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Spatio-Temporal Dynamics in Collective Frog Choruses Examined by Mathematical Modeling and Field Observations

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ABSTRACT

This paper reports theoretical and experimental studies on spatio-temporal dynamics in the choruses of male Japanese tree frogs. First, we theoretically model their calling times and positions as a system of coupled mobile oscillators. Numerical simulation of the model as well as calculation of the order parameters show that the spatio-temporal dynamics exhibits bistability between two-cluster antisynchronization and wavy antisynchronization, by assuming that the frogs are attracted to the edge of a simple circular breeding site. Second, we change the shape of the breeding site from the circle to rectangles including a straight line, and evaluate the stability of two-cluster and wavy antisynchronization. Numerical simulation shows that two-cluster antisynchronization is more frequently observed than wavy antisynchronization. Finally, we recorded frog choruses at an actual paddy field using our sound-imaging method. Analysis of the video demonstrated a consistent result with the aforementioned simulation: namely, two-cluster antisynchronization was more frequently realized.

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Spatio-temporal dynamics in a frog chorus obtained from numerical simulation with ωn = 8π, N = 20, Knm = 1, and (Lx, Ly) = (56, 4) in equations (3) and (5)–(8).(A) Spatial structure in a frog chorus. Almost equilaterally triangular patterns are generated in many of neighboring frog trios at t = 20000, because of the narrow and long geometric shape of the rectangular field characterized by (Lx, Ly) = (56, 4). (B) Disordered phase dynamics in a frog chorus. The horizontal axis represents the frog index n, and the vertical axis represents θn at t = 20000. A self-organized structure such as two-cluster or wavy antisynchronization is not realized. (C) Time series data of the order parameters, Rcluster, Rwavy, and Rin. Red, blue, green, and black lines represent the time series data of Rcluster, Rwavy for k = 1 and k = −1, and Rin, respectively. All the order parameters take considerably less values than 1.
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f5: Spatio-temporal dynamics in a frog chorus obtained from numerical simulation with ωn = 8π, N = 20, Knm = 1, and (Lx, Ly) = (56, 4) in equations (3) and (5)–(8).(A) Spatial structure in a frog chorus. Almost equilaterally triangular patterns are generated in many of neighboring frog trios at t = 20000, because of the narrow and long geometric shape of the rectangular field characterized by (Lx, Ly) = (56, 4). (B) Disordered phase dynamics in a frog chorus. The horizontal axis represents the frog index n, and the vertical axis represents θn at t = 20000. A self-organized structure such as two-cluster or wavy antisynchronization is not realized. (C) Time series data of the order parameters, Rcluster, Rwavy, and Rin. Red, blue, green, and black lines represent the time series data of Rcluster, Rwavy for k = 1 and k = −1, and Rin, respectively. All the order parameters take considerably less values than 1.

Mentions: When (Lx, Ly) = (52, 8), (54, 6), and (56, 4), both two-cluster antisynchronization and wavy antisynchronization are not or very rarely detected (Fig. 4B). Why does such instability occur at these parameter values? Figure 5A and B represents an example of the spatio-temporal dynamics at (Lx, Ly) = (56, 4). As shown in Figure 5A, almost equilaterally triangular patterns are realized in many frog trios because of the narrow and long geometric shape of the rectangular field characterized by (Lx, Ly) = (56, 4). Since the coupling strength of Γnm(θm − θn, rm − rn) in equation (7) depends on the distance between frogs, such equilateral-triangle structures cause the frog trios to interact with almost the same strength. Moreover, Γnm(θm − θn, rm − rn) in equation (7) is assumed to be a sinusoidal function. It has been theoretically shown that almost the same coupling strength with the sinusoidal function, e.g., with n = 1, 2, 3, ωn = ω, and , can strongly frustrate the calling behavior of three frogs7. We speculate that such frustration is the source of the instability of two-cluster and wavy antisynchronization at (Lx, Ly) = (52, 8), (54, 6), and (56, 4). In fact, a snapshot of the phases at t = 20000 does not show any organized structure such as the two-cluster and wavy antisynchronization (Fig. 5B), and all the order parameters Rcluster, Rwavy and Rin take considerably less values than 1 (Fig. 5C).


