Simulation results
Differences in preferred speed
We use simulations to predict the influence of differences in average speed, differences in turning ability, and social changes in speed, on the relative distributions of the two fish.
The first case is a "null case", based on what we already expect: when the two fish differ in average speed, then the faster fish will be the leader. To demonstrate the model agrees with this expected result, we simulate, using an intermediate value of social interaction, \alpha=0.8, and a realistic value for turning noise, sigma_turn = 4. And some uncorrelated changes in speed, with sigma_speed=0.1
(A) different mean speed
(B) different mean speed, plus speed noise and social changes in speed
For case (A), we find the expected result that for larger differences in average speed between the fish, the faster fish acts more as the "leader" because it is more often in front. The average front-back distance, as well as the difference in alignment and the distance apart, increase for larger differences in speed (Figure Speed-A).
(C) Same mean speed
To examine a more realistic case, we look at the same differences in average speed, but now include changes in speed based on social interactions, with the weight determined by the parameter \alpha_s = 0 (shown above), 0.4, 0.8 (i.e. consider 3 values). With social adaptation to speed changes, ...
- the actual average speed of each is shifted from the preferred average speed of \mu_i for each, because each tries to adapt with the other (Fig Speed-B)
- the leader-follower and differences in alignment due to mean preferred speed differences decrease.
FIGURE
Differences in turning and acceleration ability
Being parasitized affects the swimming ability of a fish, in particular its ability to accelerate and turn quickly. Using the simulation model, we ask if we can expect leader-follower differences if the two fish differ in their ability to turn. In the model, this is represented by the parameter \(\tau_\omega\), which determines the autocorrelation time for changes in angular velocity. A reduced ability to turn is represented by a higher value of \(\tau_\omega\), i.e. it takes longer to change angular velocity.
We examine the cases
(A) Same mean speed. Include sigma_speed = 0.1. \alpha_s = 0 (no social speed changes). then increase tau_ratio just for tau_omega
(B) Same mean speed, but include social speed changes, using the values above. Keep tau_omega and tau_s the same, but give the fish different values of alpha_S - just for acceleration?
(C) Same mean speed, and include social speed changes same as B. But now adjust both tau_s and different alpha_S_accel
Fig () shows that a reduced turning ability can indeed lead to leader-follower differentiation, although the changes in relative front-back distance are less than what is seen when the two individuals differ in speed. When the difference in tau_omega between the two fish increases, they are on average less aligned, and farther apart.
Fig ()B shows the same trend, when social changes in speed are included (? - need to do this!).
Combined factors: preferred speed, acceleration, and turning ability
The above results show that differences in leader-follower activity can be expected due to differences in turning and acceleration ability, or due to differences in speed. We now ask how these effects combine to predict leader-follow differences.
Consider:
(A) Faster fish can turn/accelerate faster. have different mu_s, and also tau_omega
(A2) also add different alpha_s_accel
(B) Slower fish can turn/accelerate faster.
(B2) also add different alpha_s_accel
Figure ()A shows that when these effects are combined, differences in acceleration and turning ability no longer lead to an increase in leader-follower differentation; leader-follower relationship is determined primarily by preferred swimming speed. Actually, the average front-back distance decreases with tau_ratio for a given value of speed differences between the pair. The average alignment again increases with tau_ratio though.
A2. when also have alpha_s_accel, ...
We now consider the opposite case, when the slower fish is the better swimming. Fig (), ..