JFSJournal of Function Spaces2314-88882314-8896Hindawi10.1155/2021/99544099954409Research ArticleAnalysis of a Predator-Prey Model with Distributed DelayChandrasekarGunasundari1https://orcid.org/0000-0003-1308-2159BoulaarasSalah Mahmoud23MurugaiahSenthilkumaran4https://orcid.org/0000-0001-6346-605XGnanaprakasamArul Joseph1https://orcid.org/0000-0002-5526-165XCherifBahri Belkacem25RagusaMaria Alessandra1Department of MathematicsCollege of Engineering and TechnologyFaculty of Engineering and TechnologySRM Institute of Science and TechnologySRM NagarKattankulathur 603203KanchipuramChennaiTamilnaduIndiasrmuniv.ac.in2Department of MathematicsCollege of Sciences and ArtsQassim UniversityAr RassSaudi Arabiaqu.edu.sa3Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO)University of Oran 1Oran31000 OranAlgeriauniv-oran1.dz4PG and Research Department of MathematicsThiagarajar CollegeMadurai 625009Indiatcarts.in5Preparatory Institute for Engineering Studies in SfaxTunisiaipeis.rnu.tn202135202120211332021224202123420213520212021Copyright © 2021 Gunasundari Chandrasekar et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we consider a predator-prey model, where we assumed that the model to be an infected predator-free equilibrium one. The model includes a distributed delay to describe the time between the predator’s capture of the prey and its conversion to biomass for predators. When the delay is absent, the model exhibits asymptotic convergence to an equilibrium. Therefore, any nonequilibrium dynamics in the model when the delay is included can be attributed to the delay’s inclusion. We assume that the delay is distributed and model the delay using integrodifferential equations. We established the well-posedness and basic properties of solutions of the model with nonspecified delay. Then, we analyzed the local and global dynamics as the mean delay varies.

1. Introduction

In applied engineering and complex system sciences, mathematical models that display deterministic chaotic dynamical behaviour are of interest. The majority of encounters in nature are admittedly delayed or isolated, as both predator and prey function stochastically in absorbing available resources. This can be used to share bandwidth and resources among network users at a bottleneck node or a leaky bucket used to track flows, for example. If we assume that network users’ behaviour is stochastic and that the accommodating segment has limited buffering space, then forwarding generated data packets can be compared to a predator-prey style interaction with limited resources characteristics during rush hours, when users interact intensively. One approach to examining a heterogeneous network susceptible to attack is modeling cyberspace as a predator-prey landscape. The predator-prey model of Gauss type is a well-known simple mathematical model describing the interaction between species. Its variations and extensions are studied in modern day population dynamics theory (see, for example, ). This model is based on the assumption that in real-world ecosystems prey populations do not grow exponentially in the absence of a predator, but rather their size is eventually limited by the absence of resources. Fan and Wolkowicz studied the effects of incorporating discrete delay in . The delay corresponds to the time lag between predator capturing the prey and its conversion to biomass for predators. Their research focused on switches of stability of the coexistence equilibrium, the occurrence of periodic solutions, and subsequent bifurcation dynamics as the length of the delay increased. Li et al. analyzed a Gause-type predator-prey model in which adult and juvenile death rates were taken to be different. In their work, the delay denoted the maturation period of the predator. They studied the dynamical behaviour of the system for the functional responses of Holling type I and Holling type II. They established the existence of stability switches due to Hopf bifurcations. These bifurcations occur in pairs that are connected and are nested. They have also shown that there is a range of parameters for which there exist two or more stable periodic solutions.

In nature, for each case, the processing delay rarely has the same duration, and instead follows a distribution of some mean value. Recently, Chaudhuri et al.  studied the following epidemic model consisting of four species, namely, sound prey, infected prey, sound predator, and infected predators. (1)dX1τdτ=X1r1X1X1+X2k~γX1X2βX4X1b1X1X3b3X1X4dX2τdτ=X2r2X2ν+γX1X2X2X1+X2k~b2X2X3b4X2X4+βX1X4dX3τdτ=X3mαX2ηX4+db1X1+b2X2dX4τdτ=X4mμ+ηX3+db3X1+b4X2+αX2X3.

