Analytical survey of the predator–prey model with fractional derivative order

Abbagari, Souleymanou; Houwe, Alphonse; Saliou, Youssoufa; Douvagaï, Douvagaï; Chu, Yu-Ming; Inc, Mustafa; Rezazadeh, Hadi; Doka, Serge Y.

doi:10.1063/5.0038826

This work addresses the analytical investigation of the prey–predator behavior modeled by nonlinear evolution equation systems with fractional derivative order. Through the New Extended Algebraic Method (NEAM), we unearthed diverse types of soliton solutions including bright, dark solitons, combined trigonometric function solutions, and singular solutions. Besides the results obtained in the work of Khater, some new complex soliton solutions are also unearthed. The NEAM can also be used like the synthesis of the two mathematical tools.

I. INTRODUCTION

The coexistence and interfering of biological species have been one of the major concerns in wildlife reserves since the dawn of time. It is in this perspective that today, the modeling of natural phenomena has become a necessity to better understand wildlife interactions. They are most often clarified by systems of linear or nonlinear equations. In most cases, investigations focused on the bio-mathematical area (see Refs. 1–3). Obviously, the implementation of mathematical models describing the behavior of these phenomena is a major asset in bio-mathematics, but the resolution of these systems remains a major concern. Recently, in this way, the famous Prey–Predator (PP) model has been investigated by adopting two integration schemes. Some other works have been followed in Ref. 2.

The advent of analytical and numerical integration methods in recent decades has facilitated the resolution of the bio-mathematics models while taking into account the constraints of their validity, e.g., the auxiliary equation method, sine-Gordon expansion method, Khater method, rational method, generalized exp[−φ(ξ)]-expansion method, sub-ODE method, sine–cosine method, sinh-expansion method, (G′/G)-expansion method, Kudryashov method, new sub-ODE method, homogenous balance method, New Extended Algebraic Method (NEAM), rational hyperbolic function method, Hirota’s bilinear method, Darboux transformation method, and homoclinic breather limit method, reproducing kernel Hilbert space method and its different modification (see Refs. 5–,51 just to list a few).

In this particular work, we are interested to the prey–predator mathematics model having a fractional derivative order as follows:

\begin{gathered} H_{t}^{α} = H_{x x}^{2 α} - β H + (κ + \frac{1}{\sqrt{δ}}) H^{2} - H^{3} - HE, \\ E_{t}^{α} = E_{x x}^{2 α} + κ HE - m E - δ E^{3}, \end{gathered}

(1)

and β, δ, κ, and m are positive parameters. The vigorous natural movement of the prey–predator mathematical model is defined by the relation between its parameters such as m − β = 0 and $κ + \frac{1}{\sqrt{δ}} - m = 1$ ⁠.

Here, H represents the prey population, and E is the predator population size. For the details of the preliminary definitions of the derivative order, see Ref. 9.

In Sec. I A, the transformation hypothesis is used to obtain the nonlinear ordinary equation (NODE). Thereafter, we apply the NEAM to investigate diverse soliton solutions to the mathematical model of the fractional derivative order of the PP system. In Sec. I B, some graphical representation and physical explanation of the obtained results are given. The conclusion is given in Sec. II.

A. Traveling wave solutions of the fractional derivative Predator–Prey (PP) model

In this section, we target diverse soliton solutions of the Predator–Prey (PP) system by employing the new extended algebraic method.^20,21 The first item is to adopt the transformation as follows:

ξ = \frac{x^{α}}{α} - \frac{v t^{α}}{α} .

(2)

Here, α ϵ (0, 1) is the fractional derivative order.^20,21 Inserting Eq. (2) into the set of coupled of nonlinear equations (1) turns to

\begin{gathered} H^{''} + v H^{'} - β H + (κ + \frac{1}{\sqrt{δ}}) H^{2} - H^{3} - HE = 0, \\ E^{''} + v E^{'} + κ HE - m E - δ E^{3} = 0 . \end{gathered}

(3)

Assume the link between the prey and predator parameters, which is given by $E = \frac{H}{\sqrt{δ}}$ ⁠, the nonlinear ordinary equation (ODE) is obtained as

H^{''} + v H^{'} - β H + κ H^{2} - H^{3} = 0 .

(4)

We know to consider the solution of Eq. (4) in the following expression:

H (ξ) = \sum_{j = - N}^{N} H_{j} Q^{j} (ξ),

(5)

where H_j ≠ 0 (j = −1, −2, −3, …, −N, 0, 1, 2, 3, …, N) are parameters to be detected and N is the positive integer. However, Q^j(ξ) is the solution of^20,21

Q^{'} (ξ) = \ln (ρ) (a + b Q (ξ) + c Q^{2} (ξ)) .

(6)

Applying the balance principle, we have N = 1. Therewith, Eq. (5) reads

H (ξ) = H_{0} + H_{1} Q (ξ) + \frac{H_{- 1}}{Q (ξ)} .

