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

Authors: Yaming Zhang; Yaya Hamadou Koura; Yanyuan Su
Journal: Mathematical Problems in Engineering
 (2020)

---

## Abstract

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.


---

## Body

## 1. 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 [1–4].In applied mathematics and engineering sciences, predator-prey models based on Lotka–Volterra are commonly used to study such systems. There 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 [5–9]. Thus, 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 [10–15].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. Their generated data packets travel through network segment and queue at accommodating segment buffer space, considered as the limiting carrying capacity of the environment. This 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 [16–28].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.


## 2. Model

ConsiderU1 and U2, 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 U1 during time interval t−1,t. This value is proportional to this user’s behavior and type and nature of his online activities. Particularly, variation over time of U1 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). This analysis holds for the second user U2. 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,U1 and U2 amount of packets traveling the segment will vary in function of interaction intensity and overall U1 or U2 amount of packets queuing at the buffer space. Mathematically, we have(1)dU1U1=α1−U1K−a1U2,α,a1,K>0,dU2U2=β1−U2K+a1b1U1,β,b1>0.In this configuration, when interaction occurs,U2 benefits from competition by occupying more space in the buffer, while U1 decreases density as system will give priority to U2 packets to be accommodated first. This 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)dU1U1=α1−U1KA−U1K−a1U2,dU2U2=β1−U2K+a1b1U1A−U2K,K>A>0.It is clear thatU1 and U2 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 U1t and U2t as functions of time. The competition model can then be expressed as(3)dU1dt=U1α1−U1K−a1U2A−U1K−a2=fU1,U2,dU2dt=U2β1−U2K+a1b1U1A−U2K−b2=gU1,U2,α,β,a1,a2,b1,b2,K,A>0,K>A;U1t,U2t<A,t⟶∞;U10,U20≥0,a1,a2,b1,b2∈0,1,where α,β represent, respectively, intrinsic growth factor of U1 and U2 in the absence of interaction. These 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 U2, denoting the fact that when meeting with U1, system will give U2 packets priority, while U1 packets will have to queue longer in the buffer if there are still rooms.b
1 represents U2 harvesting or competition efficiency. a2 and b2 are, respectively, U1 and U2 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.CoefficientA is the threshold setup to control extinction of species. This 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.


## 3. Model Analysis

### 3.1. 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+2−0. If U1<A, then, dU1/dt>0, signifying U1 will increase density only if K−U1−a1KU2U1A−U1>a2K/α>0.Similarly forU2, we have U2<A, dU2/dt>0 and U1A−U2+b1K−U2>b2K/a1β>0.It follows that, for resources not to be exhausted and for both users to increase density, the following condition must be satisfied:(4)K>U1+a1KU2.Graphical solution space of inequality (4) is illustrated in Figure 1. We can conclude that coexistence of U1 and U2 is peaceful and its evolution over time is submitted to resource availability and U2 interaction coefficient.Figure 1
Solution space of the carrying capacity equation, showing maximum value to reach for U1 and U2 when species interact.Figure 2
Phase portrait when U1 and U2 interact, and system admits two positive equilibrium points in the positive quadrant. The lower intersection point is asymptotically stable, while the upper intersection point behaves as a saddle node.In function of time,0<U1t;U2t<A for t⟶∞ and 0<U1≤K, 0<U2≤1/a1.For solutions to be unique, the following Lipschitz condition must hold:(5)∂ft,U1∂U1≤ε1,ε1∈ℝ+2,∂gt,U2∂U2≤ε2,ε2∈ℝ+2.It follows(6)∂ft,U1∂U1=αAK1−1K+αa1U2K2U1−A−a2≤αKA−a1U22U1−A−a2Kα,∂ft,U1∂U1=αA/K1−1/K+αa1U2/K2U1−A−a2A−a1U22U1−A−a2K/α≤αK,1+a1U2≠2αa1AU1U2+a2KαA.Similarly, for the predator, we have(7)∂gt,U2∂U2=a1b1AU1K+β−2a1b1U1U2K+2βU2K+b2≤a1b1AU1K+β,∂ft,U1∂U1=a1b1AU1/K+β−2a1b1U1U2/K+2βU2/K+b2a1b1AU1/β+K≤βK.We can conclude that∀t>0 and U1,U2<A<K, and solutions of system (3) are unique according to Lipschitz condition.