In , we have modified the system (1) with discrete delay. (2)dX1τdτ=X1r1X1X1+X2k~γX1X2tτ1βX4X1b1X1X3b3X1X4dX2τdτ=X2r2X2ν+γX1X2tτ1X2X1+X2k~b2X2X3b4X2X4+βX1X4dX3τdτ=X3mαX2ηX4+db1X1+b2X2dX4τdτ=X4mμ+ηX3+db3X1+b4X2+αX2X3.

They investigated the stability properties and the existence of Hopf bifurcation. In this paper, we study the effects of incorporating distributed delay in the system (1) for infected predator-free equilibrium.

In the next section, an analysis of infected predator-free equilibrium of (1) is presented. In Section 3, we established the well posedness and basic properties of the model. We investigated the stability properties for different equilibriums in Section 4. Section 5 with conclusions completes the paper.

2. Infected Predator-Free Equilibrium

Consider (1) (3)dX1τdτ=X1r1X1X1+X2k~γX1X2βX4X1b1X1X3b3X1X4dX2τdτ=X2r2X2ν+γX1X2X2X1+X2k~b2X2X3b4X2X4+βX1X4dX3τdτ=X3mαX2ηX4+eb1X1+b2X2dX4τdτ=X4mμ+ηX3+eb3X1+b4X2+αX2X3.

By introducing scaling variables x1t=θX1,x2t=ψX2,x3t=ψX3,x4t=ωX4,t=σ where σ=m,ϕ=γ/m,ω=β/m,θ=eb1/m,ψ=b1/m.

Let A=γ/eb1,B=1/K~γ,C=b3/β,C=b4/β,E=α/γ,F=η/β,G=b3/b1,X=eb2/γ,R1=r1/m,R2=r2v/m,M=μ/m. We obtain (4)ẋ1=x1R1ABx12Bx2x1x2x1x1x4x1x3Cx1x4ẋ2=x2R2x2ABx1Bx22+Ax2x1AXx3x2Dx2x4+Ax1x4ẋ3=x3+x1x3+Xx2x3Ex2x3Fx3x4ẋ4=x4Mx4+Gx1x4+GDACx2x4+GFx3x4C+EGx2x3C.

Now assume that the predator becomes disease free and for simplicity let us consider X=E=1. Then, (4) becomes (5)ẋ1=x1R1ABx12Bx2x1x2x1x1x3ẋ2=x2R2x2ABx1Bx22+Ax2x1Ax2x3ẋ3=x3+x1x3.

Now, we introduce distributed delay to (5) (6)ẋ1=x1R1ABx1B+1x2x1x1x3ẋ2=x2R2Bx2x2x1ABAAx2x3ẋ3=x3+0x1tux3tueuhudu.

Here, the function hu is the kernel of the distributed delay with the following properties (7)0hudu=10uhudu=,where is the mean delay between the capture of the prey to the conversion into the biomass of the predator.

Denote by 3, the Banach space of bounded continuous functions mapping from ,0 into R3 fitted with the uniform norm. We consider initial data Φ=Φ1,Φ2,Φ3+3=(Φ3 : ΦiΘ0, i=1,2,3,Θ0). Define int+3=(Φ3: ϕiΘ>0,i=1,2,3Θ0). Denote the solutions of (6) with initial data Φ+3 at time t by ΠΦ,t when they exist. Hence, for mentioning the positive solutions, we are referring to the solutions ΠΦ,t with Φint+3. Later, we show that each component is positive for all t>0 in this case.

3. Well Posedness and Basic Properties of the Model

Define L>0 and assume that gs=0 for all sL,. We allow L=.

Theorem 1.

Solutions of (6) exist, with initial data in +3, and for all t>0, they are unique and remain in +3.

Proof.

For each bounded functions Φ+3, there exists a unique solution of (6), ΠΦ,t such that ΠΦ,.,0=Φt,

For all t0, x1t=x2t=0 if x10=x20=0. If x10>0, then x1t will remain positive. Similarly, if x20>0,x2t will remain positive. Hence, for all t>0, there exists a unique solution.

Finally, as x3t0 on L,0, ẋ3tx3t for all 0tL. Hence, x3t0 for all LtL. By induction, x3t0, for an tLn1,Ln and nN,n>0. Hence, for all t>0, x3t0.

Proposition 2.

