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

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

Numerical simulation on the stability of two-cluster antisynchronization and wavy antisynchronization.The geometric shape of a field is assumed to be a rectangle, so as to replicate the shape of an actual paddy field in Japan. Parameters in equations (3) and (5)–(8) are fixed as ωn = 8π rad/s, N = 20, Knm = 1, and Lx + Ly = 60 m. (A) A schematic diagram describing the mathematical model. The two parameters Lx and Ly represent the length and width of the field, and  represents the vector from the origin 0, or the center of the rectangle, to the point on the edges that is nearest to the position of the nth frog rn. The nth frog is attracted to , according to Gn(rn) described by equation (3). In this simulation, Lx and Ly are varied with an interval of 2 m in the ranges of 30 ≤ Lx ≤ 60 and 0 ≤ Ly ≤ 30 under the constraint Lx + Ly = 60, and occurrences of two-cluster antisynchronization and wavy antisynchronization are calculated for 500 runs of the simulation with different initial conditions at each parameter set: namely, if only Rcluster is more than 0.9 at t = 30000, the dynamics is considered as two-cluster antisynchronization; if only Rwavy is more than 0.9 for one of k = −4, −3, −2, −1, 1, 2, 3, and 4 at t = 30000, the dynamics is considered as wavy antisynchronization. (B) Results of the numerical simulation on the stability of two-cluster and wavy antisynchronization. Red bars represent the numbers of detection of two-cluster antisynchronization, and blue bars represent those of wavy antisynchronization among 500 runs of the simulation. Two-cluster antisynchronization is more frequently observed than wavy antisynchronization, except for the cases of (Lx, Ly) = (52, 8), (54, 6) and (56, 4). The diagram of Figure 4A was drawn by I.A.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC5384602&req=5

f4: Numerical simulation on the stability of two-cluster antisynchronization and wavy antisynchronization.The geometric shape of a field is assumed to be a rectangle, so as to replicate the shape of an actual paddy field in Japan. Parameters in equations (3) and (5)–(8) are fixed as ωn = 8π rad/s, N = 20, Knm = 1, and Lx + Ly = 60 m. (A) A schematic diagram describing the mathematical model. The two parameters Lx and Ly represent the length and width of the field, and represents the vector from the origin 0, or the center of the rectangle, to the point on the edges that is nearest to the position of the nth frog rn. The nth frog is attracted to , according to Gn(rn) described by equation (3). In this simulation, Lx and Ly are varied with an interval of 2 m in the ranges of 30 ≤ Lx ≤ 60 and 0 ≤ Ly ≤ 30 under the constraint Lx + Ly = 60, and occurrences of two-cluster antisynchronization and wavy antisynchronization are calculated for 500 runs of the simulation with different initial conditions at each parameter set: namely, if only Rcluster is more than 0.9 at t = 30000, the dynamics is considered as two-cluster antisynchronization; if only Rwavy is more than 0.9 for one of k = −4, −3, −2, −1, 1, 2, 3, and 4 at t = 30000, the dynamics is considered as wavy antisynchronization. (B) Results of the numerical simulation on the stability of two-cluster and wavy antisynchronization. Red bars represent the numbers of detection of two-cluster antisynchronization, and blue bars represent those of wavy antisynchronization among 500 runs of the simulation. Two-cluster antisynchronization is more frequently observed than wavy antisynchronization, except for the cases of (Lx, Ly) = (52, 8), (54, 6) and (56, 4). The diagram of Figure 4A was drawn by I.A.

Mentions: We describe the shape of a rectangular paddy field by using two parameters Lx and Ly, which represent the length and width of the field, respectively (Fig. 4A). In addition, the summation of Lx and Ly is constrained as Lx + Ly = 60 m, for consistency with the perimeter of the circular field shown in Figures 2A and 3A. Then, Gn(rn) in equation (6) is defined as follows: where represents the vector from the origin 0 to the point on the edges that is nearest to the position of the nth frog rn, as shown in Figure 4A. We use the term Gn(rn) in equation (3), which changes its sign across the boundary condition , to explain the attraction of the male frogs towards the edges of the field.