### 3.2. Periodicity

If users interact continuously during time intervalt−1,t, then f and g are doubly continuously differentiable functions in ℝ+2, and we have then(8)∂f∂U1=αAK+3αU1∗2K2+2αa1U1∗U2∗K2−2αU1∗K+2αAU1∗K+αa1AU2∗K−a2,∂2f∂U12=−2αK−2αAK+6αU1∗K2+2αa1U2∗K,∂g∂U2=a1b1AU1∗K−2a1b1U1∗U2∗K−2βU2∗K+β−b2,∂2g∂U22=−2a1b1U1∗K−2βK,setting(9)ω1=αAK+3αU1∗2K2+2αa1U1∗U2∗K2>0,ω2=2αU1∗K+2αAU1∗K+αa1AU2∗K+a2>0,ω3=a1b1AU1∗K+β>0,ω4=2a1b1U1∗U2∗K+2βU2∗K+b2>0,divf,g=∇⋅f,g=∂f∂U1+∂g∂U2=ω1+ω3−ω2+ω4.According to Poincaré–Bendixson’s theorem, no periodic solutions exist for system (3) if ω1+ω3>ω2+ω4 or ω1+ω3<ω2+ω4. Periodic solutions or limit cycle can be found for system (3) when parameters are chosen such that ω1+ω3=ω2+ω4.


### 3.3. Existence of Equilibrium Points

System (3) critical or fixed points can be determined by solving the differential equations. We have found four equilibrium points lying in ℝ+2:(i)
The originO=0;0, corresponding to the situation where no traffic is traveling the segment or both users are offline.(ii)
System may admit two positive points lying onx-axis when only U1 is sending traffic and U2 is extinct. We have then(10)P1=K+AK±X2;0,whereX=K+AK2−4KαA−a2/α>0. To stay in the positive quadrant, the following conditions must be satisfied: K+AK2>4KαA−a2/α and K+AK>X.(iii)
System admitsP2 on y-axis corresponding to the case U1 is extinct and only U2 is present in the system. We have then(11)P2=0;Kβ−b1/β,β>b1.(iv)
When both users are simultaneously sending traffic,U1 and U2 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 P3, corresponding to the unique stable coexistence of species.System (3) zero-growth isoclines are given:(12)F:αU12−α1+AU1−αa1AU2+αa1U1U2+αA−a2K=0,G:a1b1AU1−βU2−a1b1U1U2+Kβ−b2=0.Phase portrait of the system is illustrated in Figure3 where it is shown P3 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(13)U2k=a1b1AU1k+Kβ−b2a1b1AU1k+β,k∈0,1,2…,U10=U10,U20=U20>0;∂F∂U1k=α2U1k+a1U2k−A−1;2U1k+a1U2k≠A+1;U1k+1=U1k−F∂F/∂U1k.Figure 3
Phase portrait when U1 and U2 interact, and P3 is asymptotically stable attracting all near trajectories, considered as the attracting node forα = 1; a1 = 0.12; β = 1; b1 = 0.2; A = 0.6; K = 1; b2 = 0.3; a2 = −0.12.


## 3.1. 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+2−0. If U1<A, then, dU1/dt>0, signifying U1 will increase density only if K−U1−a1KU2U1A−U1>a2K/α>0.Similarly forU2, we have U2<A, dU2/dt>0 and U1A−U2+b1K−U2>b2K/a1β>0.It follows that, for resources not to be exhausted and for both users to increase density, the following condition must be satisfied:(4)K>U1+a1KU2.Graphical solution space of inequality (4) is illustrated in Figure 1. We can conclude that coexistence of U1 and U2 is peaceful and its evolution over time is submitted to resource availability and U2 interaction coefficient.Figure 1
Solution space of the carrying capacity equation, showing maximum value to reach for U1 and U2 when species interact.Figure 2
Phase portrait when U1 and U2 interact, and system admits two positive equilibrium points in the positive quadrant. The lower intersection point is asymptotically stable, while the upper intersection point behaves as a saddle node.In function of time,0<U1t;U2t<A for t⟶∞ and 0<U1≤K, 0<U2≤1/a1.For solutions to be unique, the following Lipschitz condition must hold:(5)∂ft,U1∂U1≤ε1,ε1∈ℝ+2,∂gt,U2∂U2≤ε2,ε2∈ℝ+2.It follows(6)∂ft,U1∂U1=αAK1−1K+αa1U2K2U1−A−a2≤αKA−a1U22U1−A−a2Kα,∂ft,U1∂U1=αA/K1−1/K+αa1U2/K2U1−A−a2A−a1U22U1−A−a2K/α≤αK,1+a1U2≠2αa1AU1U2+a2KαA.Similarly, for the predator, we have(7)∂gt,U2∂U2=a1b1AU1K+β−2a1b1U1U2K+2βU2K+b2≤a1b1AU1K+β,∂ft,U1∂U1=a1b1AU1/K+β−2a1b1U1U2/K+2βU2/K+b2a1b1AU1/β+K≤βK.We can conclude that∀t>0 and U1,U2<A<K, and solutions of system (3) are unique according to Lipschitz condition.