Solutions of (6) with positive initial conditions remain positive for all t>0.

Proof.

By the previous theorem, x10>0, then x1t>0 for all t>0 and x20>0, then x2t>0 for all t>0. Assume that x3t=0 at t^. This implies that ẋ3t^0. dx3t^/dt=0x1t^ux3t^ueuhudu which is positive, a contradiction.

Lemma 3.

Solutions of (6) are bounded and limsuptx1tR1/AB, limsuptx2tR2/B and x3t1+R12/4R10euhudu.

Proof.

Note that ẋ1tx1tR1ABx1t. Also, x100, given ε>0 a T>0x1t<R1/AB+εforalltT. Therefore, limsuptx1tR1/AB. Similarly, limsuptx2tR2/B.

Consider (8)zt=x3t+0x1tueuhudu.

The derivative of zt with respect to t, (9)żt=ẋ3t+0ẋ1tueuhudu.

Now, (10)żt=x3t+0x1tux3tueuhudu+euhudu0x1tuR1ABx1tux1tux2tuB+1x1tux3tu,żt=zt+0x1tueuhudu1+R1ABx1tuB+1x2tu.

Note that 1+R1/2ABx1tu+B+1x2tu20. Therefore, (11)żtzt+1+R1240euhudu.

Also, (12)ẇt=wt+1+R1240euhudu,

has a solution (13)wt=w0et+1et1+R124R10euhudu.

For each wt with w00 and for every ε>0T>0wt1+R12/4R10euhudu+ε, for all t>T. Provided w0=z0, by the comparison theorem, we conclude that limsuptzt is also bounded by 1+R12/4R10euhudu, and therefore, so is limsuptx3t.

Set (14)X0+3:Φ20>0andΘL,0suchthatΦ2ΘΦ3Θ>0X1+2:Φ20=0X2+2:Φ20>0andΦ2ΘΦ3Θ=0,ΘL,0X=X0X1X2.

Now,

If ΦX0, since it is continuous and Φ20>0, then there exists a>0, such that for all Θa,0Φ2Θ>0

If Φ2ΘΦ3Θ=0 for all ΘL,0, then 0Φ2uΦ3ueuhudu=0

If ΦX2, since Φ is continuous, there exists a>0 such that for all Θa,0,Φ2Θ>0 and Φ3Θ=0

Denote x1t,x2t,x3t=x1t,Φt,x2t,Φt,x3t,Φt to be the solution of (6) with the initial data Φt in X0. Set TtΦΘ=x1t+Θ,x2t+Θ,x3t+Θ,Θ,0.

Lemma 4.

If the solutions of (6) have initial conditions in X0, then t>0ΘtL,tx1Θx3Θ>0. Also, M>0x1t>0,x3t>0 and 0x1tux3tueuhudu>0t>M.

Proof.

As Φ10>0 and Φ20>0, x1t>0,x2t>0 for all t>0. Consider the case Φ30>0. Then, by Theorem 1, x3t0t, and hence, ẋ3tx3t, and x3t>0t0.

Consider the case Φ30=0. Since Φ2t and Φ3t are continuous functions, then there exists t1,t2RL<t1<Θ0<t20 and Φ2tΦ3t>0tt1,t2. Therefore, there exists time TL+t1 such that ẋ3T>0 and ε>0 such that x3t>0tT,T+ε. Then, as ẋ3tx3t, x3t>0t>T, it follows that θ0t1,t2t1,T.

Lemma 5.

Sets X0,X1, and X2 are positively invariant under Tt.

Proof.

For X0, the result is true under Tt by Lemma 4. If the solution has initial conditions in X1, then, by (6), x2t=0t>0. By Theorem 1, x3t0t>0. Now, the solutions with initial conditions in X2 are taken into consideration. aS x30=0 and since x3t=0 is a solution of (6) with x30=0 and Φ2ΘΦ3Θ=0ΘL,0, then by the uniqueness of solutions, x3t=0t>0. Hence, 0x1tux3tueuhudut>0. Now, from Theorem 1, x1t0 and x2t>0t>0.

4. Stability Results with General Delay

Consider three equilibria of (6), E0=0,0,0,E1=R1/AB,00 and E+=0euhudu1,R2AR1ABAA2B0euhudu/BABA,R2ABA0euhudu)1Bx2/A.

