# Dynamical Behaviours of Stage-Structured Fractional-Order Prey-Predator Model with Crowley–Martin Functional Response

Authors: Wala’a A. Mahdi; Sadiq Al-Nassir
Journal: International Journal of Differential Equations
 (2022)

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## Abstract

In this paper, the dynamic behaviour of the stage-structure prey-predator fractional-order derivative system is considered and discussed. In this model, the Crowley–Martin functional response describes the interaction between mature preys with a predator. The existence, uniqueness, non-negativity, and the boundedness of solutions are proved. All possible equilibrium points of this system are investigated. The sufficient conditions of local stability of equilibrium points for the considered system are determined. Finally, numerical simulation results are carried out to confirm the theoretical results.


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## Body

## 1. Introduction

The subject of fractional integral and derivative is as old as classical calculus. This type of calculus is more comprehensive than the differ-integral calculus due to its worldwide applications in biology as well as in different branches of science, engineering, several ecological models, and some other interdisciplinary fields [1–4]. Many researchers have performed and investigated models of fractional derivatives [5, 6]. Some types of fractional derivatives like Caputo, Caputo–Fabrizio, Riemann–Liouville, and Marchand are powerful mathematical tools for modelling biological systems which cannot be designed by integer derivatives because fractional-order derivatives are not only dependent on the initial conditions but also on the memory of the system [7, 8]. The Lotka–Volterra equations are the first equations that described the dynamics of biological systems, which are also called the predator-prey equations. A stage-structured predator-prey model is considered and elucidated by the authors in Reference [9]. Some authors [10–16] have studied two species models with stages structured in the predators. They discussed and investigated the stability of the all equilibrium points as well as the dynamic behaviour of their systems. In recent years, the global and local dynamics behaviour of a stage-structured predator-prey model is investigated, and the optimal control of harvesting is discussed in References [17–20]. The predator-prey functional response is very important to determine the relationship between the predator and prey for that there are many types of functional responses, namely, Holling types I, II, III, and IV; Michaelis–Menten ratio-dependent type; and Beddington–DeAngelis type of them have been considered and analysed in References [21–27]. The Crowley–Martin functional response is simplified to Michaelis–Menten which has involved one more term explaining mutual interferences of predators for the case where the predator feeding rate is decreased by a higher predator density even when the prey density is high [28, 29]. The subject of the optimal harvesting is very important in managing the renewable resources due to the economic aspect and to keep population at level far from the extinct, so that many researchers and authors have widely investigated and studied this subject in their works, see [30–34] and the references therein.In this work, we have considered and developed a fractional-order with the Caputo fractional derivative model for a stage-structured prey-predator model with Crowley–Martin functional response and linear harvesting for mature prey species only. The current paper is divided as followsSection2 contains the main definitions of fractional derivatives and theories for the local stability of the equilibrium points that are used throughout this work. In Section 3, the model is formulated with the existence of all its equilibria. The existence and boundedness of its solution are proved and shown. In Section 4, the conditions for the local stability of all equilibrium points are established. Section 5 presents and confirms the numerical simulation for the theoretical results, while the conclusions and discussions are given and elucidated in Section 6.