Spatio-Temporal Dynamics in Collective Frog Choruses Examined by Mathematical Modeling and Field Observations
Spatio-temporal dynamics in a frog chorus obtained from numerical simulation with ωn = 8π, N = 20, Knm = 1, and (Lx, Ly) = (56, 4) in equations (3) and (5)–(8).(A) Spatial structure in a frog chorus. Almost equilaterally triangular patterns are generated in many of neighboring frog trios at t = 20000, because of the narrow and long geometric shape of the rectangular field characterized by (Lx, Ly) = (56, 4). (B) Disordered phase dynamics in a frog chorus. The horizontal axis represents the frog index n, and the vertical axis represents θn at t = 20000. A self-organized structure such as two-cluster or wavy antisynchronization is not realized. (C) Time series data of the order parameters, Rcluster, Rwavy, and Rin. Red, blue, green, and black lines represent the time series data of Rcluster, Rwavy for k = 1 and k = −1, and Rin, respectively. All the order parameters take considerably less values than 1.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5384602&req=5

f5: Spatio-temporal dynamics in a frog chorus obtained from numerical simulation with ωn = 8π, N = 20, Knm = 1, and (Lx, Ly) = (56, 4) in equations (3) and (5)–(8).(A) Spatial structure in a frog chorus. Almost equilaterally triangular patterns are generated in many of neighboring frog trios at t = 20000, because of the narrow and long geometric shape of the rectangular field characterized by (Lx, Ly) = (56, 4). (B) Disordered phase dynamics in a frog chorus. The horizontal axis represents the frog index n, and the vertical axis represents θn at t = 20000. A self-organized structure such as two-cluster or wavy antisynchronization is not realized. (C) Time series data of the order parameters, Rcluster, Rwavy, and Rin. Red, blue, green, and black lines represent the time series data of Rcluster, Rwavy for k = 1 and k = −1, and Rin, respectively. All the order parameters take considerably less values than 1.
Mentions: When (Lx, Ly) = (52, 8), (54, 6), and (56, 4), both two-cluster antisynchronization and wavy antisynchronization are not or very rarely detected (Fig. 4B). Why does such instability occur at these parameter values? Figure 5A and B represents an example of the spatio-temporal dynamics at (Lx, Ly) = (56, 4). As shown in Figure 5A, almost equilaterally triangular patterns are realized in many frog trios because of the narrow and long geometric shape of the rectangular field characterized by (Lx, Ly) = (56, 4). Since the coupling strength of Γnm(θm − θn, rm − rn) in equation (7) depends on the distance between frogs, such equilateral-triangle structures cause the frog trios to interact with almost the same strength. Moreover, Γnm(θm − θn, rm − rn) in equation (7) is assumed to be a sinusoidal function. It has been theoretically shown that almost the same coupling strength with the sinusoidal function, e.g., with n = 1, 2, 3, ωn = ω, and , can strongly frustrate the calling behavior of three frogs7. We speculate that such frustration is the source of the instability of two-cluster and wavy antisynchronization at (Lx, Ly) = (52, 8), (54, 6), and (56, 4). In fact, a snapshot of the phases at t = 20000 does not show any organized structure such as the two-cluster and wavy antisynchronization (Fig. 5B), and all the order parameters Rcluster, Rwavy and Rin take considerably less values than 1 (Fig. 5C).

View Article: PubMed Central - PubMed

ABSTRACT

This paper reports theoretical and experimental studies on spatio-temporal dynamics in the choruses of male Japanese tree frogs. First, we theoretically model their calling times and positions as a system of coupled mobile oscillators. Numerical simulation of the model as well as calculation of the order parameters show that the spatio-temporal dynamics exhibits bistability between two-cluster antisynchronization and wavy antisynchronization, by assuming that the frogs are attracted to the edge of a simple circular breeding site. Second, we change the shape of the breeding site from the circle to rectangles including a straight line, and evaluate the stability of two-cluster and wavy antisynchronization. Numerical simulation shows that two-cluster antisynchronization is more frequently observed than wavy antisynchronization. Finally, we recorded frog choruses at an actual paddy field using our sound-imaging method. Analysis of the video demonstrated a consistent result with the aforementioned simulation: namely, two-cluster antisynchronization was more frequently realized.

No MeSH data available.


Related in: MedlinePlus