The linearization of the system (6) around an equilibrium E+=x1,x2,x3 is given by (15)Ẋt=AXt+B0euhuXtudu.

Here, (16)A=R12ABx1x2B+1x3x1B+1x1A1Bx2R22Bx2+x1AABAx3Ax2001,B=000000x30x1.

4.1. Stability at <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M196"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>

At E0 (15) becomes (17)ẋ1tẋ2tẋ3t=R1000R20001x1tx2tx3t.

Since two of the eigen values are positive E0 is an unstable saddle point.

Lemma 6.

E0 is globally asymptotically stable with initial data in X1.

Proof.

We know that x1t and x2t is equal to 0 for all t>0. Now, consider L=. If Φ2ΘΦ3Θ=0Θ,0 and therefore 0Φ2uΦ3ueuhudu, then since x1t=0t>0, 0x1tux3tueuhudu=0t>0. Hence, ẋ3t=x3tt>0. Hence, limtx3t=0.

If Θ,0t Φ2ΘΦ3Θ>0, then T>0, 0x1tux3tueuhudu>0t>T. Also, as x1t=0t>0, and hence, (18)ẋ3t=x3t+0tx1tux3tueuhudu+tx1tux3tueuhudu=x3t+tx1tux3tueuhudu.

The limit of tx1tux3tueuhudu as t is 0. Therefore, limtx3t=0.When L<, as x1t=0 for t>0, then tx1tux3tueuhudu=0t>L. Then, ẋ3t=x3tt>L.

4.2. Stability at <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M247"><mml:msub><mml:mrow><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>

The linearization around E1 takes the form (19)ẋ1tẋ2tẋ3t=R1R1ABB+1R1AB0R2+R1ABAAB0001x1tx2tx3t+00000000R1AB0x1tueuhudu0x2tueuhudu0x3tueuhudu.

The characteristic equation takes the form (20)λ+10e1+λuhuduλ2+λR2+R1ABAABR1+R1R2+R1ABAAB=0.

Theorem 7.

E1 is locally asymptotically stable if 0euhudu<1 and unstable if either inequality is reversed.

Proof.

The term in the square brackets has roots R2+R1/ABAAB,R1 which are both negative iff R2>R1/ABAAB. The stability of E1 is determined by the sign of the real parts of the roots of mλ=λ+10e1+λuhudu.

Substituting λ=β+iγ,γ0 in mλ and separating real and imaginary parts, we obtain (21)Lβ=β+1=0e1+λucosγuhudu=Rβ,(22)γ=0e1+λusinγuhudu=0.

First, we show that if 0euhudu>1, then mλ has a positive real root. Note that if γ=0, then (22) is satisfied. In this case in (21), L0<R0, Rβ is a decreasing function of β and Lβ is an increasing function of β and limβLβ=+. Therefore, mλ has a real root which is positive and E1 is unstable. Also, if 0euhudu<1, L0>R0, Rβ is decreasing and Lβ is increasing, and (21) can never be satisfied for β>0. Hence, E1 is locally asymptotically stable.

Lemma 8.

E1 is globally asymptotically stable with initial data in X2.

Proof.

We know that x3t=0t>0. Then, (6) becomes an ODE model. By Lemma 6 in , this lemma is true.

4.3. Properties of the Model when <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M282"><mml:msub><mml:mrow><mml:mi mathvariant="bold-italic">E</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msub></mml:math></inline-formula> Exists

The characteristic equation around E+=x1,x2,x3 is (23)λ3+Aλ2+Bλ+C+0e1+λuhuduDλ2+λE+F=0,

where A=1x1A+x2+3x1AB+3x2B+x3+Ax3R1R2,(24)B=x1A+x2+3x1AB2x12A2B+3x2B+2x22B+2x12A2B2+4x1Ax2B2+2x22B2+x3+AX3x1x3A+Ax2x3+x1x3AB+2x1x3A2B+2x2x3B+ABx2x3+Ax32R1+x1AR1x1ABR12x2BR1Ax3R1R2x2R22x1ABR2x2BR2x3R2+R1R2,C=2x12A2B+2x22B+2x12A2B2+4x1x2AB2+2x22B2x1x3A+x2x3A+x2x3AB+2x1A2Bx3+2Bx2x3+ABx2x3+Ax32+x1AR1x1ABR12x2BR1Ax3R1x2R22x1ABR2x2BR2x3R2+R1R2D=x1,E=x12Ax2x13x12AB3x1x2BAx1x3+R1x1+R2x1,F=2x13A2B2x23B2x13A2B24x12x2AB22x1x22B22Ax1x2x32x12x3A2B2ABx1x2x3x12AR1+x12ABR1+2x1x2BR1+Ax1x3R1+x1x2R2+2x12ABR1+x1x2BR2x1R1R2.