## 3.2. Periodicity

If users interact continuously during time intervalt−1,t, then f and g are doubly continuously differentiable functions in ℝ+2, and we have then(8)∂f∂U1=αAK+3αU1∗2K2+2αa1U1∗U2∗K2−2αU1∗K+2αAU1∗K+αa1AU2∗K−a2,∂2f∂U12=−2αK−2αAK+6αU1∗K2+2αa1U2∗K,∂g∂U2=a1b1AU1∗K−2a1b1U1∗U2∗K−2βU2∗K+β−b2,∂2g∂U22=−2a1b1U1∗K−2βK,setting(9)ω1=αAK+3αU1∗2K2+2αa1U1∗U2∗K2>0,ω2=2αU1∗K+2αAU1∗K+αa1AU2∗K+a2>0,ω3=a1b1AU1∗K+β>0,ω4=2a1b1U1∗U2∗K+2βU2∗K+b2>0,divf,g=∇⋅f,g=∂f∂U1+∂g∂U2=ω1+ω3−ω2+ω4.According to Poincaré–Bendixson’s theorem, no periodic solutions exist for system (3) if ω1+ω3>ω2+ω4 or ω1+ω3<ω2+ω4. Periodic solutions or limit cycle can be found for system (3) when parameters are chosen such that ω1+ω3=ω2+ω4.


## 3.3. Existence of Equilibrium Points

System (3) critical or fixed points can be determined by solving the differential equations. We have found four equilibrium points lying in ℝ+2:(i)
The originO=0;0, corresponding to the situation where no traffic is traveling the segment or both users are offline.(ii)
System may admit two positive points lying onx-axis when only U1 is sending traffic and U2 is extinct. We have then(10)P1=K+AK±X2;0,whereX=K+AK2−4KαA−a2/α>0. To stay in the positive quadrant, the following conditions must be satisfied: K+AK2>4KαA−a2/α and K+AK>X.(iii)
System admitsP2 on y-axis corresponding to the case U1 is extinct and only U2 is present in the system. We have then(11)P2=0;Kβ−b1/β,β>b1.(iv)
When both users are simultaneously sending traffic,U1 and U2 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 P3, corresponding to the unique stable coexistence of species.System (3) zero-growth isoclines are given:(12)F:αU12−α1+AU1−αa1AU2+αa1U1U2+αA−a2K=0,G:a1b1AU1−βU2−a1b1U1U2+Kβ−b2=0.Phase portrait of the system is illustrated in Figure3 where it is shown P3 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(13)U2k=a1b1AU1k+Kβ−b2a1b1AU1k+β,k∈0,1,2…,U10=U10,U20=U20>0;∂F∂U1k=α2U1k+a1U2k−A−1;2U1k+a1U2k≠A+1;U1k+1=U1k−F∂F/∂U1k.Figure 3
Phase portrait when U1 and U2 interact, and P3 is asymptotically stable attracting all near trajectories, considered as the attracting node forα = 1; a1 = 0.12; β = 1; b1 = 0.2; A = 0.6; K = 1; b2 = 0.3; a2 = −0.12.


