Asymptotic Behavior of a Predator-Prey Model with Allee Threshold Applied to Online Social Network Users’ Data Forwarding

We consider a predator-prey relationship in a fair system in which interacting species have diﬀerent needs of resources to survive. We analyzed qualitatively the outcome of interaction using a modiﬁed logistic predator-prey model with Allee threshold in both predator and prey equations. We showed that the system had very rich dynamical behavior as stability around ﬁxed points and periodic solutions could be obtained at certain conditions. Interaction outcome is highly submitted to initial conditions, species behavior, and the threshold applied. Numerical results suggested adapting resource allocation and the threshold value to optimize ecosystem sustainability.


Introduction
In an ecosystem where resource accessibility for interactive species is fair, competition intensity, which can be assimilated to each species ability to harvest, is a key parameter in controlling system stability and optimizing ecosystem sustainability. In addition, when shared resource, for instance, niche or habitat is extremely limited, one needs to monitor species birth rate, death rate, and overall population size to avoid overdomination and extinction that can lead to chaos [1][2][3][4].
In applied mathematics and engineering sciences, predator-prey models based on Lotka-Volterra are commonly used to study such systems. ere exist numerous models adjusted to formalize the particular relationship between individuals of the same or different species sharing one or many types of resources. Models with logistic growth, which implies limiting capacity, are of interest as they are admitted to be more realistic in studying underlying relationship between entities. However, classical Lotka-Volterra systems by supposing unlimited resources for interacting species and focusing mainly on interaction outcome fail to formalize small variations happening in species intrinsic growth rate due to resource abundance, competition, or delay in assimilation of harvested resource, for example. As many research studies pointed out, in nature and in complex systems, such as social networks, most of interactions occur in limited resource environment [5][6][7][8][9]. us, models that incorporate threshold or refuge terms in predator or prey equations present special interest for implementing better control mechanism. Such models are widely spread and intensively studied in related literature. Particularly, models with Allee-threshold extinction control mechanism present the advantage of capturing such variations of species growth rate by setting a maximum value to reach for population size to not go extinct [10][11][12][13][14][15].
Furthermore, systems that exhibit rich dynamical behavior, including periodic solutions, stable limit cycles, bifurcation, and chaos have many applications in economics, management sciences, engineering sciences, and so forth.
In this article, we are interested in analyzing an ecosystem with limited resources in which two interacting species with different needs and resource accessibility are competing for their growth and survival. We assimilate this type of dynamic and relationship to online social network users characterized by their traffic profile during peak hours. eir generated data packets travel through network segment and queue at accommodating segment buffer space, considered as the limiting carrying capacity of the environment. is investigation is motivated by current network congestion control issues and aimed to provide a new angle of analysis for decision-making process by incorporating users' behavior in the traffic shaping and priority to give to certain flows to avoid latency and congestion at peak hours [16][17][18][19][20][21][22][23][24][25][26][27][28].
By applying stability theory and solving model differential equations, we found that, at certain conditions, the proposed system could be stable and species coexistence could be obtained when resource accessibility is fair. Competition outcome depends on initial conditions, the Allee threshold, and species ability to harvest and consume available resources. Results of numerical simulations suggested adopting hybrid resource allocation and priority at certain saturated nodes to avoid latency and poor quality of service by enhancing segment reliability.

Model
Consider U 1 and U 2 , respectively, as prey and predator population size at time t > 0 and K, the buffering capacity or carrying capacity of the environment. Prey population size represents the number of individuals of this species present in the system. Here, it is assimilated to amount of data packets generated by U 1 during time interval [t − 1, t]. is value is proportional to this user's behavior and type and nature of his online activities. Particularly, variation over time of U 1 amount of packets is proportional to system configuration in terms of amount of packets generated per unit time and the proportion of packets that have left the system (corrupted, dropped, and transmitted). is analysis holds for the second user U 2 . We assume the maximum to reach for both species is K.
Predator-prey principle formalizes variation over time of population density due to the presence or absence of other species in the ecosystem, representing the effect of increase or decrease via interaction coefficients. Based on this principle, U 1 and U 2 amount of packets traveling the segment will vary in function of interaction intensity and overall U 1 or U 2 amount of packets queuing at the buffer space. Mathematically, we have In this configuration, when interaction occurs, U 2 benefits from competition by occupying more space in the buffer, while U 1 decreases density as system will give priority to U 2 packets to be accommodated first. is scenario will lead to overdomination of low priority users who will experience very poor quality of service. To be more realistic, let us incorporate in system (1) a threshold to control extinction. It follows (2) It is clear that U 1 and U 2 will increase density as far as A is not reached. However, as users' behavior is random and network segment reliability and availability to respond may vary from time to time, it would be more realistic to consider analyzing this particular dynamic incorporating a decay factor and consider U 1 (t) and U 2 (t) as functions of time.
e competition model can then be expressed as where α, β represent, respectively, intrinsic growth factor of U 1 and U 2 in the absence of interaction. ese parameters can be explained by the ratio of each user's generated packets per unit time. In the next section of this article, we will consider α, β as positive constant number for analysis purpose. a 1 represents interaction coefficient of U 2 , denoting the fact that when meeting with U 1 , system will give U 2 packets priority, while U 1 packets will have to queue longer in the buffer if there are still rooms. b 1 represents U 2 harvesting or competition efficiency. a 2 and b 2 are, respectively, U 1 and U 2 decay factors formalizing rate of decrease in density when respective packets leave the system for any reason related to delay, latency, congestion, successful transmission,and so forth.
Coefficient A is the threshold setup to control extinction of species.
is parameter has to be fixed smaller than segment maximum carrying capacity such that, at any t > 0, users' generated amount of packets never exceed this value.