Lemma 9.

If =0, E+ is locally asymptotically stable.

Proof.

Since =0, then hu=δ0u. 0eαuhudu=1 and E+ becomes

1,R2AR1ABAA2B/BABA,R2ABABx2/A. Then, (23) becomes (25)λ3+Aλ2+Bλ+C+Dλ2+λE+F=0.

Simplify (25) to the following equation (26)λ3+A+Dλ2+B+Eλ+C+F=0.

By Routh hurwitz criterion if A+DB+EC+F>0, E+ is locally asymptotically stable.

Theorem 10.

As increases from zero, if a root appears on or crosses the imaginary axis as increases from 0, then the number of roots of (23) with a positive real part can change.

Proof.

For gλ=Aλ2+Bλ+C+0e1+λuhuduDλ2+λE+F, it is easy that limsupλλ3gλ=0<1.

Hence, none of the root of (23) with positive real part can enter from as bifurcates from 0. As Lemma 6 holds, the result follows.

Also, if 0euhudu>1, then E+ is locally asymptotically stable and if 0euhudu>1, then (23) has no positive real roots.

4.3.1. Global DynamicsLemma 11.

If 0euhudu>1 and =0, then E+ is globally asymptotically stable with respect to the solutions of (6) with x10>0,x20>0 and x30>0.

Proof.

Since =0, then hu=δ0u, and therefore, system (6) reduces to its ODE prototype (27)ẋ1t=x1R1ABx1x2B+1x3ẋ2t=x2R2Bx2x1ABAAx3ẋ2t=x3+x1x3.

Solutions with positive initial conditions will remain positive for all t>0. Using the Dulac criterion, we observe that there are no periodic solutions lying in intR+2. Observe that only the solutions with initial conditions on the y-axis converge to E0, while solutions on the x-axis, not including the origin, converge to E1. On the other hand, E1 repels the solutions with initial data not on the x-axis. Using straightforward phase plane argument, one can see that neither E0, nor E1 is in the ω-limit set of solutions with initial data in intR+2. Then, by the Poincare-Bendixson theorem , E+ is globally asymptotically stable.

Lemma 12.

Consider the solutions of (6) with initial data in X0X2. There is no positive monotonically increasing sequence tn, with tn as n such that x1tn,x2tn,x3tn converges to E0.

Here, for every solution with initial data in X0X2, xt>0t0. We prove this theorem by contradiction.

Assume that a monotonically increasing sequence tn which is positive, with tn such that ẋtn0 and x1tn,x2tn,x3tn converges to E0 as n. For every ε>0, T>0x1tn<ε, x2tn<ε and x3tn<ε, for all tn>T. Set ε<1/2. Then, ẋtn>xtn12lε>0, for sufficiently large n, which is a contradiction. Also, if 0euhudu>1, then no solution of (6) with initial data in X0 converges to E1.

5. Conclusion

Through evolution, nature has developed natural propensities in complex systems (including animalia and plants) that enable survival through adaptation. Malicious agents, such as viruses, worms, and denial-of-service attacks, plague the Internet and the vast array of networks and applications that link to it. For example, using the Internet as an environment, the malicious attacks described above (viruses) can be viewed as predators, with their interactions with the ecosystem (servers) resembling a predator-prey relationship. A predator-prey model with distributed delay is considered in this paper. For infected predator-free equilibrium, we established properties of the system such as positivity and boundedness and conditions for global asymptotic stability of some equilibria for the general delay. We were particularly interested in the dynamics when E+ exists. We showed that solutions with positive initial data remain positive for all time. Moreover, we determined the set of initial data such that the solutions eventually become positive.

Data Availability

No data were used to support the study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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