Spatio-Temporal Dynamics in Collective Frog Choruses Examined by Mathematical Modeling and Field Observations
Numerical simulation on the stability of two-cluster antisynchronization and wavy antisynchronization.The geometric shape of a field is assumed to be a rectangle, so as to replicate the shape of an actual paddy field in Japan. Parameters in equations (3) and (5)–(8) are fixed as ωn = 8π rad/s, N = 20, Knm = 1, and Lx + Ly = 60 m. (A) A schematic diagram describing the mathematical model. The two parameters Lx and Ly represent the length and width of the field, and  represents the vector from the origin 0, or the center of the rectangle, to the point on the edges that is nearest to the position of the nth frog rn. The nth frog is attracted to , according to Gn(rn) described by equation (3). In this simulation, Lx and Ly are varied with an interval of 2 m in the ranges of 30 ≤ Lx ≤ 60 and 0 ≤ Ly ≤ 30 under the constraint Lx + Ly = 60, and occurrences of two-cluster antisynchronization and wavy antisynchronization are calculated for 500 runs of the simulation with different initial conditions at each parameter set: namely, if only Rcluster is more than 0.9 at t = 30000, the dynamics is considered as two-cluster antisynchronization; if only Rwavy is more than 0.9 for one of k = −4, −3, −2, −1, 1, 2, 3, and 4 at t = 30000, the dynamics is considered as wavy antisynchronization. (B) Results of the numerical simulation on the stability of two-cluster and wavy antisynchronization. Red bars represent the numbers of detection of two-cluster antisynchronization, and blue bars represent those of wavy antisynchronization among 500 runs of the simulation. Two-cluster antisynchronization is more frequently observed than wavy antisynchronization, except for the cases of (Lx, Ly) = (52, 8), (54, 6) and (56, 4). The diagram of Figure 4A was drawn by I.A.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f4: Numerical simulation on the stability of two-cluster antisynchronization and wavy antisynchronization.The geometric shape of a field is assumed to be a rectangle, so as to replicate the shape of an actual paddy field in Japan. Parameters in equations (3) and (5)–(8) are fixed as ωn = 8π rad/s, N = 20, Knm = 1, and Lx + Ly = 60 m. (A) A schematic diagram describing the mathematical model. The two parameters Lx and Ly represent the length and width of the field, and represents the vector from the origin 0, or the center of the rectangle, to the point on the edges that is nearest to the position of the nth frog rn. The nth frog is attracted to , according to Gn(rn) described by equation (3). In this simulation, Lx and Ly are varied with an interval of 2 m in the ranges of 30 ≤ Lx ≤ 60 and 0 ≤ Ly ≤ 30 under the constraint Lx + Ly = 60, and occurrences of two-cluster antisynchronization and wavy antisynchronization are calculated for 500 runs of the simulation with different initial conditions at each parameter set: namely, if only Rcluster is more than 0.9 at t = 30000, the dynamics is considered as two-cluster antisynchronization; if only Rwavy is more than 0.9 for one of k = −4, −3, −2, −1, 1, 2, 3, and 4 at t = 30000, the dynamics is considered as wavy antisynchronization. (B) Results of the numerical simulation on the stability of two-cluster and wavy antisynchronization. Red bars represent the numbers of detection of two-cluster antisynchronization, and blue bars represent those of wavy antisynchronization among 500 runs of the simulation. Two-cluster antisynchronization is more frequently observed than wavy antisynchronization, except for the cases of (Lx, Ly) = (52, 8), (54, 6) and (56, 4). The diagram of Figure 4A was drawn by I.A.
Mentions: We describe the shape of a rectangular paddy field by using two parameters Lx and Ly, which represent the length and width of the field, respectively (Fig. 4A). In addition, the summation of Lx and Ly is constrained as Lx + Ly = 60 m, for consistency with the perimeter of the circular field shown in Figures 2A and 3A. Then, Gn(rn) in equation (6) is defined as follows: where represents the vector from the origin 0 to the point on the edges that is nearest to the position of the nth frog rn, as shown in Figure 4A. We use the term Gn(rn) in equation (3), which changes its sign across the boundary condition , to explain the attraction of the male frogs towards the edges of the field.

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