(7)

Using Eqs. (7) and (6) into Eq. (4), thereafter collecting all the terms of Q^j(ξ) and set to zero, we obtain the following results:

Case 1: For b² − 4ac < 0 and c ≠ 0,

\begin{gathered} H_{0} = \frac{1}{2} (\sqrt{2} b + \sqrt{- 8 a c + 2 b^{2}}) \ln (p), v = \frac{(4 {(\ln (p))}^{2} a c - {(\ln (p))}^{2} b^{2} + β) (\sqrt{2} b + \sqrt{- 8 a c + 2 b^{2}})}{\ln (p) (- 4 \sqrt{2} a c + \sqrt{2} b^{2} + b \sqrt{- 8 a c + 2 b^{2}})}, \\ H_{1} = 0, H_{- 1} = \sqrt{2} a \ln (p), m = m, κ = - \frac{1}{2} \frac{(\sqrt{2} b + \sqrt{- 8 a c + 2 b^{2}}) (8 {(\ln (p))}^{2} a c - 2 {(\ln (p))}^{2} b^{2} - β) \sqrt{2}}{\ln (p) (- 4 \sqrt{2} a c + \sqrt{2} b^{2} + b \sqrt{- 8 a c + 2 b^{2}})}, \end{gathered}

(8)

\begin{gathered} H_{0} = \frac{1}{2} \sqrt{2} b \ln (p) + \frac{1}{2} \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β}, H_{1} = 0, H_{- 1} = \sqrt{2} a \ln (p), m = m, \\ v = \frac{Ω_{1}}{- 4 {(\ln (p))}^{2} c \sqrt{2} a + b^{2} {(\ln (p))}^{2} \sqrt{2} + b \ln (p) \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β} + β \sqrt{2}}, \\ κ = - \frac{\sqrt{2} Ω_{1}}{- 4 {(\ln (p))}^{2} c \sqrt{2} a + b^{2} {(\ln (p))}^{2} \sqrt{2} + b \ln (p) \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β} + β \sqrt{2}}, \end{gathered}

(9)

\begin{gathered} H_{0} = \frac{1}{2} (\sqrt{2} b + \sqrt{- 8 a c + 2 b^{2}}) \ln (p), v = - \frac{(4 {(\ln (p))}^{2} a c - {(\ln (p))}^{2} b^{2} + β) (\sqrt{2} b + \sqrt{- 8 a c + 2 b^{2}})}{\ln (p) (- 4 \sqrt{2} a c + \sqrt{2} b^{2} + b \sqrt{- 8 a c + 2 b^{2}})}, m = m, \\ H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, κ = - \frac{1}{2} \frac{(\sqrt{2} b + \sqrt{- 8 a c + 2 b^{2}}) (8 {(\ln (p))}^{2} a c - 2 {(\ln (p))}^{2} b^{2} - β) \sqrt{2}}{\ln (p) (- 4 \sqrt{2} a c + \sqrt{2} b^{2} + b \sqrt{- 8 a c + 2 b^{2}})}, \end{gathered}

(10)

\begin{gathered} H_{0} = \frac{1}{1} \sqrt{2} b \ln (p) + \frac{1}{2} \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β}, H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, m = m, \\ v = - \frac{Ω_{1}}{- 4 {(\ln (p))}^{2} c \sqrt{2} a + b^{2} {(\ln (p))}^{2} \sqrt{2} + b \ln (p) \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β} + β \sqrt{2}}, \\ Ω_{1} = \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β} (4 {(\ln (p))}^{2} a c - {(\ln (p))}^{2} b^{2} - β) + \sqrt{2} (4 {(\ln (p))}^{2} a c - {(\ln (p))}^{2} b^{2} - 2 β) \ln (p) b, \\ κ = - \frac{\sqrt{2} Ω_{1}}{- 4 {(\ln (p))}^{2} c \sqrt{2} a + b^{2} {(\ln (p))}^{2} \sqrt{2} + b \ln (p) \sqrt{2 {(\ln (p))}^{2} b^{2} - 8 {(\ln (p))}^{2} a c + 4 β} + β \sqrt{2}} . \end{gathered}

(11)

Consequently, we unearth the following two families of soliton solutions of the prey population size:

From Eq. (8) or Eq. (9), we construct the general solution as follows:

H_{1, 1} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 σ} + \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \tan_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{2} ξ)},

(12)

H_{1, 2} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \cot_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{2} ξ)},

(13)

H_{1, 3} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} + \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \tan_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ) \pm \frac{\sqrt{- p q (b^{2} - 4 a c)}}{2 σ} \sec_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ)},

(14)

H_{1, 4} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \cot_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ) \pm \frac{\sqrt{- p q (b^{2} - 4 a c)}}{2 c} \csc_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ)},

(15)

H_{1, 5} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} + \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \tan_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{4} ξ) - \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \cot_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{4} ξ)} .

(16)

(ii)
Using Eq. (10) or Eq. (11), the related soliton solutions to the prey population size give

\begin{align} H_{1, 6} (ξ) = H_{0} + H_{1} [- \frac{b}{2 σ} + \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \tan_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{2} ξ)], \end{align}

(17)

\begin{align} H_{1, 7} (ξ) = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \cot_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{2} ξ)], \end{align}

(18)

\begin{align} H_{1, 8} (ξ) & = H_{0} + H_{1} [- \frac{b}{2 c} + \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \tan_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ)) \\ \pm (\frac{\sqrt{- p q (b^{2} - 4 a c)}}{2 σ} \sec_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ)], \end{align}

(19)

\begin{align} H_{1, 9} (ξ) & = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \cot_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ)) \\ \pm (\frac{\sqrt{- p q (b^{2} - 4 a c)}}{2 c} \csc_{ρ} (\sqrt{- (b^{2} - 4 a c)} ξ)], \end{align}

(20)

\begin{align} H_{1, 10} (ξ) & = H_{0} + H_{1} [- \frac{b}{2 c} + \frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \tan_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{4} ξ)) \\ - (\frac{\sqrt{- (b^{2} - 4 a c)}}{2 c} \cot_{ρ} (\frac{\sqrt{- (b^{2} - 4 a c)}}{4} ξ)], \end{align}

(21)

and

ξ = \frac{x^{α}}{α} - \frac{v t^{α}}{α} .