## 2. Main Concepts

Definition 1.
(see [35, 36]). The fractional-order derivatives in the meaning of Caputo are defined as follows:(1)Dtqft=In−qfnt,q>0,where n is the least integer which is not less than q,Imis the Riemann–Liouville operator of order m which is given by:(2)Im=1Γm∫0tt−τm−1dτ,m>0,Γm is a gamma function.
Some results for the fractional-order derivatives are found in References [21, 25, 37] that are needed throughout this paper.Lemma 1.
Assume thatft∈Ca,b.(1)
Daqft∈a,bwith0<q≤1,then,ft=fa+1/ΓqDaqfζt−aq.wherea≤ζ≤sfor alls∈a,b(2)
IfDtqft∈Ca,bwith0<q≤1, then we have:(i)
IfDaqft≥0,for allt∈a,b,thenftis a nondecreasing function for all t∈a,b.(ii)
IfDaqft≤0for allt∈a,b then ftis a nonincreasing function for allt∈a,b.Lemma 2.
Assume the Cauchy problem.(3)Daqht=λht+ft,ha=bb∈ℝ,with0<q≤1 and λ∈ℝ, then the solution of (4) is given by(4)ht=bEqλt−aq+∫att−cq−1Eq,qλt−cqfcdc.
In the autonomous case:(5)Daqht=λht,ha=bb∈ℝ.
Then, the solution isht=bEqλt−aq and Eq is the Mittag-Leffler function.Lemma 3 (see [21]).
Letwt be a continuous function on t0,∞ and satisfies
Daqwt≤−λwt+μ, then the form of the solution of this equation is given by:(6)wt≤wt0−μλEq−λt−t0q+μλ,where 0<q<1,λ,μ∈ℝ2,λ≠0, and t0≥0 and t0 is the initial time.
The next lemma is found in Reference [38] which gives the uniqueness of the solution of fractional-order system.Lemma 4.
LetDtqut=ft,u,t>t0 be a system fractional-q-order derivatives with the initial condition ut0,where0<q≤1,f:t0,∞×Ω⟶ℝn. Then, the above system has a unique solution on t0,∞×Ω if ft,u satisfies the locally Lipchitz condition concerning u.
To examine the stability of the fractional-order system, we need the following.Definition 2.
(see [39]). Let fx=xn+vn−1xn−1+vn−2xn−2+……+v0 be a polynomial of degree n, then the discriminant of fx denoted by Df is defined by Df=−1nn−1/2Rf,f′, where f′ is the derivative of f and gx=xn+kn−1xl−1+kn−2xl−2+……+k0 and Rf,g are n+l⊗n+l determinants.Lemma 5 (see [39]).
Consider a characteristic polynomial equation,fλ = λn+vn−1λn−1+vn−2λn−2+……..+v0, then:(1)
If n=2, then the conditions for argλi>qπ/2, fori = 1,2 q∈0,1 are either Routh–Hurwitz conditions or v1<0,4v0>v12 and(7)tan−14v0−v12v1>qπ2.(2)
If n=3, then we have the following cases:(i)
Df>0v2>0,v0>0v1v2>v0,thenargλi>qπ/2,i+1,2,3forq∈0,1,whereDf=18v2v1v0+v2v12−4v0v23−4v13−27v02(ii)
IfDf<0,v2≥0,v1≥0,v0>0,thenargλi>qπ/2i = 1, 2, 3 for q<2/3.