Uniqueness of Solutions.
As all parameters of system (3) are positive constant numbers, when interaction is occurring at peak hours, we can consider f and g as continuous and differentiable functions in R 2 It follows that, for resources not to be exhausted and for both users to increase density, the following condition must be satisfied: Graphical solution space of inequality (4) is illustrated in Figure 1. We can conclude that coexistence of U 1 and U 2 is peaceful and its evolution over time is submitted to resource availability and U 2 interaction coefficient.
For solutions to be unique, the following Lipschitz condition must hold: It follows Similarly, for the predator, we have We can conclude that ∀t > 0 and U 1 , U 2 < A < K, and solutions of system (3) are unique according to Lipschitz condition.

3.2.
Periodicity. If users interact continuously during time interval [t − 1, t], then f and g are doubly continuously differentiable functions in R 2 + , and we have then setting Mathematical Problems in Engineering According to Poincaré-Bendixson's theorem, no periodic solutions exist for system (3) if ω 1 + ω 3 > (ω 2 + ω 4 ) or ω 1 + ω 3 < (ω 2 + ω 4 ). Periodic solutions or limit cycle can be found for system (3) when parameters are chosen such that ω 1 + ω 3 � (ω 2 + ω 4 ). (3) critical or fixed points can be determined by solving the differential equations. We have found four equilibrium points lying in x-axis when only U 1 is sending traffic and U 2 is extinct. We have then

Existence of Equilibrium Points. System
where X � (K + AK) 2 − (4K(αA − a 2 )/α) > 0. To stay in the positive quadrant, the following conditions must be satisfied: (K + AK) 2 > (4K(αA− a 2 )/α) and (K + AK) > �� X √ . (iii) System admits P 2 on y-axis corresponding to the case U 1 is extinct and only U 2 is present in the system. We have then (iv) When both users are simultaneously sending traffic, U 1 and U 2 zero-growth isoclines may intersect at more than one point, depending on parameters value and accommodating segment state, making system (3) to admit several positive solutions in the positive quadrant as portrayed in Figure 2. We will restrict our analysis on the most interesting case when system zero-growth isoclines intersect at P 3 , corresponding to the unique stable coexistence of species.
System (3) zero-growth isoclines are given: Phase portrait of the system is illustrated in Figure 3 where it is shown P 3 has asymptotical stable behavior. Using Newton-Raphson iterative method, one can approximate positive solutions of system (3) in case of coexistence of species for a given set of parameters value and initial conditions. We have then

Model Stability Analysis
We study in this section of the article local stability conditions of equilibrium points based on system (3) linearized matrix. We have   (i) At O � (0; 0), we have

Mathematical Problems in Engineering
Roots of the characteristic polynomial equation are given as follows: Case 1: β � b 2 , T � (αA/K) − a 2 . e origin is a saddle unstable node if αA > a 2 K. is point will behave as a nodal saddle stable if αA < a 2 K.
, the origin is a source repelling node for all closer enough trajectories if αA > a 2 K. If T 2 < 4D, then the origin is a spiral source node, and all nearby trajectories will orbit around this equilibrium point in clockwise direction. Case 3: β < b 2 , αA > a 2 K, the origin is a saddle. For αA < a 2 K, this equilibrium point is behaving as a sink node sucking all trajectories. For αA � a 2 K, this equilibrium point is a saddle stable node of the system. Case 4: αA + βK � K(a 2 + b 2 ), system bifurcates at the origin, and roots of the polynomial equation are complex conjugate purely imaginary numbers expressed as O is a center for the system if β ≠ b 2 , trajectories are closed curves and solutions are periodic. (ii) At P 1 , suppose (K + AK) 2 > (4K(αA − a 2 )/α) holds; then, we have where System is stable at the neighborhood of P 1 only if is could be fulfilled if m 1 > m 2 and m 5 > b 2 or m 1 < m 2 and m 5 < b 2 . System will bifurcate at P 1 only if m 1 + m 5 System will undergo saddle node bifurcation at P 1 if (20) holds. (iii) At P 2 , we have