(22)

Case 2: For b² − 4ac > 0 and c ≠ 0,

Substituting Eq. (8) or Eq. (9) into Eq. (7), soliton solutions to the prey population size give

H_{2, 1} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \tanh_{ρ} (\frac{\sqrt{(b^{2} - 4 a c)}}{2} ξ)},

(23)

H_{2, 3} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \coth_{ρ} (\frac{\sqrt{(b^{2} - 4 a c)}}{2} ξ)},

(24)

H_{2, 4} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \tanh_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ) \pm ı \frac{\sqrt{p q (b^{2} - 4 a c)}}{2 c} {sech}_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ)},

(25)

H_{2, 5} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \coth_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ) \pm \frac{\sqrt{p q (b^{2} - 4 a c)}}{2 c} {csch}_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ)},

(26)

H_{2, 6} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{b}{2 c} - \frac{\sqrt{- (b^{2} - 4 a c)}}{4 c} \tanh_{ρ} (\frac{\sqrt{(b^{2} - 4 a c)}}{4} ξ) - \frac{\sqrt{(b^{2} - 4 a c)}}{4 c} \coth_{ρ} (\frac{\sqrt{(b^{2} - 4 a c)}}{4} ξ)} .

(27)

(ii)
Substituting Eq. (10) or Eq. (11) into Eq. (7), the obtained soliton solutions to the prey population size give

\begin{align} H_{2, 7} (ξ) = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \tanh_{ρ} (\frac{\sqrt{(b^{2} - 4 a c)}}{2} ξ)], \end{align}

(28)

\begin{align} H_{2, 8} (ξ) = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \coth_{ρ} (\frac{\sqrt{(b^{2} - 4 a c)}}{2} ξ)], \end{align}

(29)

\begin{align} H_{2, 9} (ξ) & = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \tanh_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ)) \\ \pm (ı \frac{\sqrt{p q (b^{2} - 4 a c)}}{2 c} {sech}_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ)], \end{align}

(30)

\begin{align} H_{2, 10} (ξ) & = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{(b^{2} - 4 a c)}}{2 c} \coth_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ)) \\ \pm (\frac{\sqrt{p q (b^{2} - 4 a c)}}{2 c} {csch}_{ρ} (\sqrt{(b^{2} - 4 a c)} ξ)], \end{align}

(31)

\begin{align} H_{2, 11} (ξ) & = H_{0} + H_{1} [- \frac{b}{2 c} - \frac{\sqrt{- (b^{2} - 4 a c)}}{4 c} \tanh_{ρ}) \\ (\times (\frac{\sqrt{(b^{2} - 4 a c)}}{4} ξ) - \frac{\sqrt{(b^{2} - 4 a c)}}{4 c} \coth_{ρ}) \\ (\times (\frac{\sqrt{(b^{2} - 4 a c)}}{4} ξ)], \end{align}

(32)

while

ξ = \frac{x^{α}}{α} - \frac{v t^{α}}{α} .

Case 3: For b = 0, ac > 0, it is obtained,

\begin{gathered} H_{0} = \sqrt{- 2 {(\ln (p))}^{2} a c + β}, H_{1} = 0, H_{- 1} = \sqrt{2} a \ln (p), \\ v = \frac{(2 {(\ln (p))}^{2} a c - β) \sqrt{2}}{\sqrt{- 2 {(\ln (p))}^{2} a c + β}}, m = m, \\ κ = - 2 \frac{2 {(\ln (p))}^{2} a c - β}{\sqrt{- 2 {(\ln (p))}^{2} a c + β}}, \end{gathered}

(33)

\begin{gathered} H_{0} = \sqrt{- 2 a c} \ln (p), H_{1} = 0, H_{- 1} = \sqrt{2} a \ln (p), \\ v = \frac{1}{2} \frac{(4 {(\ln (p))}^{2} a c + β) \sqrt{2}}{\sqrt{- 2 a c} \ln (p)}, m = m, κ = - \frac{1}{2} \frac{8 {(\ln (p))}^{2} a c - β}{\sqrt{- 2 a c} \ln (p)}, \end{gathered}

(34)

\begin{gathered} H_{0} = \sqrt{- 2 a c} \ln (p), H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, \\ v = - \frac{1}{2} \frac{(4 {(\ln (p))}^{2} a c + β) \sqrt{2}}{\sqrt{- 2 a c} \ln (p)}, m = m, κ = - \frac{1}{2} \frac{8 {(\ln (p))}^{2} a c - β}{\sqrt{- 2 a c} \ln (p)}, \end{gathered}

(35)

\begin{gathered} H_{0} = \sqrt{- 2 {(\ln (p))}^{2} a c + β}, H_{1} = \sqrt{2} c \ln (p), \\ H_{- 1} = 0, v = - \frac{(2 {(\ln (p))}^{2} a c - β) \sqrt{2}}{\sqrt{- 2 {(\ln (p))}^{2} a c + β}}, m = m, \\ κ = - 2 \frac{2 {(\ln (p))}^{2} a c - β}{\sqrt{- 2 {(\ln (p))}^{2} a c + β}} . \end{gathered}