## 3. The Fractional-Order Model

In Reference [40], the author considered the following model:(8)ddTX1T=RX2T−D1X1T−BX1TddTX2T=BX1T−D2X2T−B1X22T−A1X2TY2T1+αX2T1+βY2TddTY1T=A2X2TY2T1+αX2T1+βY2T−D3Y1T−H1Y1TddTY2T=H1Y1T−D4Y1T−B2Y22T..X1T,X2T are the densities of immature and mature prey species, respectively. Y1TandY2T represent the densities of immature and mature predator species respectively. The parameter R is the birth rate of immature prey; B and H1 indicate maturity rate of immature prey and immature predator, respectively; B1 and B2 express the competition rate between a mature prey population and mature predator population, respectively; D1, D2, D3,D4 are the death rates of immature and mature prey, the death rates of immature and mature predators, respectively. A1 is the maximum value which per capita reduction rate of the mature prey can attain, while A2 is the predator conversation rate. α and β are measures of the half-saturation of prey species and the coefficient of interference among predators at a high density of mature prey, respectively. All that parameters are positive.In this work, the system (8) is modified and then fractional order derivative is introduced in the modified system with the Caputo-type derivative. We also consider the harvesting in the mature prey only. The modified system is given as follows:(9)ddTX1T=RX2T−D1X1T−BX1TddTX2T=BX1T−D2X2T−A1X2TX3T1+αX2T1+βX3T−HX2TddTX3T=A2X2TX3T1+αX2T1+βX3T−D3X3T.The parameterH denotes the harvesting rate of mature prey species. Next, we reduce the parameters of the modified system (9) by using the following nondimensional transformation:(10)x1=BX1RD2,x2=αX2,x3=βX3,t=D1T.Therefore, we obtain a new system which has the following form:(11)ddtx1t=rx2t−1+bx1tddtx2t=d1x1t−d2x2t−a1x2tx3t1+x2t1+x3tddtx3t=a2x2tx3t1+x2t1+x3t−d3x3t,where(12)r=BαD2D1,b=BD1,d1=RD2αD1,d2=D2D1,a1=A1D1β,d3=D3D1,a2=A2D1α.Now, we present the fractional-order derivativesq in model (3) with the Caputo-type derivatives and then the system (11) will be as follows:(13)Dtqx1t=rx2t−1+bx1tDtqx2t=d1x1t−d2x2t−a1x2tx3t1+x2t1+x3t−hx2tDtqx3t=a2x2tx3t1+x2t1+x3t−d3x3t.Withx1t≥0,x2t≥0,x3t≥0.The following theorem gives the boundedness and non-negativity of the solution of the system (13).Theorem 1.
For the fractional-order system (13), we have the following:(1)
All solutions that start inF+=x1,x2,x3∈ℝ3|x1≥0,x2≥0,x3≥0 are non-negative solutions.(2)
If1+b>rd1/d2, then all solutions that start in F+ are bounded.Proof.
(1)
To prove the solution of the fractional-order system (13) is nonnegative, we start with x10>0fort=0 assuming that x1t≥0,fort≥0 is not true; therefore, there exists t∗>0 such that(14)x1t>0,0≤t<t∗x1t=0,t=t∗x1t<0,t>t∗.So, from the first equation of the system (13), we obtain the following:(15)Dtqx1t|t=t∗=0.According to part (1) of Lemma1, we get x1t∗+=0 and this is contradiction because x1t∗+<0,that isx1t<0,t>t∗.Therefore,x1t≥0,for allt≥0. In the same manner, we can prove that x2t≥0,x3t≥0,for allt≥0.(2)
LetF+=x1,x2,x3∈ℝ3|x1≥0,x2≥0,x3≥0, and x1, x2, and x3 in ℝ3, define a function Vt=x1t+r/d2x2t+ra1/d2a2ra1/d2a2x3t, it follows that(16)DqVt=Dqx1t+rd2Dqx2t+ra1d2a2Dqx3t=rx2−1+bx1+rd2d1x1−d2+hx2−a1x2x31+x21+x3+ra1d2a2a2x2x31+x21+x3−d3x3,=rd1d2−1+bx1−rd2hx2−ra1d2a2d3x3.Now, for eachk> 0, we have(17)DqVt+kVt=rd1d2−1+bx1−rd2hx2−ra1d2a2d3x3+kx1+krd2x2+kra1d2a2x3=k−1+b−rd1d2x1+rd2k−hx2+ra1d2a2k−d3x3.If we choosek<min1+b−rd1/d2,h,d3, then we get DqVt+kVt≤ε for some ε>0.
Then, from Lemma3, we obtain that Vt≤V0−ε/kEq−kt−0q+v/K. Therefore, we have 0≤Vt≤ε/k as t⟶∞ and all solutions of system (13) are bounded.
The next theorem gives the existence and uniqueness of the solution system (13).Theorem 2.
The existence and uniqueness solution of system (4) are given for each non-negative initial condition:Proof.
Let the region be defined byF × (0, T], T<∞.
WhereF=x1,x2,x3∈ℝ+3:maxx1,x2,x3≤M.
We assume thatX=x1,x2,x3,X˙ = x1˙,x2˙,x3˙ and BX=B1X,B2X,B3X be a mapping, such that(18)B1X=rx2−1+bx1,B2X=d1x1−d2+hx2−a1x2x31+x21+x3,B3X=a2x2x31+x21+x3−d3x3.
ForX,X˙∈F, one can get the following:(19)BX−BX˙=B1X−B1X˙+B2X−B2X˙+B3X−B3X˙=rx2−x2˙−1+bx1−x1˙+d1x1−x1˙−d2+hx2−x2˙−a1x2x31+x2˙1+x3˙−x2˙x3˙1+x21+x31+x21+x31+x2˙1+x3˙+−d3x3−x3˙+a2x2x31+x2˙1+x3˙−x2˙x3˙1+x21+x31+x21+x31+x2˙1+x3˙.
Since1/1+x21+x31+x2˙1+x3˙≤1,maxx1,x2,x3≤M, we have.
WhereL=max1+b+d1,r+d2+h+a1+a2M1+M,d3+a1+a2M1+M. Therefore, BX satisfies the Lipchitz condition, so that the system (4) has a unique solution.


