Analysis of a Predator-Prey Model with Distributed Delay

Department of Mathematics, College of Engineering and Technology, Faculty of Engineering and Technology, SRM Institute of Science and Technology, SRM Nagar, Kattankulathur 603203, Kanchipuram, Chennai, Tamilnadu, India Department of Mathematics, College of Sciences and Arts, Qassim University, Ar Rass, Saudi Arabia Laboratory of Fundamental and Applied Mathematics of Oran (LMFAO), University of Oran 1, Oran, 31000 Oran, Algeria PG and Research Department of Mathematics, Thiagarajar College, Madurai 625009, India Preparatory Institute for Engineering Studies in Sfax, Tunisia


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, [1][2][3][4][5][6][7][8][9][10][11][12][13][14]). 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 [15]. 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. [16] 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. [17] studied the following epidemic model consisting of four species, namely, sound prey, infected prey, sound predator, and infected predators.
In [4], we have modified the system (1) with discrete delay.
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.

Infected Predator-Free Equilibrium
By introducing scaling variables x 1 ðtÞ = θX 1 ð℘Þ, x 2 ðtÞ = ψX 2 ð℘Þ, x 3 ðtÞ = ψX 3 ð℘Þ, x 4 ðtÞ = ωX 4 Now assume that the predator becomes disease free and for simplicity let us consider X = E = 1. Then, (4) becomes Now, we introduce distributed delay to (5) Here, the function hðuÞ is the kernel of the distributed delay with the following properties 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 ð−∞, 0 into R 3 fitted with the uniform norm. We consider initial data Denote the solutions of (6) with initial data Φ ∈ ≀ 3 + at time t by ΠðΦ, tÞ when they exist. Hence, for mentioning the positive solutions, we are referring to the solutions ΠðΦ, tÞ with Φ ∈ int ≀ 3 + . Later, we show that each component is positive for all t > 0 in this case.

Well Posedness and Basic Properties of the Model
Define L > 0 and assume that gðsÞ = 0 for all s ∈ ðL,∞Þ. We allow L = ∞.

Journal of Function Spaces
Theorem 1. Solutions of (6) exist, with initial data in ≀ 3 + , and for all t > 0, they are unique and remain in ≀ 3 + .
Finally, as Proposition 2. Solutions of (6) with positive initial conditions remain positive for all t > 0.

The linearization of the system (6) around an equilibrium
Here,

Stability at
Since two of the eigen values are positive E 0 is an unstable saddle point. Lemma 6. E 0 is globally asymptotically stable with initial data in X 1 .

4.2.
Stability at E 1 . The linearization around E 1 takes the form The characteristic equation takes the form Theorem 7. E 1 is locally asymptotically stable if Ð ∞ 0 e −u hðuÞd u < 1 and unstable if either inequality is reversed.
Proof. The term in the square brackets has roots −ðR 2 + R 1 /A BðA − ABÞÞ, −R 1 which are both negative iff R 2 > R 1 /ABðA − ABÞ. The stability of E 1 is determined by the sign of the real parts of the roots of mðλÞ = ðλ + 1 − Ð ∞ 0 e ð−1+λÞu hðuÞduÞ. Substituting λ = β + iγ, γ ≥ 0 in mðλÞ and separating real and imaginary parts, we obtain First, we show that if Ð ∞ 0 e −u hðuÞdu > 1, then mðλÞ has a positive real root. Note that if γ = 0, then (22) is satisfied. In this case in (21), Lð0Þ < Rð0Þ, RðβÞ is a decreasing function of β and LðβÞ is an increasing function of β and lim β⟶∞ L ðβÞ = +∞. Therefore, mðλÞ has a real root which is positive and E 1 is unstable. Also, if Ð ∞ 0 e −u hðuÞdu < 1, Lð0Þ > Rð0Þ, R ðβÞ is decreasing and LðβÞ is increasing, and (21) can never be satisfied for β > 0. Hence, E 1 is locally asymptotically stable. 4 Journal of Function Spaces Lemma 8. E 1 is globally asymptotically stable with initial data in X 2 .

Properties of the Model when
Lemma 9. If ℘ = 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.
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.
Proof. Since ℘ = 0, then hðuÞ = δ 0 ðuÞ, and therefore, system (6) reduces to its ODE prototype 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 int R 2 + . Observe that only the solutions with initial conditions on the y-axis converge to E 0 , while solutions on the x-axis, not including the origin, converge to E 1 . On the other hand, E 1 repels the solutions with initial data not on the x-axis. Using straightforward phase plane argument, one can see that neither E 0 , nor E 1 is in the ω-limit set of solutions with initial data in int R 2 + . Then, by the Poincare-Bendixson theorem [10], E + is globally asymptotically stable.

Lemma 12.
Consider the solutions of (6) with initial data in X 0 ∪ X 2 . There is no positive monotonically increasing sequence ft n g, with t n ⟶ ∞ as n ⟶ ∞ such that ðx 1 ðt n Þ, x 2 ðt n Þ, x 3 ðt n ÞÞ converges to E 0 .
Here, for every solution with initial data in X 0 ∪ X 2 , xðtÞ > 0∀ t ≥ 0. We prove this theorem by contradiction.

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.