(36)

Consequently, the analytical survey soliton solutions of the prey population are given by the following:

Plug Eq. (32) or Eq. (33) into Eq. (7), we set out the soliton solutions to the prey population size,

H_{3, 1} (ξ) = H_{0} + \frac{H_{- 1}}{\sqrt{\frac{a}{c}} \tan_{ρ} (\sqrt{a c} ξ)},

(37)

H_{3, 2} (ξ) = H_{0} + \frac{H_{- 1}}{- \sqrt{\frac{c}{σ}} \cot_{ρ} (\sqrt{a c} ξ)},

(38)

H_{3, 3} (ξ) = H_{0} + \frac{H_{- 1}}{\sqrt{\frac{a}{σ}} \tan_{ρ} (2 \sqrt{a c} ξ) \pm \sqrt{p q \frac{a}{c}} \sec_{ρ} (2 \sqrt{a c} ξ)},

(39)

H_{3, 4} (ξ) = H_{0} + \frac{H_{- 1}}{- \sqrt{\frac{a}{c}} \cot_{ρ} (2 \sqrt{a c} ξ) \pm \sqrt{p q \frac{a}{c}} \csc_{ρ} (2 \sqrt{a c} ξ)},

(40)

H_{3, 5} (ξ) = H_{0} + \frac{H_{- 1}}{\frac{1}{2} \sqrt{\frac{a}{c}} \tan_{ρ} (\frac{\sqrt{a c}}{2} ξ) - \frac{1}{2} \sqrt{\frac{a}{c}} \cot_{ρ} (\frac{\sqrt{a c}}{2} ξ)} .

(41)

(ii)
Using Eq. (34) or Eq. (35), we set out the soliton solutions to the prey population size,

H_{3, 6} (ξ) = H_{0} + H_{1} \sqrt{\frac{a}{c}} \tan_{ρ} (\sqrt{a c} ξ),

(42)

H_{3, 7} (ξ) = H_{0} - H_{1} [\sqrt{\frac{c}{σ}} \cot_{ρ} (\sqrt{a c} ξ)],

(43)

H_{3, 8} (ξ) = H_{0} + H_{1} [\sqrt{\frac{a}{σ}} \tan_{ρ} (2 \sqrt{a c} ξ) \pm \sqrt{p q \frac{a}{c}} \sec_{ρ} (2 \sqrt{a c} ξ)],

(44)

H_{3, 9} (ξ) = H_{0} + H_{1} [- \sqrt{\frac{a}{c}} \cot_{ρ} (2 \sqrt{a c} ξ) \pm \sqrt{p q \frac{a}{c}} \csc_{ρ} (2 \sqrt{a c} ξ)],

(45)

H_{3, 10} (ξ) = H_{0} + H_{1} [\frac{1}{2} \sqrt{\frac{a}{c}} t a n_{ρ} (\frac{\sqrt{a c}}{2} ξ) - \frac{1}{2} \sqrt{\frac{a}{c}} \cot_{ρ} (\frac{\sqrt{a c}}{2} ξ)],

(46)

with

ξ = \frac{x^{α}}{α} - \frac{v t^{α}}{α} .

Case 4: ac < 0 and b = 0 together with Eqs. (32)--(35), we unearth

Use Eq. (32) or Eq. (33), we set out the soliton solutions of the prey population,

H_{4, 1} (ξ) = H_{0} + \frac{H_{- 1}}{- \sqrt{\frac{- a}{c}} \tanh_{ρ} (\sqrt{- a c} ξ)},

(47)

H_{4, 2} (ξ) = H_{0} + \frac{H_{- 1}}{- \sqrt{\frac{- a}{c}} \coth_{ρ} (\sqrt{- a c} ξ)},

(48)

H_{4, 3} (ξ) = H_{0} + \frac{H_{- 1}}{- \sqrt{\frac{- a}{c}} \tanh_{ρ} (2 \sqrt{a c} ξ) \pm ı \sqrt{\frac{- p q a}{c}} {sech}_{A} (2 \sqrt{- a c} ξ)},

(49)

H_{4, 4} (ξ) = H_{0} + \frac{H_{- 1}}{- \sqrt{\frac{- a}{c}} \coth_{ρ} (2 \sqrt{- a c} ξ) \pm \sqrt{- p q \frac{a}{c}} {csch}_{ρ} (2 \sqrt{- a c} ξ)},

(50)

H_{4, 5} (ξ) = H_{0} + \frac{H_{- 1}}{- \frac{1}{2} \sqrt{\frac{- a}{c}} \tanh_{ρ} (\frac{\sqrt{- a c}}{2} ξ) - \frac{1}{2} \sqrt{\frac{- a}{c}} \coth_{ρ} (\frac{\sqrt{- a c}}{2} ξ)} .

