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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.

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, [

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. [

In [

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 (

In the next section, an analysis of infected predator-free equilibrium of (

Consider (

By introducing scaling variables

Let

Now assume that the predator becomes disease free and for simplicity let us consider

Now, we introduce distributed delay to (

Here, the function

Denote by

Define

Solutions of (

For each bounded functions

For all

Finally, as

Solutions of (

By the previous theorem,

Solutions of (

Note that

Consider

The derivative of

Now,

Note that

Also,

has a solution

For each

Set

Now,

If

If

If

Denote

If the solutions of (

As

Consider the case

Sets

For

Consider three equilibria of (

The linearization of the system (

Here,

At

Since two of the eigen values are positive

We know that

If

The limit of

The linearization around

The characteristic equation takes the form

The term in the square brackets has roots

Substituting

First, we show that if

We know that

The characteristic equation around

where

If

Since

Simplify (

By Routh hurwitz criterion if

As

For

Hence, none of the root of (

Also, if

If

Since

Solutions with positive initial conditions will remain positive for all

Consider the solutions of (

Here, for every solution with initial data in

Assume that

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

No data were used to support the study.

The authors declare that they have no conflicts of interest.