## 4. 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(14)JU¯1;U¯2=Bαa1U1∗K23αU1∗−Aa1b1U2∗KA−2U2∗β+a1b1AU1∗K−2a1b1U1∗U2∗K+2βU2∗K+b2,B=αAK+3αU1∗2K2+2αa1U1∗U2∗K2−2αU1∗K+2αAU1∗K+αa1AU2∗K+a2.(i)
AtO=0;0, we have(15)J0;0=αAK−a200β−b2,T=αA+βK−Ka2+b2K,D=β−b2αA−a2KK.Roots of the characteristic polynomial equation are given as follows:(16)λ1,2=T±T2−4D2.Case 1:β=b2, T=αA/K−a2. The origin is a saddle unstable node if αA>a2K. This point will behave as a nodal saddle stable if αA<a2K.Case 2:β>b2, D,T>0, then if T2≥4D, the origin is a source repelling node for all closer enough trajectories if αA>a2K. If T2<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<b2, αA>a2K, the origin is a saddle. For αA<a2K, this equilibrium point is behaving as a sink node sucking all trajectories. For αA=a2K, this equilibrium point is a saddle stable node of the system.Case 4:αA+βK=Ka2+b2, system bifurcates at the origin, and roots of the polynomial equation are complex conjugate purely imaginary numbers expressed as(17)λ1,2=±−D=±iβ−b2.O is a center for the system if β≠b2, trajectories are closed curves and solutions are periodic.(ii)
AtP1, suppose K+AK2>4KαA−a2/α holds; then, we have(18)JK1+A+X2;0=BCEF,where(19)B=m1−m2,C=m3−m4,E=0,F=m5−b2,m1=3αKK1+A+X22+αAK,m2=αK1+AK1+A+X+a2,m3=2αa1KK1+A+X22,m4=αa1AKK1+A+X2,m5=β+a1b1AKK1+A+X2.System is stable at the neighborhood ofP1 only if m1+m5<m2+b2 and m1−m2m5−b2>0. This could be fulfilled if m1>m2 and m5>b2 or m1<m2 and m5<b2.System will bifurcate atP1 only if m1+m5=m2+b2 and m1−m2m5−b2>0. This implies m5=m1+b2−m2. It follows(20)λ1,2=±m1−m2.System will undergo saddle node bifurcation atP1 if (20) holds.(iii)
AtP2, we have(21)J0;Kβ−b1β=αAK−αa1Aβ−b1β−a20a1b1Aβ−b1β−2a1b1Kβ−b12β2b2−b1.Setting(22)m6=αAK+αa1b2Aβ+b2,m7=αa1b2Aβ+β+a2,we can write(23)T=m6−m7,D=b2−βm6−b2−m7−β.System is stable at the neighborhood ofP2 only if m6−b2>m7−β,b2>β and m6<m7, which cannot be fulfilled; therefore, P2 is unstable.Small perturbations can be applied to invert or modifythis dynamic and make system bifurcates atP2. For that to happen, the following condition must be satisfied: m6=m7. This implies(24)λ1,2=±β−b2.System will undergo saddle node bifurcation atP2 as far as (24) holds.(iv)
AtP3, we have(25)Jp;s=V1V2V3V4=Bαa1pK23αp−Aa1b1sKA−2sβ+a1b1ApK−2a1b1psK+2βsK+b2,B=αAK+αpK23p+2a1s−αK2p+2Ap+a1As−a2,wherep,s>0 represent coordinates of the stable lower positive intersection point of the zero-growth isoclines in the positive quadrant.We have(26)T=μ1−μ2,D=μ3−μ4μ5−μ6−V2V3,where(27)μ1=αAK+αpK23p+2a1s+β+a1b1ApK,μ2=αK2p+2Ap+a1As+2a1b1psK+2βsK+a2+b2,μ3=αAK+αpK23p+2a1s,μ4=αK2p+2Ap+a1As+a2μ5=β+a1b1ApK,μ6=2a1b1psK+2βsK+b2.System is stable, and all nearby trajectories will be attracted byP3 only if μ1<μ2 and μ3−μ4μ5−μ6>V2V3. This can happen if μ3>μ4,μ5>μ6 or μ3<μ4,μ5<μ6. Both U1 and U2 will grow at relative speed depending on available resources and parameters value. This 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.


## 5. 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 assumeU1 generates less packets compared with U2, 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 Figure4 (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. For predator, more prey individuals signify more resources to harvest. This dynamic is submitted to initial conditions, the threshold value, interaction parameters, and system state with respect to congestion control, queuing discipline, and segment saturation.Figure 4
Dynamic of interaction (a) and phase space (b) when threshold is applied, coexistence is peaceful, and system is stable forα = 1.18; a1 = 0.2; a2 = 0.13; β = 1.15; b1 = 0.18; A = 2; K = 3; b2 = 0.25.
(a)
(b)Figure 5
Dynamic of interaction (a) and phase space (b) when no threshold is applied, system is stable, and U1 shrinks and dies out forα = 2.18; a1 = 0.2; a2 = 0.13; β = 3.15; b1 = 0.18; K = 30; b2 = 0.25.
(a)
(b)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 Figure6, when the threshold is fixed at more than 80 percent of the maximum carrying capacity, U1 stands the competition and gets its packets accommodated at relatively high speed benefiting from U2 significantly larger decay factor (b2 = 0.8). Allocating more resources to users with lower priority has an impact on interaction outcome and system dynamic. This phenomenon is identifiable to weaker Allee effect that results in relatively lower extinction occurrences.Figure 6
Dynamic of interaction (a) and phase space (b) when system is stable accommodating both users packets and U1 dominates forα = 3.18; a1 = 0.2; a2 = 0.6; β = 1.15; b1 = 0.18; A = 2.5; K = 3; b2 = 0.8.
(a)
(b)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 Figures7 and 8. This is consistent with theoretical analysis as all available resources will be consumed depending on interaction intensity, users’ behavior, system state, and initial conditions. This 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.Figure 7
Dynamic of interaction (a) and phase space (b) when no threshold is applied, solutions are periodic, and system is dynamically stable forα = 1.6; a1 = 0.28; a2 = 0.15; β = 0.028; b1 = 0.19; K = 80; b2 = 0.159.
(a)
(b)Figure 8
Dynamic of interaction (a) and phase space (b) when no threshold is applied and solutions are periodic, system is dynamically stable forα = 1.6; a1 = 0.28; a2 = 0.15; β = 0.028; b1 = 0.19; K = 80; b2 = 0.159.
(a)
(b)


## 6. 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.



---

*Source: 8549247-2020-03-19.xml*