(51)

Furthermore,

(ii)
Employ Eq. (34) or Eq. (35), we set out the soliton solutions of the prey population,

H_{4, 6} (ξ) = H_{0} - H_{1} \sqrt{\frac{- a}{c}} \tanh_{ρ} (\sqrt{- a c} ξ),

(52)

H_{4, 7} (ξ) = H_{0} - H_{1} \sqrt{\frac{- a}{c}} \coth_{ρ} (\sqrt{- a c} ξ),

(53)

\begin{align} H_{4, 8} (ξ) & = H_{0} + H_{1} [- \sqrt{\frac{- a}{c}} \tanh_{ρ} (2 \sqrt{a c} ξ)) \\ \pm (ı \sqrt{\frac{- p q a}{c}} {sech}_{A} (2 \sqrt{- a c} ξ)], \end{align}

(54)

\begin{align} H_{4, 9} (ξ) & = H_{0} + H_{1} [- \sqrt{\frac{- a}{c}} \coth_{ρ} (2 \sqrt{- a c} ξ)) \\ \pm (\sqrt{- p q \frac{a}{c}} {csch}_{ρ} (2 \sqrt{- a c} ξ)], \end{align}

(55)

\begin{align} H_{4, 10} (ξ) & = H_{0} + H_{1} [- \frac{1}{2} \sqrt{\frac{- a}{c}} \tanh_{ρ} (\frac{\sqrt{- a c}}{2} ξ)) \\ - (\frac{1}{2} \sqrt{\frac{- a}{c}} \coth_{ρ} (\frac{\sqrt{- a c}}{2} ξ)] . \end{align}

(56)

Hence,

ξ = \frac{x^{α}}{α} - \frac{v t^{α}}{α} .

Case 5: For b = 0, a = −c, we obtain the following set of results:

\begin{gathered} H_{0} = \sqrt{2} c \ln (p), H_{1} = 0, H_{- 1} = \sqrt{2} c \ln (p), \\ v = \frac{1}{2} \frac{4 {(\ln (p))}^{2} c^{2} - β}{\ln (p) c}, m = m, κ = \frac{1}{4} \frac{(8 {(\ln (p))}^{2} c^{2} + β) \sqrt{2}}{\ln (p) c}, \end{gathered}

(57)

\begin{gathered} H_{0} = \sqrt{2 {(\ln (p))}^{2} c^{2} + β}, H_{1} = 0, H_{- 1} = \sqrt{2} c \ln (p), \\ v = \sqrt{2 {(\ln (p))}^{2} c^{2} + β} \sqrt{2}, m = m, κ = 2 \sqrt{2 {(\ln (p))}^{2} c^{2} + β}, \end{gathered}

(58)

\begin{gathered} H_{0} = \sqrt{2} c \ln (p), H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, \\ v = \frac{4 {(\ln (p))}^{2} c^{2} - β}{2 \ln (p) c}, m = m, κ = \frac{(8 {(\ln (p))}^{2} c^{2} + β) \sqrt{2}}{4 \ln (p) c}, \end{gathered}

(59)

\begin{gathered} H_{0} = \sqrt{2 {(\ln (p))}^{2} c^{2} + β}, H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, \\ v = \sqrt{2 {(\ln (p))}^{2} c^{2} + β} \sqrt{2}, m = m, κ = 2 \sqrt{2 {(\ln (p))}^{2} c^{2} + β} . \end{gathered}

(60)

Thereafter,

utilize Eq. (56) or Eq. (57), we set out the soliton solutions of the prey population,

H_{5, 1} (ξ) = H_{0} - \frac{H_{- 1}}{\tanh_{ρ} (a ξ)},

(61)

H_{5, 2} (ξ) = H_{0} - \frac{H_{- 1}}{\coth_{ρ} (a ξ)},

(62)

H_{5, 3} (ξ) = H_{0} + \frac{H_{- 1}}{- \tanh_{ρ} (2 a ξ) \pm ı \sqrt{p q} {sech}_{ρ} (2 a ξ)},

(63)

H_{5, 4} (ξ) = H_{0} + \frac{H_{- 1}}{- \coth_{ρ} (2 a ξ) \pm \sqrt{p q} {csch}_{ρ} (2 a ξ)},

(64)

H_{5, 5} (ξ) = H_{0} + \frac{H - 1}{- \frac{1}{2} (\tanh_{ρ} (\frac{a}{2} ξ) + \coth_{ρ} (\frac{a}{2} ξ))} .

(65)

Hence, with the help of

(ii)
using Eq. (58) or Eq. (59), we set out the soliton solutions of the prey population,

H_{5, 6} (ξ) = H_{0} - H_{1} \tanh_{ρ} (a ξ),

(66)

H_{5, 7} (ξ) = H_{0} - H_{1} \coth_{ρ} (a ξ),

(67)

H_{5, 8} (ξ) = H_{0} + H_{1} [- \tanh_{ρ} (2 a ξ) \pm ı \sqrt{p q} {sech}_{ρ} (2 a ξ)],

(68)

H_{5, 9} (ξ) = H_{0} + H_{1} [- \coth_{ρ} (2 a ξ) \pm \sqrt{p q} {csch}_{ρ} (2 a ξ)],

(69)

H_{5, 10} (ξ) = H_{0} + H_{1} [- \frac{1}{2} (\tanh_{ρ} (\frac{a}{2} ξ) + \coth_{ρ} (\frac{a}{2} ξ))] .