Mathematical Problems in Engineering
Setting we can write System is stable at the neighborhood of P 2 only if (m 6 − b 2 ) > (m 7 − β), b 2 > β and m 6 < m 7 , which cannot be fulfilled; therefore, P 2 is unstable. Small perturbations can be applied to invert or modifythis dynamic and make system bifurcates at P 2 . For that to happen, the following condition must be satisfied: m 6 � m 7 . is implies System will undergo saddle node bifurcation at P 2 as far as (24) holds. (iv) At P 3 , we have where p, s > 0 represent coordinates of the stable lower positive intersection point of the zero-growth isoclines in the positive quadrant.
We have where System is stable, and all nearby trajectories will be attracted by P 3 only if μ 1 < μ 2 and (μ 3 − μ 4 )( is can happen if μ 3 > μ 4 , μ 5 > μ 6 or μ 3 < μ 4 , μ 5 < μ 6 . Both U 1 and U 2 will grow at relative speed depending on available resources and parameters value. is dynamic is highly sensitive to initial conditions, interaction intensity, respective amount of packets traveling the segment, users' behavior, and whether system undergoes weaker or stronger Allee effects.

Numerical Results
In this section, we test the proposed model applicability and predictability by simulating two interacting social network users defined by their online activities in terms of amount of data packets generated per unit time. We assume U 1 generates less packets compared with U 2 , respectively, considered as prey and predator. Generated packets are handled and travel through the accommodating segment. Based on system configuration, when the accommodating segment is heavily loaded and there is no room in the buffer space, all incoming packets will be discarded. In this simulation, we consider the case all active sources are sending traffic continuously during peak hours. Packets may be corrupted during their travel due to factors related to congestion, latency, errors, and so forth. If that happens, considered packets will be dropped and leave the segment, decreasing respective users' density.
Buffering capacity, queuing discipline, and queuing time are important in maintaining system stability. In heavy load situation, as shown in Figure 4 (with threshold applied) and Figure 5 (no threshold applied), optimizing resource allocation is crucial to reduce latency. When the system has a threshold mechanism, prey abundance may affect its overall population size as self-competition will become intensive.

Mathematical Problems in Engineering
For predator, more prey individuals signify more resources to harvest. is dynamic is submitted to initial conditions, the threshold value, interaction parameters, and system state with respect to congestion control, queuing discipline, and segment saturation. When resources are limited, adjusting the output link transmission rate to free up buffer space may be necessary to stabilize the system and optimize throughput. In Figure 6, when the threshold is fixed at more than 80 percent of the maximum carrying capacity, U 1 stands the competition and gets its packets accommodated at relatively high speed benefiting from U 2 significantly larger decay factor (b2 = 0.8). Allocating more resources to users with lower priority has an impact on interaction outcome and system dynamic. is phenomenon is identifiable to weaker Allee effect that results in relatively lower extinction occurrences.
Periodic trajectories appear if no threshold is applied, and all resources are accessible to interacting entities for a given set of parameter value and initial conditions, as shown in Figures 7 and 8. is is consistent with theoretical analysis as all available resources will be consumed depending on interaction intensity, users' behavior, system state, and initial conditions. is result suggests that applying a threshold to control congestion, avoid extinction, and manage resources has a significant impact on both system dynamics and interaction outcome.

Conclusion
A two-species modified Lotka-Volterra predator system with threshold in both prey and predator equations has been proposed in this paper. We have analyzed the interaction phenomenon occurring by focusing on resource availability and the Allee threshold applied to control extinction of species. We applied the proposed model to online social network users sending traffic at peak hours and characterized by their traffic profile in terms of amount of packets generated per unit time. System can be stable at certain conditions if chosen some right values for model parameters. System dynamic is highly submitted to interaction intensity, available resources, system state, initial conditions, threshold value, and users' behavior. Results suggested adapting resource allocation in function of users' behavior by optimizing accessibility to guarantee quality of service.

Data Availability
No data have been used.

Conflicts of Interest
e authors declare that there are no conflicts of interest regarding the publication of this paper.