Analysis of a Predator-Prey Model with Distributed Delay

Abstract

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, [1–14]). 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.

2. Infected Predator-Free Equilibrium

Consider (1)

By introducing scaling variables where .

Let . We obtain

Now assume that the predator becomes disease free and for simplicity let us consider . Then, (4) becomes

Now, we introduce distributed delay to (5)

Here, the function 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 , the Banach space of bounded continuous functions mapping from into fitted with the uniform norm. We consider initial data : , . Define : . Denote the solutions of (6) with initial data at time by when they exist. Hence, for mentioning the positive solutions, we are referring to the solutions with . Later, we show that each component is positive for all in this case.

3. Well Posedness and Basic Properties of the Model

Define and assume that for all . We allow .

Theorem 1. Solutions of (6) exist, with initial data in , and for all , they are unique and remain in .

Proof. For each bounded functions , there exists a unique solution of (6), such that ,
For all , if . If , then will remain positive. Similarly, if will remain positive. Hence, for all , there exists a unique solution.
Finally, as on , for all . Hence, for all . By induction, , for an and ,. Hence, for all , .

Proposition 2. Solutions of (6) with positive initial conditions remain positive for all .

Proof. By the previous theorem, , then for all and , then for all . Assume that at . This implies that . which is positive, a contradiction.

Lemma 3. Solutions of (6) are bounded and , and .

Proof. Note that Also, , given a . Therefore, . Similarly, .
Consider The derivative of with respect to , Now, Note that . Therefore, Also, has a solution For each with and for every for all . Provided , by the comparison theorem, we conclude that is also bounded by , and therefore, so is
Set Now, (1)If , since it is continuous and , then there exists , such that for all (2)If for all , then (3)If , since is continuous, there exists such that for all and Denote to be the solution of (6) with the initial data in . Set .

Lemma 4. If the solutions of (6) have initial conditions in , then . Also, and .

Proof. As and , for all . Consider the case . Then, by Theorem 1, , and hence, , and .
Consider the case . Since and are continuous functions, then there exists and . Therefore, there exists time such that and such that . Then, as , , it follows that

Lemma 5. Sets and are positively invariant under .

Proof. For , the result is true under by Lemma 4. If the solution has initial conditions in , then, by (6), . By Theorem 1, . Now, the solutions with initial conditions in are taken into consideration. aS and since is a solution of (6) with and , then by the uniqueness of solutions, . Hence, . Now, from Theorem 1, and .

4. Stability Results with General Delay

Consider three equilibria of (6), and .

The linearization of the system (6) around an equilibrium is given by

Here,

4.1. Stability at

At (15) becomes

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

Lemma 6. is globally asymptotically stable with initial data in .

Proof. We know that and is equal to for all . Now, consider If and therefore , then since , . Hence, . Hence,
If t , then , . Also, as , and hence, The limit of as is 0. Therefore, .When , as for , then . Then, .

4.2. Stability at

The linearization around takes the form

The characteristic equation takes the form

Theorem 7. is locally asymptotically stable if and unstable if either inequality is reversed.

Proof. The term in the square brackets has roots which are both negative iff . The stability of is determined by the sign of the real parts of the roots of .
Substituting in and separating real and imaginary parts, we obtain First, we show that if , then has a positive real root. Note that if , then (22) is satisfied. In this case in (21), , is a decreasing function of and is an increasing function of and . Therefore, has a real root which is positive and is unstable. Also, if , , is decreasing and is increasing, and (21) can never be satisfied for . Hence, is locally asymptotically stable.

Lemma 8. is globally asymptotically stable with initial data in .

Proof. We know that . Then, (6) becomes an ODE model. By Lemma 6 in [9], this lemma is true.

4.3. Properties of the Model when Exists

The characteristic equation around is

where

Lemma 9. If , is locally asymptotically stable.

Proof. Since , then . and becomes
. Then, (23) becomes Simplify (25) to the following equation By Routh hurwitz criterion if , is locally asymptotically stable.

Theorem 10. As increases from zero, if a root appears on or crosses the imaginary axis as increases from , then the number of roots of (23) with a positive real part can change.

Proof. For , it is easy that .
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 , then is locally asymptotically stable and if , then (23) has no positive real roots.

4.3.1. Global Dynamics

Lemma 11. If and , then is globally asymptotically stable with respect to the solutions of (6) with and .

Proof. Since , then , and therefore, system (6) reduces to its ODE prototype Solutions with positive initial conditions will remain positive for all . Using the Dulac criterion, we observe that there are no periodic solutions lying in . Observe that only the solutions with initial conditions on the -axis converge to , while solutions on the -axis, not including the origin, converge to . On the other hand, repels the solutions with initial data not on the -axis. Using straightforward phase plane argument, one can see that neither , nor is in the -limit set of solutions with initial data in . Then, by the Poincare-Bendixson theorem [10], is globally asymptotically stable.

Lemma 12. Consider the solutions of (6) with initial data in . There is no positive monotonically increasing sequence , with as such that converges to .

Here, for every solution with initial data in , . We prove this theorem by contradiction.

Assume that a monotonically increasing sequence which is positive, with such that and converges to as . For every , , and , for all . Set . Then, , for sufficiently large , which is a contradiction. Also, if , then no solution of (6) with initial data in converges to .

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 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.