(70)

Case 6: For b = 0 a = c, the set of results is obtained,

\begin{gathered} H_{0} = \sqrt{- 2 {(\ln (p))}^{2} c^{2} + β}, H_{1} = 0, H_{- 1} = \sqrt{2} c \ln (p), \\ v = - \sqrt{- 2 {(\ln (p))}^{2} c^{2} + β} \sqrt{2}, m = m, κ = 2 \sqrt{- 2 {(\ln (p))}^{2} c^{2} + β}, \end{gathered}

(71)

\begin{gathered} H_{0} = \sqrt{- 2 {(\ln (p))}^{2} c^{2} + β}, H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, \\ v = \sqrt{- 2 {(\ln (p))}^{2} c^{2} + β} \sqrt{2}, m = m, κ = 2 \sqrt{- 2 {(\ln (p))}^{2} c^{2} + β} . \end{gathered}

(72)

Insert Eq. (70) into Eq. (7), we emphasize soliton solutions of the prey population,

H_{6, 1} (ξ) = H_{0} + \frac{H_{- 1}}{\tan_{ρ} (a ξ)},

(73)

H_{6, 2} (ξ) = H_{0} - \frac{H_{- 1}}{\cot_{ρ} (a ξ)},

(74)

H_{6, 3} (ξ) = H_{0} + \frac{H_{- 1}}{\tan_{ρ} (2 a ξ) \pm \sqrt{p q} \sec_{ρ} (2 a ξ)},

(75)

H_{6, 4} (ξ) = H_{0} + \frac{H_{- 1}}{- \cot_{ρ} (2 a ξ) \pm \sqrt{p q} \csc_{ρ} (2 a ξ)},

(76)

H_{6, 5} (ξ) = H_{0} + \frac{H_{- 1}}{\frac{1}{2} (\tan_{ρ} (\frac{a}{2} ξ) - \cot_{ρ} (\frac{a}{2} ξ))} .

(77)

Thereupon,

plug Eq. (71) into Eq. (7), we emphasize soliton solutions of the prey population,

H_{6, 6} (ξ) = H_{0} + H_{1} \tan_{ρ} (a ξ),

(78)

H_{6, 7} (ξ) = H_{0} - H_{1} \cot_{ρ} (a ξ),

(79)

H_{6, 7} (ξ) = H_{0} + H_{1} [\tan_{ρ} (2 a ξ) \pm \sqrt{p q} \sec_{ρ} (2 a ξ)],

(80)

H_{6, 8} (ξ) = H_{0} + H_{1} [- \cot_{ρ} (2 a ξ) \pm \sqrt{p q} \csc_{ρ} (2 a ξ)],

(81)

H_{6, 8} (ξ) = H_{0} + H_{1} [\frac{1}{2} (\tan_{ρ} (\frac{a}{2} ξ) - \cot_{ρ} (\frac{a}{2} ξ))] .

(82)

Case 7: For b ≠ 0 and a = 0, it is unearthed

\begin{gathered} H_{0} = 0, H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, v = - \frac{{(\ln (p))}^{2} b^{2} - β}{\ln (p) b}, \\ m = m, κ = - \frac{1}{2} \frac{(2 {(\ln (p))}^{2} b^{2} + β) \sqrt{2}}{\ln (p) b}, \end{gathered}

(83)

\begin{gathered} H_{0} = \sqrt{2} b \ln (p), H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, \\ v = \frac{{(\ln (p))}^{2} b^{2} - β}{\ln (p) b}, m = m, κ = \frac{1}{2} \frac{(2 {(\ln (p))}^{2} b^{2} + β) \sqrt{2}}{\ln (p) b}, \end{gathered}

(84)

\begin{gathered} H_{0} = \frac{1}{4} \sqrt{2} (2 \ln (p) b + \sqrt{2} \sqrt{2 {(\ln (p))}^{2} b^{2} + 4 β}), \\ v = \frac{2 {(\ln (p))}^{2} b^{2} + \sqrt{2} \ln (p) b \sqrt{2 {(\ln (p))}^{2} b^{2} + 4 β} + 4 β}{2 \ln (p) b + \sqrt{2} \sqrt{2 {(\ln (p))}^{2} b^{2} + 4 β}}, m = m, \\ H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, \\ κ = \frac{\sqrt{2} (2 {(\ln (p))}^{2} b^{2} + \sqrt{2} \ln (p) b \sqrt{2 {(\ln (p))}^{2} b^{2} + 4 β} + 4 β)}{2 \ln (p) b + \sqrt{2} \sqrt{2 {(\ln (p))}^{2} b^{2} + 4 β}} . \end{gathered}

(85)

Connect Eqs. (82)-(84) into Eq. (7), we obtain soliton solutions of the prey population,

H_{7, 1} (ξ) = H_{0} + \frac{p b H_{1}}{c (\cosh_{ρ} (b ξ) - \sinh_{ρ} (b ξ) - p)},

(86)

H_{7, 1} (ξ) = H_{0} - \frac{b H_{1} (\sinh_{ρ} (b ξ) + \cosh_{ρ} (b ξ))}{c (\cosh_{ρ} (b ξ) - \sinh_{ρ} (b ξ) + q)} .