## 4. Local Stability Analysis

In this section, we investigate the existence of the equilibrium points of the system (4). We also give the conditions for the local stability of its equilibria.Theorem 3 (see [41, 42]).
Consider the following fractional-order differential system:(20)Dtqxt⟶=f⟶x⟶.
x0⟶=x0⟶, q∈0,1, and x⟶∈ℝn. The point x∗⟶ that satisfies f⟶x∗⟶=0 is called the equilibrium point of the system (20). It is called locally asymptotically stable if argλi>qπ/2 for i=1,2,…..n, where λi are the eigenvalues of the Jacobian matrix which are evaluated at x∗⟶. Otherwise, it is called an unstable point.
To find all possible equilibrium points of the system (4), we have to solve the following equations:(21)Dtqx1t=0,Dtqx2t=0,Dtqx3t=0.
Therefore, the system (4) has three possible equilibrium points, namely:(1)
The trivial equilibrium pointe0=0,0,0 always exists.(2)
The free predator pointe1=x1∗,x2∗,0 exists if(22)x2∗=1+brx1∗,r=d2+h+d2b+hbd1.(3)
The interiorequilibrium pointe2=x3∗,x3∗,x3∗ exists only if(23)d1r1+b>d2+h,a1x3∗1+x3∗d1r/1+b−d2−h>1.wherex2∗=a1x3∗/1+x3∗d1r/1+b−d2−h−1,x1∗ = (r/1+bx2∗,x3∗ is the positive root of the following equation:(24)x2∗+ax2∗+b1=0.Here,a=a1d3−a1a2+a2d1r/1+b–d2−h/a1d3 and b1=a2d1r/1+b−d2−h/a1d3.
Sinceb1>0ifa<0 then, equation (24) may have two positive real roots.
The Jacobian matrix of fractional-order system (4) associated with arbitrary fixed point (x1, x2, x3) is given by(25)Jx1,x2,x3=−1+br0d1−d2+h−a1x31+x31+x22−a1x21+x31+x220a2x31+x31+x22a2x21+x21+x32−d3.
So, the general characteristic equation of (25) is as follows:(26)fλ=λ3+p2λ2+p1λ+p0=0.where(27)p2=1+b+d2+h+a1x31+x31+x22−a2x21+x21+x32−d3,(28)p1=a1a2x3x21+x231+x33−rd1−d2+h+a1x31+x31+x22a2x21+x21+x32−d3−1+bd2+h+a1x31+x31+x22+a2x21+x21+x32−d3,p0=1+ba1x21+x21+x32a2x31+x31+x22+rd1a2x21+x21+x32−d3−1+bd2+h+a1x31+x31+x22a2x21+x21+x32−d3.
To analyse the local stability of the fixed pointse0, e1,e2, we give the following theorem.Theorem 4.
For the system (4), we have the following:(1)
Ifd2+h+bd2+bh>rd1, then the point e0 is a locally stable point.(2)
The free predator pointe1 is always an unstable point.(3)
The interiorequilibrium pointe2 is locally asymptotically stable if one of the following conditions hold.(i)
IfDf>0, p2>0,p0>0 and p1p2>p0for q∈0,1.(ii)
IfDf<0,p2≥0,p1≥0,p0>0 for q<2/3, where p2,p1, and p0 are defined in equation(29).Proof.
(1)
For the pointe0, the Jacobian matrix Je0 is given as follows:(29)Je0=−1+br0d1−d2+h000−d3.So, the characteristic equation isPλ=−d3−λλ2+u1λ+u0.Whereu1=1+b+d2+h and u0=d2+h+bd2+bh−rd1. The roots of the above characteristic equation are λ1=−d3. It is clear that argλ1=π>qπ/2.Sinceu1>0 and if d2+h+bd2+bh>rd1 that means u0>0, then the Routh–Hurwitz conditions are satisfied.According to part 1 of Lemma5, e0 is stable.(2)
It is clear that the Jacobian matrixJe1 is as follows:(30)Je1=−1+b,d2+h+d2b+hbd1,0,d1,−d2+h,−a11+b/rx1∗1+1+b/rx1∗,0,0,a21+b/rx1∗1+1+b/rx1∗−d3..And the corresponding characteristic equation is as follows:(31)Pλ=a21+b/rx1∗1+1+b/rx1∗−d3−λλλ+1+b+d2+h=0.Therefore, one of the eigenvalues is equal to zero. Thus, the free predatorpointe1 is unstable.(3)
The Jacobian matrix at the interiorequilibrium pointe2 is given as follows:(32)Je2=−1+br0d1−d2+h−a1x3∗1+x2∗1+x3∗2−a1x2∗1+x2∗1+x3∗20a2x3∗1+x2∗1+x3∗2a2x2∗1+x2∗1+x3∗2−d3.So, the characteristic polynomial ofJe2 can be obtained:(33)fλ=λ3+p2λ2+p1λ+p0,where(34)p2=1+b+d2+h+a1x3∗1+x3∗1+x2∗2+d3−a2x3∗1+x2∗1+x3∗2,p1=a1a2x3∗x2∗1+x3∗31+x3∗3−rd1−d2+h+a1x3∗1+x3∗1+x2∗2a2x2∗1+x2∗1+x3∗2−d3+1+bd2+h+a1x3∗1+x3∗1+x2∗2+d3−a2x2∗1+x2∗1+x3∗2,p0=1+ba1x3∗1+x2∗1+x3∗2a2x3∗1+x3∗1+x3∗2+rd1a2x3∗1+x3∗1+x3∗2−d3−1+bd2+h+a1x3∗1+x3∗1+x2∗2a2x2∗1+x2∗1+x3∗2−d3.Then, the discriminantDf of the cubic polynomial fλ is the following:(35)Df=−1p2p101p232p2p1p00p1p00000302p23p12p20p1=18p2p1p0+p2p12−4p0p23−4p13−27p02.
According to Lemma5 (2), if Df>0, p2>0,p0>0 and p1p2>p0for q∈0,1, or Df<0,p2≥0,p1≥0,p0>0 for q<2/3.
Then, the interior fixed pointe2 is locally asymptotically stable.


