Two-actor conflict with time delay: A dynamical model

Recent mathematical dynamical models of the conflict between two different actors, be they nations, groups, or individuals, have been developed that are capable of predicting various outcomes depending on the chosen feedback strategies, initial conditions, and the previous states of the actors. In addition to these factors, this paper examines the effect of time delayed feedback on the conflict dynamics. Our analysis shows that under certain initial and feedback conditions, a stable neutral equilibrium of conflict may destabilize for some critical values of time delay, and the two actors may evolve to new emotional states. We investigate the results by constructing critical delay surfaces for different sets of parameters and analyzing results from numerical simulations. These results provide new insights regarding conflict and conflict resolution and may help planners in adjusting and assessing their strategic decisions.

Figure 1

Stability surfaces for the system described by Eq. (7). The critical stability surfaces define the minimal delay value separating stable and unstable regions of the neutral state ($x_1=0,x_2=0$) as a function of the feedback strength $c$ and the intrinsic inertia parameter $m$ for (a) symmetric response strengths ($C_{12}=C=C_{21}$) representing the cooperation and competition feedback and (b) asymmetric response strengths ($C_{12}=C=-C_{21}$) representing the mixed (positive-negative) feedback. The region denoted by S shows the stable areas while U shows the regions where the system is always unstable independent of time delay $\tau$. SU shows the regions where the system is unstable above the critical surface and stable below it.

Figure 2

Stability surface for the neutral state of the system described by Eq. (7) with delay at a fixed value of the inertia parameter $m=-0.1$ (this value is chosen only for a better visual representation although all values $m<0$ return similar results). The surface defines the minimal delay value as a function of the feedback strengths of the two actors, $C_{12}$ and $C_{21}$. The region denoted by S shows the stable regions while U shows the regions where the system is always unstable. SU shows the regions where the system is unstable above the critical surface and stable below it. The right panel is the top view of the surface shown in the left panel.

Figure 3

Time series of emotional states of the two actors, $x_1\left(t\right)$ (solid line) and $x_2\left(t\right)$ (dashed line), with weak feedback strengths ($|C_{12}|=|C|=|C_{21}|=0.5<|m|=0.9$) for all types of feedback. Numerical integration is used to calculate $x_1\left(t\right)$ and $x_2\left(t\right)$ as the emotional states of the two actors evolve from their initial values of $x_1\left(0\right)=1$ and $x_2\left(0\right)=2$ for the three feedback types (rows) when the feedback strength ($\left|C\right|$) is less than the threshold equal to the absolute value of the inertia to change, that is, $\left|C\right|<\left|m\right|$. The first column shows the results when there is no time delay and the second column with time delay, $\tau =1$. Note that the two actors will evolve to the neutral state $x_1\left(t\right)=0$ and $x_2\left(t\right)=0$ in all feedback types for any value of time delay.

Figure 4

Time series of emotional states of the two actors, $x_1\left(t\right)$ (solid line) and $x_2\left(t\right)$ (dashed line), with strong positive-positive feedbacks ($C_{12}=C_{21}=3$) at different time delay values (columns) with $m=-0.9$. The values of $x_1\left(t\right)$ and $x_2\left(t\right)$ are shown as the states of the two actors evolve from different initial values (rows). The values of $x_1\left(t\right)$ and $x_2\left(t\right)$ always evolve toward the stable states where either both are positive or both are negative for any value of time delay $\tau \ge 0$. When the initial conditions are different, both actors will oscillate out of phase as they evolve towards the same final state. The actor of stronger initial emotional state (larger magnitude) will drag the other towards the new state that is closer to its initial side.

Figure 5

Time series of emotional states of the two actors, $x_1\left(t\right)$ (solid line) and $x_2\left(t\right)$ (dashed line), with strong negative-negative feedbacks ($C_{12}=C_{21}=-3$) at different time delay values (columns) with $m=-0.9$. The values of $x_1\left(t\right)$ and $x_2\left(t\right)$ are shown as the states of the two actors evolve from different initial values (rows). In the absence of time delay, both actors, with the same parameters and the same initial conditions, will evolve to the neutral state. If time delay is less than a critical value, the two actors will oscillate about the neutral state with decaying amplitude. As time delay exceeds the critical value, both actors will oscillate in phase about the neutral state but with growing amplitudes. When the initial conditions are not the same, the two actors will oscillate for a short period of time and move away from each other toward two opposite states. The larger the time delay, the longer the period with significant oscillatory behavior.

Figure 6

Time series of emotional states of the two actors, $x_1\left(t\right)$ (solid line) and $x_2\left(t\right)$ (dashed line), with strong positive-negative feedbacks ($C_{12}=3$, $C_{21}=-3$) at different time delay values (columns) with $m=-0.9$. The values of $x_1\left(t\right)$ and $x_2\left(t\right)$ exhibit out-of-phase oscillation about the neutral state as they evolve from different initial values (rows). As time delay increases, the amplitude of the out-of-phase oscillation becomes larger, and eventually changes from being damped to undamped.

Figure 7

The effect of time delay when one actor unilaterally changes the sign of its feedback in a conflict with negative-negative feedback. In the first row: at time $t=6$, actor $x_1$ temporarily switches its strategy to positive feedback for duration $D=0.75$. For all time delay values, both actors will evolve to their previous fixed states. In the second row, the duration is larger: $D=1.5$. In this case, for any time delay less than a certain value, the two actors will evolve to two states opposite their respective signs before the switch. As time delay increases, the reverse of the states cannot be achieved. This effect is highly sensitive to the initial conditions, moment of switch, duration of switch, and the time delay. The initial conditions of the two states shown are $x_1\left(0\right)=1.0,x_2\left(0\right)=0.5$, and the inertia $m=-0.9$.

Figure 8

The effect of one actor responding unilaterally with delay and the effect of symmetry breaking in time delay values on the stability of the neutral state. The three columns represent the three different types of feedback: cooperation, competition, and mixed, respectively. The first row shows the previously obtained results of the time series of the emotional states of the two actors, $x_1\left(t\right)$ (solid line) and $x_2\left(t\right)$ (dashed line), with instantaneous (undelayed) responses. The second row is the time series of $x_1\left(t\right)$ and $x_2\left(t\right)$ with the same nonzero delay values, ${\tau }_1={\tau }_2=1$. The third row shows the effect of one actor who unilaterally responds with delay while the other actor responds instantaneously. The last row shows the effect of a small symmetry breaking between time delay values on the stability of the neutral state of the two actors. The initial conditions are the same for each type. The inertia value $m=-0.9$.