(87)

Case 8: For b² = 4ac, it is revealed,

\begin{gathered} H_{0} = \sqrt{a c} \sqrt{2} \ln (p) + \sqrt{β}, v = \frac{4 {(a c)}^{3 / 2} {(\ln (p))}^{3} - 4 \sqrt{a c} a c {(\ln (p))}^{3} - β^{3 / 2} \sqrt{2} - 4 \sqrt{a c} β \ln (p)}{2 \sqrt{a c} \sqrt{β} \sqrt{2} \ln (p) + β}, m = m, \\ H_{1} = 0, H_{- 1} = \sqrt{2} a \ln (p), κ = 2 (\frac{{(a c)}^{3 / 2} \sqrt{2} {(\ln (p))}^{3} - {(\ln (p))}^{3} \sqrt{a c} c \sqrt{2} a + 2 \sqrt{a c} \sqrt{2} β \ln (p) + β^{3 / 2}}{2 \sqrt{a c} \sqrt{β} \sqrt{2} \ln (p) + β}), \end{gathered}

(88)

\begin{gathered} H_{0} = \sqrt{a c} \sqrt{2} \ln (p) + \sqrt{β}, v = - \frac{4 {(a c)}^{3 / 2} {(\ln (p))}^{3} - 4 \sqrt{a c} a c {(\ln (p))}^{3} - β^{3 / 2} \sqrt{2} - 4 \sqrt{a c} β \ln (p)}{2 \sqrt{a c} \sqrt{β} \sqrt{2} \ln (p) + β}, m = m, \\ H_{1} = \sqrt{2} c \ln (p), H_{- 1} = 0, κ = 2 (\frac{{(a c)}^{3 / 2} \sqrt{2} {(\ln (p))}^{3} - {(\ln (p))}^{3} \sqrt{a c} c \sqrt{2} a + 2 \sqrt{a c} \sqrt{2} β \ln (p) + β^{3 / 2}}{2 \sqrt{a c} \sqrt{β} \sqrt{2} \ln (p) + β}) . \end{gathered}

(89)

Substitute Eq. (87) into Eq. (7), it is carried out soliton solutions of the prey population,

H_{8, 1} (ξ) = H_{0} - \frac{H_{- 1} b^{2} ξ \ln (ρ)}{2 a (b ξ \ln (ρ) + 2)} .

(90)

Therewith,

(ii)
connect Eq. (88) into Eq. (7), it is pick up soliton solutions of the prey population,

H_{8, 2} (ξ) = H_{0} - \frac{2 H_{1} a (b ξ \ln (ρ) + 2)}{b^{2} ξ \ln (ρ)},

(91)

where

ξ = \frac{x^{α}}{α} - \frac{v t^{α}}{α} .

The generalized hyperbolic and triangular function are given in Refs. 21 and 22.

B. Graphical representation and physical explanation

We demonstrate our results in Figs. 1–3. Figures 1 and 2 show the behavior of the prey population with the effect of the fractional derivative order. By the chosen appropriate parameters of the direct extended algebraic method associated with the PP parameters (β), the dark soliton solution of the PP population size is affected for certain values of α [see 1(A₂), 1(A₃), and 1(A₄) in Figs. 1 and 2]. Meanwhile, taking the imaginary part of Eq. (29), we depict the bright soliton solutions associated with the prey population size. It is also set out the effect of the fractional derivative. For the negative values of the velocity of the soliton, the wave move from right to left (see Figs. 4 and 5). However, the spatiotemporal plot 3D (Fig. 6) is depicted for different values of α. We emphasize on the behavior of the prey population under the control of the fractional derivative order. The obtained results have set out some new cases of the traveling wave compared to Refs. 1 and 4. For example, Eqs. (14)–(16), (19)–(21), (24)–(26), (29)–(31), (42)–(45), and (48)–(50) were not obtained in Ref. 1. These results could probably in the future explain the coexistence of interfering biological species such as PP. On the other hand, the NEAM can be used like the synthesis of the Kather method and the exp[−φ(ξ)]-expansion method.

FIG. 1.

$FIG. 1. Spatiotemporal plot of the kink-like soliton of |H2.7(x, t)|2 under the effect of the fractional derivative order (A1) α1 = 1, (A2) α1 = 0.9, (A3) α1 = 0.85, and (A4) α1 = 0.8 for β2 = 0.0015, b = 2.5, a = −0.4, c = 0.072, v = −1.0072, p = q = 1, ρ = e.$

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Spatiotemporal plot of the kink-like soliton of |H_2.7(x, t)|² under the effect of the fractional derivative order (A₁) α₁ = 1, (A₂) α₁ = 0.9, (A₃) α₁ = 0.85, and (A₄) α₁ = 0.8 for β₂ = 0.0015, b = 2.5, a = −0.4, c = 0.072, v = −1.0072, p = q = 1, ρ = e.

FIG. 2.

$FIG. 2. Plot evolution 2D of the kink-like soliton of |H2.7(x, t)|2 under the effect of the fractional derivative order (α = 0.95) and the velocity of the soliton (A5) v = −1.1263, β = 0.100 15, (A6) v = −1.1662, β = 0.300 15, (A7) v = −1.1766, β = 0.400 15, and (A8) v = −1.2061, β0.500 15 for b = 2.5, a = −0.4, c = 0.072, p = q = 1, ρ = e.$

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Plot evolution 2D of the kink-like soliton of |H_2.7(x, t)|² under the effect of the fractional derivative order (α = 0.95) and the velocity of the soliton (A₅) v = −1.1263, β = 0.100 15, (A₆) v = −1.1662, β = 0.300 15, (A₇) v = −1.1766, β = 0.400 15, and (A₈) v = −1.2061, β0.500 15 for b = 2.5, a = −0.4, c = 0.072, p = q = 1, ρ = e.