## 5. Numerical Simulations

In this section, we give numerical simulations to confirm the theoretical results that are employed in Section4. For the equilibrium point e0, the parameter values in Table 1 are considered to carry out that e0 is locally asymptotically stable. Figure 1 shows that e0 is locally stable according to conditions in Theorem 4.Table 1
Parameter values of the equilibrium pointe0.
ParameterParameter valued10.1b0.2825h0.1d30.5133r6.2416a20.5851d20.9872a10.6621q1,0.98, 0.88, 0.78x100.4175x201.9799x300.837Figure 1
Shows the local stability of the equilibrium pointe0 at which x (t) represents immature prey species, y (t) represents mature prey species, and z (t) represents predator species. The values in Table 1 are used with different values of fractional-order(a) q=1, (b) q=0.98, (c) q=0.88, and (d) q=0.78.
(a)(b)(c)(d)For the equilibrium pointe2, the values of parameter are chosen and set in Table 2. According to the conditions in Theorem 4, the point e2 is locally stable. Figure 2 illustrates the stability of the positive equilibrium point e2.Table 2
Parameter values of the equilibrium pointe2.
ParameterParameter valued10.3299b0.2825h0.6168d30.5133r6.2416a20.5851d20.9872a10.6621q1, 0.98, 0.88, 0.78x1041.6x208x300.2Figure 2
Shows the local stability of the equilibrium pointe3 at which x (t) represents immature prey species, y (t) represents mature prey species, and z (t) represents predator species. The values in Table 2 are used with different values of fractional-order(a) q=1, (b) q=0.98, (c) q=0.88, and (d) q=0.78.
(a)(b)(c)(d)


## 6. Conclusions

The fractional-order differential system has been successfully applied in mathematical biology. In this paper, we have discussed the dynamical behaviour of a stage-structure prey-predator with Crowley–Martian functional response and a linear harvesting rate in a fractional-order state. The existences, uniqueness, non-negativity, and the boundedness solutions of the fractional-order system are presented and proved. It is found that the considered system has at least three equilibrium points. One of them are unstable points. For the other points, conditions are set to guarantee their local stability.



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