpha m_{1} m_{2} ^{2} }}{{2d^{2} }}e^{{(r - 2d)t}} {\text{ }} + \frac{{\alpha m_{1} ^{2} m_{2} }}{{dk}}e^{{(r - d)t}} - \frac{{\alpha ^{2} m_{1} m_{2} ^{2} }}{{d^{2} }}e^{{(r - d)t}} \hfill \\ {\text{ }} + \frac{{\alpha m_{1} ^{2} m_{2} }}{{(r - d)k}}e^{{(2r - d)t}} + \frac{{\alpha m_{1} m_{2} ^{2} }}{{2d^{2} }}e^{{(r - 2d)t}} \hfill \\ \end{gathered}$$

(C.16)

$$\begin{gathered} y_{2} (t) = \left( \begin{gathered} \frac{{\alpha ^{2} m^{2} m_{{_{1} }} ^{2} m_{2} }}{{r^{2} }} + \frac{{\delta \alpha mm_{1} m_{2} }}{{rd}} - \frac{{\alpha ^{2} m^{2} m_{1} ^{2} m_{2} }}{{2r^{2} }} - \frac{{m\alpha \delta m_{1} m_{2} ^{2} }}{{d(r - d)}} - \frac{{m\alpha m_{1} ^{2} m_{2} }}{{kr}} + \frac{{m\alpha ^{2} m_{1} m_{2} ^{2} }}{{rd}} \hfill \\ + \frac{{\alpha mm_{1} ^{2} m_{2} }}{{2rk}} - \frac{{m\alpha ^{2} m_{1} m_{2} ^{2} }}{{d(r - d)}} + \frac{{2\delta \alpha mm_{1} m_{2} ^{2} }}{{rd}} + \frac{{2\delta ^{2} m_{2} ^{3} }}{{d^{2} }} + \frac{{2\delta \alpha mm_{1} m_{2} ^{2} }}{{r(r - d)}} - \frac{{\delta ^{2} m_{2} ^{3} }}{{d^{2} }} \hfill \\ \end{gathered} \right)e^{{ - {\text{d}}t}} \hfill \\ {\text{ }} - \frac{{\alpha ^{2} m^{2} m_{{_{1} }} ^{2} m_{2} }}{{r^{2} }}e^{{(r - d)t}} {\text{ }} - \frac{{\delta \alpha mm_{1} m_{2} }}{{rd}}e^{{(r - d)t}} + \frac{{\alpha ^{2} m^{2} m_{1} ^{2} m_{2} }}{{2r^{2} }}e^{{(2r - d)t}} + \frac{{m\alpha \delta m_{1} m_{2} ^{2} }}{{d(r - d)}}e^{{(r - 2d)t}} \hfill \\ {\text{ }} + \frac{{m\alpha m_{1} ^{2} m_{2} }}{{kr}}e^{{(r - d)t}} - \frac{{m\alpha ^{2} m_{1} m_{2} ^{2} }}{{rd}}e^{{(r - d)t}} {\text{ }} - \frac{{\alpha mm_{1} ^{2} m_{2} }}{{2rk}}e^{{(2r - d)t}} + \frac{{m\alpha ^{2} m_{1} m_{2} ^{2} }}{{d(r - d)}}e^{{(r - 2d)t}} \hfill \\ {\text{ }} - \frac{{2\delta \alpha mm_{1} m_{2} ^{2} }}{{rd}}e^{{ - 2dt}} - \frac{{2\delta ^{2} m_{2} ^{3} }}{{d^{2} }}e^{{ - 2{\text{d}}t}} - \frac{{2\delta \alpha mm_{1} m_{2} ^{2} }}{{r(r - d)}}e^{{(r - 2d)t}} + \frac{{\delta ^{2} m_{2} ^{3} }}{{d^{2} }}e^{{ - 3{\text{d}}t}} \hfill \\ \end{gathered}$$

(C.17)

According to the HPM, we can finalize that

$$x(t) = \mathop {\lim x(t)}\limits_{{p \to 1}} = x_{0} + x_{1} + x_{2} + .......................$$

(C.18)

$$y(t) = \mathop {\lim y(t)}\limits_{{p \to 1}} = y_{0} + y_{1} + y_{2} + .......................$$

(C.19)

After substituting Eqs. (C.11–C.16) into Eqs. (C.18) and (C.19), we obtain the final solutions, which can be described in the equation in the text.

Appendix D

We drive the general solution of nonlinear equations using the VIM.