FIG. 3.

$FIG. 3. Contour plot evolution of the bright soliton |ImH2.9(x,t)|2 [Eq. (29)] under the effect of the fractional derivative order (A9) α = 1, (A10) α = 0.98, (A11) α = 0.9, and (A12) α = 0.85 for v = −0.1765, b = 0.75, a = −0.0074, c = −0.0002, β = −0.700 001 5, p = q = 1, ρ = e.$

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Contour plot evolution of the bright soliton $| Im H_{2.9} (x, t) |^{2}$ [Eq. (29)] under the effect of the fractional derivative order (A₉) α = 1, (A₁₀) α = 0.98, (A₁₁) α = 0.9, and (A₁₂) α = 0.85 for v = −0.1765, b = 0.75, a = −0.0074, c = −0.0002, β = −0.700 001 5, p = q = 1, ρ = e.

FIG. 4.

$FIG. 4. Plot evolution of the bright soliton |ImH2.9(x,t)|2 [Eq. (29)] with the effect of the fractional derivative order (α = 0.98) at (D1) t = 0 ms, (D2) t = 20 ms, (D3) t = 40 ms, (D4) t = 60 ms, and (D5) t = 80 ms for v = −0.6364, b = 0.55, a = −0.0074, c = −0.0002, β = −0.200 001 5, p = q = 1, ρ = e.$

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Plot evolution of the bright soliton $| Im H_{2.9} (x, t) |^{2}$ [Eq. (29)] with the effect of the fractional derivative order (α = 0.98) at (D₁) t = 0 ms, (D₂) t = 20 ms, (D₃) t = 40 ms, (D₄) t = 60 ms, and (D₅) t = 80 ms for v = −0.6364, b = 0.55, a = −0.0074, c = −0.0002, β = −0.200 001 5, p = q = 1, ρ = e.

FIG. 5.

$FIG. 5. Plot evolution of the bright soliton |ImH2.9(x,t)|2 [Eq. (29)] with the effect of the fractional derivative order (α = 0.75) at (D6) t = 0 ms, (D7) t = 20 ms, (D8) t = 40 ms, (D9) t = 60 ms, and (D10) t = 80 ms for v = −0.6364, b = 0.55, a = −0.0074, c = −0.0002, β = −0.200 001 5, p = q = 1, ρ = e.$

View large Download slide

Plot evolution of the bright soliton $| Im H_{2.9} (x, t) |^{2}$ [Eq. (29)] with the effect of the fractional derivative order (α = 0.75) at (D₆) t = 0 ms, (D₇) t = 20 ms, (D₈) t = 40 ms, (D₉) t = 60 ms, and (D₁₀) t = 80 ms for v = −0.6364, b = 0.55, a = −0.0074, c = −0.0002, β = −0.200 001 5, p = q = 1, ρ = e.

FIG. 6.

$FIG. 6. Spatiotemporal plot evolution of the bright soliton |ImH2.9(x,t)|2 [Eq. (29)] with the effect of the fractional derivative order (B1) α = 0.7, (B2) α = 0.75, (B3) α = 0.8, and (B4) α = 0.85 for v = −0.6364, b = 0.55, a = −0.0074, c = −0.0002, β = −0.200 001 5, p = q = 1, ρ = e.$

View large Download slide

Spatiotemporal plot evolution of the bright soliton $| Im H_{2.9} (x, t) |^{2}$ [Eq. (29)] with the effect of the fractional derivative order (B₁) α = 0.7, (B₂) α = 0.75, (B₃) α = 0.8, and (B₄) α = 0.85 for v = −0.6364, b = 0.55, a = −0.0074, c = −0.0002, β = −0.200 001 5, p = q = 1, ρ = e.

II. CONCLUSION

In this work, we investigated the analytical survey of the fractional derivative mathematical model of the PP system. To reach the main goal, we use the famous powerful mathematic method, namely, the NEAM. The behavior of the effect of the fractional order derivative (α) on the bright and dark soliton solutions has been obtained, and their profiles have been plotted in 2D and 3D. In addition, diverse solutions have also been obtained, such as trigonometric function solutions, singular soliton solutions, bright soliton dark soliton, combined trigonometric solutions, and complex solutions. Some of the obtained results are new in the bio-mathematics field and can be useful to better explain the effects of anti-predator behavior in the PP system. The next work will focus on the numerical simulation of the obtained results to better emphasize the phenomenon of the PP interference.

DATA AVAILABILITY

The data that support the findings of this study are available within the article.

ACKNOWLEDGMENTS

This work was supported by the Natural Science Foundation of China (Grant Nos. 61673169, 11301127, 11701176, 11626101, and 11601485).

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Author(s)

Analytical survey of the predator–prey model with fractional derivative order

I. INTRODUCTION

A. Traveling wave solutions of the fractional derivative Predator–Prey (PP) model

B. Graphical representation and physical explanation

II. CONCLUSION

DATA AVAILABILITY

ACKNOWLEDGMENTS

REFERENCES

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Analytical survey of the predator–prey model with fractional derivative order

I. INTRODUCTION

A. Traveling wave solutions of the fractional derivative Predator–Prey (PP) model

B. Graphical representation and physical explanation

II. CONCLUSION

DATA AVAILABILITY

ACKNOWLEDGMENTS

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

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