Implementation of the model without harvesting

We are given the following nonlinear differential equation:

$$L[x(t)] + N[x(t)] = g(t)$$

(D.1)

Here, L(t) is a linear operator, N(t) is a nonlinear operator, and g(t) is a given function. The variational iteration method can be established and analysed using a correct functional as follows:

$$x_{{n + 1}} (t) = x_{n} (t) + \int\limits_{0}^{t} {\varphi \{ \left[ {L[x_{n} (\tau )]} \right]} + [N[\mathop {x_{n} }\limits^{\sim } (\tau )]] - g[\tau ]\} d\tau$$

(D.2)

where $\varphi$ is a general Lagrange multiplier, $u_{n}$ is the net order approximate solution, and $\tilde{u}_{n}$ denotes a restricted variation,

$${\text{i}}.{\text{e}}.,\,\psi u_{n} = 0.$$

(D.3)

$$x_{{n + 1}} (t) = x_{n} (t) + \int\limits_{0}^{t} {\varphi _{1} \{ x_{n} ^{'} (\chi ) - r} x_{n} (\chi ) + \frac{r}{k}\mathop {\left[ {x_{n} ^{2} (\chi )} \right]}\limits^{{\sim \sim \sim \sim }} + \mathop {\left[ {\alpha x_{n} (\chi )y_{n} (\chi )} \right]}\limits^{{\sim \sim \sim \sim \sim \sim \sim \sim \sim \sim }} \} d\chi$$

(D.4)

$$y_{{n + 1}} (t) = y_{n} (t) + \int\limits_{0}^{t} {\varphi _{2} \{ y_{n} ^{'} (\chi ) - \mathop {\left[ {m\alpha x_{n} (\chi )y_{n} (\chi )} \right]}\limits^{{\sim \sim \sim \sim \sim \sim \sim \sim \sim \sim }} + d} y_{n} (\chi ) + \mathop {\left[ {\delta y_{n} ^{2} (\chi )} \right]}\limits^{{\sim \sim \sim \sim }} \} d\chi$$

(D.5)

$$\psi x_{{n + 1}} (t) = \psi x_{n} (t) + \psi \int\limits_{0}^{t} {\varphi _{1} \{ x_{n} ^{'} (\chi ) - r} x_{n} (\chi ) + \frac{r}{k}\mathop {\left[ {x_{n} ^{2} (\chi )} \right]}\limits^{{\sim \sim \sim \sim }} + \mathop {\left[ {\alpha x_{n} (\chi )y_{n} (\chi )} \right]}\limits^{{\sim \sim \sim \sim \sim \sim \sim \sim \sim \sim }} \} d\chi$$

(D.6)

$$\psi y_{{n + 1}} (t) = \psi y_{n} (t) + \psi \int\limits_{0}^{t} {\varphi _{2} \{ y_{n} ^{'} (\chi ) - \mathop {\left[ {m\alpha x_{n} (\chi )y_{n} (\chi )} \right]}\limits^{{\sim \sim \sim \sim \sim \sim \sim \sim \sim \sim }} + d} y_{n} (\chi ) + \mathop {\left[ {\delta y_{n} ^{2} (\chi )} \right]}\limits^{{\sim \sim \sim \sim }} \} d\chi$$

(D.7)

where $\varphi _{1}$ and $\varphi _{2}$ are general Lagrange multipliers and $x_{0}$ and $y_{0}$ are initial approximation functions.

$x_{n} ^{2} (\chi ),y_{n} ^{2} (\chi ){\text{ }}and{\text{ }}m\alpha x_{n} (\chi )y_{n} (\chi )$ are restricted variations,

i.e., $\psi \mathop {x_{n} }\limits^{\sim } = 0,\psi \mathop {y_{n} }\limits^{\sim } = 0,\,and\,\psi \mathop {x_{n} }\limits^{\sim } \mathop {y_{n} }\limits^{\sim } = 0$$,\psi x_{n} (0) = 0,\psi y_{n} (0) = 0\,and\,\psi x_{n} (0)y_{n} (0) = 0.$

$$\psi x_{n} :1 + \varphi _{1} (\chi )|_{{\chi = t}} = 0,\psi y_{n} :1 + \varphi _{2} (\chi )|_{{\chi = t}} = 0,$$

(D.8)

$$\psi x_{n} : - \varphi _{1} ^{'} (\chi ) - r\varphi _{1} (\chi )|_{{\chi = t}} = 0,\psi y_{n} : - \varphi _{2} ^{'} (\chi ) + d\varphi _{1} (\chi )|_{{\chi = t}} = 0$$

(D.9)

$$\varphi _{1} (\chi ) = - e^{{ - r(\chi - t)}}$$

(D.10)

$$\varphi _{2} (\chi ) = - e^{{d(\chi - t)}}$$

(D.11)

By substituting the Lagrangian multipliers and n=0 in the iteration formula,

we obtain

$$x_{1} (t) = x_{0} (t) - \int\limits_{0}^{t} {e^{{ - r(\chi - t)}} \left[ {x_{0} ^{'} (\chi ) - rx_{0} (\chi ) + \frac{r}{k}x_{0} ^{2} (\chi ) + \alpha x_{0} (\chi )y_{0} (\chi )} \right]} d\chi$$

(D.12)

$$y_{1} (t) = y_{0} (t) - \int\limits_{0}^{t} {e^{{d(\chi - t)}} \left[ {y_{0} ^{'} (\chi ) - m\alpha x_{0} (\chi )y_{0} (\chi ) + dy_{0} (\chi ) + \delta y_{0} ^{2} (\chi )} \right]} d\chi$$

(D.13)

Assuming that the initial approximate solution that satisfies the initial conditions has the form

$$x_{0} (t) = m_{1} e^{{ - rt}}$$

(D.14)

$$y_{0} (t) = m_{2} e^{{ - dt}}$$

(D.15)

by the iteration formulas (D.12) and (D.13), we derive Eqs. (10) and (11) in the text.